a brief introduction to data mining and analysis for insurance regulators

8/14/2019 A Brief Introduction to Data Mining and Analysis for Insurance Regulators

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© 2005 National Association of Insurance Commissioners

making progress . . . together

NAI C

BrenResearch

MO DBrent.Kabler@(573)

A Brief Introduct ion to DM ining and A nalysis

For

InsuranceRegulators




D ata M ining - D efinit i

Data mining inductively seeks to identify hidd

sequences, and relationships in (typically) very larg

Inductive analysis – analysis without theory-driven

Automated algorithms – search functions, gr

clustering methods, pattern recognition routines

search for patterns in very large datasets.




D ata M ining - C rit ic“Data mining is what IS staff call statistics.”

“Data mining is a set of techniques designed to tortu

until it confesses.”

“Data mining refers to the exaggerated claims of

and/or forecasting precision generated by the selecti

of results obtained when the structure of the

determined experimentally by repeated applicatio

procedures as regression to the same body of data…




D ata M ining – A U se

Powerful Tool – with c“There have always been a considerable number of

busy themselves examining the last thousand num

have appeared on a roulette wheel, in search of som

pattern. Sadly enough, they have usually found it.”

Fred Schwed, 1940, Where are the Customers’ Yacht

Some statisticians argue that data mining fail to ob

strict rules of inference associated with the

hypothetical-deductive method, such as strict htesting and theory building, and can do more

researchers astray as to enlighten them.




C aveats (C ont)In the prior example of finding patterns in a roulettrandom process, such as a roulette wheel or a flip of

appear to produce meaningful or non-random outco

to false inferences (i.e. that the wheel is not random)

For example, it is very likely that 1 million spins ofwheel will produce apparently non-random sequences

“five red, five black,” or “1, 2, 3, 4, 5, 6, 7, 8, 9, 10…

Considerable substantive and methodological expertisto avoid mistakes – false inferences




PositivesMy opinion: data mining can be, and has proven

extremely valuable tool for analysts confronted

volumes of data, in situations in which prior kn

low.

Advantages:1. Can produce counterintuitive and unexpected

more traditional methods might have missed

2. Largely automated, less labor intensive than

analysis




D ata mining tools

Association –looking for correlations. Event or at

typically present when event or attribute B is presen

Sequence – temporal association. Event B usual

Event A

Clustering or grouping – identifying classes th

relatively unique characteristics; identifying “

unobservable structures through observable patterns

Prediction – assessing probability of a future outcknown present conditions




A valuable regulatory t

Data mining represents a potentially powerful

directly facilitates regulators’ core mission:

1. Predicting likelihood of non-compliance

2. Assessing likely consumer harm associated compliance

3. Identifying appropriate interventions best

minimize risk of harm

4. Evaluation of the effectiveness of alternative instrategies




Target ing ResourcesPareto Rule – A small number of causes are responsi

great number of effects.

Examples – most companies generate greatest profit

proportionately small percentage of customer base

rule)

It is likely that the majority of consumer harm from

conduct issues come from a small percentage of fir

small number of particular industry practices.

Data mining can assist in identifying areas where re

intervention is most needed, and likely to have t

impact.




Examples of Regulato

SuccessOSHA, a pilot project in Maine during

developed sophisticated data analysis of the so

workplace injuries: “identify important conce

of risk through systematic data analysis and the

resources on those risks in an attempt to mitigate t

Result: identified a significant concentration

associated with just 200 companies, represent

1% of the market. Focused intervention on th

risk companies.




Regulatory Success –

C onsumer Product Safety C ouImplemented extensive data collections, and

analytical resources to analyze them. Res

counterintuitive, and suggested that resour

misdirected to less severe problems:

1. 411,689 Americans were injured by th

mattresses, or pillows seriously enough to

emergency room treatment, far outweighing th

injured by skateboards (48,186) or tra

(82,722)2. More people were hurt by sound recording e

(38,956) than by chainsaws (29,684)




A G rowing Pract ice

A recent GAO survey of 128 federal agencies

that 52 agencies are using or are planning

data mining techniques to exploit vast amou

data at their disposal.

Also, widespread in private sector – credit scor




Lack of quality data

ETS and RIRS databases are subject to different

standards between the states, incomplete par

subject to error, can’t be integrated, a

elaborate coding mechanisms that would maksuitable for statistical analysis.

These data could prove invaluable as a m

regulatory outcomes, to assess differing

practices and interventions.

NAIC is currently devoting considerable attentio

issues.

Problems




Lack of sophisticated methodologies specific

regulation.

Data mining and other analytical approac

address this deficiency relatively quickly.

Problems




Targeting Market Interventions

1. Do current targeting techniques have a “hit ra

significantly better than one could do by chanc

2. Can more elaborate statistical profiling techn

richer data perform better than current

techniques?

Example of C oncrete Pro




M issouri ResearchMissouri preliminary study to assess exam target

states: can we predict which companies are mo

exhibit serious market behavior problems?

Predicted Group

Actual Group N No Fine Fine

No Fine 122

100%

83

68.0%

39

32.0%

Fine 39

100%

9

23.1%

30

76.9%

Predicting Exam Outcomes




Percent of all cases correctly classified: 70.2%. If model held

cases, could have increased the “hit rate” from 24% to 43%, o

higher depending on how the overall market compared with th

It seems likely that percentage could probably increase significbetter data and additional research: data mining can be a powe

technique to further this purpose.

M issouri Research




ResourcesWeka Toolkit – freeware downloadable software th

many data mining algorithms.

See http://www.cs.waikato.ac.nz/~ml/weka/index.htm

Also, see SAS and SPSS websites for two prominent

statistical packages with recent additions of data mi

modules.

SAS – www.sas.com

SPSS – www.spss.com



© Osmar R. Zaïane, 1999 CMPUT690 Principles of Knowledge Discovery in Databases

University of Alberta page 1 Department of Computing Science

Chapter I: Introduction to Data Mining

We are in an age often referred to as the information age. In this information age,because we believe that information leads to power and success, and thanks tosophisticated technologies such as computers, satellites, etc., we have been collecting

tremendous amounts of information. Initially, with the advent of computers and meansfor mass digital storage, we started collecting and storing all sorts of data, counting on thepower of computers to help sort through this amalgam of information. Unfortunately,these massive collections of data stored on disparate structures very rapidly becameoverwhelming. This initial chaos has led to the creation of structured databases anddatabase management systems (DBMS). The efficient database management systemshave been very important assets for management of a large corpus of data and especiallyfor effective and efficient retrieval of particular information from a large collectionwhenever needed. The proliferation of database management systems has alsocontributed to recent massive gathering of all sorts of information. Today, we have farmore information than we can handle: from business transactions and scientific data, to

satellite pictures, text reports and military intelligence. Information retrieval is simply notenough anymore for decision-making. Confronted with huge collections of data, we havenow created new needs to help us make better managerial choices. These needs areautomatic summarization of data, extraction of the “essence” of information stored, andthe discovery of patterns in raw data.

What kind of information are we collecting?

We have been collecting a myriad of data, from simple numerical measurements and textdocuments, to more complex information such as spatial data, multimedia channels, andhypertext documents. Here is a non-exclusive list of a variety of information collected in

digital form in databases and in flat files.

• Business transactions: Every transaction in the business industry is (often)“memorized” for perpetuity. Such transactions are usually time related and canbe inter-business deals such as purchases, exchanges, banking, stock, etc., orintra-business operations such as management of in-house wares and assets. Largedepartment stores, for example, thanks to the widespread use of bar codes, storemillions of transactions daily representing often terabytes of data. Storage space isnot the major problem, as the price of hard disks is continuously dropping, but theeffective use of the data in a reasonable time frame for competitive decision-making is definitely the most important problem to solve for businesses that

struggle to survive in a highly competitive world.• Scientific data: Whether in a Swiss nuclear accelerator laboratory counting

particles, in the Canadian forest studying readings from a grizzly bear radio collar,on a South Pole iceberg gathering data about oceanic activity, or in an Americanuniversity investigating human psychology, our society is amassing colossalamounts of scientific data that need to be analyzed. Unfortunately, we can captureand store more new data faster than we can analyze the old data alreadyaccumulated.





• Medical and personal data: From government census to personnel and customerfiles, very large collections of information are continuously gathered aboutindividuals and groups. Governments, companies and organizations such ashospitals, are stockpiling very important quantities of personal data to help themmanage human resources, better understand a market, or simply assist clientele.

Regardless of the privacy issues this type of data often reveals, this information iscollected, used and even shared. When correlated with other data this informationcan shed light on customer behaviour and the like.

• Surveillance video and pictures: With the amazing collapse of video cameraprices, video cameras are becoming ubiquitous. Video tapes from surveillancecameras are usually recycled and thus the content is lost. However, there is atendency today to store the tapes and even digitize them for future use andanalysis.

• Satellite sensing: There is a countless number of satellites around the globe:some are geo-stationary above a region, and some are orbiting around the Earth,but all are sending a non-stop stream of data to the surface. NASA, which controls

a large number of satellites, receives more data every second than what all NASAresearchers and engineers can cope with. Many satellite pictures and data aremade public as soon as they are received in the hopes that other researchers cananalyze them.

• Games: Our society is collecting a tremendous amount of data and statistics aboutgames, players and athletes. From hockey scores, basketball passes and car-racinglapses, to swimming times, boxer’s pushes and chess positions, all the data arestored. Commentators and journalists are using this information for reporting, buttrainers and athletes would want to exploit this data to improve performance andbetter understand opponents.

• Digital media: The proliferation of cheap scanners, desktop video cameras and

digital cameras is one of the causes of the explosion in digital media repositories.In addition, many radio stations, television channels and film studios aredigitizing their audio and video collections to improve the management of theirmultimedia assets. Associations such as the NHL and the NBA have alreadystarted converting their huge game collection into digital forms.

• CAD and Software engineering data: There are a multitude of ComputerAssisted Design (CAD) systems for architects to design buildings or engineers toconceive system components or circuits. These systems are generating atremendous amount of data. Moreover, software engineering is a source of considerable similar data with code, function libraries, objects, etc., which needpowerful tools for management and maintenance.

• Virtual Worlds: There are many applications making use of three-dimensionalvirtual spaces. These spaces and the objects they contain are described withspecial languages such as VRML. Ideally, these virtual spaces are described insuch a way that they can share objects and places. There is a remarkable amountof virtual reality object and space repositories available. Management of theserepositories as well as content-based search and retrieval from these repositoriesare still research issues, while the size of the collections continues to grow.





• Text reports and memos (e-mail messages): Most of the communications withinand between companies or research organizations or even private people, arebased on reports and memos in textual forms often exchanged by e-mail. Thesemessages are regularly stored in digital form for future use and reference creatingformidable digital libraries.

• The World Wide Web repositories: Since the inception of the World Wide Webin 1993, documents of all sorts of formats, content and description have beencollected and inter-connected with hyperlinks making it the largest repository of data ever built. Despite its dynamic and unstructured nature, its heterogeneouscharacteristic, and its very often redundancy and inconsistency, the World WideWeb is the most important data collection regularly used for reference because of the broad variety of topics covered and the infinite contributions of resources andpublishers. Many believe that the World Wide Web will become the compilationof human knowledge.

What are Data Mining and Knowledge Discovery?

With the enormous amount of data stored in files, databases, and other repositories, it isincreasingly important, if not necessary, to develop powerful means for analysis andperhaps interpretation of such data and for the extraction of interesting knowledge thatcould help in decision-making. Data Mining, also popularly known as Knowledge Discovery in Databases (KDD), refersto the nontrivial extraction of implicit, previously unknown and potentially usefulinformation from data in databases. While data mining and knowledge discovery indatabases (or KDD) are frequently treated as synonyms, data mining is actually part of the knowledge discovery process. The following figure (Figure 1.1) shows data mining asa step in an iterative knowledge discovery process.

ata

Cleaning

ata Integration

Databases

Data

Warehouse

Task-relevant

Data

Selection and

Transformation

Pattern

Evaluation

Figure 1.1: Data Mining is the core of Knowledge Discovery process





The Knowledge Discovery in Databases process comprises of a few steps leading fromraw data collections to some form of new knowledge. The iterative process consists of

the following steps:

• Data cleaning: also known as data cleansing, it is a phase in which noise data andirrelevant data are removed from the collection.

• Data integration: at this stage, multiple data sources, often heterogeneous, maybe combined in a common source.

• Data selection: at this step, the data relevant to the analysis is decided on andretrieved from the data collection.

• Data transformation: also known as data consolidation, it is a phase in which theselected data is transformed into forms appropriate for the mining procedure.

• Data mining: it is the crucial step in which clever techniques are applied to

extract patterns potentially useful.• Pattern evaluation: in this step, strictly interesting patterns representing

knowledge are identified based on given measures.

• Knowledge representation: is the final phase in which the discovered knowledgeis visually represented to the user. This essential step uses visualizationtechniques to help users understand and interpret the data mining results.

It is common to combine some of these steps together. For instance, data cleaning anddata integration can be performed together as a pre-processing phase to generate a datawarehouse. Data selection and data transformation can also be combined where theconsolidation of the data is the result of the selection, or, as for the case of data

warehouses, the selection is done on transformed data.

The KDD is an iterative process. Once the discovered knowledge is presented to the user,the evaluation measures can be enhanced, the mining can be further refined, new data canbe selected or further transformed, or new data sources can be integrated, in order to getdifferent, more appropriate results.

Data mining derives its name from the similarities between searching for valuableinformation in a large database and mining rocks for a vein of valuable ore. Both implyeither sifting through a large amount of material or ingeniously probing the material toexactly pinpoint where the values reside. It is, however, a misnomer, since mining for

gold in rocks is usually called “gold mining” and not “rock mining”, thus by analogy,data mining should have been called “knowledge mining” instead. Nevertheless, datamining became the accepted customary term, and very rapidly a trend that evenovershadowed more general terms such as knowledge discovery in databases (KDD) thatdescribe a more complete process. Other similar terms referring to data mining are: datadredging, knowledge extraction and pattern discovery.





What kind of Data can be mined?

In principle, data mining is not specific to one type of media or data. Data mining shouldbe applicable to any kind of information repository. However, algorithms and approachesmay differ when applied to different types of data. Indeed, the challenges presented by

different types of data vary significantly. Data mining is being put into use and studiedfor databases, including relational databases, object-relational databases and object-oriented databases, data warehouses, transactional databases, unstructured and semi-structured repositories such as the World Wide Web, advanced databases such as spatialdatabases, multimedia databases, time-series databases and textual databases, and evenflat files. Here are some examples in more detail:

• Flat files: Flat files are actually the most common data source for data miningalgorithms, especially at the research level. Flat files are simple data files in textor binary format with a structure known by the data mining algorithm to beapplied. The data in these files can be transactions, time-series data, scientific

measurements, etc.

• Relational Databases: Briefly, a relational database consists of a set of tablescontaining either values of entity attributes, or values of attributes from entityrelationships. Tables have columns and rows, where columns represent attributesand rows represent tuples. A tuple in a relational table corresponds to either anobject or a relationship between objects and is identified by a set of attributevalues representing a unique key. In Figure 1.2 we present some relationsCustomer , Items, and Borrow representing business activity in a fictitious videostore OurVideoStore. These relations are just a subset of what could be adatabase for the video store and is given as an example.

customerID date itemID # …

C1234 99/09/06 98765 1 …

Borrow

. . .

customerID name address birthdatepassword family_income group …

C1234 John Smith 120 main street Marty 1965/10/10 $45000 A …

Customer

. . .

itemID type title categorymedia Value # …

98765 Video Titanic DVD Drama 15.00 2 …

Items

. . .

Figure 1.2: Fragments of some relations from a relational database for OurVideoStore.





The most commonly used query language for relational database is SQL, whichallows retrieval and manipulation of the data stored in the tables, as well as thecalculation of aggregate functions such as average, sum, min, max and count. Forinstance, an SQL query to select the videos grouped by category would be:SELECT count(*) FROM Items WHERE type=video GROUP BY category.

Data mining algorithms using relational databases can be more versatile than datamining algorithms specifically written for flat files, since they can take advantageof the structure inherent to relational databases. While data mining can benefitfrom SQL for data selection, transformation and consolidation, it goes beyondwhat SQL could provide, such as predicting, comparing, detecting deviations, etc.

• Data Warehouses: A data warehouse as a storehouse, is a repository of datacollected from multiple data sources (often heterogeneous) and is intended to beused as a whole under the same unified schema. A data warehouse gives the

option to analyze data from different sources under the same roof. Let us supposethat OurVideoStore becomes a franchise in North America. Many video storesbelonging to OurVideoStore company may have different databases and differentstructures. If the executive of the company wants to access the data from all storesfor strategic decision-making, future direction, marketing, etc., it would be moreappropriate to store all the data in one site with a homogeneous structure thatallows interactive analysis. In other words, data from the different stores would beloaded, cleaned, transformed and integrated together. To facilitate decision-making and multi-dimensional views, data warehouses are usually modeled by amulti-dimensional data structure. Figure 1.3 shows an example of a threedimensional subset of a data cube structure used for OurVideoStore data

warehouse.

DramaComedy

Horror

Category

Sum

Group By

Sum

Aggregate

DramaComedy

Horror

Q4Q1

By Time

By Category

Sum

Cross Tab

Q3Q2

Q1Q2Red Deer

Edmonton

Drama

Comedy

Horror

By Category

By Time & Category

By Time & City

By Category & City

By Time By City

Sum

The Data Cube and The Sub-Space Aggregates

LethbridgeCalgary

Q3Q4

Figure 1.3: A multi-dimensional data cube structure commonly used in data for data warehousing.





The figure shows summarized rentals grouped by film categories, then a crosstable of summarized rentals by film categories and time (in quarters). The datacube gives the summarized rentals along three dimensions: category, time, andcity. A cube contains cells that store values of some aggregate measures (in thiscase rental counts), and special cells that store summations along dimensions.

Each dimension of the data cube contains a hierarchy of values for one attribute.

Because of their structure, the pre-computed summarized data they contain andthe hierarchical attribute values of their dimensions, data cubes are well suited forfast interactive querying and analysis of data at different conceptual levels, knownas On-Line Analytical Processing (OLAP). OLAP operations allow the navigationof data at different levels of abstraction, such as drill-down, roll-up, slice, dice,etc. Figure 1.4 illustrates the drill-down (on the time dimension) and roll-up (onthe location dimension) operations.

• Transaction Databases: A transaction database is a set of records representingtransactions, each with a time stamp, an identifier and a set of items. Associatedwith the transaction files could also be descriptive data for the items. Forexample, in the case of the video store, the rentals table such as shown in Figure1.5, represents the transaction database. Each record is a rental contract with acustomer identifier, a date, and the list of items rented (i.e. video tapes, games,VCR, etc.). Since relational databases do not allow nested tables (i.e. a set asattribute value), transactions are usually stored in flat files or stored in twonormalized transaction tables, one for the transactions and one for the transactionitems. One typical data mining analysis on such data is the so-called market

Red Deer

Q1 Q2 Q4

Drama

Horror

Sci. Fi..

Comedy

Time (Quarters)

Location(city, AB)

Category

Q3Edmonton

Calgary Lethbridge

Red Deer

Jul Se

Drama

Horror

Sci. Fi..

Comedy

Time (Months, Q3)

Location(city, AB)

Category

AugEdmonton

Calgary Lethbridge

Drill down on Q3

Roll-up on Location

Prairies

Q1 Q2 Q4

Drama

HorrorSci. Fi..

Comedy

Time (Quarters)

Location(province,

Canada) Category

Q3Maritimes

Quebec Ontario

Western Pr

Figure 1.4: Summarized data from OurVideoStore before and after drill-down and roll-up operations.





basket analysis or association rules in which associations between items occurringtogether or in sequence are studied.

• Multimedia Databases: Multimedia databases include video, images, audio andtext media. They can be stored on extended object-relational or object-oriented

databases, or simply on a file system. Multimedia is characterized by its highdimensionality, which makes data mining even more challenging. Data miningfrom multimedia repositories may require computer vision, computer graphics,image interpretation, and natural language processing methodologies.

• Spatial Databases: Spatial databases are databases that, in addition to usual data,store geographical information like maps, and global or regional positioning. Suchspatial databases present new challenges to data mining algorithms.

•Time-Series Databases: Time-series databases contain time related data suchstock market data or logged activities. These databases usually have a continuousflow of new data coming in, which sometimes causes the need for a challengingreal time analysis. Data mining in such databases commonly includes the study of trends and correlations between evolutions of different variables, as well as theprediction of trends and movements of the variables in time. Figure 1.7 showssome examples of time-series data.

transactionID date time customerID itemList

T12345 99/09/06 19:38 C1234 {I2, I6, I10, I45 …}

Rentals

. . .

Figure 1.5: Fragment of a transaction database for the rentals at OurVideoStore.

Figure 1.6: Visualization of spatial OLAP (from GeoMiner system)





• World Wide Web: The World Wide Web is the most heterogeneous and dynamicrepository available. A very large number of authors and publishers arecontinuously contributing to its growth and metamorphosis, and a massivenumber of users are accessing its resources daily. Data in the World Wide Web isorganized in inter-connected documents. These documents can be text, audio,

video, raw data, and even applications. Conceptually, the World Wide Web iscomprised of three major components: The content of the Web, whichencompasses documents available; the structure of the Web, which covers thehyperlinks and the relationships between documents; and the usage of the web,describing how and when the resources are accessed. A fourth dimension can beadded relating the dynamic nature or evolution of the documents. Data mining inthe World Wide Web, or web mining, tries to address all these issues and is oftendivided into web content mining, web structure mining and web usage mining.

What can be discovered?

The kinds of patterns that can be discovered depend upon the data mining tasksemployed. By and large, there are two types of data mining tasks: descriptive data mining tasks that describe the general properties of the existing data, and predictive data mining tasks that attempt to do predictions based on inference on available data.

The data mining functionalities and the variety of knowledge they discover are brieflypresented in the following list:

Figure 1.7: Examples of Time-Series Data (Source: Thompson Investors Group)





• Characterization: Data characterization is a summarization of general features of objects in a target class, and produces what is called characteristic rules. The datarelevant to a user-specified class are normally retrieved by a database query andrun through a summarization module to extract the essence of the data at differentlevels of abstractions. For example, one may want to characterize the

OurVideoStore customers who regularly rent more than 30 movies a year. Withconcept hierarchies on the attributes describing the target class, the attribute-

oriented induction method can be used, for example, to carry out datasummarization. Note that with a data cube containing summarization of data,simple OLAP operations fit the purpose of data characterization.

• Discrimination: Data discrimination produces what are called discriminant rules and is basically the comparison of the general features of objects between twoclasses referred to as the target class and the contrasting class. For example, onemay want to compare the general characteristics of the customers who rentedmore than 30 movies in the last year with those whose rental account is lowerthan 5. The techniques used for data discrimination are very similar to the

techniques used for data characterization with the exception that datadiscrimination results include comparative measures.

• Association analysis: Association analysis is the discovery of what arecommonly called association rules. It studies the frequency of items occurringtogether in transactional databases, and based on a threshold called support ,identifies the frequent item sets. Another threshold, confidence, which is theconditional probability than an item appears in a transaction when another itemappears, is used to pinpoint association rules. Association analysis is commonlyused for market basket analysis. For example, it could be useful for theOurVideoStore manager to know what movies are often rented together or if thereis a relationship between renting a certain type of movies and buying popcorn or

pop. The discovered association rules are of the form: P→Q [s,c], where P and Qare conjunctions of attribute value-pairs, and s (for support) is the probability thatP and Q appear together in a transaction and c (for confidence) is the conditionalprobability that Q appears in a transaction when P is present. For example, thehypothetic association rule:

RentType(X, “game”) ∧ Age(X, “13-19”) → Buys(X, “pop”) [s=2% ,c=55%] would indicate that 2% of the transactions considered are of customers agedbetween 13 and 19 who are renting a game and buying a pop, and that there is acertainty of 55% that teenage customers who rent a game also buy pop.

• Classification: Classification analysis is the organization of data in given classes.Also known as supervised classification, the classification uses given class labels

to order the objects in the data collection. Classification approaches normally usea training set where all objects are already associated with known class labels.The classification algorithm learns from the training set and builds a model. Themodel is used to classify new objects. For example, after starting a credit policy,the OurVideoStore managers could analyze the customers’ behaviours vis-à-vistheir credit, and label accordingly the customers who received credits with threepossible labels “safe”, “risky” and “very risky”. The classification analysis would





generate a model that could be used to either accept or reject credit requests in thefuture.

• Prediction: Prediction has attracted considerable attention given the potentialimplications of successful forecasting in a business context. There are two majortypes of predictions: one can either try to predict some unavailable data values or

pending trends, or predict a class label for some data. The latter is tied toclassification. Once a classification model is built based on a training set, the classlabel of an object can be foreseen based on the attribute values of the object andthe attribute values of the classes. Prediction is however more often referred to theforecast of missing numerical values, or increase/ decrease trends in time relateddata. The major idea is to use a large number of past values to consider probablefuture values.

• Clustering: Similar to classification, clustering is the organization of data inclasses. However, unlike classification, in clustering, class labels are unknownand it is up to the clustering algorithm to discover acceptable classes. Clustering isalso called unsupervised classification, because the classification is not dictated

by given class labels. There are many clustering approaches all based on theprinciple of maximizing the similarity between objects in a same class (intra-class

similarity) and minimizing the similarity between objects of different classes(inter-class similarity).

• Outlier analysis: Outliers are data elements that cannot be grouped in a givenclass or cluster. Also known as exceptions or surprises, they are often veryimportant to identify. While outliers can be considered noise and discarded insome applications, they can reveal important knowledge in other domains, andthus can be very significant and their analysis valuable.

• Evolution and deviation analysis: Evolution and deviation analysis pertain to thestudy of time related data that changes in time. Evolution analysis models

evolutionary trends in data, which consent to characterizing, comparing,classifying or clustering of time related data. Deviation analysis, on the otherhand, considers differences between measured values and expected values, andattempts to find the cause of the deviations from the anticipated values.

It is common that users do not have a clear idea of the kind of patterns they can discoveror need to discover from the data at hand. It is therefore important to have a versatile andinclusive data mining system that allows the discovery of different kinds of knowledgeand at different levels of abstraction. This also makes interactivity an important attributeof a data mining system.

Is all that is discovered interesting and useful?

Data mining allows the discovery of knowledge potentially useful and unknown. Whetherthe knowledge discovered is new, useful or interesting, is very subjective and dependsupon the application and the user. It is certain that data mining can generate, or discover,a very large number of patterns or rules. In some cases the number of rules can reach themillions. One can even think of a meta-mining phase to mine the oversized data miningresults. To reduce the number of patterns or rules discovered that have a high probability





to be non-interesting, one has to put a measurement on the patterns. However, this raisesthe problem of completeness. The user would want to discover all rules or patterns, but

only those that are interesting. The measurement of how interesting a discovery is, oftencalled interestingness, can be based on quantifiable objective elements such as validity of the patterns when tested on new data with some degree of certainty, or on some

subjective depictions such as understandability of the patterns, novelty of the patterns, orusefulness.

Discovered patterns can also be found interesting if they confirm or validate a hypothesissought to be confirmed or unexpectedly contradict a common belief. This brings the issueof describing what is interesting to discover, such as meta-rule guided discovery thatdescribes forms of rules before the discovery process, and interestingness refinementlanguages that interactively query the results for interesting patterns after the discoveryphase. Typically, measurements for interestingness are based on thresholds set by theuser. These thresholds define the completeness of the patterns discovered.

Identifying and measuring the interestingness of patterns and rules discovered, or to bediscovered, is essential for the evaluation of the mined knowledge and the KDD processas a whole. While some concrete measurements exist, assessing the interestingness of discovered knowledge is still an important research issue.

How do we categorize data mining systems?

There are many data mining systems available or being developed. Some are specializedsystems dedicated to a given data source or are confined to limited data miningfunctionalities, other are more versatile and comprehensive. Data mining systems can becategorized according to various criteria among other classification are the following:

• Classification according to the type of data source mined: this classificationcategorizes data mining systems according to the type of data handled such asspatial data, multimedia data, time-series data, text data, World Wide Web, etc.

• Classification according to the data model drawn on: this classificationcategorizes data mining systems based on the data model involved such asrelational database, object-oriented database, data warehouse, transactional, etc.

• Classification according to the king of knowledge discovered: thisclassification categorizes data mining systems based on the kind of knowledgediscovered or data mining functionalities, such as characterization, discrimination,association, classification, clustering, etc. Some systems tend to be

comprehensive systems offering several data mining functionalities together.• Classification according to mining techniques used: Data mining systems

employ and provide different techniques. This classification categorizes datamining systems according to the data analysis approach used such as machinelearning, neural networks, genetic algorithms, statistics, visualization, database-oriented or data warehouse-oriented, etc. The classification can also take intoaccount the degree of user interaction involved in the data mining process such asquery-driven systems, interactive exploratory systems, or autonomous systems. A





comprehensive system would provide a wide variety of data mining techniques tofit different situations and options, and offer different degrees of user interaction.

What are the issues in Data Mining?

Data mining algorithms embody techniques that have sometimes existed for many years,but have only lately been applied as reliable and scalable tools that time and againoutperform older classical statistical methods. While data mining is still in its infancy, itis becoming a trend and ubiquitous. Before data mining develops into a conventional,mature and trusted discipline, many still pending issues have to be addressed. Some of these issues are addressed below. Note that these issues are not exclusive and are notordered in any way.

Security and social issues: Security is an important issue with any data collection that isshared and/or is intended to be used for strategic decision-making. In addition, when datais collected for customer profiling, user behaviour understanding, correlating personal

data with other information, etc., large amounts of sensitive and private informationabout individuals or companies is gathered and stored. This becomes controversial giventhe confidential nature of some of this data and the potential illegal access to theinformation. Moreover, data mining could disclose new implicit knowledge aboutindividuals or groups that could be against privacy policies, especially if there is potentialdissemination of discovered information. Another issue that arises from this concern isthe appropriate use of data mining. Due to the value of data, databases of all sorts of content are regularly sold, and because of the competitive advantage that can be attainedfrom implicit knowledge discovered, some important information could be withheld,while other information could be widely distributed and used without control.

User interface issues: The knowledge discovered by data mining tools is useful as longas it is interesting, and above all understandable by the user. Good data visualizationeases the interpretation of data mining results, as well as helps users better understandtheir needs. Many data exploratory analysis tasks are significantly facilitated by theability to see data in an appropriate visual presentation. There are many visualizationideas and proposals for effective data graphical presentation. However, there is still muchresearch to accomplish in order to obtain good visualization tools for large datasets thatcould be used to display and manipulate mined knowledge. The major issues related touser interfaces and visualization are “screen real-estate”, information rendering, andinteraction. Interactivity with the data and data mining results is crucial since it providesmeans for the user to focus and refine the mining tasks, as well as to picture the

discovered knowledge from different angles and at different conceptual levels.

Mining methodology issues: These issues pertain to the data mining approaches appliedand their limitations. Topics such as versatility of the mining approaches, the diversity of data available, the dimensionality of the domain, the broad analysis needs (when known),the assessment of the knowledge discovered, the exploitation of background knowledgeand metadata, the control and handling of noise in data, etc. are all examples that candictate mining methodology choices. For instance, it is often desirable to have different





data mining methods available since different approaches may perform differentlydepending upon the data at hand. Moreover, different approaches may suit and solveuser’s needs differently.

Most algorithms assume the data to be noise-free. This is of course a strong assumption.

Most datasets contain exceptions, invalid or incomplete information, etc., which maycomplicate, if not obscure, the analysis process and in many cases compromise theaccuracy of the results. As a consequence, data preprocessing (data cleaning andtransformation) becomes vital. It is often seen as lost time, but data cleaning, as time-consuming and frustrating as it may be, is one of the most important phases in theknowledge discovery process. Data mining techniques should be able to handle noise indata or incomplete information.

More than the size of data, the size of the search space is even more decisive for datamining techniques. The size of the search space is often depending upon the number of dimensions in the domain space. The search space usually grows exponentially when the

number of dimensions increases. This is known as the curse of dimensionality. This“curse” affects so badly the performance of some data mining approaches that it isbecoming one of the most urgent issues to solve.

Performance issues: Many artificial intelligence and statistical methods exist for dataanalysis and interpretation. However, these methods were often not designed for the verylarge data sets data mining is dealing with today. Terabyte sizes are common. This raisesthe issues of scalability and efficiency of the data mining methods when processingconsiderably large data. Algorithms with exponential and even medium-order polynomialcomplexity cannot be of practical use for data mining. Linear algorithms are usually thenorm. In same theme, sampling can be used for mining instead of the whole dataset.However, concerns such as completeness and choice of samples may arise. Other topicsin the issue of performance are incremental updating, and parallel programming. Thereis no doubt that parallelism can help solve the size problem if the dataset can besubdivided and the results can be merged later. Incremental updating is important formerging results from parallel mining, or updating data mining results when new databecomes available without having to re-analyze the complete dataset.

Data source issues: There are many issues related to the data sources, some are practicalsuch as the diversity of data types, while others are philosophical like the data glutproblem. We certainly have an excess of data since we already have more data than wecan handle and we are still collecting data at an even higher rate. If the spread of databasemanagement systems has helped increase the gathering of information, the advent of datamining is certainly encouraging more data harvesting. The current practice is to collect asmuch data as possible now and process it, or try to process it, later. The concern iswhether we are collecting the right data at the appropriate amount, whether we knowwhat we want to do with it, and whether we distinguish between what data is importantand what data is insignificant. Regarding the practical issues related to data sources, thereis the subject of heterogeneous databases and the focus on diverse complex data types.We are storing different types of data in a variety of repositories. It is difficult to expect a





data mining system to effectively and efficiently achieve good mining results on all kindsof data and sources. Different kinds of data and sources may require distinct algorithmsand methodologies. Currently, there is a focus on relational databases and datawarehouses, but other approaches need to be pioneered for other specific complex datatypes. A versatile data mining tool, for all sorts of data, may not be realistic. Moreover,

the proliferation of heterogeneous data sources, at structural and semantic levels, posesimportant challenges not only to the database community but also to the data miningcommunity.

References

• M. S. Chen, J. Han, and P. S. Yu. Data mining: An overview from a database

perspective. IEEE Trans. Knowledge and Data Engineering, 8:866-883, 1996.

• U. M. Fayyad, G. Piatetsky-Shapiro, P. Smyth, and R. Uthurusamy.

Advances in Knowledge Discovery and Data Mining. AAAI/MIT Press, 1996.

• W. J. Frawley, G. Piatetsky-Shapiro and C. J. Matheus, Knowledge

Discovery in Databases: An Overview. In G. Piatetsky-Shapiro et al. (eds.),

Knowledge Discovery in Databases. AAAI/MIT Press, 1991.

• J. Han and M. Kamber. Data Mining: Concepts and Techniques. Morgan

Kaufmann, 2000.

• T. Imielinski and H. Mannila. A database perspective on knowledge

discovery. Communications of ACM, 39:58-64, 1996.

• G. Piatetsky-Shapiro, U. M. Fayyad, and P. Smyth. From data mining toknowledge discovery: An overview. In U.M. Fayyad, et al. (eds.), Advances in

Knowledge Discovery and Data Mining, 1-35. AAAI/MIT Press, 1996.

• G. Piatetsky-Shapiro and W. J. Frawley. Knowledge Discovery in Databases.

AAAI/MIT Press, 1991.



1

An Introduction to Cluster Analysis for Data Mining

10/02/2000 11:42 AM

1. INTRODUCTION......................................................................................... 4

1.1. Scope of This Paper ..........................................................................................................................4

1.2. What Cluster Analysis Is..................................................................................................................4

1.3. What Cluster Analysis Is Not...........................................................................................................5

2. OVERVIEW................................................................................................. 5

2.1. Definitions..........................................................................................................................................5 2.1.1. The Data Matrix.......................................................................................................................... 5 2.1.2. The Proximity Matrix..................................................................................................................6

2.1.3. The Proximity Graph .................................................................................................................. 7 2.1.4. Some Working Definitions of a Cluster...................................................................................... 7

2.2. Measures (Indices) of Similarity and Dissimilarity.............. ........... ........... ............ ........... ........... .. 9 2.2.1. Proximity Types and Scales........................................................................................................ 9 2.2.2. Requirements on Similarity and Dissimilarity Measures........... ........... ........... .......... ........... .... 10 2.2.3. Common Proximity Measures .................................................................................................. 11

2.2.3.1. Distance Measures ............................................................................................................11 2.2.3.2. Ordinal Measures .............................................................................................................. 12 2.2.3.3. Similarity Measures Between Binary Vectors ..................................................................12 2.2.3.4. Cosine Measure................................................................................................................. 13

3. BASIC CLUSTERING TECHNIQUES ...................................................... 13

3.1. Types of Clustering.........................................................................................................................13

3.2. Objective Functions: Clustering as an Optimization Problem ........... ........... ............ ........... ..... 15

4. SPECIFIC CLUSTERING TECHNIQUES................................................. 15

4.1. Center-Based Partitional Clustering............................................................................................. 15 4.1.1. K-means Clustering .................................................................................................................. 16

4.1.1.1. Basic Algorithm ................................................................................................................16 4.1.1.2. Time and Space Complexity .............................................................................................16 4.1.1.3. Choosing initial centroids ................................................................................................. 17 4.1.1.4. Updating Centroids Incrementally .................................................................................... 18 4.1.1.5. Different Definitions of Centroid...................................................................................... 19 4.1.1.6. Pre and Post Processing .................................................................................................... 20 4.1.1.7. Limitations and problems..................................................................................................22

4.1.2. K-medoid Clustering.................................................................................................................22 4.1.3. CLARANS................................................................................................................................ 23

4.2. Hierarchical Clustering.................................................................................................................. 24 4.2.1. Agglomeration and Division..................................................................................................... 24



2

4.2.2. Simple Divisive Algorithm (Minimum Spanning Tree (MST) )......... ........... ........... ........... .... 24 4.2.3. Basic Agglomerative Hierarchical Clustering Algorithm......................................................... 25 4.2.4. Time and Space Complexity ..................................................................................................... 25 4.2.5. MIN or Single Link................................................................................................................... 26 4.2.6. MAX or Complete Link or CLIQUE........................................................................................26 4.2.7. Group Average..........................................................................................................................26 4.2.8. Ward’s Method and Centroid methods .....................................................................................27 4.2.9. Key Issues in Hierarchical Clustering.......................................................................................27

4.2.9.1. Lack of a Global Objective Function ................................................................................27 4.2.9.2. Cluster Proximity and Cluster Representation ..................................................................28 4.2.9.3. The Impact of Cluster Size................................................................................................28 4.2.9.4. Merging Decisions are Final .............................................................................................28

4.2.10. Limitations and problems..........................................................................................................29 4.2.11. CURE........................................................................................................................................29 4.2.12. Chameleon ................................................................................................................................ 30

4.3. Density Based Clustering................................................................................................................32 4.3.1. DBSCAN.................................................................................................................................. 32 4.3.2. DENCLUE................................................................................................................................ 33 4.3.3. WaveCluster..............................................................................................................................34

4.4. Graph-Based Clustering................................................................................................................. 35 4.4.1. Historical techniques................................................................................................................. 35 4.4.1.1. Shared Nearest Neighbor Clustering..................................................................................... 35 4.4.1.2. Mutual Nearest Neighbor Clustering .................................................................................... 36 4.4.2. Hypergraph-Based Clustering................................................................................................... 36 4.4.3. ROCK .......................................................................................................................................38 4.4.4. Sparsification ............................................................................................................................39

4.4.4.1. Why Sparsify?...................................................................................................................39 4.4.4.2. Sparsification as a Pre-processing Step.............. .......... ........... ........... .......... ........... .......... 40 4.4.4.3. Sparsification Techniques.................................................................................................41 4.4.4.4. Keeping the Proximity Matrix Symmetric? ...................................................................... 41 4.4.4.5. Description of Sparsification Techniques ........................................................................ 41

4.4.4.6. Evaluating Sparsification Results...................................................................................... 42 4.4.4.7. Tentative Conclusions....................................................................................................... 43

5. ADVANCED TECHNIQUES FOR SPECIFIC ISSUES ............................. 43

5.1. Scalability ........................................................................................................................................43 5.1.1. BIRCH...................................................................................................................................... 43 5.1.2. Bubble and Bubble-FM.............................................................................................................45

5.2. Clusters in Subspaces .....................................................................................................................46 5.2.1. CLIQUE.................................................................................................................................... 46 5.2.2. MAFIA (pMAFIA) ................................................................................................................... 47

6. EVALUATIONS ........................................................................................ 47

6.1. Requirements for Clustering Analysis for Data Mining............... ........... ........... ............ ........... .. 47

6.2. Typical Problems and Desired Characteristics ............................................................................47 6.2.1. Scalability ................................................................................................................................. 47 6.2.2. Independence of the order of input ........................................................................................... 48 6.2.3. Effective means of detecting and dealing with noise or outlying points.............. ........... .......... 48



3

6.2.4. Effective means of evaluating the validity of clusters that are produced........... ........... .......... .. 48 6.2.5. Easy interpretability of results .................................................................................................. 48 6.2.6. The ability to find clusters in subspaces of the original space. .......... ........... ............ ........... ..... 48 6.2.7. The ability to handle distances in high dimensional spaces properly................. ........... ............ 48 6.2.8. Robustness in the presence of different underlying data and cluster characteristics............. .... 49 6.2.9. An ability to estimate any parameters.......................................................................................50 6.2.10. Ability to function in an incremental manner ...........................................................................50

7. DATA MINING FROM OTHER PERSPECTIVES..................................... 50

7.1. Statistics-Based Data Mining.........................................................................................................50 7.1.1. Mixture Models......................................................................................................................... 50

7.2. Text Mining and Information Retrieval ....................................................................................... 51 7.2.1. Uses of Clustering for Documents ............................................................................................51 7.2.2. Latent Semantic Indexing .........................................................................................................52

7.3. Vector Quantization........................................................................................................................52

7.4. Self-Organizing Maps.....................................................................................................................53 7.4.1. SOM Overview .........................................................................................................................53 7.4.2. Details of a Two-dimensional SOM .........................................................................................53 7.4.3. Applications.............................................................................................................................. 55

7.5. Fuzzy Clustering ............................................................................................................................. 56

8. RELATED TOPICS................................................................................... 59

8.1. Nearest-neighbor search (Multi-Dimensional Access Methods) ........... ........... ........... ........... ..... 59

8.2. Pattern Recognition........................................................................................................................61

8.3. Exploratory Data Analysis .............................................................................................................61

9. SPECIAL TOPICS .................................................................................... 62

9.1. Missing Values................................................................................................................................. 62

9.2. Combining Attributes..................................................................................................................... 62 9.2.1. Monothetic or Polythetic Clustering .........................................................................................62 9.2.2. Assigning Weights to features ..................................................................................................62

10. CONCLUSIONS........................................................................................ 62

LIST OF ARTICLES AND BOOKS FOR CLUSTERING FOR DATA MINING.. 64



4

1. Introduction

1.1. Scope of This Paper

Cluster analysis divides data into meaningful or useful groups (clusters). If

meaningful clusters are the goal, then the resulting clusters should capture the “natural”structure of the data. For example, cluster analysis has been used to group related

documents for browsing, to find genes and proteins that have similar functionality, and to provide a grouping of spatial locations prone to earthquakes. However, in other cases,

cluster analysis is only a useful starting point for other purposes, e.g., data compression

or efficiently finding the nearest neighbors of points. Whether for understanding or utility, cluster analysis has long been used in a wide variety of fields: psychology and

other social sciences, biology, statistics, pattern recognition, information retrieval,

machine learning, and data mining.

The scope of this paper is modest: to provide an introduction to cluster analysis inthe field of data mining, where we define data mining to be the discovery of useful, but

non-obvious, information or patterns in large collections of data. Much of this paper isnecessarily consumed with providing a general background for cluster analysis, but wealso discuss a number of clustering techniques that have recently been developed

specifically for data mining. While the paper strives to be self-contained from a

conceptual point of view, many details have been omitted. Consequently, manyreferences to relevant books and papers are provided.

1.2. What Cluster Analysis Is

Cluster analysis groups objects (observations, events) based on the information

found in the data describing the objects or their relationships. The goal is that the objects

in a group will be similar (or related) to one other and different from (or unrelated to) the

objects in other groups. The greater the similarity (or homogeneity) within a group, andthe greater the difference between groups, the “better” or more distinct the clustering.

The definition of what constitutes a cluster is not well defined, and, in manyapplications clusters are not well separated from one another. Nonetheless, most cluster

analysis seeks as a result, a crisp classification of the data into non-overlapping groups.

Fuzzy clustering, described in section 7.5, is an exception to this, and allows an object to

partially belong to several groups.To better understand the difficulty of deciding what constitutes a cluster, consider

figures 1a through 1d, which show twenty points and three different ways that they can

be divided into clusters. If we allow clusters to be nested, then the most reasonableinterpretation of the structure of these points is that there are two clusters, each of which

has three subclusters. However, the apparent division of the two larger clusters into threesubclusters may simply be an artifact of the human visual system. Finally, it may not beunreasonable to say that the points form four clusters. Thus, we stress once again that the

definition of what constitutes a cluster is imprecise, and the best definition depends on

the type of data and the desired results.



5

Figure 1a: Initial points.

Figure 1b: Two clusters.

Figure 1c: Six clusters

Figure 1d: Four clusters.

1.3. What Cluster Analysis Is Not

Cluster analysis is a classification of objects from the data, where by classification

we mean a labeling of objects with class (group) labels. As such, clustering does not use

previously assigned class labels, except perhaps for verification of how well theclustering worked. Thus, cluster analysis is distinct from pattern recognition or the areas

of statistics know as discriminant analysis and decision analysis, which seek to find rules

for classifying objects given a set of pre-classified objects.

While cluster analysis can be useful in the previously mentioned areas, either

directly or as a preliminary means of finding classes, there is much more to these areasthan cluster analysis. For example, the decision of what features to use when

representing objects is a key activity of fields such as pattern recognition. Cluster analysis typically takes the features as given and proceeds from there.

Thus, cluster analysis, while a useful tool in many areas (as described later), is

normally only part of a solution to a larger problem which typically involves other stepsand techniques.

2. Overview

2.1. Definitions

2.1.1. The Data MatrixObjects (samples, measurements, patterns, events) are usually represented as

points (vectors) in a multi-dimensional space, where each dimension represents a distinctattribute (variable, measurement) describing the object. For simplicity, it is normally

assumed that values are present for all attributes. (Techniques for dealing with missing

values are described in section 9.1.)Thus, a set of objects is represented (at least conceptually) as an m by n matrix,

where there are m rows, one for each object, and n columns, one for each attribute. This

matrix has different names, e.g., pattern matrix or data matrix, depending on the

particular field. Figure 2, below, provides a concrete example of some points and their

corresponding data matrix.The data is sometimes transformed before being used. One reason for this is that

different attributes may be measured on different scales, e.g., centimeters and kilograms.In cases where the range of values differs widely from attribute to attribute, these

differing attribute scales can dominate the results of the cluster analysis and it is common

to standardize the data so that all attributes are on the same scale.The following are some common approaches to data standardization:



6

(In what follows, xi, is the ith object, xij is the value of the jth attribute of the ith

object, and xij ′ is the standardized attribute value.)

a)ij

i

ij

ij x

x x

max=′ . Divide each attribute value of an object by the maximum observed

absolute value of that attribute. This restricts all attribute values to lie between –1 and

1. Often all values are positive, and thus, all transformed values lie between 0 and 1.

b) j

jij

ij

x x

σ

µ −=′ . For each attribute value subtract off the mean, µ j, of that attribute and

then divide by the attribute’s standard deviation, σ j. If the data are normally

distributed, then most attribute values will lie between –1 and 1 [KR90].

( µ j = å=

m

i

ij xm 1

1is the mean of the jth

feature, σ j = ( )

2

1

1å=

−m

i

jij xm

µ is the standard

deviation of the jthfeature.)

c) A j

jij

ij

x

x σ

µ −

=′ . For each attribute value subtract off the mean of that attribute and

divide by the attribute’s absolute deviation, σ j A

, [KR90]. Typically, most attribute

values will lie between –1 and 1. (σ j A

= å=

−m

i

jij xm 1

1 µ is the absolute standard

deviation of the jthfeature.)

The first approach, (a), may not produce good results unless the attributes are

uniformly distributed, while both the first and second approaches, (a) and (b), are

sensitive to outliers. The third approach, (c), is the most robust in the presence of outliers, although an attempt to eliminate outliers is sometimes made before cluster

analysis begins.Another reason for initially transforming the data is to reduce the number of

dimensions, particularly if the initial number of dimensions is large. (The problems

caused by high dimensionality are discussed more fully in section 6.2.7.) This can be

accomplished by discarding features that show little variation or that are highly correlated

with other features. (Feature selection is a complicated subject in its own right.)Another approach is to project points from a higher dimensional space to a lower

dimensional space. Typically this is accomplished by applying techniques from linear

algebra, which are based on the eigenvalue decomposition or the singular valuedecomposition of a data matrix or normalized data matrix. Examples of such techniques

are Principal Component Analysis [DJ88] or Latent Semantic Indexing [FO95] (section

7.2.2).

2.1.2. The Proximity MatrixWhile cluster analysis sometimes uses the original data matrix, many clustering

algorithms use a similarity matrix, S, or a dissimilarity matrix, D. For convenience, bothmatrices are commonly referred to as a proximity matrix, P. A proximity matrix, P, is anm by m matrix containing all the pairwise dissimilarities or similarities between the

objects being considered. If xi and x j are the ithand jth

objects, respectively, then the entry



7

at the ith row and jth column of the proximity matrix is the similarity, sij, or the

dissimilarity, d ij, between xi and x j. For simplicity, we will use pij to represent either sij or d ij. Figure 2 shows four points and the corresponding data and proximity (distance)matrices. More examples of dissimilarity and similarity will be provided shortly.

point x y

p1 0 2

p2 2 0

p3 3 1

p4 5 1

p1 p2 p3 p4

p1 0.000 2.828 3.162 5.099

p2 2.828 0.000 1.414 3.162

p3 3.162 1.414 0.000 2.000

p4 5.099 3.162 2.000 0.000

Figure 2. Four points and their corresponding data and proximity (distance) matrices.

For completeness, we mention that objects are sometimes represented by more

complicated data structures than vectors of attributes, e.g., character strings. Determining

the similarity (or differences) of two objects in such a situation is more complicated, butif a reasonable similarity (dissimilarity) measure exists, then a clustering analysis can still

be performed. In particular, clustering techniques that use a proximity matrix areunaffected by the lack of a data matrix.

2.1.3. The Proximity GraphA proximity matrix defines a weighted graph, where the nodes are the points

being clustered, and the weighted edges represent the proximities between points, i.e., the

entries of the proximity matrix. While this proximity graph can be directed, which

corresponds to an asymmetric proximity matrix, most clustering methods assume anundirected graph. Relaxing the symmetry requirement can be useful for clustering or

pattern recognition, but we will assume undirected proximity graphs (symmetric

proximity matrices) in our discussions.

From a graph point of view, clustering is equivalent to breaking the graph intoconnected components, one for each cluster. Likewise, many issues related to clustering

can be cast in graph-theoretic terms, e.g., the issues of cluster cohesion and the degree of

coupling with other clusters can be measured by the number and strength of links between and within clusters. Also, many clustering techniques, e.g., minimum spanning

tree (section 4.2.1), single link (section 4.2.5), and complete link (section 4.2.6), are most

naturally described using a graph representation.Some clustering algorithms, e.g., Chameleon (section 4.2.12), first sparsify the

proximity graph. Sparsification reduces the connectivity of nodes by breaking some of

the links of the proximity graph. (This corresponds to setting a proximity matrix entry to

0 or ∞, depending on whether we are using similarities or dissimilarities, respectively.)

See section 4.4.4 for details.

2.1.4. Some Working Definitions of a Cluster As mentioned above, the term, cluster, does not have a precise definition.

However, several working definitions of a cluster are commonly used.

0

1

2

3

0 1 2 3 4 5 6

p1

p2

p3 p4



8

1) Well-Separated Cluster Definition: A cluster is a set of points such that any point in

a cluster is closer (or more similar) to every other point in the cluster than to any

point not in the cluster. Sometimes a threshold is used to specify that all the points ina cluster must be sufficiently close (or similar) to one another.

Figure 3: Three well-separated clusters of 2 dimensional points.

However, in many sets of data, a point on the edge of a cluster may be closer (or more

similar) to some objects in another cluster than to objects in its own cluster.Consequently, many clustering algorithms use the following criterion.

2) Center-based Cluster Definition: A cluster is a set of objects such that an object in acluster is closer (more similar) to the “center” of a cluster, than to the center of any

other cluster. The center of a cluster is often a centroid, the average of all the points

in the cluster, or a medoid, the most “representative” point of a cluster.

Figure 4: Four center-based clusters of 2 dimensional points.

3) Contiguous Cluster Definition (Nearest neighbor or Transitive Clustering): A

cluster is a set of points such that a point in a cluster is closer (or more similar) to oneor more other points in the cluster than to any point not in the cluster.

Figure 5: Eight contiguous clusters of 2 dimensional points.



9

4) Density-based definition: A cluster is a dense region of points, which is separated by

low-density regions, from other regions of high density. This definition is more often

used when the clusters are irregular or intertwined, and when noise and outliers are present. Note that the contiguous definition would find only one cluster in figure 6. Also

note that the three curves don’t form clusters since they fade into the noise, as does the

bridge between the two small circular clusters.

Figure 6: Six dense clusters of 2 dimensional points.

5) Similarity-based Cluster definition: A cluster is a set of objects that are “similar”,and objects in other clusters are not “similar.” A variation on this is to define a cluster

as a set of points that together create a region with a uniform local property, e.g.,

density or shape.

2.2. Measures (Indices) of Similarity and Dissimilarity

2.2.1. Proximity Types and ScalesThe attributes of the objects (or their pairwise similarities and dissimilarities) can

be of different data types and can be measured on different data scales.

The different types of attributes are

1) Binary (two values)

2) Discrete (a finite number of values)

3) Continuous (an effectively infinite number of values)

The different data scales are

1) Qualitative

a) Nominal – the values are just different names.

Example 1: Zip Codes

Example 2: Colors: white, black, green, red, yellow, etc.Example 3: Sex: male, female

Example 4: 0 and 1 when they represent Yes and No.

b) Ordinal – the values reflect an ordering, nothing more.Example 1: Good, Better, Best

Example 2: Colors ordered by the spectrum.

Example 3: 0 and 1 are often considered to be ordered when used as Boolean

values (false/true), with 0 < 1.



10

2) Quantitativea) Interval – the difference between values is meaningful, i.e., a unit of

measurement exits.Example 1: On a scale of 1 to 10 rate the following potato chips.

b) Ratio – the scale has an absolute zero so that ratios are meaningful.

Example 1: The height, width, and length of an object.Example 2: Monetary quantities, e.g., salary or profit.

Example 3: Many physical quantities like electrical current, pressure, etc.

Data scales and types are important since the type of clustering used oftendepends on the data scale and type.

2.2.2. Requirements on Similarity and Dissimilarity MeasuresProximities are normally subject to a number of requirements [DJ88]. We list

these below. Recall that pij is the proximity between points, xi and x j.

1) (a) For a dissimilarity: pii = 0 for all i . (Points aren’t different from themselves.)(b) For a similarity: pii > max pij (Points are most similar to themselves.)

(Similarity measures are often required to be between 0 and 1, and in that case pii = 1, is the requirement.)

2) pij = p ji (Symmetry)

This implies that the proximity matrix is symmetric. While this is typical, it is notalways the case. An example is a confusion matrix, which implicitly defines a

similarity between entities that is measured by how often various entities, e.g.,

characters, are misclassified as one another. Thus, while 0’s and O’s are confusedwith each other, it is not likely that 0’s are mistaken for O’s at the same rate that O’s

are mistaken for 0’s.

3) pij > 0 for all i and j (Positivity)Additionally, if the proximity measure is real-valued and is a true metric in a

mathematical sense, then the following two conditions also hold in addition to

conditions 2 and 3.

4) pij = 0 only if i = j .

5) pik <<<< pij + p jk for all i, j, k . (Triangle inequality.)

Dissimilarities that satisfy conditions 1-5 are called distances, while

“dissimilarity” is a term commonly reserved for those dissimilarities that satisfy onlyconditions 1-3.



11

2.2.3. Common Proximity Measures

2.2.3.1. Distance MeasuresThe most commonly used proximity measure, at least for ratio scales (scales with

an absolute 0) is the Minkowski metric, which is a generalization of the normal distance

between points in Euclidean space. It is defined as

where, r is a parameter, d is the dimensionality of the data object, and xik and x jk are,

respectively, the k th

components of the ith

and jth

objects, xi and x j.

The following is a list of the common Minkowski distances for specific values of

r .

1) r = 1. City block (Manhattan, taxicab, L1 norm) distance.

A common example of this is the Hamming distance, which is just the number of bitsthat are different between two binary vectors.

2) r = 2. Euclidean distance. The most common measure of the distance between two

points.

3) r → ∞. “supremum” (Lmax norm, L∞ norm) distance.This is the maximum difference between any component of the vectors.

Figure 7 gives the proximity matrices for L1, L2 and L∞ , respectively using the given

data matrix which we copied from an earlier example.

point x y

p1 0 2

p2 2 0

p3 3 1

p4 5 1

L1 p1 p2 p3 p4

p1 0.000 4.000 4.000 6.000

p2 4.000 0.000 2.000 4.000

p3 4.000 2.000 0.000 2.000

p4 6.000 4.000 2.000 0.000

L2 p1 p2 p3 p4

p1 0.000 2.828 3.162 5.099

p2 2.828 0.000 1.414 3.162

p3 3.162 1.414 0.000 2.000

p4 5.099 3.162 2.000 0.000

L∞ p1 p2 p3 p4

p1 0.000 2.000 3.000 5.000

p2 2.000 0.000 1.000 3.000

p3 3.000 1.000 0.000 2.000

p4 5.000 3.000 2.000 0.000

Figure 7. Data matrix and the L1, L2, and L∞ proximity matrices.

The r parameter should not be confused with the dimension, d . For example,

Euclidean, Manhattan and supremum distances are defined for all values of d , 1, 2, 3, …,

and specify different ways of combining the differences in each dimension (attribute) intoan overall distance.

r r d

k

jk ik ij x x p

/1

1÷÷

ø

ö

çç

è

æ −= å

=



12

2.2.3.2. Ordinal MeasuresAnother common type of proximity measure is derived by ranking the distances

between pairs of points from 1 to m * (m - 1) / 2. This type of measure can be used withmost types of data and is often used with a number of the hierarchical clustering

algorithms that are discussed in section 4.2. Figure 8 shows how the L2 proximity matrix

of figure 7 would appear as an ordinal proximity matrix.

L2 p1 p2 p3 p4

p1 0.000 2.828 3.162 5.099

p2 2.828 0.000 1.414 3.162

p3 3.162 1.414 0.000 2.000

p4 5.099 3.162 2.000 0.000

ordinal p1 p2 p3 p4

p1 0 3 4 5

p2 3 0 1 4

p3 4 1 0 2

p4 5 4 2 0

Figure 8. An L2 proximity matrix and the corresponding ordinal proximity matrix.

2.2.3.3. Similarity Measures Between Binary VectorsThere are many measures of similarity between binary vectors. These measures

are referred to as similarity coefficients, and have values between 0 and 1. A value of 1indicates that the two vectors are completely similar, while a value of 0 indicates that the

vectors are not at all similar. There are many rationales for why one coefficient is better

than another in specific instances [DJ88, KR90], but we will mention only a few.

The comparison of two binary vectors, p and q, leads to four quantities:M01 = the number of positions where p was 0 and q was 1

M10 = the number of positions where p was 1 and q was 0

M00 = the number of positions where p was 0 and q was 0M11 = the number of positions where p was 1 and q was 1

The simplest similarity coefficient is the simple matching coefficient

SMC = (M11 + M00) / (M01 + M10 + M11 + M00)

Another commonly used measure is the Jaccard coefficient.

J = (M11) / (M01 + M10 + M11)

Conceptually, SMC equates similarity with the total number of matches, while Jconsiders only matches on 1’s to be important. It is important to realize that there are

situations in which both measures are more appropriate.For example, consider the following pairs of vectors and their simple matching

and Jaccard similarity coefficients

a = 1 0 0 0 0 0 0 0 0 0 SMC = .8

b = 0 0 0 0 0 0 1 0 0 1 J = 0

If the vectors represent student’s answers to a True-False test, then both 0-0 and1-1 matches are important and these two students are very similar, at least in terms of the

grades they will get.



13

Suppose instead that the vectors indicate whether 10 particular items are in the

“market baskets” of items purchased by two shoppers. Then the Jaccard measure is a

more appropriate measure of similarity since it would be very odd to say that two market baskets that don’t contain any similar items are the same. For calculating the similarity of

market baskets, and in many other cases, the presence of an item is far more important

than absence of an item.

2.2.3.4. Cosine MeasureFor purposes of information retrieval, e.g., as in Web search engines, documents

are often stored as vectors, where each attribute represents the frequency with which a particular term (word) occurs in the document. (It is much more complicated than this, of

course, since certain common words are ignored and normalization is performed to

account for different forms of the same word, document length, etc.) Such vectors canhave thousands or tens of thousands of attributes.

If we wish to compare the similarity of two documents, then we face a situation

much like that in the previous section, except that the vectors are not binary vectors.

However, we still need a measure like the Jaccard measure, which ignores 0-0 matches.The motivation is that any two documents are likely to “not contain” many of the same

words, and thus, if 0-0 matches are counted most documents will be highly similar to

most other documents. The cosine measure, defined below, is the most common measureof document similarity. If d 1 and d 2 are two document vectors, then

cos( d 1 , d 2 ) = (d 1 • d 2) / ||d 1|| ||d 2|| ,

where • indicates vector dot product and ||d || is the length of vector d .

For example,

d 1 = 3 2 0 5 0 0 0 2 0 0

d 2 = 1 0 0 0 0 0 0 1 0 2

d 1 • d 2= 3*1 + 2*0 + 0*0 + 5*0 + 0*0 + 0*0 + 0*0 + 2*1 + 0*0 + 0*2 = 5||d 1|| = (3*3 + 2*2 + 0*0 + 5*5 + 0*0 +0*0 + 0*0 + 2*2 + 0*0 + 0*0)

0.5= 6.480

||d 2|| = (1*1 + 0*0 + 0*0 + 0*0 + 0*0 + 0*0 + 0*0 + 1*1 + 0*0 + 2*2)0.5

= 2.236

cos( d 1 , d 2 ) = .31

3. Basic Clustering Techniques

3.1. Types of Clustering

Many different clustering techniques that have been proposed over the years.

These techniques can be described using the following criteria [DJ88]:

1) Hierarchical vs. partitional (nested and unnested). Hierarchical techniques

produce a nested sequence of partitions, with a single, all inclusive cluster at the top



14

and singleton clusters of individual points at the bottom. Each intermediate level can

be viewed as combining (splitting) two clusters from the next lower (next higher)

level. (While most hierarchical algorithms involve joining two clusters or splitting acluster into two sub-clusters, some hierarchical algorithms join more than two clusters

in one step or split a cluster into more than two sub-clusters.)

The following figures indicate different ways of graphically viewing thehierarchical clustering process. Figures 9a and 9b illustrate the more “traditional”

view of hierarchical clustering as a process of merging two clusters or splitting one

cluster into two. Figure 9a gives a “nested set” representation of the process, while

Figure 9b shows a “tree’ representation, or dendogram. Figure 9c and 9d show adifferent, hierarchical clustering, one in which points p1 and p2 are grouped together

in the same step that also groups points p3 and p4 together.

.

Figure 9a. Traditional nested set. Figure 9b. Traditional dendogram

Figure 9c. Non-traditional nested set Figure 9d. Non-traditional dendogram.

Partitional techniques create a one-level (unnested) partitioning of the data points. If K is the desired number of clusters, then partitional approaches typically

find all K clusters at once. Contrast this with traditional hierarchical schemes, which bisect a cluster to get two clusters or merge two clusters to get one. Of course, a

hierarchical approach can be used to generate a flat partition of K clusters, and

likewise, the repeated application of a partitional scheme can provide a hierarchical

clustering.

p4

p1

p2

p3

p4 p1 p2 p3

p4

p1

p2

p3

p4 p1 p2 p3



15

2) Divisive vs. agglomerative. Hierarchical clustering techniques proceed either from

the top to the bottom or from the bottom to the top, i.e., a technique starts with one

large cluster and splits it, or starts with clusters each containing a point, and thenmerges them.

3) Incremental or non-incremental. Some clustering techniques work with an item ata time and decide how to cluster it given the current set of points that have already

been processed. Other techniques use information about all the points at once. Non-

incremental clustering algorithms are far more common.

3.2. Objective Functions: Clustering as an Optimization Problem

Many clustering techniques are based on trying to minimize or maximize a globalobjective function. The clustering problem then becomes an optimization problem,

which, in theory, can be solved by enumerating all possible ways of dividing the points

into clusters and evaluating the “goodness” of each potential set of clusters by using the

given objective function. Of course, this “exhaustive” approach is computationally

infeasible (NP complete) and as a result, a number of more practical techniques for optimizing a global objective function have been developed.

One approach to optimizing a global objective function is to rely on algorithms,which find solutions that are often good, but not optimal. An example of this approach is

the K-means clustering algorithm (section 4.1.1) which tries to minimize the sum of the

squared distances (error) between objects and their cluster centers.Another approach is to fit the data to a model. An example of such techniques is

mixture models (section 7.1.1), which assume that the data is a “mixture” of a number of

underlying statistical distributions. These clustering algorithms seek to find a solution to

a clustering problem by finding the maximum likelihood estimate for the statistical parameters that describe the clusters.

Still another approach is to forget about global objective functions. In particular,hierarchical clustering procedures proceed by making local decisions at each step of theclustering process. These ‘local’ or ‘per-step’ decisions are also based on an objective

function, but the overall or global result is not easily interpreted in terms of a global

objective function This will be discussed further in one of the sections on hierarchicalclustering techniques (section 4.2.9).

4. Specific Clustering Techniques

4.1. Center-Based Partitional Clustering

As described earlier, partitional clustering techniques create a one-level partitioning of the data points. There are a number of such techniques, but we shall onlydescribe two approaches in this section: K-means (section 4.1.1) and K-medoid (section

4.1.2).

Both these techniques are based on the idea that a center point can represent a

cluster. For K-means we use the notion of a centroid, which is the mean or median pointof a group of points. Note that a centroid almost never corresponds to an actual data

point. For K-medoid we use the notion of a medoid, which is the most representative



16

(central) point of a group of points. By its definition a medoid is required to be an actual

data point.

Section 4.1.3 introduces CLARANS, a more efficient version of the basicK-medoid that was applied to spatial data mining problems. BIRCH and Bubble are

clustering methods for data mining that are based on center-based partitional approaches.

(BIRCH is more like K-means and Bubble is more like K-medoid.) However, both of BIRCH and Bubble use a wide variety of techniques to help enhance their scalability, i.e.,

their ability to handle large data sets, and consequently, their discussion is postponed to

sections 5.1.1 and section 5.1.2, respectively.

4.1.1. K-means Clustering

4.1.1.1. Basic AlgorithmThe K-means clustering technique is very simple and we immediately begin with

a description of the basic algorithm. We elaborate in the following sections.

Basic K-means Algorithm for finding K clusters.

1. Select K points as the initial centroids.

2. Assign all points to the closest centroid.

3. Recompute the centroid of each cluster.

4. Repeat steps 2 and 3 until the centroids don’t change.

In the absence of numerical problems, this procedure always converges to a

solution, although the solution is typically a local minimum. The following diagram

gives an example of this. Figure 10a shows the case when the cluster centers coincide

with the circle centers. This is a global minimum. Figure 10b shows a local minima.

Figure 10a. A globally minimal clustering solution

Figure 10b. A locally minimal clustering solution.

4.1.1.2. Time and Space ComplexitySince only the vectors are stored, the space requirements are basically O(mn),

where m is the number of points and n is the number of attributes. The time requirementsare O(I*K*m*n), where I is the number of iterations required for convergence. I is

typically small (5-10) and can be easily bounded as most changes occur in the first few



17

iterations. Thus, K-means is linear in m, the number of points, and is efficient, as well as

simple, as long as the number of clusters is significantly less than m.

4.1.1.3. Choosing initial centroidsChoosing the proper initial centroids is the key step of the basic K-means

procedure. It is easy and efficient to choose initial centroids randomly, but the results are

often poor. It is possible to perform multiple runs, each with a different set of randomlychosen initial centroids – one study advocates 30 - but this may still not work depending

on the data set and the number of clusters sought.

We start with a very simple example of three clusters and 16 points. Figure 11a

indicates the “natural” clustering that results when the initial centroids are “well”distributed. Figure 11b indicates a “less natural” clustering that happens when the initial

centroids are poorly chosen.

Figure 11a: Good starting centroids and a “natural” clustering.

Figure 11b: Bad starting centroids and a “less natural” clustering.

We have also constructed the artificial data set, shown in figure 12a as another

illustration of what can go wrong. The figure consists of 10 pairs of circular clusters,

where each cluster of a pair of clusters is close to each other, but relatively far from theother clusters. The probability that an initial centroid will come from any given cluster is

0.10, but the probability that each cluster will have exactly one initial centroid is 10!/1010



18

= 0.00036. (Technical note: This assumes sampling with replacement, i.e., that two

initial centroids could be the same point.)

There isn’t any problem as long as two initial centroids fall anywhere in a pair of clusters, since the centroids will redistribute themselves, one to each cluster, and so

achieve a globally minimal error,. However, it is very probable that one pair of clusters

will have only one initial centroid. In that case, because the pairs of clusters are far apart,the K-means algorithm will not redistribute the centroids between pairs of clusters, and

thus, only a local minima will be achieved. When starting with an uneven distribution of

initial centroids as shown in figure 12b, we get a non-optimal clustering, as is shown in

figure 12c, where different fill patterns indicate different clusters. One of the clusters issplit into two clusters, while two clusters are joined in a single cluster.

Figure 12a: Data distributed in 10 circular regions

Figure 12b: Initial Centroids

Figure 12c: K-means clustering result

Because random sampling may not cover all clusters, other techniques are often

used for finding the initial centroids. For example, initial centroids are often chosen from

dense regions, and so that they are well separated, i.e., so that no two centroids arechosen from the same cluster.

4.1.1.4. Updating Centroids IncrementallyInstead of updating the centroid of a cluster after all points have been assigned to

clusters, the centroids can be updated as each point is assigned to a cluster. In addition,

the relative weight of the point being added may be adjusted. The goal of thesemodifications is to achieve better accuracy and faster convergence. However, it may be

difficult to make a good choice for the relative weight. These update issues are similar to

those of updating weights for artificial neural nets.

Incremental update also has another advantage – empty clusters are not produced.(All clusters start with a single point and if a cluster ever gets down to one point, then

that point will always be reassigned to that cluster.) Empty clusters are often observed

when centroid updates are performed only after all points have been assigned to clusters.



19

This imposes the need for a technique to choose a new centroid for an empty cluster, for

otherwise the squared error will certainly be larger than it would need to be. A common

approach is to choose as the new centroid the point that is farthest away from any currentcenter. If nothing else, this eliminates the point that currently contributes the most to the

squared error.

Updating points incrementally may introduce an order dependency problem,which can be ameliorated by randomizing the order in which the points are processed.

However, this is not really feasible unless the points are in main memory. Updating the

centroids after all points are assigned to clusters results in order independence approach.

Finally, note that when centers are updated incrementally, each step of the processmay require updating two centroids if a point switches clusters. However, K-means

tends to converge rather quickly and so the number of points switching clusters will tend

to be small after a few passes over all the points.

4.1.1.5. Different Definitions of CentroidThe K-means algorithm is derived by asking how we can obtain a partition of the

data into K clusters such that the sum of the squared distance of a point from the “center”of the cluster is minimized. In mathematical terms we seek to minimize

Error = åå= ∈

−K

1i C x

2

i

i

c xH

HH

= ååå= ∈ =

− K

i C x

d

j

ij j

i

c x1 1

2)(H

where xH

is a vector (point) and icH

is the “center” of the cluster, d is the dimension of xH

and icH

and x j and cij are the components of xH

and icH

.)

We can solve for the pthcluster, pc

H

, by solving for each component, c pk , 1 ≤ k ≤ d

by differentiating the Error , setting it to 0, and solving as the following equations

indicate.

ååå= ∈ =

−∂

∂=

∂

∂ K

i C x

d

j

ij j

pk pk i

c xc

Error c 1 1

2)(

H

ååå= ∈ =

−∂

∂=

K

i C x

d

j

ij j

pk i

c xc1 1

2)(H

å∈

=−= pC x

pk k c xH

0)(*2

ååå ∈∈∈=Þ=Þ=−

p p p C x

k

p

pk

C x

k pk p

C x

pk k xnc xcnc xHHH

1

0)(*2

Thus, we find that å∈

=pC x

p xcH

HH

pn

1, the mean of the points in the cluster. This is

pleasing since it agrees with our intuition of what the center of a cluster should be.



20

However, we can instead ask how we can obtain a partition of the data into K

clusters such that the sum of the distance of a point from the “center” of the cluster is

minimized. In mathematical terms, we now seek to minimize the equation

Error = åå= ∈

−K

1i C xi

i

c xH

HH

This problem has been extensively studied in operations research under the name

of the multi-source Weber problem [Ta98] and has obvious applications to the location of warehouses and other facilities that need to be centrally located. We leave it as an

exercise to the reader to verify that the centroid in this case is still the mean.

Finally, we can instead ask how we can obtain a partition of the data into K

clusters such that the sum of the L1 distance of a point from the “center” of the cluster isminimized. In mathematical terms, we now seek to minimize the equation

Error = ååå= ∈ =

−K

1i C x i

H

d

1 j

ij j cx

We can solve for the pth

cluster, pcH

, by solving for each component, c pk , 1 ≤ k ≤ d

by differentiating the Error , setting it to 0, and solving as the following equations

indicate.

ååå= ∈ =

−∂

∂=

∂

∂ K

i C x

d

j

ij j

pk pk i

c xc

Error c 1 1

H

.

ååå= ∈ =

−∂

∂=

K

i C x

d

j

ij j

pk i

c xc1 1

H

.

01

=−∂

∂= åå∈ = pC x

d

j

pk k

pk c xcH

.

0)(01

=−Þ=−∂

∂ååå∈∈ = p p C x

pk k

C x

d

j

pk k

pk

c x signc xc HH

Thus, if we solve for pcH

, we find that pp C xc ∈=HH

median , the point whose

coordinates are the median values of the corresponding coordinate values of the points in

the cluster. While the median of a group of points is straightforward to compute, this

computation is not as efficient as the calculation of a mean.

In the first case we say we are attempting to minimize the within cluster squarederror, while in the second case and third cases we just say that we are attempting to

minimize the absolute within cluster error, where the error may either the L1 or L2distance.

4.1.1.6. Pre and Post ProcessingSometimes the quality of the clusters that are found can be improved by pre-

processing the data. For example, when we use the squared error criteria, outliers can



21

unduly influence the clusters that are found. Thus, it is common to try to find such values

and eliminate them with a preprocessing step.

Another common approach uses post-processing steps to try to fix up the clustersthat have been found. For example, small clusters are often eliminated since they

frequently represent groups of outliers. Alternatively, two small clusters that are close

together can be merged. Finally, large clusters can be split into smaller clusters.While pre- and post-processing techniques can be effective, they are based on

heuristics that may not always work. Furthermore, they often require that the user choose

values for a number of parameters.

ISODATA [DJ88] is the most famous of such techniques and is still used today,especially in image processing. After choosing initial centroids in a manner that

guarantees that they will be well separated, ISODATA proceeds in the following way:

1) Points are assigned to the closest centroid and cluster centroids are recomputed. Thisstep is repeated until no points change clusters.

2) Clusters with “too few” points are discarded.

3) Clusters are merged (lumped) or split.

a) If there are “too many” clusters, then clusters are split.b) If there are “too few” clusters compared to the number desired, then clusters are

merged.

c) Otherwise the cluster splitting and merging phases alternate. Clusters are merged if their centers are close, while clusters are split if the are

“too loose.”

4) Steps 1,2 and 3 are repeated until the results are “acceptable” or until some pre-determined number of iterations has been exceeded.

The splitting phase of ISODATA is an example of strategies that decrease thetotal error by increasing the number of clusters. Two general strategies are given below:

a) Split a cluster. Often the cluster with the largest error is chosen, but in the case of

ISODATA the cluster with the largest standard deviation in some particular attributeis chosen.

b) Introduce a new cluster center. Often the point that is farthest from any cluster center

is chosen.

Likewise, the merging phase of ISODATA is just one example of strategies thatdecrease the number of clusters. (This will increase the error.) Two general strategies are

given below:

a) Disperse a cluster, i.e. remove the center that corresponds to the center and reassignthe points to other centers. Ideally the cluster which is dispersed should be the one

that increases the error the least. A good heuristic for determining this is based on

keeping track of the second closest center of each point.

b) Merge two clusters. In ISODATA, the clusters with the closest centers are chosen,

although another, perhaps better, approach is to merge the two clusters that result in

the smallest increase in error. These two merging strategies are, respectively, the

same strategies that are used in the hierarchical clustering techniques known as theCentroid method and Ward’s method. Both methods are discussed in section 4.2.8.



22

Finally, it is important to realize that there are certain clustering applications,

where outliers cannot be eliminated. In data compression every point must be clustered,

and in financial analysis, apparent outliers are often the most interesting points, e.g.,unusually profitable customers.

4.1.1.7. Limitations and problemsK-means attempts to minimize the squared or absolute error of points with respect

to their cluster centroids. While this is sometimes a reasonable criterion and leads to a

simple algorithm, K-means has a number of limitations and problems. In particular,

Figures 13a and 13b show the problems that result when clusters have widely differentsizes or have convex shapes.

Figure 13a: Different sizes Figure 13b: Convex shapes

The difficulty in these two situations is that the K-means objective function is amismatch for the kind of clusters we are trying to find. The K-means objective function

is minimized by globular clusters of equal size or by clusters that are well separated.

The K-means algorithm is also normally restricted to data in Euclidean spaces because in many cases the required means and medians do not make sense. (This is not

true in all cases, e.g., documents.) A related technique, K-medoid clustering, does not

have this restriction and is discussed in the next section.

4.1.2. K-medoid ClusteringThe objective of K-medoid clustering [KR90] is to find a non-overlapping set of

clusters such that each cluster has a most representative point, i.e., a point that is most

centrally located with respect to some measure, e.g., distance. These representative

points are called medoids. Once again, the algorithm is conceptually simple.

Basic K-medoid Algorithm for finding K clusters.

1) Select K initial points. These points are the candidate medoids and are intended to be the most central points of their clusters.

2) Consider the effect of replacing one of the selected objects (medioids) with one of

the non-selected objects. Conceptually, this is done in the following way. Thedistance of each non-selected point from the closest candidate medoid is calculated,

and this distance is summed over all points. This distance represents the “cost” of the

current configuration. All possible swaps of a non-selected point for a selected one

are considered, and the cost of each configuration is calculated.

3) Select the configuration with the lowest cost. If this is a new configuration, then

repeat step 2.



23

4) Otherwise, associate each non-selected point with its closest selected point

(medoid) and stop.

Step 3 requires more explanation. The ithmedoid is evaluated by usingå

=

in

1 jij p ,

where pij, is the proximity between the ith

medoid and the jth

point in the cluster. For

similarities (dissimilarities) we want this sum to be as large (small) as possible. It can beseen that such an approach is not restricted to Euclidean spaces and is likely to be more

tolerant of outliers. However, finding a better medoid requires trying all points that are

currently not medoids and is computationally expensive.

4.1.3. CLARANSCLARANS [NH94] was one of the first clustering algorithms that was developed

specifically for use in data mining – spatial data mining. CLARANS itself grew out of two clustering algorithms, PAM and CLARA [KR90], that were developed in the field of

statistics.

PAM (Partitioning Around Medoids) is a “ K -medoid” based clustering algorithm

that attempts to cluster a set of m points into K clusters by performing the steps describedin section 4.1.2.

CLARA (Clustering LARge Applications) is an adaptation of PAM for handlinglarger data sets. It works by repeatedly sampling a set of data points, calculating the

medoids of the sample, and evaluating the cost of the configuration that consists of these

“sample-derived” medoids and the entire data set. The set of medoids that minimizes thecost is selected. As an optimization, the set of medoids from one iteration is usually

included in the sample for the next iteration. As presented, CLARA normally samples 40

+ 5 K objects and uses 5 iterations.

CLARANS uses a randomized search approach to improve on both CLARA andPAM. Conceptually, CLARANS does the following.

1) Randomly pick K candidate medoids. 2) Randomly consider a swap of one of the selected points for a non-selected point.

3) If the new configuration is better, i.e., has lower cost, then repeat step 2 with the new

configuration. 4) Otherwise, repeat step 2 with the current configuration unless a parameterized limit

has been exceeded. (This limit was set to max (250, K *(m - K )).

5) Compare the current solution with any previous solutions and keep track of the best. 6) Return to step 1 unless a parameterized limit has been exceeded. (This limit was set

to 2.) Thus, CLARANS differs from PAM in that, given the current configuration, it

does not consider all possible swaps of medoid and non-medoid points, but rather, only a

random selection of them and only until it finds a better configuration. Unlike CLARA,CLARANS works with all data points.

CLARANS was used in two spatial data mining tools, SD(CLARANS) and

NSD(CLARANS). These tools added some capabilities related to cluster analysis.In [EKX95], a number of techniques were proposed for making CLARANS more

efficient. The basic idea was to use an R* trees to store the data being considered. R*

trees are a type of nearest-neighbor tree (see section 8.1) that try to store data points in



24

the same disk page if the data points are “close” to one another. [EKX95] proposes to

make use of this in three ways to improve CLARANS.

1) Focus on representative objects. For all the objects in a single page of the R* tree,find the representative object, i.e., the object which is closest to the calculated center

of the objects in the page. Use CLARANS only on these representative objects. This

amounts to an intelligent sampling of the database. 2) Focus on relevant clusters. When considering a possible swap of a non-medoid

object for a medoid object, the change in the cost from the old configuration to the

new configuration can be determined by only considering the clusters to which the

medoid and non-medoid objects belong.

3) Focus on the cluster. It is possible to efficiently construct a bounding Voronoi

polyhedron for a cluster by looking only at the medoids. This polyhedron is just the

set of all points that are closer to the medoid of the cluster than to any other medoid.This polyhedron can be translated into a range query which can be efficiently

implemented by the R* tree.

These improvements improve the speed of CLARANS by roughly two orders of

magnitude, but reduce the clustering effectiveness by only a few percent.

4.2. Hierarchical Clustering

In hierarchical clustering the goal is to produce a hierarchical series of nested

clusters, ranging from clusters of individual points at the bottom to an all-inclusive

cluster at the top. A diagram called a dendogram graphically represents this hierarchyand is an inverted tree that describes the order in which points are merged (bottom-up

view) or clusters are split (top-down view).

One of the attractions of hierarchical techniques is that they correspond to

taxonomies that are very common in the biological sciences, e.g., kingdom, phylum,genus, species, … . (Some cluster analysis work occurs under the name of “mathematical

taxonomy.”) Another attractive feature is that hierarchical techniques do not assume any particular number of clusters. Instead any desired number of clusters can be obtained by“cutting” the dendogram at the proper level. Finally, hierarchical techniques are thought

to produce better quality clusters [DJ88].

4.2.1. Agglomeration and DivisionThere are two basic approaches to generating a hierarchical clustering:

a) Agglomerative: Start with the points as individual clusters and, at each step, mergethe closest pair of clusters. This requires defining the notion of cluster proximity.

b) Divisive: Start with one, all-inclusive cluster and, at each step, split a cluster until

only singleton clusters of individual points remain. In this case, we need to decide

which cluster to split at each step.

Divisive techniques are less common, and we will mention only one example

before focusing on agglomerative techniques.

4.2.2. Simple Divisive Algorithm (Minimum Spanning Tree (MST) )

1) Compute a minimum spanning tree for the proximity graph.



25

2) Create a new cluster by breaking the link corresponding to the largest distance

(smallest similarity).

3) Repeat step 2 until only singleton clusters remain.

This approach is the divisive version of the “single link” agglomerative technique

that we will see shortly.

4.2.3. Basic Agglomerative Hierarchical Clustering AlgorithmMany hierarchical agglomerative techniques can be expressed by the following

algorithm, which is known as the Lance-Williams algorithm.

Basic Agglomerative Hierarchical Clustering Algorithm

1) Compute the proximity graph, if necessary. (Sometimes the proximity graph is all that is available.)

2) Merge the closest (most similar) two clusters.

3) Update the proximity matrix to reflect the proximity between the new cluster and theoriginal clusters. 4) Repeat steps 3 and 4 until only a single cluster remains.

The key step of the previous algorithm is the calculation of the proximity between

two clusters, and this is where the various agglomerative hierarchical techniques differ.

Any of the cluster proximities that we discuss in this section can be viewed as a choice of different parameters (in the Lance-Williams formula) for the proximity between clusters

Q and R, where R is formed by merging clusters A and B.

p(R, Q) = α A p(A, Q) + α B p(B, Q) + β p(A, Q) + γ | p(A, Q) – p(B, Q) |

In words, this formula says that after you merge clusters A and B to form cluster R, then the distance of the new cluster, R, to an existing cluster, Q, is a linear function of the distances of Q from the original clusters A and B.

Any hierarchical technique that can be phrased in this way does not need the

original points, only the proximity matrix, which is updated as clustering occurs.However, while a general formula is nice, it is often easier to understand the different

hierarchical methods by looking directly at the definition of cluster distance that each

method uses, and that is the approach that we shall take here. [DJ88] and [KR90] both

give a table that describes each method in terms of the Lance-Williams formula.

4.2.4. Time and Space ComplexityHierarchical clustering techniques typically use a proximity matrix. This requires

the computation and storage of m2 proximities, a factor that limits the size of data sets that

can be processed. (It is possible to compute the proximities on the fly and save space, but

this increases computation time.) Once the proximity matrix is available, the timerequired for hierarchical clustering is O(m2 ).



26

1 2 3 4 5

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00

4.2.5. MIN or Single LinkFor the single link or MIN version of hierarchical clustering, the proximity of two

clusters is defined to be minimum of the distance (maximum of the similarity) betweenany two points in the different clusters. (The technique is called single link because, if

you start with all points as singleton clusters and add links between points, strongest links

first, then these single links combine the points into clusters.) Single link is good athandling non-elliptical shapes, but is sensitive to noise and outliers.

The following table gives a sample similarity matrix for five items (I1 – I5) and

the dendogram shows the series of merges that result from using the single link

technique.

4.2.6. MAX or Complete Link or CLIQUEFor the complete link or MAX version of hierarchical clustering, the proximity of

two clusters is defined to be maximum of the distance (minimum of the similarity)

between any two points in the different clusters. (The technique is called complete link

because, if you start with all points as singleton clusters, and add links between points,strongest links first, then a group of points is not a cluster until all the points in it are

completely linked, i.e., form a clique.) Complete link is less susceptible to noise and

outliers, but can break large clusters, and has trouble with convex shapes.

The following table gives a sample similarity matrix and the dendogram showsthe series of merges that result from using the complete link technique.

4.2.7. Group AverageFor the group average version of hierarchical clustering, the proximity of two

clusters is defined to be the average of the pairwise proximities between all pairs of

points in the different clusters. Notice that this is an intermediate approach between MINand MAX. This is expressed by the following equation:

1 2 3 4 5

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5 0.20 0.50 0.30 0.80 1.00



27

proximity (cluster1, cluster2) =)2(*)1(

),(

21

21

2

1

cluster sizecluster size

p p proximity

cluster pcluster p

å∈∈

The following table gives a sample similarity matrix and the dendogram shows

the series of merges that result from using the group average approach. The hierarchicalclustering in this simple case is the same as produced by MIN.

4.2.8. Ward’s Method and Centroid methodsFor Ward’s method the proximity between two clusters is defined as the increase

in the squared error that results when two clusters are merged. Thus, this method uses the

same objective function as is used by the K-means clustering. While it may seem that

this makes this technique somewhat distinct from other hierarchical techniques, somealgebra will show that this technique is very similar to the group average method when

the proximity between two points is taken to be the square of the distance between them.

Centroid methods calculate the proximity between two clusters by calculating thedistance between the centroids of clusters. These techniques may seem similar to K-

means, but as we have remarked, Ward’s method is the correct hierarchical analogue.

Centroid methods also have a characteristic – often considered bad – that other

hierarchical clustering techniques don’t posses: the possibility of inversions. In other words, two clusters that are merged may be more similar than the pair of clusters that

were merged in a previous step. For other methods the similarity of clusters merged

decreases (the distance between merged clusters increases) as we proceed from singletonclusters to one all inclusive clusters.

4.2.9. Key Issues in Hierarchical Clustering

4.2.9.1. Lack of a Global Objective FunctionIn Section 3.2 we mentioned that hierarchical clustering cannot be viewed as globally

optimizing an objective function. Instead, hierarchical clustering techniques use variouscriteria to decide “locally” at each step which clusters should be joined (or split for divisive approaches). For agglomerative hierarchical techniques, the criteria is typically

to merge the “closest” pair of clusters, where “close” is defined by a specified measure of

cluster proximity.This approach yields clustering algorithms that avoid the difficulty of trying to

solve a hard combinatorial optimization problem. (As previously, the general clustering

problem for objective functions like squared error is NP complete.) Furthermore, such

1 2 3 4 5

I1 I2 I3 I4 I5

I1 1.00 0.90 0.10 0.65 0.20

I2 0.90 1.00 0.70 0.60 0.50

I3 0.10 0.70 1.00 0.40 0.30

I4 0.65 0.60 0.40 1.00 0.80

I5

0.20

0.50

0.30

0.80

1.00



28

approaches do not have problems with local minima or difficulties in choosing initial

points. Of course, the time and space complexity is O(m2 ), which is prohibitive in many

cases.

4.2.9.2. Cluster Proximity and Cluster Representation

Cluster proximity is typically defined by a conceptual view of the clusters. For example, if we view a cluster as being represented by all the points, then we can take the

proximity between the closest (most similar) two points in different clusters as being the

proximity between the two clusters. This defines the MIN technique (section 4.2.5).

Likewise, we can also take the proximity between the farthest (least similar) two points indifferent clusters as our definition of cluster proximity. This defines the MAX technique

(section 4.2.6). Finally, we can average the pairwise proximities of all pairs of two points

from different clusters. This yields the Group Average technique (section 4.2.7).If instead, we chose to represent each cluster by a centroid, then we find that

different definitions of cluster proximity are more natural. In one case, the cluster

distance is defined as the distance between cluster centroids. This defines the Centroid

clustering technique (section 4.2.8). Ward’s method (section 4.2.8), is conceptuallyrelated to Centroid clustering since it merges the two clusters that result in the smallest

increase in the total squared error, i.e., it minimizes the sum of the squared distance of

points from their centroids.However, Ward’s method can also be viewed as an example of cluster methods

that take an “all points” view of cluster representation, but define proximity in terms of

preserving desired cluster properties. For Ward’s method, the desired cluster property isa low squared error, i.e., minimum squared distance of all points from the cluster center.

Chameleon (section 4.2.12) uses the concepts of cluster closeness and connectivity to

define cluster properties that must be preserved.As we will see later, CURE (section 4.2.11) is a hybrid approach that mixes both

the centroid and “all points” view. It does this by using a well-chosen subset of the

points in a cluster to represent the cluster. Depending on parameter settings, thisapproach may approximate the MIN technique or the Centroid technique, or something in

between.

4.2.9.3. The Impact of Cluster SizeAnother aspect of hierarchical agglomerative clustering that should be considered

is how to treat the relative sizes of the pairs of clusters that may be merged. There are

basically two schemes: weighted and unweighted. Weighted schemes treat each cluster

equally and thus, objects in smaller clusters effectively have larger weight. Unweightedschemes treat all objects in the clusters equally. Note that this discussion only applies to

cluster proximity schemes that involve sums, i.e., Centroid and Group Average.

Unweighted schemes seem to be more popular, and in our further discussions about theGroup Average and Centroid schemes, we refer only to the unweighted versions.

4.2.9.4. Merging Decisions are FinalHierarchical clustering tends to make good local decisions about combining two

clusters since it has the entire proximity matrix available. However, once a decision is

made to merge two clusters, the hierarchical scheme does not allow for that decision to be



29

changed. This prevents a local optimization criterion from becoming a global

optimization criterion.

For example, in Ward’s method, the “minimize squared error” criteria from K-means is used in deciding which clusters to merge. However, this does not result in a

clustering that could be used to solve the K-means problem. Even though the local, per-

step decisions try to minimize the squared error, the clusters produced on any level,except perhaps the very lowest levels, do not represent an optimal clustering from a

“minimize global squared error” point of view. Furthermore, the clusters are not even

“stable,” in the sense that a point in a cluster may be closer to the centroid of some other

cluster than to the centroid of its current cluster.However, Ward’s method can be used as a robust method of initializing a K-

means clustering. Thus, a local “minimize squared error” objective function seems to

have some connection with a global “minimize squared error” objective function.Finally it is possible to attempt to fix up the hierarchical clustering produced by

hierarchical clustering techniques. One idea is to move branches of the tree around so as

to improve some global objective function [Fi96]. Another idea [KHK99b] is to refine

the clusters produced by a hierarchical technique by using an approach similar to thatused for the multi-level refinement of graph partitions.

4.2.10. Limitations and problemsWe summarize the problems with agglomerative hierarchical clustering.

1) No global objective function is being optimized.

2) Merging decisions are final.

3) Good local merging decisions may not result in good global results. 4) Agglomerative hierarchical clustering techniques have trouble with one of more of

the following: noise and outliers, non-convex shapes, and a tendency to break largeclusters.

4.2.11. CURECURE (Clustering Using Representatives) [GRS98] is a clustering algorithm that

uses a variety of different techniques to create an approach which can handle large data

sets, outliers, and clusters with non-spherical shapes and non-uniform sizes.

CURE represents a cluster by using multiple “representative” points from thecluster. These points will, in theory, capture the geometry and shape of the cluster. The

first representative point is chosen to be the point furthest from the center of the cluster,while the remaining points are chosen so that they are farthest from all the previously

chosen points. In this way, the representative points are naturally relatively welldistributed. The number of points chosen, is a parameter, c, but it was found that a value

of 10 or more worked well.

Once the representative points are chosen, they are shrunk toward the center by a

factor, α. This helps moderate the effect of outliers, which are usually farther away fromthe center and thus, are “shrunk” more. For example, a representative point that was a

distance of 10 units from the center would move by 3 units (for α = .7), while arepresentative point at a distance of 1 unit would only move .3 units.

CURE uses an agglomerative hierarchical scheme to perform the actual

clustering. The distance between two clusters is the minimum distance between any two



30

representative points (after they are shrunk toward their respective centers). While this

scheme is not exactly like any other hierarchical scheme that we have seen, it is

equivalent to centroid based hierarchical clustering if α = 0, and roughly the same as

single link hierarchical clustering if α = 1. Notice that while a hierarchical clusteringscheme is used, the goal of CURE is to find a given number of clusters as specified by

the user.CURE takes advantage of certain characteristics of the hierarchical clustering

process to eliminate outliers at two different points in the clustering process. First, if a

cluster is growing slowly, then this may mean that it consists mostly of outliers, since by

definition outliers are far from others and will not be merged with other points very often.

In CURE this first phase of outlier elimination typically occurs when the number of clusters is 1/3 the original number of points. The second phase of outlier elimination

occurs when the number of clusters is on the order of K , the number of desired clusters.

At this point small clusters are again eliminated.Since the worst case complexity of CURE is O(m2log m), it cannot be applied

directly to large data sets directly. For this reason, CURE uses two techniques for

speeding up the clustering process. The first technique takes a random sample and performs hierarchical clustering on the sampled data points. This is followed by a final

pass that assigns each remaining point in the data set to one of the clusters by choosing

the cluster with closest representative point.In some cases, the sample required for clustering is still too large and a second

additional technique is required. In this situation, CURE partitions the sample data into p

partitions of size m/p and then clusters the points in each partition until there are m/pq

clusters in each partition. This pre-clustering step is then followed by a final clustering of the m/q intermediate clusters. Both clustering passes uses CURE’s hierarchical clustering

algorithm and are followed by a final pass that assigns each point in the data set to one of

the clusters.

We summarize our description of CURE by explicitly listing the different steps:

1) Draw a random sample from the data set. The CURE paper is notable for explicitly deriving a formula for what the size of this sample should be in order to

guarantee, with high probability, that all clusters are represented by a minimum

number of points.

2) Partition the sample into p equal sized partitions.

3) Cluster the points in each cluster using the hierarchical clustering algorithm to

obtain m/pq clusters in each partition and a total of m/q clusters. Note that some

outlier elimination occurs during this process.

4) Eliminate outliers. This is the second phase of outlier elimination.

5) Assign all data to the nearest cluster to obtain a complete clustering.

4.2.12. ChameleonChameleon [KHK99A] is a clustering algorithm that combines an initial

partitioning of the data using an efficient graph partitioning algorithm with a novelhierarchical clustering scheme that dynamically models clusters. The key idea is that two

clusters will be merged only if the resulting cluster is similar to the original clusters, i.e.,

self-similarity is preserved. While applicable to a wide range of data, Chameleon has



31

specifically been shown to be superior to DBSCAN (section 4.3.1) and CURE (section

4.2.11) for clustering spatial data.

The first step in Chameleon is to generate a k-nearest neighbor graph.Conceptually, such a graph is derived from the proximity graph, and contains links only

between a point and its k-nearest neighbors, i.e., the points to which it is closest or most

similar. Working with a sparsified proximity graph instead of the full proximity graphcan significantly reduce the effects of noise and outliers. See section 4.4.4 for more

discussion.

Once a sparsified graph has been obtained, an efficient multi-level graph-

partitioning algorithm [KK98a, KK98b] can be used to partition the data set. Chameleonstarts with an all inclusive cluster and then splits the largest current cluster until no

cluster has more than MIN_SIZE points, where MIN_SIZE is a user specified parameter.

This process results in a large number of almost equally sized groups of well-connectedvertices (highly similar data points). Graph partitioning takes into account global

structure and produces partitions that are sub-clusters, i.e., contain points mostly from

one “true” cluster.

To recombine the sub-clusters a novel agglomerative hierarchical algorithm isused. Two clusters are combined if the resulting cluster shares certain properties with the

constituent clusters, where two key properties are used to model cluster similarity:

• Relative Interconnectivity: The absolute interconnectivity of two clustersnormalized by the internal connectivity of the clusters. In words, we will combine

two clusters if the points in the resulting cluster are almost as “strongly” connected as

points in each of the original clusters. Mathematically,

RI =( ) ( )( ) ji

ji

C EC C EC

C C EC

+2

1

),(, where EC(Ci, C j)is the sum of the edges of k-nearest

neighbor graph that connect clusters Ci and C j, EC(Ci) is the minimum sum of the

“cut” edges if we bisect cluster Ci, and EC(C j) is the minimum sum of the “cut” edgesif we bisect cluster C j. (EC stands for edge cut.)

• Relative Closeness: The absolute closeness of two clusters normalized by theinternal closeness of the clusters. In words, we will combine two clusters only if the

points in the resulting cluster are almost as “close” to each other as in the originalcluster. Mathematically,

RC =

( ) ( ) ji

ji

C EC

ji

j

C EC

ji

i

C C EC

S C C

C S

C C

C

S

++

+

),(, where instead of using the edge cut, i.e., the sum

of the edges that are cut to bisect a graph, we use the average similarity of the edges

involved in the edge cut.Chameleon has two alternative merging schemes. For the first scheme

Chameleon uses two thresholds, TRI and TRC to select candidate pairs of clusters to merge.

If more than one pair of clusters has RI and RC values higher than these thresholds, thenthe pair to merge is chosen by taking the one with the largest value of RI, i.e., by

choosing the pair of clusters that are most interconnected. For the second scheme,

Chameleon chooses the pair of clusters that maximizes RI(Ci, C j) * RC(Ci, C j)α

, where α is a user specified parameter, which is typically greater than 1.



32

Unlike traditional hierarchical clustering techniques, Chameleon will merge more

than one pair of clusters during a single pass. Another difference (for the first merging

scheme) is that Chameleon will reach a point at which it will stop merging clusters.(Recall that traditional agglomerative hierarchical algorithms continue merging until only

one all-inclusive cluster remains.) If desired, the thresholds can be relaxed and the

clustering resumed. The second merging scheme will automatically merge to a singlecluster.

If m is the number of data points and p is the number of partitions the complexity

of Chameleon is O(mp + mlog m + p2log p). However, this depends on being able to

build the k-nearest neighbor graph in O(mlog m) time. This, in turn, requires the use of ak-d tree or a similar type of data structure. Such data structures only work well for low-

dimensional data sets, and thus, for higher dimensional data sets the complexity of

Chameleon becomes O(m2 ).

In summary, the steps of the Chameleon algorithms are:

1) Build a k-nearest neighbor graph.

2) Partition the graph into partitions using a multilevel graph-partitioningalgorithm.

3) Perform a hierarchical clustering starting with the partitions. This hierarchical

clustering will merge the clusters which best preserve the cluster self-similarity withrespect to relative interconnectivity and relative closeness.

4.3. Density Based Clustering

CLIQUE and MAFIA are also density based clustering schemes, but because they

are specifically designed for handling clusters in high-dimensional data, we discuss them

in sections 5.2.1 and 5.2.2, respectively.

4.3.1. DBSCANDBSCAN [EKSX96] is a density based clustering algorithm that works with a

number of different distance metrics. When DBSCAN has processed a set of data points,a point will either be in a cluster or will be classified as noise.

DBSCAN is based on the concepts of a point being “density reachable” and

“density connected.” These concepts are explained in detail in [EKSX96]. What followsis a more basic description.

Conceptually, data points fall into three classes:a) Core points. These are points that are at the interior of a cluster. A point is an

interior point if there are enough points in its neighborhood, i.e., if the number of

points within a given neighborhood around the point, as determined by the distance

function and a supplied distance parameter, exceeds a certain threshold - also asupplied parameter. If two core points belong to each other’s neighborhoods, then the

core points belong to the same cluster.

b) Border points. A border point is a point that is not a core point, i.e., there are notenough points in its neighborhood, but it falls within the neighborhood of a core

point. Note that a border point may fall within the neighborhoods of core points from

several different clusters. If so, the cluster that the border point is grouped with willdepend on the details of the DBSCAN algorithm, and is order dependent.



33

c) Noise points. A noise point is any point that is not a core point or a border point.

Thus, for DBSCAN, a cluster is the set of all core points whose neighborhoodstransitively connect them together, along with some border points.

4.3.2. DENCLUEDENCLUE (DENsity CLUstEring) [HK98] is a density clustering approach that

takes a more formal approach to density based clustering by modeling the overall density

of a set of points as the sum of “influence” functions associated with each point. The

resulting overall density function will have local peaks, i.e., local density maxima, andthese local peaks can be used to define clusters in a straightforward way. Specifically,

for each data point, a hill climbing procedure finds the nearest peak associated with that

point, and the set of all data points associated with a particular peak (called a localdensity attractor) becomes a (center-defined) cluster. However, if the density at a local

peak is too low, then the points in the associated cluster are classified as noise and

discarded. Also, if a local peak can be connected to a second local peak by a path of data

points, and the density at each point on the path is above a minimum density threshold, ξ,then the clusters associated with these local peaks are merged. Thus, clusters of any

shape can be discovered.

DENCLUE is based on a well-developed area of statistics and pattern recognitionwhich is know as “kernel density estimation” [DH73]. The goal of kernel density

estimation (and many other statistical techniques as well) is to describe the distribution of

the data by a function. For kernel density estimation, the contribution of each point to theoverall density function is expressed by an “influence” (kernel) function. The overall

density is then merely the sum of the influence functions associated with each point.

Typically the influence or kernel function is symmetric (the same in all directions)

and its value (contribution) decreases as the distance from the point increases. For

example, for a particular point, x, the Gaussian function, K ( x) =2

2

2

),(-distance

σ

y x

e , is often used

as a kernel function. (σ is a parameter which governs how quickly the influence of point

drops off.) Figure 14a shows how a Gaussian function would look for a single two

dimensional point, while figure 14c shows what the overall density function produced bythe Gaussian influence functions of the set of points shown in 14b.

a) Gaussian Kernel b) Set of points c) Overall density function



34

Figure 14: Example of the Gaussian influence (kernel) function and an overall

density function. (σ = 0.75)

The DENCLUE algorithm has two steps, a preprocessing step and a clustering

step. In the pre-clustering step, a grid for the data is created by dividing the minimal bounding hyper-rectangle into d -dimensional hyper-rectangles with edge length 2σ. Therectangles that contain points are then determined. (Actually, only the occupied hyper-

rectangles are constructed.) The hyper-rectangles are numbered with respect to a

particular origin (at one edge of the bounding hyper-rectangle and these keys are stored in

a search tree to provide efficient access in later processing. For each stored cell, thenumber of points, the sum of the points in the cell, and connections to neighboring

population cubes are also stored.

For the clustering step DENCLUE, considers only the highly populated cubes andthe cubes that are connected to them. For each point, x, the local density function is

calculated only by considering those points that are from clusters which are a) in clusters

that are connected to the one containing the point and b) have cluster centroids within adistance of k σ of the point, where k = 4. As an efficiency measure, each point, x′ , on the

path from x to its density attractor is assigned to the same cluster as x if dist ( x, x′ ) ≤ σ/2.

As mentioned above, DENCLUDE discards clusters associated with a density attractor

whose density is less than ξ. Finally, DENCLUE merges density attractors that can be

joined by a path of points, all of which have a density greater than ξ.DENCLUE can be parameterized so that it behaves much like DBSCAN, but is

much more efficient that DBSCAN. DENCLUE can also behave like K-means by

choosing σ appropriately and by omitting the step that merges center-defined clusters into

arbitrary shaped clusters. Finally, by doing repeated clusterings for different values of σ,a hierarchical clustering can be obtained.

4.3.3. WaveCluster

WaveCluster [SCZ98] is a clustering technique that interprets the original data as

a two-dimensional signal and then applies signal processing techniques ( the wavelet

transform) to map the original data to a new space where cluster identification is morestraightforward. More specifically, WaveCluster defines a uniform two-dimensional grid

on the data and represents the points in each grid cell by the number of points. Thus, a

collection of two-dimensional data points becomes an image, i.e., a set of “gray-scale” pixels, and the problem of finding clusters becomes one of image segmentation.

While there are a number of techniques for image segmentation, wavelets have a

couple of features that make them an attractive choice. First, the wavelet approachnaturally allows for a multiscale analysis, i.e., the wavelet transform allows features, and

hence, clusters, to be detected at different scales, e.g., fine, medium, and coarse.

Secondly, the wavelet transform naturally lends itself to noise elimination.

The basic algorithm of WaveCluster is as follows:



35

1) Create a grid and assign each data object to a cell in the grid. The grid is

uniform, but the grid size will vary for different scales of analysis. Each grid cell

keeps track of the statistical properties of the points in that cell, but for waveclustering only the number of points in the cell is used.

2) Transform the data to a new space by applying the wavelet transform. This

results in 4 “subimages” at several different levels of resolutionan “average” image,an image that emphasizes the horizontal features, an image that emphasizes vertical

features and an image that emphasizes corners.

3) Find the connected components in the transformed space. The average subimage

is used to find connected clusters, which are just groups of connected “pixels,” i.e., pixels which are connected to one another horizontally, vertically, or diagonally.

4) Map the clusters labels of points in the transformed space back to points in the

original space. WaveCluster creates a lookup table that associates each point in theoriginal with a point in the transformed space. Assignment of cluster labels to the

original points is then straightforward.

In summary, the key features of WaveCluster are order independence, no need tospecify a number of clusters (although it is helpful to know this in order to figure out the

right scale to look at, speed (linear), the elimination of noise and outliers, and the ability

to find arbitrarily shaped clusters. While the WaveCluster approach can theoretically beextended to more than two dimensions, it seems unlikely that WaveCluster will work

well (efficiently and effectively) for medium or high dimensions.

4.4. Graph-Based Clustering

The hierarchical clustering algorithms that we discussed in section 4.2 can be

viewed as operating on a proximity graph. However, they are most commonly viewed interms of merging or splitting clusters, and often there is no mention of graph related

concepts. There are some clustering techniques, however, that are explicitly cast in terms

of a graph or a hypergraph. Many of these algorithms are based on the idea of looking at

the nearest neighbors of a point. We begin first with a couple of old, but still quiterelevant clustering techniques.

4.4.1. Historical techniquesBoth approaches we consider use what could be called the nearest neighbor

principle, which is also used in classification.

Nearest Neighbor Principle: A point probably belongs to the same class as the points it

is closest to, i.e., with high probability a point belongs to the same class as it’s nearest

neighbors.

4.4.1.1. Shared Nearest Neighbor ClusteringShared nearest neighbor clustering is described in [JP73].

1) First the k-nearest neighbors of all points are found. In graph terms this can beregarded as breaking all but the k strongest links from a point to other points in the

proximity graph.

2) All pairs of points are compared and if



36

a) any two points share more than k t ≤≤≤≤ k neighbors, and

b) The two points being compared are among the k-nearest neighbors of each,

Then the two points and any cluster they are part of are merged.

This approach has a number of nice properties. It can handle clusters of differentdensities since the nearest neighbor approach is self-scaling. This approach is transitive,i.e., if point, p, shares lots of near neighbors with point, q, which in turn shares lots of

near neighbors with point, r , then points p, q and r all belong to the same cluster. This

allows this technique to handle clusters of different sizes and shapes.However, transitivity can also join clusters that shouldn’t be joined, depending on

the k and k t parameters. Large values for both of these parameters tend to prevent these

spurious connections, but also tend to favor the formation of globular clusters.

4.4.1.2. Mutual Nearest Neighbor ClusteringMutual nearest neighbor clustering is described in [GK77]. It is based on the idea

of the “mutual neighborhood value (mnv)” of two points, which is the sum of the ranks of the two points in each other’s sorted nearest-neighbor lists. Two points are then said to

be mutual nearest neighbors if they are the closest pair of points with that mnv.

Clusters are built up by starting with points as singleton clusters and then mergingthe closest pair of clusters, where close is defined in terms of the mnv. The mnv between

two clusters is the maximum mnv between any pair of points in the combined cluster. If

there are ties in mnv between pairs of clusters, they are resolved by looking at the originaldistances between points. Thus, the algorithm for mutual nearest neighbor clustering

works in the following way.

a) First the k-nearest neighbors of all points are found. In graph terms this can be

regarded as breaking all but the k strongest links from a point to other points in the

proximity graph. b) For each of the k points in a particular point’s k-nearest neighbor list, calculate

the mnv value for the two points. It can happen that a point is in one point’s k-nearest neighbor list, but not vice-versa. In that case, set the mnv value to some value

larger than 2k.

c) Merge the pair of clusters having the lowest mnv (and the lowest distance in case

of ties).

d) Repeat step (c) until the desired number of clusters is reached or until the only

clusters remaining cannot be merged. The latter case will occur when no points in

different clusters are k-nearest neighbors of each other.

The mutual nearest neighbor technique has behavior similar to the shared nearest

neighbor technique in that it can handle clusters of varying density, size, and shape.However, it is basically hierarchical in nature while the shared nearest neighbor approach

is partitional in nature.

4.4.2. Hypergraph-Based ClusteringHypergraph-based clustering [HKKM97] is an approach to clustering in high

dimensional spaces that uses hypergraphs. Hypergraphs are an extension of regular



37

graphs, which relax the restriction that an edge can only join two vertices. Instead an

edge can join many vertices.

Hypergraph-based clustering consists of the following steps:

1) Define the condition for connecting a number of objects (which will be the vertices of

the hypergraph) by a hyperedge. 2) Define a measure of the strength or weight of a hyperedge.

3) Use a graph-partitioning algorithm [KK98b] to partition the hypergraph into two parts

in such a way that the weight of the hyperedges cut is minimized.

4) Continue the partitioning until a fixed number of partitions are achieved, or until a“fitness” condition for the goodness of a cluster indicates that a current partition is a

good cluster.

In [HKKM97], the data being classified is “market basket” data. With this kind

of data there are a number of items and a number of “baskets”, or transactions, each of

which contains a subset of all possible items. (A prominent example of market basket

data is the subset of store items purchased by customers in individual transactions – hence the name market basket data.) This data can be represented by a set of binary

vectors – one for each transaction. Each item is associated with a dimension (variable),

and a value of 1 indicates that the item was present in the transaction, while a value of 0indicates that the item was not present.

The individual items are the vertices of the hypergraph. The hyperedges are

determined by determining frequent itemsets [AS97]. These are subsets of items thatfrequently occur together. For example, baby formula and diapers are often purchased

together.

The strength of the hyperedges is determined in the following manner. If thefrequent itemset being considered is of size n, and the items of the frequent itemset are i1,

i2… i

n, then the strength of a hyperedge is given by the following formula:

Less formally, we find the strength of each hyperedge by the following steps:

1) Consider each individual item, i j, in the frequent itemset. 2) Determine what fraction of the market baskets that contain the other n - 1 items also

contain i j. This (estimate of the) conditional probability that i j occurs when the other

items do is a measure of the strength of the association between the items. 3) Average these conditional probabilities together.

In [HKKM97] this is explained in slightly different terms using the concept of

association rules [AS97]. An association “rule” is an expression of the formi1 ,…,i j-1 ,i j+1 ,…,in → i j , where the i’s are individual items as above. (More generally, theconsequent can consist of more than one item.) The “confidence” of a rule is the(estimated) conditional probability as described in point 2 of the preceding paragraph.

The quality of a cluster is measured by a fitness measure. This measure is the

ratio of the weight of the edges that connect only vertices in the cluster to the weight of the edges that include at least one vertex in the cluster. Similarly, the degree to which a

vertex is associated with a cluster is measured by a connectivity measure. This measure

å=

+−

n

j

n j j j iiiii probn 1

111 ),,,...,|(1



38

is the ratio of number of edges that contain the vertex and are in the cluster to the number

of edges that contain the vertex.

4.4.3. ROCKROCK (RObust Clustering using linKs ) [GRS99] is a clustering algorithm for

data with categorical and Boolean attributes. It redefines the distances between points to be the number of shared neighbors whose strength is greater than a given threshold and

then uses a hierarchical clustering scheme to cluster the data.

As with hypergraph-based clustering discussed in section 4.4.2, ROCK deals primarily with market basket data. Traditional Euclidean distance measures are not

appropriate for such data and instead, ROCK uses the Jaccard coefficient (section 2.2.3.3)

to measure similarity. This rules out clustering approaches such as K-means or Centroid

based hierarchical clustering. (However, K-means can actually be modified to work wellfor some non-Euclidean data, e.g., documents.)

While the Jaccard coefficient provides a reasonable measure of the distance

between points, clusters are sometimes not well separated and so a new measure of

similarity between points was introduced that reflects the neighborhood of a point. If sim(pi , p j ) is the similarity between points, pi and p j, and 0 ≤ θ ≤ 1 is a parameter, then

link(pi , p j ) = | {q : sim(pi , q) ≥ θ } ∩ {q : sim(p j , q) ≥ θ }|

In words, link(pi , p j ) is the number of shared neighbors of pi and p j. The idea is

that two points will be “close” only if they share a relatively large number of neighbors.Such a strategy is intended to handle the problem of “border” points, which are close to

each other, but belong to different clusters. (This approach is really a form of

sparsification as is described in section 4.4.4).

ROCK also introduces a new objective function that is to be maximized:

å å= ∈

+=

k

i C p p f

i

r qi

ir qn

p plink n E 1 ,

)(21),(*

θ

Thus, we try to minimize the sum of the “link” similarity, i.e., the number of shared neighbors, between pairs of points in a cluster, subject to some scaling by the size

of the cluster. (This scaling factor is just the expected number of links.) This criterion

can be used to derive a criterion for merging clusters via a hierarchical agglomerativescheme by merging the two clusters that lead to the largest increase in E .

ROCK samples the data set in the same manner as CURE in order to avoid using

a hierarchical clustering algorithm on a large number of points. This is followed by an

assignment step where each remaining points is assigned to a cluster. A fraction of pointsis selected from each cluster and a calculation is performed to determine the number of

those points that are neighbors of the point to be assigned. This quantity is scaled by theexpected number of neighbors (based on the size of the cluster) and the point is assignedto cluster with the maximum number of neighbors (after scaling).

The basic steps of ROCK are as follows:

1) Obtain a sample of points from the data set.



39

2) Compute the link value for each set of points, i.e., transform the original similarities

(computed by the Jaccard coefficient) into similarities that reflect the number of

shared neighbors between points.

3) Perform an agglomerative hierarchical clustering on the data using the “number of

shared neighbors” similarities and the “maximize the shared neighbors” objective

function defined above. 4) Assign the remaining points to the clusters that have been found.

4.4.4. SparsificationThe sparsification of the proximity graph as a part of clustering has been

discussed briefly before. This section provides a more detailed look at this technique.

4.4.4.1. Why Sparsify?

Many clustering techniques, e.g., hierarchical techniques, use a similarity matrix

as their starting point. There are compelling advantages for sparsifying the similarity

matrix before beginning the actual clustering process. In particular:

The amount of data that needs to be processed is drastically reduced. Sparsification can eliminate more than 99% of the entries in a similarity matrix. This has

two beneficial effects:

• The amount of time required to cluster the data is drastically reduced.

• The size of the problems that can be handled is increased. Clustering may work better. Sparsification techniques keep the connections to

the most similar (nearest) neighbors of a point while breaking the connections to less

similar points. (This is in keeping with the nearest neighbor principle that the nearestneighbors of a point tend to belong to the same class as the point itself.) This reduces the

impact of noise and outliers and sharpens the distinction between clusters.

Sparsification facilitates the use of graph partitioning algorithms (or

algorithms inspired by graph partitioning algorithms). Such algorithms often work best on sparse graphs. The METIS family of graph partitioning algorithms [KK98a,

KK98b] are well known and widely used.

There are some techniques that use sparsification with good results. We provide examples of some specific techniques later, but for now we present a simple

example. When we applied one form of sparsification to spatial data and then plotted the

results, we were able to clearly differentiate core points, boundary points, and noise points. In the following figures (4 and 5), we show, from left to right, a) the initial spatial

data (all points), b) points with low connectivity (noise points), c) points with medium

connectivity (boundary points), and d) points with high connectivity (core points).The DBSCAN clustering algorithm described in [EKSX96] can also make a

classification of points into core, boundary and noise (outlier) points. In this case, theclassification was intentional, and is the basis of the clustering algorithm.



40

Figure 15: Separation of Spatial Data Points into Noise, Boundary, and Core

Points.

Figure 16: Separation of Spatial Data Points into Noise, Boundary, and CorePoints.

4.4.4.2. Sparsification as a Pre-processing StepSparsification of the proximity graph must be regarded as an initial step before the

use of real clustering algorithms. In theory, a perfect sparsification could leave the proximity matrix split into connected components corresponding to the desired clusters,

but, in practice, this rarely happens. It is very easy for a single edge to link two clusters

or for a single cluster to be split into several disconnected subclusters. Also, asmentioned, users often desire a hierarchical clustering.

Therefore, the real goal of sparsification should be the efficient and effective

identification of the core (strongly connected) points belonging to clusters andsubclusters. Sparsification discards outliers and/or boundary points, and may result in a

cluster being broken into a number of “good,” but separate subclusters. The actual

clustering process that follows sparsification should be designed to take the core pointsand the original points as input and produce clusters or hierarchies of clusters as output.

The following diagram gives a view of a complete clustering process that uses

sparsification.

Figure 17. The Complete Clustering Process



41

4.4.4.3. Sparsification TechniquesIn this section we discuss some basic issues of sparsification and review some

sparsification techniques.

4.4.4.4. Keeping the Proximity Matrix Symmetric?

As mentioned earlier, most clustering techniques work with symmetric proximitymatrices. Conceptually, it is convenient to think of the sparsification process as

consisting of two parts: a first step that sparsifies the matrix, and a second step that makes

the matrix symmetric again. (In practice, both steps are often carried outsimultaneously.)

Two symmetrically corresponding entries, pij and p ji, may both still be equal after

sparsification, i.e., either both values are sparsified or both retain their original and

symmetric value. (We are considering only proximity matrices that were originallysymmetric.) If not, then there are two options for keeping the proximity matrix

symmetric. We can sparsify both entries, or we can set both entries to their original

value. The first option is called sparsification with inclusion, while the second option is

sparsification with exclusion. Sparsification with exclusion is sometimes referred to as a“reciprocal” or “mutual” approach. The same sparsification procedure can have quite

different properties, depending on whether inclusion or exclusion is used.

4.4.4.5. Description of Sparsification Techniquesa) Global Thresholds [Kow97]. All entries of the similarity (dissimilarity) matrix that

are less than (greater than) the given threshold are set to 0 (∞). While this approachcan be effective, members of different classes often have different levels of similarity

or dissimilarity. For example, members of class A might have similarities ranging

from 0.3 to 0.5, while members of class B may have similarities ranging from 0.6 to0.8. Thus, a global threshold normally doesn’t work well. (However, this procedure

is frequently used.) b) Nearest neighbor techniques. These techniques, while not explicitly cast as matrix

sparsification techniques [JP73, GK78], use conditions involving the nearestneighbors of a point to sparsify the similarity matrix. These techniques have many

variations, but we consider only three of the most straightforward approaches.

1. Adaptive Thresholds [KHK99]. The average similarity of a point’s two nearest

neighbors is calculated and is multiplied by a fixed constant, α, 0 ≤ α ≤ 1. This

adaptively calculated threshold is used to sparsify the entries in the row of the

proximity matrix that corresponds to this point. This technique can be used with

inclusion or with exclusion.

2. K Nearest Neighbors [KHK99b]. For the row corresponding to a particular

point, all entries except those of the K nearest neighbors are sparsified. Thistechnique can be used with inclusion or with exclusion.

3. Shared Nearest Neighbors [JP73]. First we apply K nearest neighbors

sparsification (with exclusion). The remaining entries are sparsified further by

requiring that the entries of a row and the point corresponding to the row must

share at least K 1 ≤ K neighbors. These two requirements automatically produce asymmetric matrix. In the degenerate case, when K 1 = 0, this approach is just K

nearest neighbors (with exclusion).



42

Two additional comments about nearest neighbor approaches:

• Early work that used nearest neighbor techniques to sparsify proximity matrices[JP73, GK78] used the connected components of the sparsified proximity matrix to

define the clusters. However, a bad link can ruin the clustering and this is one reason

that additional steps are needed following the sparsification process. • The ROCK algorithm [GRS99] is a new hierarchical clustering algorithm for

categorical data, whose initial step redefines the proximity of two points to be the

number of nearest neighbors that they share after sparsification with a global

threshold. While using a shared nearest neighbors technique for defining proximity isa worthwhile approach - it has also been used in [JP73] and [GK78] - global

thresholds do not work as well as some of the other techniques.

4.4.4.6. Evaluating Sparsification ResultsIf the “true” classes that points belong to are available, then the results of a

sparsification can be evaluated by looking at the percentage of points that fall into

different five categories.

a) Percentage of isolated points. This is the percentage of points that do not have

connections to any points after the sparsification. Normally, isolating outliers is

good, but isolating other points is bad.

b) Percentage of really bad points, i.e., the percentage of points isolated from all

points of the same class. This is the percentage of points that are not isolated, but do

not have any connections to points within their own classes. Ideally, this percentage

should be low.

c) Percentage of really good points, i.e., the percentage of points isolated from all

points of different classes. This is the percentage of points that are not isolated and

do not have any connections to points in other classes. Ideally, this percentage should be high, since points with neighbors of only the same class should, intuitively, be

easier to cluster correctly.

d) Percentage of mostly bad points, i.e., the percentage of points that have a

stronger connection to points of other classes. This is the percentage of points that

are not isolated, but whose connections to points in other classes have more weight

than do its connections to points of the same class. Ideally, this percentage should be

low.

e) Percentage of mostly good points, i.e., the percentage of points that have astronger connection to points of the same class. This is the percentage of points

that are not isolated, but whose connections to points of the same class have moreweight than do its connections to points in other classes. Ideally, this percentage

should be high. The five percentages just described will sum to 100%, and can be plotted

simultaneously to provide an easily understood measure of how specific sparsification

techniques distribute points into the different categories described above.

By summing the percentage of mostly good points and the percentage of really

good points, we can obtain the percentage of good points. This measure provides anoverall indication of how well the sparsification is working. If all classes have large

percentages of points whose connections are predominantly to the same class, then

clustering “should” be easier. Conversely, if some or many classes have large numbers of



43

points which are isolated or which have neighbors that are predominantly connected to

points of other classes, then clustering will be hard.

4.4.4.7. Tentative ConclusionsSparsification has not been intensively investigated and the results that we present

here are based on unpublished research.

1) Sparsification is best used as a pre-processing step. Sparsification often breaks

“natural” clusters and should be used with an algorithm that can recombine these sub-clusters in an intelligent way. Chameleon is the best example of such an algorithm.

2) Global thresholds don’t work very well. Not surprisingly, there are two problems:

a) The effect of the global threshold varies widely for each class, and

b) Each class has a different threshold that is best.

3) Adaptive thresholds work better than global thresholds. Again this is not

surprising since such approaches model the data more accurately.

4) Nearest neighbor techniques are the best of all. K-nearest neighbor or shared

nearest neighbor techniques appear to model the data more accurately than adaptivethreshold techniques.

5) There is a certain amount of flexibility in the choice of the nearest neighbor

sparsification scheme and the parameters. In the research done for Chameleon it

was discovered that a number of different nearest neighbor approaches and

parameters worked about as well. However, Chameleon a) uses a graph-partitioning

algorithms to efficiently divide the points into strongly connected groups, and b)merges sub-clusters back together in a way that undoes the cluster breakage caused

by sparsification.

5. Advanced Techniques for Specific Issues

5.1. Scalability

For data mining, the scalability of an algorithm to large data sets is key. The

following two related techniques combine a variety of techniques to achieve that goal.

5.1.1. BIRCHBIRCH (Balanced and Iterative Reducing and Clustering using Hierarchies)

[ZRL96] is a highly efficient clustering technique for data in Euclidean vector spaces,

i.e., data for which averages make sense. BIRCH can efficiently cluster such data with a

single pass and can improve that clustering in additional passes. BIRCH also can alsodeal effectively with outliers.

BIRCH is based on the notion of a clustering feature (CF) and a CF tree. The ideais that a cluster of data points (vectors) can be represented by a triple of numbers (N, LS,

SS), where N is the number of points in the cluster, LS is the linear sum of the points, andSS is the sum of squares of the points. These are common statistical quantities and a

number of different inter-cluster distance measures can be derived from them.

A CF tree is a height-balanced tree. Each interior node has entries of the form[CFi, childi], where childi is a pointer to the i

thchild node. The space that each entry

takes and the page size, P, determine the number of entries, B, in an interior node. The



44

space of each entry is, in turn, determined by the size of the points, i.e., by the

dimensionality of the points.

Leaf nodes consist of a sequence of clustering features, CFi, where each clusteringfeature represents a number of points that have been previously scanned. Leaf nodes are

subject to the restriction that each leaf node must have a diameter that is less than a

parameterized threshold, T. The space that each entry takes and the page size, P,determine the number of entries, L, of a leaf.

By adjusting a threshold parameter, T, the height of the tree can be controlled. T

controls the fineness of the clustering, i.e., the extent to which the data in the original setof data is reduced. The goal is to keep the CF tree in main memory by adjusting the T

parameter as necessary.

A CF tree is built as the data is scanned. As each data point is encountered, the

CF tree is traversed, starting from the root and choosing the closest node at each level.When the closest “leaf” cluster for the current data point is finally identified, a test is

performed to see if adding the data item to the candidate cluster will result in a new

cluster with a diameter greater than the given threshold, T. If not, then the data point is

“added” to the candidate cluster by updating the CF information. The cluster informationfor all nodes from the leaf to the root is also updated.

If the new cluster would have a diameter greater than T, then a new entry iscreated if the leaf node is not full. Otherwise the leaf node must be split. The two entries

(clusters) that are farthest apart are selected as “seeds” and the remaining entries are

distributed to one of the two new leaf nodes, based on which leaf node contains theclosest “seed” cluster. Once the leaf node has been split, the parent node is updated, and

split if necessary, i.e., if the parent node is full. This process may continue all the way to

the root node.

BIRCH follows each split with a merge step. At the interior node where the splitstops, the two closest entries are found. If these entries do not correspond to the two

entries that just resulted from the split, then an attempt is made to merge these entries and

their corresponding child nodes. This step is intended to increase space utilization andavoid problems with skewed data input order.

BIRCH also has a procedure for removing outliers. When the tree needs to be

rebuilt, because it has run out of memory, then outliers can optionally be written to disk.(An outlier is defined to be node that has “far fewer” data points than average.) At

certain points in the process, outliers are scanned to see if they can be absorbed back into

the tree without causing the tree to grow in size. If so, they are re-absorbed. If not, theyare deleted.

BIRCH consists of a number of phases beyond the initial creation of the CF tree.

The phases of BIRCH are as follows:

1) Load the data into memory by creating a CF tree that “summarizes” the data.

2) Build a smaller CF tree if it is necessary for phase 3. T is increased, and then the

leaf node entries (clusters) are reinserted. Since T has increased, some clusters will

be merged.

3) Perform global clustering. Different forms of global clustering (clustering which

uses the pairwise distances between all the clusters) can be used. However, an

agglomerative, hierarchical technique was selected. Because the clustering features

store summary information that is important to certain kinds of clustering, the global



45

clustering algorithm can be applied as if it were being applied to all the points in

cluster represented by the CF.

4) Redistribute the data points using the centroids of clusters discovered in step 3

and thus, discover a new set of clusters. This overcomes certain problems that can

occur in the first phase of BIRCH. Because of page size constraints and the T

parameter, points that should be in one cluster are sometimes split, and points thatshould be in different clusters are sometimes combined. Also, if the data set contains

duplicate points, these points can sometimes be clustered differently, depending on

the order in which they are encountered. By repeating this phase, multiple times, the process converges to a (possibly local) minimum.

5.1.2. Bubble and Bubble-FMBubble and Bubble-FM [GRGPF99] are scalable clustering algorithms that are

extensions of BIRCH to general metric spaces, i.e., collections of data points where only

distances are meaningful. (In Euclidean space, points (vectors) can be added and

concepts such as a mean or median of a group of points make sense. This is not the case

for general metric spaces, although there are exceptions, e.g., documents.) Bubble andBubble-FM can efficiently cluster general metric data with a single pass.

(A good example of non-Euclidean data is strings. An “edit” distance can bedefined between two strings which is the minimum number of character insertions,

deletions, and transpositions that would be needed to transform one string into another.

While these distances form a metric space, they are not an Euclidean vector space.)

Bubble and Bubble-FM are based on BIRCH*, which generalizes the notion of a cluster

feature that was introduced in BIRCH.

Conceptually, this new CF is a quintuple:

a) The number of objects in the cluster. b) The objects the cluster.

c) The row sum of each object in the clusters. The row sum is the sum of the squared

distance of an object to each other object in the cluster.

d) The clustroid of the cluster. The clustroid is the object in the cluster that has thesmallest row sum. The clustroid represents a generalization of the idea of a centroid

to general metric spaces. Note that the clustroid is always an actual object, and thus,

is like a medoid.

e) The radius of the cluster, which is defined as( )

n

clustroid object d n

iiå =1

2 ,, where d is the

distance function and n is the number of objects in the cluster.

However, from an efficiency point of view not all objects and row sums can bemaintained in memory. Once a cluster becomes large enough, only a representative set of

objects (and their corresponding row sums) is kept and updates to the CF are calculated

approximatelyAs with BIRCH, these cluster features are organized into a CF tree, which is a

height-balanced tree similar to an R* tree. Each interior node has entries of the form

[CFi, childi], where childi is a pointer to the ith

child node. Cluster features for non-leaf

nodes are similar to those of leaf nodes, where the representative objects are chosen from



46

the representative objects of the node’s children. However, a condition is imposed that at

least one representative object be chosen from each child.

The basic steps of BIRCH* are the same as BIRCH, although the details aresomewhat different. One important parameter governs the way in which the distance of

point being added to a CF node is calculated. For Bubble, the point is considered to be a

cluster and the distance to non-leaf nodes is calculated as the average distance of the point to each representative point in the cluster. For leaf nodes, the distance is calculated

as the distance of the point to the clustroid.

When a new data point is processed, the data point is treated like a singletoncluster. The CF tree is traversed, starting from the root and choosing the closest node at

each level. When the closest “leaf” cluster for the current data point is finally identified,

a test is performed to see if adding the data item to the candidate cluster will result in a

new cluster with a radius greater than a given threshold, T. If not, then the data point is“added” to the candidate cluster by updating the CF information. The cluster information

for all nodes from the leaf to the root is also updated. If the new cluster would have a

radius greater than T, then a new entry is created.

Bubble-FM uses an algorithm called Fast Map [FL95] to map the representativeobjects of a non-leaf node into Euclidean space, called image space, in a way that

preserves the distances between the points. The clustroid is then replaced by the centroidof the mapped representative points. When a point is added to the CF tree and needs to be

compared to a particular non-leaf node, it is mapped into the image space of that cluster

and its distance to the centroid is calculated and compared to the threshold. Distancecomputations to the leaf nodes are still performed as with Bubble.

The mapping adds a lot of complexity and does not give as accurate an estimate

of the distance of a point from a cluster, but it requires fewer distance computations. This

can be important in many cases, e.g., string comparisons, where the distance calculationsare time consuming. Distances to leaf nodes are not calculated using the mapping,

because it is important not to make a mistake in adding a point to the wrong cluster. It is

better to start a new cluster, and the frequent updates at leaf nodes would require toomany re-mappings.

As with BIRCH, Bubble and Bubble-FM both follow the initial clustering by a

hierarchical clustering to refine the clusters and another pass through the data thatredistributes the points to the clusters.

5.2. Clusters in Subspaces

In high dimensional spaces, clusters often lie in a subspace. The following two

related techniques have been devised to handle that situation.

5.2.1. CLIQUECLIQUE [AGGR98] is a clustering algorithm that finds high-density regions by

partitioning the data space into cells (hyper-rectangles) and finding the dense cells.

Clusters are found by taking the union of all adjacent, high-density cells. For simplicityand ease of use, clusters are described by expressing the cluster as a DNF (disjunctive

normal form) expression and then simplifying the expression.

CLIQUE is based on the following simple property of clusters: Since a cluster represents a dense region in some subspace of the feature space, there will be dense areas



47

corresponding to the cluster in all lower dimensional subspaces. Thus, CLIQUE starts by

finding all the dense areas in the one-dimensional spaces corresponding to each attribute.

CLIQUE then generates the set of two-dimensional cells that might possibly be dense, bylooking at dense one-dimensional cells, as each two-dimensional cell must be associated

with a pair of dense one-dimensional cells. More generally, CLIQUE generates the

possible set of k-dimensional cells that might possibly be dense by looking at dense(k - 1) dimensional cells, since each k-dimensional cell must be associated with a set of k

dense (k - 1)-dimensional cells.

The CLIQUE algorithm is similar to the APRIORI algorithm [AS97] for findingfrequent itemsets.

5.2.2. MAFIA (pMAFIA)MAFIA (Merging of Adaptive Finite Intervals (And more than a CLIQUE))

[GHC99] is a modification of CLIQUE that runs faster (40+) and finds better quality

clusters. pMAFIA is the parallel version.

The main modification is to use an adaptive grid. Initially, each dimension

(attribute) is partitioned into a fixed number of cells. A histogram is generated that showsthe number of data points in each cell. Groups of five adjacent cells are grouped into

windows, and each window is assigned the maximum value of the number of points in itsfive cells. Adjacent windows are grouped together if the values of the two windows are

close – within 20%. As a special case, if all windows are combined into one window, the

dimension is partitioned into a fixed number of cells and the threshold for beingconsidered a dense unit is increased for this dimension. A relatively uniform distribution

of data in a particular dimension normally indicates that no clustering exists in this

dimension.

6. Evaluations

In this section we evaluate some of the clustering techniques for data mining thathave been discussed earlier. (Actual reviews of the specific techniques are omitted in this preliminary version of the paper.)

6.1. Requirements for Clustering Analysis for Data Mining

6.2. Typical Problems and Desired Characteristics

The desired characteristics of a clustering algorithm depend on the particular

problem under consideration. The following is a list. (Note however, that there are other

general requirements for data mining that have also been described [CHY96].)

6.2.1. ScalabilityClustering techniques for large sets of data must be scalable, both in terms of

speed and space. It is not unusual for a database to contain millions of records, and thus,

any clustering algorithm used should have linear or near linear time complexity to handle

such large data sets. (Even algorithms that have complexity of O(m2) are not practical for

large data sets.) Some clustering techniques use statistical sampling. Nonetheless, there

are cases, e.g., situations where relatively rare points have a dramatic effect on the final

clustering, where a sampling is insufficient.



48

Furthermore, clustering techniques for databases cannot assume that all the data

will fit in main memory or that data elements can be randomly accessed. These

algorithms are, likewise, infeasible for large data sets. Accessing data points sequentiallyand not being dependent on having all the data in main memory at once are important

characteristics for scalability.

6.2.2. Independence of the order of inputSome clustering algorithms are dependent on the order of the input, i.e., if the

order in which the data points are processed changes, then the resulting clusters maychange. This is unappealing since it calls into question the validity of the clusters that

have been discovered. They may just represent local minimums or artifacts of the

algorithm.

6.2.3. Effective means of detecting and dealing with noise or outlyingpoints

A point which is noise or is simply an atypical point (outlier) can often distort a

clustering algorithm. By applying tests that determine if a particular point really belongsto a given cluster, some algorithms can detect noise and outliers and delete them or

otherwise eliminate their negative effects. This processing can occur either while theclustering process is taking place or as a post-processing step.

However, in some instances, points cannot be discarded and must be clustered as

well as possible. In such cases, it is important to make sure that these points do not

distort the clustering process for the majority of the points.

6.2.4. Effective means of evaluating the validity of clusters that areproduced.

It is common for clustering algorithms to produce clusters that are not “good”

clusters when evaluated later.

6.2.5. Easy interpretability of resultsMany clustering methods produce cluster descriptions that are just lists of the

points belonging to each cluster. Such results are often hard to interpret. A descriptionof a cluster as a region may be much more understandable than a list of points. This may

take the form of a hyper-rectangle or a center point with a radius.

Also, data clustering is sometimes preceded by a transformation of the original

data space – often into a space with a reduced number of dimensions. While this can behelpful for finding clusters, it can make the results very hard to interpret.

6.2.6. The ability to find clusters in subspaces of the original space.Clusters often occupy a subspace of the full data space. (Hence, the popularity of

dimensionality reduction techniques.) Many algorithms have difficulty finding, for

example, a 5 dimensional cluster in a 10 dimensional space.

6.2.7. The ability to handle distances in high dimensional spacesproperly.

High-dimensional spaces are quite different from low dimensional spaces. In

[BGRS99], it is shown that the distances between the closest and farthest neighbors of a



49

point may be very similar in high dimensional spaces. Perhaps an intuitive way to see this

is to realize that the volume of a hyper-sphere with radius, r , and dimension, d , is

proportional to r d , and thus, for high dimensions a small change in radius, means a large

change in volume. Distance based clustering approaches may not work well in such

cases.

If the distance between points in a high dimensional space are plotted, then thegraph will often show two peaks: a “small” distance representing the distance between

points in clusters, and a “larger” distance representing the average distance between

points. (See [Bri95].) If only one peak is present or if the two peaks are close, thenclustering via distance based approaches will likely be difficult.

Yet another set of problems has to do with how to weight the different

dimensions. If different aspects of the data are being measured in different scales, then a

number of difficult issues arise. Most distance functions will weight dimensions withgreater ranges of data more highly. Also, clusters that are determined by using only

certain dimensions may be quite different from the clusters determined by using different

dimensions. Some techniques are based on using the dimensions that result in the

greatest differentiation between data points. Many of these issues are related to the topicof feature selection, which is an important part of pattern recognition.

6.2.8. Robustness in the presence of different underlying data and cluster characteristics

A robust clustering technique must be take into account the following

characteristics of the data and the clusters:

1) Dimensionality ( Discussed separately in sections 6.2.6 and 6.2.7) 2) Noise and Outliers ( Discussed separately in section 6.2.3)

3) Statistical Distribution. Some data is has a Gaussian (normal) or uniformdistribution, but this is frequently not the case.

4) Cluster Shape. In some cases clusters are regularly shaped regions, e.g., rectangular

or globular, but irregularly shaped, non-convex clusters are common.

5) Cluster Size. Some clustering methods, e.g., K-means don’t work well in the

presence of different size clusters.

6) Cluster Density. Some clustering methods, e.g., K-means don’t work well in the presence of clusters that have different density.

7) Cluster Separation. In some cases the clusters are well-separated, but in many other

cases the clusters may touch or overlap.

8) Type of data space, e.g., Euclidean or non-Euclidean. Some clustering techniquescalculate means or use other vector operations that often only make sense in

Euclidean space.

9) Many and Mixed Attribute Types, e.g., (temporal, continuous, categorical). A mixof attribute types is usually handled by having a proximity function that can combine

all the different attributes in an “intelligent” way.

Many clustering techniques have assumption about the data and cluster

characteristics and do not work well if those assumptions are violated. In such cases the

clustering technique may miss clusters, split clusters, merge clusters, or just produce poor

clusters.



50

6.2.9. An ability to estimate any parameters(For example, the number of clusters, the size of clusters, or the density of

clusters.) Many clustering algorithms take the number of clusters as a parameter. This can

be a useful feature in certain instances, e.g., when using a clustering algorithm to create a

balanced tree for nearest neighbor lookup, or when using clustering for compression.However, this is not generally good, since the number of clusters parameterized may not

match the “real” number of clusters.

An algorithm that requires the number of clusters up front can always be runmultiple times. Assuming that there is some way to compare the quality of the clusters

produced, it is then possible to empirically determine the best number of clusters. Of

course, this increases the amount of computation required.

Likewise, it is often difficult to estimate the proper values for other parameters of clustering algorithms. In general, the parameters of a clustering algorithm may identify

areas of weakness. In the best case, the results produced by a clustering algorithm will be

relatively insensitive to modest changes in the parameter values.

6.2.10. Ability to function in an incremental manner In certain cases, e.g., data warehouses, the underlying data used for the original

clustering can change over time. If the clustering algorithm can incrementally handle the

addition of new data or the deletion of old data, then this is usually much more efficient

than re-running the algorithm on the new data set.

7. Data Mining From Other Perspectives

7.1. Statistics-Based Data Mining

Cluster Analysis has a long history in statistics and statisticians developed some

of the techniques that have been previously discussed. We will not discuss most of thesetechniques any further here. However, there is a class of techniques, mixture models,

which deserve additional attention.

7.1.1. Mixture ModelsThe basic idea of mixture models is to view the data as a mixture of observations

from a number of different probability distributions. The probability distributions areusually taken to be multivariate normal, since this type of distribution is well-understood,

relatively easy to work with, and has been shown to work well in many instances. These

types of distributions yield hyper-ellipsoidal clusters.There are a number of different assumptions that can be made when taking a

mixture model approach, i.e., whether all clusters have the same shape, volume, andorientation. These assumptions are reflected in the number of statistical parameters (mean

vectors and covariance matrices), that need to be estimated. Of course, the more flexiblethe model, the more difficult and time consuming it is to find a solution.

Some form of the EM (expectation maximization) algorithm is typically used to

perform the parameter estimation. This algorithm can be slow, is not practical for modelswith large numbers of components, and does not work well when clusters contain only a

few data points or if the data points are nearly co-linear. There is also a problem in



51

estimating the number of clusters or, more generally, in choosing the exact form of the

model to use. This problem has typically been dealt with by applying a Bayesian

approach, which roughly speaking, gives the odds of one model versus another, based onan estimate derived from the data. Mixture models may also have difficulty with noise

and outliers, although work has been done to deal with this problem.

In spite of all the apparent difficulties with mixture models, they do have theadvantage of having a sound mathematical foundation. Also, a model-based approach

provides a disciplined way of eliminating some of the complexity associated with data.

(To see the patterns in data it often necessary to simplify the data, and fitting the data to amodel is a good way to do that, at least if the model is a good match for the data.)

Furthermore, it is easy to characterize the clusters produced, since they can be described

by a small number of parameters. Finally, many sets of data are indeed the result of

random processes and thus, should satisfy the statistical assumptions of these models.AutoClass [CS96] is one of the most prominent of the mixture-model based

programs and has been used for data mining to some extent. A readable and current

technical report on clustering using mixture models is given by [FR98]. This source also

provides a pointer to state-of-the-art mixture modeling software. An approach to usingmixture models for co-occurrence data (frequent itemsets of size 2) is given in [HP98].

7.2. Text Mining and Information Retrieval

Information retrieval is the study the problems and techniques involved in

automatically indexing documents and retrieve collections of documents. A commontask in information retrieval is retrieving relevant documents based on keywords

specified by a user, e.g., web search engines.

Text mining is the discovery of useful and non-obvious information in collectionsof text documents. Some people regard text mining as separate from data mining, since

they view data mining as restricted to databases, but many of the concepts and techniques

used for data mining and text mining are quite similar. It is true that some text miningtechniques make use of natural language processing or the structure of documents, butmany techniques do not.

For both text mining and information retrieval, documents are typically

represented by vectors that indicate the frequency of keywords in the documents. In suchcases, many of the clustering techniques that we have previously described, i.e., K-means

and agglomerative hierarchical clustering, can be directly applied. (This is so even

though document vectors are a very high dimensional space - thousands or tens of thousands of dimensions.) In what follows, we will restrict ourselves to the assumption

that we are dealing with document vectors.

7.2.1. Uses of Clustering for DocumentsClustering has been used in three principal ways with documents:

1) Document Retrieval. Early researchers hoped that clustering documents would lead

to quicker and better document retrieval. These hopes have not been realized as other (non-cluster based) techniques for document retrieval continue to be superior.

2) Document browsing. Document clustering can provide a way of summarizing a

large number of documents into a small number of groups that are suitable for browsing. The collection of documents being browsed might be the entire collection



52

of documents or it might be just the documents returned by a user query. The

documents can be reclustered to match a user’s interest.

3) Automatic Creation of Document Hierarchies. Current hierarchical classificationsof documents are created mostly by hand. There has been some work on trying to

automate or partially automate this process.

Many of the techniques used for clustering documents are ones that we have seen

in previous sections. Hierarchical techniques were often used in the past, but they do not

scale well to the large number of documents often found in collections of documentstoday, e.g., consider the number of Web pages. Consequently more recent studies have

often focused on using variations of K-means. [SKK00]

7.2.2. Latent Semantic IndexingLSI is a dimension reduction technique that can be used as a preliminary to a

clustering algorithm in the same way that principal component analysis is sometimes

used. (Recall that principal component analysis (PCA) is a technique for dimensionality

reduction commonly used in statistics.)However, Latent Semantic Indexing is a technique that is mostly used for

information retrieval. Given a number of keywords, n, and a number of documents, m, an

m by n matrix is constructed, such that the ijth

entry of the matrix contains the number of

times that the jth

keyword occurs in the ith

document. Using standard linear algebra

techniques, an alternative form of this matrix, the singular value decomposition (SVD), is

found. This form can be simplified (see [BD95] and [FO95]). Conceptually, the onlyinformation that is kept is the information that reflects the most significant connections

between keywords and documents. The results of a query consisting of a keyword vector

can be computed using linear algebra operations.

7.3. Vector Quantization

There has been a considerable amount of cluster analysis work done in the area of vector quantization. The idea is the following: given a set of vectors representing, for

example an image, find a set of reference vectors that will be used instead of the full set

of vectors. The idea is that in video and voice, many of the data points that describe a particular image or sound are similar, and, hence, not much information will be lost if

each vector is replaced by its closest reference vector. In addition, the information can be

represented in a much smaller space, i.e. a high degree of compression is possible.Vector quantization may also aid pattern recognition by significantly reducing the

number of patterns that need to be considered.

The challenge is to minimize the error that results from approximating each vector by a reference vector and to maximize the possible compression by minimizing the

number of reference vectors needed.

Various clustering techniques have been used to perform this function. In

particular, the common K-means algorithm [DJ88] has been used. However, thisalgorithm requires specifying the number of clusters, and more recently, other

information (entropy) based techniques [BK93] have been used that can determine both

the reference vectors (the clusters) and the best value for K, the number of clusters.



53

This type of approach has been extended to more general clustering problems

[BH97]. However, these approaches are very mathematically and computationally

intense, using concepts similar to those used in statistical mechanics and informationtheory, and relying on simulated and deterministic annealing algorithms.

An interesting thing to note about vector quantization is that there is not any

notion of an outlier or noise. All points must be mapped to some reference vector.

7.4. Self-Organizing Maps

7.4.1. SOM OverviewThe Kohonen Self-Organizing Feature Map (SOFM or SOM) [Koh97] is a

clustering and data visualization technique based on neural networks. The “self-

organizing” part of the name derives from the ability of neural nets to learn their configuration from training data. The “map” part of the name comes from the way that

SOM neural nets work, i.e., when an input (a point) is fed into a SOM neural network, the

output is the label of a particular neuron in the network, and thus, a SOM neural network can be viewed as mapping a point to neuron. Finally, the “feature” portion of the name

comes from observation that, in a trained SOM network, a small set of neurons are often

associated with a particular data feature.A detailed explanation of how SOM works is provided below, but conceptually,

SOM can be viewed as a vector quantization technique where the reference vectors are

learned by training a neural network. Thus, like vector quantization, the goal of SOM is

to find a set of reference vectors and to assign each point in the data set to the “closest”reference vector. With the SOM approach, each reference vector is associated with a

particular neuron, and the components of that reference vector become the “weights” of

that neuron. As with other neural network systems, a SOM neural net is trained byconsidering the data points one at a time and adjusting the weights (reference vectors) so

as to best fit the data.

The final output of the SOM technique is a set of reference vectors whichimplicitly defines clusters. (Each cluster consists of the points closest to a particular

reference vector.) However, while SOM might seem very similar to K-means or other

vector quantization approaches, there is a fundamental conceptual difference. During thetraining process, SOM uses each data point to update the closest reference vector and thereference vectors of nearby neurons. In this way, SOM produces an “ordered” set of

reference vectors. In other words, the reference vectors of neurons which are close to

each other in the SOM neural net will be more closely related to each other than to thereference vectors of neurons that are farther away in the neural net. Because of this

constraint, the reference vectors in SOM can be viewed as lying on a smooth, elastic two-

dimensional surface in m dimensional space, a surface which tries to fit the mdimensional data as well as possible. (Regression is a statistical techniques that tries to

find the hyper-plane that best fits a set of data, and thus, you could think of the SOM

reference vectors as the result of a nonlinear regression with respect to the data points.)

7.4.2. Details of a Two-dimensional SOMThere are many types of SOM neural networks, but we restrict our discussion to a simple

variant where the neurons are organized as a two-dimensional lattice as shown in Figure17.



54

Figure 17. Two-dimensional 3 by 3 rectangular SOM neural network.

Note that each neuron (node) is assigned a pair of coordinates (i, j). Often such a

network is drawn with links between adjacent neurons, but that is misleading since theinfluence of one neuron on another is via a neighborhood which is defined in terms of

coordinates, not links. Each neuron is associated with an m dimensional reference vector,

where m is the dimensionality of the data points.At a high level, clustering using the SOM techniques consists of the following steps:

1) Initialize the reference vectors. 2) While the training is not complete do the following:

a. Select the next training sample (point).

b. Determine the neuron whose reference vector is closest to the current point. c. Update this reference vector and the reference vectors of all neurons which are close,

i.e., in a specified neighborhood.

3) When the training is complete, assign all points to the closest reference vectors and

return the reference vectors and clusters.

Initialization can be performed in a number of ways. One approach is to choose

each component of a reference vector randomly from the range of values observed in the

data for that component. While this approach works, it is not necessarily the bestapproach, especially for producing a rapid convergence to an equilibrium state. Another

approach is to randomly choose the initial reference vectors form the available data

points. (This is very much like one of the techniques for initializing K-means centroids.)There are is also a more sophisticated technique which uses the autocorrelation matrix of

the data and which is described in [Koh97].

The second step is the training or the convergence step. Selection of the next

training sample is fairly straightforward, although there are a few issues. Since trainingmay require a 100,000 steps, each data point may be used multiple times, especially if the

number of points is small. However, if the number of points is large, then not every point

must be used. Also, it is possible to enhance the influence of certain groups of points,e.g., a less frequent class of points, by increasing their frequency in the training set.

(However, this becomes more like supervised classification.)

The determination of closest reference vector is also straightforward, although itdoes require the specification of a distance metric. The Euclidean distance metric is often

used, although a dot product metric is often used as well. When using the dot product

distance, the data vectors are often normalized beforehand and the reference vectors are

(0,0)

(2,0)

(1,0)

(0,1)

(2,1)

(1,1)

(0,2)

(2,2)

(1,2)



55

normalized at each step. In such cases dot product metric is equivalent to using the

cosine measure.

The update step is the most complicated. Let m1, …, mk be the reference vectorsassociated with each neuron. (For a rectangular grid, note that

k = number of rows * number of columns.) For time step t , let s be the current training

point and assume that the closest reference vector to s is mc, the reference vector associated with neuron c. Then, for time t + 1, the ith reference vector is updated by using

the following equation. (We will see shortly that the update is really restricted to

reference vectors whose neurons are in a small neighborhood of neuron c.)

mi(t +1) = mi(t ) + hci(t) ( s(t ) – mi(t ))

Thus, at time t , a reference vector, mi(t ), is updated by adding a term, hci(t) ( s(t ) –

mi(t )), which is proportional to the difference, s(t ) – mi(t ), between the reference vector,

mi(t ), and the training point, x. hci(t) determines the effect that the difference, s(t ) – mi(t ),will have and is chosen so that a) it diminishes with time and b) it enforces a

neighborhood effect, i.e., the effect of a point is strongest on the neurons closest to theneuron c. Typically, hci(t) is chosen to be one of the following two functions:

hci(t ) = α(t ) exp(-||r c – r i ||2

/ (2σ2(t)) (Gaussian function)

hci(t ) = α(t ) if ||r c – r i || ≤ threshold , 0 otherwise (step function)

These function require quite a bit of explanation. α(t ) is a learning rate

parameter, 0 < α(t ) < 1, which decreases monotonically and which controls the rate of

convergence. r i = (xi, yi) is the two-dimensional point which gives the grid coordinates of

the ith

neuron. ||r c – r i || is the Euclidean distance of the two neurons, i.e.,

22 )()( cici y y x x −+− . Thus, we see that for neurons that are “far” from neuron c, the

influence of training point s will be either greatly diminished or non-existent. Finally,

note that σ is the typical Gaussian “variance” parameter and controls the width of the

neighborhood, i.e., a small σ will yield a small neighborhood, while a large σ will yield awide neighborhood. The threshold used for the step function also controls theneighborhood size.

(It is important to emphasize that it is the neighborhood updating technique that

enforces a relationship (ordering) between reference vectors associated with neighboring

neurons.)Deciding when training is complete is an important issue. Ideally, the training

procedure should continue until convergence occurs, i.e., until the reference vectors are

not changing much. The rate of convergence will depend on a number of factors, e.g.,

the data and α(t ). We will not discuss these issues further (see [Koh97]) except to

mention that the most important issue, as with neural network in general, is the fact that

convergence is often slow.

7.4.3. ApplicationsOnce the SOM vectors are found, they can be used for many purposes besides

clustering. For example, with a two-dimensional SOM it is possible to associate various



56

quantities at the grid points associated with each neuron and visualize the results via

various types of plots. For example, plotting the number of points associated with each

reference vector (neuron) yields a surface plot that reveals the distribution of pointsamong neurons (clusters). (A two-dimensional SOM is a non-linear projection of the

original probability distribution function into two dimensions. This projection attempts to

preserve topological features and thus, SOM has been compared to the process of “pressing a flower.”)

Besides data exploration, SOM and its supervised learning variant, LVQ

(Learning Vector Quantization), have been used for many useful tasks, e.g., imagesegmentation, organization of document files, speech processing, etc. There are

thousands of papers that deal with SOM neural networks and the basic techniques have

been extended in many directions. For a good introduction and more than 2000

references see [Koh97].

7.5. Fuzzy Clustering

If data points are distributed in well-separated groups, then a crisp classification

of the points into disjoint clusters seems like an ideal approach. However, in most cases,the points in a data set cannot be partitioned into well-separated clusters and thus, there

will be a certain arbitrariness in assigning a point to a particular cluster, e.g., consider a point that lies near the boundary of two clusters, but is just slightly closer to one of the

two clusters. Thus, for each point, xi, and each cluster, C j, it would be beneficial to

determine a weight, wij, that indicates the “degree” to which xi belongs to C j.For probabilistic approaches, e.g., mixture models, wij = pij, the probability that xi

belongs to c j. While this approach is useful in many situations, there are significant

limitations to probabilistic models, i.e., times when it is difficult to determine anappropriate statistical model, and thus, it would be useful to have non-probabilistic

clustering techniques that provide the same “fuzzy” capabilities. Fuzzy clustering

techniques [HKKR99] are based on fuzzy set theory [KY95] and provide a naturaltechnique for producing a clustering where membership weights (the wij) have a natural(but not probabilistic) interpretation. In this section we will briefly describe the general

approach of fuzzy clustering and provide a specific example in terms of fuzzy c-means

(fuzzy K-means).Lofti Zadeh introduced fuzzy set theory and fuzzy logic in 1965 [Zah65] as a way

to deal with imprecision and uncertainty. Briefly, fuzzy set theory allows an object to

partially belong to a set with a degree of membership between 0 and 1, while fuzzy logicallows a statement to be true with a degree of certainty between 0 and 1. (Traditional set

theory and logic are special cases of their fuzzy counterparts which restrict the degree of

set membership or the degree of certainty to be either 0 or 1.) Fuzzy concepts have beenapplied to many different areas, including control, pattern recognition, and data analysis

(classification and clustering).

Consider the following example of fuzzy logic. The degree of truth of the

statement “It is cloudy” can be defined to be the percentage of cloud cover in the sky, i.e.,if the sky is 50% covered by clouds, then we would assign “It is cloudy” a degree of truth

of 0.5. If we have two sets, “cloudy days” and “non-cloudy days,” then we can similarly

assign each day a degree of membership in the two sets. Thus, if a day were 25% cloudy,



57

it would have a 25% degree of membership in “cloudy days” and a 75% degree of

membership in “non-cloudy” days.

We now define what fuzzy clusters are. Consider a set of data points X = { x1, …,

xm}, where each point, xi, is an m dimensional point, i.e., xi = ( xi1, …, xim). We define a

collection of fuzzy clusters, C1, C2 , …, Ck to be a subset of all possible fuzzy subsets of X.

(This simply means that the membership weights (degrees), wij, have been assignedvalues between 0 and 1 for each point, xi, and each cluster, C j.) However, we also want

to impose the following “reasonable conditions on the clusters:

1. All the weights for a given point add up to 1.

å=

=k

j

ijw1

1

2. Each cluster (fuzzy subset) “contains” at least one point, but not all of the points.

å=

<<m

i

ij mw1

0

These two conditions ensure that the clusters form what is called a fuzzy psuedo- partition.

While there are many types of fuzzy clustering – indeed, many data analysis

algorithms can be “fuzzified” – we will only consider the fuzzy version of K-means,

which is called fuzzy c-means. (In the clustering literature, the version of K-means whichdoes not use incremental updates of cluster centroids is sometimes referred to as c-means

and this was the term adapted by the fuzzy people for the fuzzy version of K-means.)

We will discuss some preliminaries and then present the fuzzy c-means algorithm.First notice that we can also extend the definition of a centroid of a cluster to

fuzzy sets in a straightforward way. For a cluster, C j, the corresponding centroid, c j, is

defined by the following equation:

å

å

=

==m

i

m

ij

m

i

i

m

ij

j

w

xw

c

1

1 , where m is a parameter, 1 < m < ∞, that

determines the influence of the weights (the membership degrees). Thus, the fuzzy

centroid definition is much like the traditional definition except that all points are

considered (any point can belong to any cluster, at least somewhat) and the contributionof each point to the centroid is weighted by its membership degree. In the case of

traditional crisp sets, i.e., all wij are either 0 or 1, this definition reduces to the traditional

definition of a centroid.

Furthermore, we can also modify the definition of overall cluster error.

Error(C1, C2 , …, Ck ) = ji

k

j

m

i

m

ij c xw −åå= =1 1

, which is just the weighted version of the

typical K-means error.



58

Lastly notice that during the c-means clustering process we will need to update

the weights wij associated with each point and cluster. This can be accomplished by

using the following equation.

ijw′ =

1

1

1

12

2

−

−

=÷÷÷÷

ø

ö

çççç

è

æ

÷÷÷

ø

ö

ççç

è

æ

−−å

m

k

p pi

ji

c x

c x

However, if 2

ji c x − = 0 for one or more values of j, 1 ≤ j ≤ k , then we need to

use a different method of updating the weights associated with xi. Since this corresponds

to a situation where a point is the same as one or more centroids (hopefully centroids are

unique, but they don’t have to be), we want to set ijw′ = 0 for any cluster j where xi ≠c j and

split the weight of the point (which is 1) between the ijw′ where xi =c j. In the normal case,

a vector will only be the same as one centroid and thus, we can just set all weights to 0,except for the ijw′ where xi =c j. In that case, we set ijw′ = 1.

After preceding discussion the description of the fuzzy c-means algorithm is

straightforward.

Fuzzy c-means Clustering Algorithm

1. Select an initial fuzzy fuzzy psuedo-partition, i.e., assign values to all the wij.

(Like usual this can be done randomly or in a variety of ways.)

2. Update the cluster centroids.

3. Update the wij. 4. If the change in the error is below a specified threshold or the absolute change in

any wij is below a given threshold, then stop. Otherwise, go to step 2.

This algorithm is similar in structure to the K-means algorithm and also behaves

in a similar way. In particular, fuzzy c-means works best for globular clusters, which areroughly the same size and density.

There are a few considerations when choosing the value of m. Choosing m = 2

simplifies the weight update formula. However, if m is chosen to be near 1, then fuzzy c-

means behaves like traditional K-means. Going in the other direction, as m gets larger allthe cluster centroids approach the centroid of the data. In other words, the partition

becomes fuzzier as m increases.

We close this discussion of fuzzy clustering with a cautionary note. Fuzzy

clustering, as presented here, deals well with situations where points are on the boundaryof one or more clusters. It does not deal well with the situation where a point is an

outlier, i.e., is not close to any cluster. The reason for this is that we assumed that all points have a weight of 1, a weight which is split between the different clusters. Thus, if

we have a situation, as illustrated in figure 18, a point can have a relatively high weight

with respect to one or more clusters, but not really be a “true” member of any cluster.



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Figure 18: An outlier that is not really associated with either cluster.

In this case the point should have low membership weights for both clusters, but

then the weights do not add up to 1. However, with such an approach it would be

possible to eliminate noise and outliers effectively, i.e., just eliminate points that do not

have high membership weights in any cluster. Indeed, there is a variation of fuzzy c-means that takes this approach [HKKR99], but it has other problems, e.g., a tendency for

all the centroids to become very close together.

8. Related Topics

The following sections deal with topics that are usefully related to data mining.

8.1. Nearest-neighbor search (Multi-Dimensional Access Methods)

Given a set of complicated data, e.g., images, text strings, DNA sequences, polygons, etc., two problems arise:

1) How can items be located efficiently? While a “representative” vector is

commonly used to “index” these objects, these data points will normally be verysparse in the space, and thus, it is not feasible to use an array to store the data. Also,

many sets of data do not have any features that constitute a “key” that would allow

the data to be accessed using standard and efficient database techniques.

2) How can similarity queries be conducted? Many applications require the nearest

neighbor (or the k nearest neighbors) of a point. For example, in a database of

photographs, a user may want to find all the pictures similar to a particular

photograph. To do this photographs are represented as vectors and a “similarity”search is conducted. As another example, some of the clustering techniques

discussed earlier (CURE, Chameleon) depended on data structures and algorithms

that allowed the nearest neighbors of a point to be located efficiently.

Techniques for nearest-neighbor search are often discussed in papers about multi-

dimensional access methods or spatial access methods. Strictly speaking the topic of multi-dimensional access methods is broader than nearest-neighbor search since it

addresses all of the many different types of queries and operations that a user might want

to perform on multi-dimensional data.A large amount of work has been done in the area of nearest neighbor search (and

multi-dimensional access methods). Recent developments include the SS-tree [JW96],

the SR-tree [KS97], the X-tree [BKK96], the GNAT tree [Bri95], the M-tree [CPZ97],

and the “pyramid technique” [BBK98]. (The X-tree is a descendant of the older, butfrequently mentioned R* tree, which is in turn a descendant of the R tree.) A good

survey of nearest-neighbor search, albeit from the slightly more general perspective of

multi-dimensional access methods is given by [GG98].



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The basic approach is to create tree-based structures, such that the “closeness” of

the data increases as the tree is traversed from top to bottom. Thus, the nodes at the

bottom of the tree represent “clusters” of data that are relatively cohesive. Indeed, one of the simplest techniques for generating a nearest neighbor tree is to cluster the data into K clusters and then, recursively break each cluster into subclusters until the subclusters

consist of individual points. The nearest-neighbor tree consists of the clusters andsubclusters generated along the way.

A k-nearest-neighbor query is conducted as follows:

1) Add the children of the root node to a search queue.

2) While the search queue is not empty, take a node off the search queue. a) Evaluate whether that node and any of its descendants are “close enough” to be

considered for the search.

b) If not, then discard the subtree represented by this node. c) Otherwise, if the node is not a leaf node, i.e., a data point, add the children of this

node to a search queue.

d) If the node is a leaf node, then add the node to a list of possible solutions. 3) Sort the list of possible solutions by distance from the query point and return the k-

nearest neighbors.

This sounds fairly simple and it seems as though nearest neighbor trees would be

an easy way to cluster data and/or to make practical use of a clustering technique.However, there are significant problems.

1) One of the goals of many nearest-neighbor tree techniques is to serve as efficient

secondary storage based access methods for non-traditional databases, e.g., spatialdatabases, multimedia databases, document databases, etc. Because of requirements

related to page size and efficient page utilization, “natural” clusters may be split

across pages or nodes. However, data is normally highly clustered and this can beused for “real” clustering as shown by the improvements made to CLARANS

[EKX95] by the use of an R* tree.

2) Because of the nature of nearest-neighbor trees, the tree search is a branch-and-bound

technique and needs to search large parts of the tree, i.e., at any particular level, many

children and their descendants may need to be examined.

To see this, consider a point and all points that are within a given distance of it. This hyper-sphere (or hyper-rectangle in the case of multi-dimensional range

queries) may very well cut across a number of nodes (clusters) - particularly if the

point is on the edge of a cluster and/or the query distance being considered is greater than the distance between clusters.

More specifically, it is “hard” for the algorithms that construct nearest

neighbor trees to avoid a significant amount of overlap in the volumes represented bydifferent nodes in the tree. In [BKK96], it has been shown that the degree of overlap

in a frequently used nearest neighbor tree, the R* tree, approaches 100% as the

dimensionality of the vectors exceeds 5. Even in two dimensions the overlap wasabout 40%. Thus, the X-tree (proposed as a successor to the R* tree in [BKK96])



61

basically becomes linear search for high dimensional data. Other nearest neighbor

search techniques suffer from the same problem.

3) The nature of high dimensional spaces. It has been shown that there are significant

problems that arise with nearest neighbor searches in high-dimensional space.

In [BGRS99] it was shown that the concept of “nearest neighbor” is notmeaningful in many situations. Minimum and maximum distances in high

dimensional space tend to be very similar. Thus, unless there is significant clustering

in the data and/or the query ranges stay within individual clusters, then the pointsreturned by nearest neighbor queries are not much closer to the query point than to

the points not returned. In this latter case, then (to use the terminology of [BGRS99])

the nearest neighbor query is “unstable.”

Recently, in [BBK98], there has been some work on developing techniquesthat avoid this problem. However, this “pyramid” technique was restricted to hyper-

cube shaped queries, and many similarity queries are spherical.

In some cases, a linear scan can work better than more sophisticated techniques.

4) It is not possible to discard outliers. All data points must be stored, and if some of the data points do not fit particularly well in clusters, then this often has deleterious

effects on lookups of other data points.

8.2. Pattern Recognition

Clustering has been used in a number of different ways in pattern recognition.[DJ88] is a clustering book that is strongly oriented toward the area of pattern recognition

giving a number of different and extensive examples of the use of clustering in pattern

recognition. This paper only mentions some examples briefly.In certain cases, e.g., the segmentation of an image into different regions, it is

hard to “train” a recognition system to behave properly because there is such a wide

variation in the possible input, and clear cut classes are not readily available. In such

instances, the use of the appropriate clustering technique can prove very effective.As was noted before, if clear-cut classes are available, then other techniques, e.g.,

discriminant analysis, are more appropriate. However, in some cases clustering analysis

may be used to find a preliminary assignment of classes that are then refined by a domainexpert.

The use of clustering to reduce the number of patterns that need to be considered

in pattern recognition is discussed below.

8.3. Exploratory Data Analysis

Cluster analysis is a very important part of exploratory data analysis, althoughcluster analysis is just one of the tools used in exploratory data analysis to understand

patterns that may occur in data.

When cluster analysis is used for exploratory data analysis, some of the moreuser-oriented factors assume additional importance. In particular, it is important for the

user to be able to easily interpret the results of the cluster analysis. As mentioned above,



62

various clustering techniques differ in the kinds of results that are produced. Some

produce lists of objects, some produce hyper-ellipsoids, and some produce hyper-

rectangles.In some cases, techniques (like multi-dimensional scaling [DJ88]) that map

medium and high dimensional data to 1, 2, or 3 dimensions can be very helpful.

Unfortunately, the distortions introduced by such mappings sometimes render the processuseless. These techniques are more likely to work better for medium dimensionalities

than for high dimensionalities.

Finally, exploratory data analysis is often an integral part of a clustering analysis.A user looks at the results of an initial clustering and, as a result, changes the parameters

of a clustering algorithm or chooses a new clustering algorithm. Numerous iterations

may be required before a suitable clustering is produced.

9. Special Topics

9.1. Missing Values

Omitted from preliminary version.

9.2. Combining Attributes

9.2.1. Monothetic or Polythetic ClusteringThe features of an object can be used for clustering in an incremental or non-

incremental way. While some techniques use one variable at a time (monothetic), most

approaches use all features at once (polythetic) to calculate the distance or similarity.

.([KR90] provides an example of Monothetic clustering algorithm.)

9.2.2. Assigning Weights to featuresIn some cases, the attributes are weighted so as to give more weight to certain

attributes, but this typically only affects how the proximity of points is calculated and nothow the clustering actually works. In some sense, the use of attribute weights can be

regards as a more general version of feature selection.

10. Conclusions

The following conclusions seem most important:

• Cluster analysis is still an active field of development. In the areas of statistics(mixture models), computer science (Data Mining, machine learning, nearestneighbor search), pattern recognition, and vector quantitization, there is still a lot of

work being done.

• Many cluster analysis techniques do not have a strong formal basis. While sometechniques make use of formal mathematical methods, they often do not work better

than more informal methods.

• Cluster analysis is a rather ad-hoc field. Almost all techniques have a number of arbitrary parameters that can be “adjusted” to improve results. It remains to be seen,

whether this represents a temporary situation, or is an unavoidable use of problem and



63

domain specific heuristics.

• There are a wide variety of clustering techniques. Some would argue that the widerange of subject matter, size and type of data, and differing user goals makes this

inevitable, and that cluster analysis is really a collection of different problems that

require a variety of techniques for their solution. The relationships between thedifferent types of problems and solutions are often not clear.

• Comparisons among different clustering techniques are difficult. While everyarticle that presents a new clustering technique shows its superiority to other

techniques, it is hard to judge how well the technique will really do.

• There don’t seem to be any standard benchmarks. Thus, the authors of newtechniques are reduced to devising their own comparisons to previous techniques.Usually authors obtain a few of the data sets used by previous authors and run

comparisons based on those. Even the criteria to be measured are not standard.

• All techniques seem to impose a certain structure on the data and yet few

authors describe the type of limitations being imposed. In [DJ88], the authors

mention that the performance of a clustering algorithm on uniformly distributed datacan sometimes be very informative. In general, it might be useful to understand whatkind of artifacts particular clustering techniques introduce.

• In spite of all these problems, clustering analysis is a useful (and interesting)

field. As mentioned in the introduction, many people use cluster analysis for a wide

variety of useful tasks.



64

List of Articles and Books for Clustering for Data Mining

Note that web addresses are given for many of the articles. This usually takes you

to a home page of one of the authors or to a list of publications. In the first case, it is

usually straightforward to find the article by looking at the person’s publications,although in a few cases a reference is all that is provided. Additionally, the proceedings

of some conferences can be found online: SIGMOD and KDD (99 onward) at ACM

Digital Library, www.acm.org, VLDB at the Very Large Database home page,www.vldb.org, ICDE (99 onward) at http://www.computer.org/proceedings/. A good

place to locate the home pages of authors is at http://ftp.ust.hk/dblp/db/index.html.

Finally, another good general site is the Research Index site at NEC,http://citeseer.nj.nec.com/cs.

The following three books provide an introduction to and overview of

Cluster Analysis

[DJ88] Richard C. Dubes and Anil K. Jain, (1988), Algorithms for Clustering Data,

Prentice Hall.

A survey and introduction to cluster analysis that is frequently referenced in theliterature. It devotes considerable attention to cluster validity. It is somewhat oriented

toward pattern recognition and has a whole chapter devoted to the use of clustering in

pattern recognition. It also has appendices that summarize various areas related to cluster analysis.

[KR90] L. Kaufman and P. J. Rousseeuw, (1990), Finding Groups in Data: an Introduction to Cluster Analysis, John Wiley and Sons.

Probably the most commonly referenced text on clustering that I encountered,

perhaps because it is still in print. Among other things, this text explains clustering based

on medoids, which are objects that are the most representative of the other objects in acluster. The clustering techniques described in this book are implemented in the S-Plus

statistical system.

[OD74] Patrick L. Odell and Benjamin S. Duran, (1974), Cluster Analysis: A Survey,Springer-Verlag.

This book is somewhat dated, but is fairly short and basic. Also, since a lot of basicclustering techniques were in place by the time this book was written, most of the

material it covers is still applicable.



65

The following papers describe specific techniques, algorithms, or systems

used for clustering in Data Mining. The techniques described in these papers are

described in section 4 of this document. Only very brief comments will be made

here.

[AGGR98] Rakesh Agrawal, Johannes Gehrke, Dimitrios Gunopoulos, and Prabhakar Raghavan, (1998), Automatic Subspace Clustering of High Dimensional Data for Data Mining Applications, Technical Report, IBM Almaden Research

Center.

http://www.almaden.ibm.com/cs/people/ragrawal/

Describes CLIQUE, a density-based algorithm that has a number of good

properties, including the ability to find lower dimensional clusters in high dimensionalsubspaces. This algorithm is similar in flavor to the APRIORI algorithm used for finding

frequent itemsets, and finds higher dimensional regions by only considering those regions

that are associated with lower dimensional regions of high density.

[CHY96] Ming-Syan Chen, Jiawei Han, and Philip, S. Yu, (1996 ), Data Mining: AnOverview from Database Perspective, IEEE Transactions on Knowledge andData Engineering, 8(6): 866-883.

http://db.cs.sfu.ca/sections/publication/kdd/kdd.html

Provides a general overview of data mining techniques. Included is an overviewof CLARANS, enhanced CLARANS, and BIRCH. This paper has an extensive

discussion of the characteristics that data mining techniques should have.

[EKSWX98] Martin Ester, Hans-Peter Kriegel, Jorg Sander, Michael Wimmer, XiaoweiXu, (1998), Incremental Clustering for Mining in a Data Warehousing Environment , Proceedings of 1998 International Conference on Very Large

Data Bases (VLDB'98), New York, USA.http://www.dbs.informatik.uni-muenchen.de/index_e.html

Describes how the DBSCAN clustering algorithm can be made incremental for use in a data warehouse environment.

[EKSX96] Martin Ester, Hans-Peter Kriegel, Jorg Sander, Xiaowei Xu, (1996), A Density-Based Algorithm for Discovering Clusters in Large Spatial

Databases with Noise, Proceedings of 2nd International Conference onKnowledge Discovery and Data Mining, Portland, OR, pp. 226-231.

http://www.dbs.informatik.uni-muenchen.de/index_e.html

Describes, DBSCAN, a density-based clustering algorithm that handles noise

well. While this algorithm is presented and evaluated in the context of spatial databases,there is no reason that it could not be used in a more general setting.



66

[EKX95] Martin Ester, Hans-Peter Kriegel, and Xiaowei Xu, (1995 ) Knowledge Discovery in Large Spatial Databases: Focusing Techniques for Efficient Class Identification, Proceedings of the 4th International Symposium onLarge Spatial Databases (SSD'95), Portland, ME, 1995, in: Lecture Notes in

Computer Science, Vol. 591, Springer, 1995, pp. 67-82.


Describes a number of techniques for making CLARANS more efficient. The

basic idea is to use an R* trees to store the data being considered. . (R* trees are a typeof nearest-neighbor tree that try to store data points in the same page if the data points are

“close” to one another.)

[GHC99] Sanjay Goil, Harasha Nagesh, and Alok Choudhary, (1999), MAFIA: Efficient and Scalable Subspace Clustering for Very Large Data Sets,Technical Report Number CPDC-TR-9906-019, Center for Parallel and

Distributed Computing, Northwestern University, June 1999.http://www.ece.nwu.edu/~sgoil/

Describes MAFIA (Merging of Adaptive Finite Intervals (and more than aCLIQUE), which is an modification of CLIQUE that runs faster and finds better quality

clusters. pMAFIA is the parallel version.

[GRGPF99] Venkatesh Ganti, Raghu Ramakrishnan, Johannes Gehrke, Allison L.

Powell, James C. French, (1999 ), Clustering Large Datasets in ArbitraryMetric Spaces, ICDE 1999, pp. 502-511.http://www.cs.wisc.edu/~raghu/

Describes Bubble and Bubble-FM, which are clustering algorithms based on

BIRCH*, an extension to the BIRCH framework to handle non-metric data. One of thekey ideas is to define a new cluster feature based on the idea of the representative points

of a cluster. Bubble-FM also makes used of a technique, multi-dimensional scaling, to

map non-metric data into a metric space. This permits fewer and cheaper distancecalculation in those cases where distance calculations are very expensive.

[GRS98] Sudipto Guha, Rajeev Rastogi, Kyuseok Shim, (1998 ), CURE: An Efficient Clustering Algorithm for Large Databases, ACM SIGMOD Conference,

1998, pp. 73-84.

http://www.bell-labs.com/user/rastogi/

Describes a clustering algorithms that describes a cluster by widely scattered

representative points that are shrunk by a constant factor toward the center of a cluster to

reduce the effect of outliers. The basic clustering technique used is similar to single link and sampling and partitioning are used to make the method scalable to large data sets.

CURE was shown to be effective at finding 2-D spatial clusters that were of non-circular

shapes and which had widely varying sizes.



67

[GRS99] Sudipto Guha, Rajeev Rastogi, and Kyuseok Shim, (1998 ), ROCK: A Robust Clustering Algorithm for Categorical Attributes, In Proceedings of the 15th

International Conference on Data Engineering, 1999.http://www.bell-labs.com/user/rastogi/

Describes an approach for clustering algorithm for data with categorical andBoolean attributes. It redefines the distances between points to be the number of shared

neighbors whose strength is greater than a given threshold and then uses a hierarchical

clustering scheme to cluster the data.

[HK98] Alexander Hinneburg Daniel. A. Keim and: An Efficient Approach toClustering in Large Multimedia Databases with Noise, Proceedings of the 4th

International Conference on Knowledge Discovery and Data Mining, AAAIPress.

http://www.informatik.uni-halle.de/~keim/

Describes a density-based approach to clustering data which models the influenceof each data point via influence (kernel) functions and then finds clusters using the

resulting global density function. Initially, each data point is associated with a localmaxima, forming a center based clusters. Clusters whose local maxima is “not dense

enough” are classified as noise and discarded, while remaining clusters are merged if

there is a path connecting their local maxima which is of “high enough” density.

[KHK99a] George Karypis, Eui-Hong Han, and Vipin Kumar, (1999) CHAMELEON: A Hierarchical Clustering Algorithm Using Dynamic Modeling , IEEE

Computer, Vol. 32, No. 8, August, 1999. pp. 68-75.http://www-users.cs.umn.edu/~karypis/

Describes a clustering algorithm that combines an initial partitioning of the datausing an efficient graph-partitioning algorithm with a novel hierarchical clustering

scheme that dynamically models clusters. The key idea is that two clusters will be

merged only if the resulting cluster is similar to the original clusters in, i.e., self-similarityis preserved. While applicable to a wide range of data, Chameleon has specifically been

shown to be superior to DBSCAN and CURE for clustering spatial data.

[KHK99b] George Karypis, Eui-Hong Han, and Vipin Kumar, (1999 ) Multi-level Refinement for Hierarchical Clustering . (Unpublished)

http://www-users.cs.umn.edu/~karypis/

Describes an approach to improving hierarchical clustering that is related to the

technique of multi-level refinement that is commonly used to improve the solution to

graph partitioning problems.



68

[HKKM97] Eui-Hong Han, George Karypis, Vipin Kumar, and Bomshad Mobasher,

(1997), Clustering In a High-Dimensional Space Using Hypergraph Models,

Technical Report TR-97-063, Department of Computer Science, Universityof Minnesota, Minneapolis, Minnesota.

http://www-users.cs.umn.edu/~karypis/

Describes the technique of hypergraph based clustering in the context of market

basket data. This technique uses frequent itemsets and their related association rules to

construct weighted hyperedges between the items being considered for clustering. Ahypergraph based clustering algorithm (HMETIS) is then used to create the clusters. This

algorithm naturally finds lower dimensional clusters in a higher dimensional space and

has straightforward measures for detecting outliers and valid clusters.

[NH94] Raymond Ng and Jiawei Han, (1994), Efficient and Effective Clustering Method for Spatial Data Mining , Proceedings of 1994 International

Conference on Very Large Data Bases (VLDB'94), Santiago, Chile,September 1994, pp. 144-155.

http://db.cs.sfu.ca/sections/publication/kdd/kdd.html

Describes CLARANS, a medoid based clustering algorithm for spatial databases.

This was one of the first clustering algorithms specifically oriented towards data mining.

[SCZ98] Gholamhosein Sheikholeslami, Surojit Chatterjee and Aidong Zhang (1998),

Wavecluster: A multi-resolution clustering approach for very large spatial databases. In Proceedings of the 24th VLDB Conference.http://www.cs.buffalo.edu/~surojit/

WaveCluster is a grid-based clustering algorithm for low-dimensional data.

WaveCluster transforms the data so that it looks like image data, i.e., a rectangular grid of pixels each of which has an integer values (the count of data points in the grid cell) and

then using image processing techniques (wavelets) to segment the “image” and produce a

clustering of the points.

[ZRL96] Tian Zhang, Raghu Ramakrishnan, and Miron Livny, (1996), BIRCH: An Efficient Data Clustering Method for Very Large Databases, in Proceedings

of ACM SIGMOD International Conference on Data Management, June1996, Canada.

http://www.cs.wisc.edu/~raghu/

Describes BIRCH, a very efficient clustering technique based on a CF (ClusteringFeature), tree. This tree is a summary of the data in the form of a hierarchical clustering.

This rough clustering can be refined in later steps. This basic CF tree structure can beused with a number of underlying clustering algorithms.

The following papers discuss nearest neighbor clustering techniques.



69

[GK78] K. C. Gowda and G. Krishna, (1978), Agglomerative Clustering Using theConcept of Mutual Nearest Neighborhood, Pattern Recognition, Vol. 10, pp.

105-112.

[JP73] R. A. Jarvis and E. A. Patrick, (1973), Clustering Using a Similarity Measure

Based on Shared Nearest Neighbors, IEEE Transactions on Computers, Vol.C-22, No. 11, November, 1973.

The following papers discuss association rules.

[AS97] Rakesh Agrawal, and Ramakrishnan Srikant, (1997), Fast Algorithms for Mining Association Rules, In Proceedings of the 20

thVLDB Conference,

pages 487-499, Santiago, Chile.


Describes the APRIORI algorithm (and a couple variants), a very efficient

algorithm for finding frequent itemsets and association rules.

[AS97b] Rakesh Agrawal and Ramakrishnan Srikant, (1997), Mining Generalized Association Rules, Technical Report, IBM Almaden Research Center.


Describes the mining of generalized association rules when items are grouped into

a taxonomy. Also contains an interest measure for rules that can be used to prune

uninteresting rules.

[BMS97] Sergey Brin, Rajeev Motwani, and Craig Silverstein, (1997 ), Beyond Market Baskets: Generalizing Association Rules to Correlations, In Proceedings of

1997 ACM-SIGMOD International Conference on Management of Data,Tucson Arizona.

http://theory.stanford.edu/people/motwani/motwani.html

Discusses the limitations of association rules and proposes using correlations

instead.

The following papers are more statistically oriented.

[FR98] C. Fraley and A. E. Raferty, (1998), How Many Clusters? Which Clustering Method? Answers Via Model-Based Cluster Analysis, Technical Report No.

329, Department of Statistics, University of Washington, Seattle,Washington.http://www.stat.washington.edu/fraley/

Contains a readable description of a mixture model based clustering techniquethat can determine both the clusters and the number of clusters. The technique described

is available as software that works with the S-Plus statistical package.



70

[CHS90] Peter Cheeseman, Robin Hanson, and John Stutz, (1990), BayesianClassification Theory, Technical Report FIA-90-12-7-01, NASA Ames

Research Center, Moffet Field, California.http://ic-www.arc.nasa.gov/ic/projects/bayes-

group/people/cheeseman/home.html

Contains a description of the theory on which the clustering program, AutoClass

is based.

[CS96] Peter Cheeseman and John Stutz, (1996), Bayesian Classification(AutoClass): Theory and Results, In U. M. Fayyad, G. Piatetsky-Shapiro, P.

Smith, and R. Uthurusamy, editors, Advances in Knowledge Discovery and Data Mining , pages 153-180, AAAI/MIT Press.http://ic-www.arc.nasa.gov/ic/projects/bayes-

group/people/cheeseman/home.html

Describes the AutoClass clustering program and its use in analyzing sensor and

DNA data. AutoClass uses a mixture model based approach and was one of the first

programs used for clustering in databases.

[HP98] Thomas Hofmann and Jan Puzicha, (1998), Statistical Models for Co-occurrence Data, AI Memo No. 1625, Artificial Intelligence Laboratory,

Massachusetts Institute of Technology.

http://www.ai.mit.edu/people/hofmann/

Describes a mixture model approach to modeling co-occurrence data, i.e., data of

the form (x,y), where x and y are any members of two sets, X and Y. This approach has anumber of possible applications, including data mining and information retrieval.

The following papers and books are related to fuzzy clustering.

[HKKR99] Frank Hoppner, Frank Klawonn, Rudolf Kruse, and Thomas Runkler, (1999),

Fuzzy Cluster Analysis: Methods for Classification, Data Analysis, and Image Recognition, John Wiley & Sons.

http://www.fuzzy-clustering.de/

A fairly advanced book that provides a discussion of fuzzy clustering techniques

including some interesting approaches for finding clusters that are shells, polygons or lines. Fuzzy clustering software is available from the website listed above.

[KY95] George J. Klir and Bo Yuan, (1995), Fuzzy Sets and Fuzzy Logic, PrenticeHall PTR.

A comprehensive introduction to fuzzy set theory and fuzzy logic which covers a broad range of topics. The books includes a brief, but clear description of fuzzy c-means

and also discussion an approach to fuzzy hierarchical clustering.



71

[Zah65] Lofti Zadeh, (1965), Fuzzy Sets, Information and Control, Vol. 8, 338:353.

This is the original paper on fuzzy sets.

The following papers are related to vector quantization and maximum

entropy related clustering approaches.

[BK93] Joachim Buhmann and Hans Kuhnel, (1993), Vector Quantization withComplexity Costs, IEEE Transactions on Information Theory 39:1133-1145.http://www-dbv.informatik.uni-bonn.de/papers.html

Describes a maximum entropy approach to vector quantization that can

simultaneously determine the clusters and the number of clusters. Also contains acomparison to the K-means algorithm in the field of image compression.

[BH97] Joachim M. Buhmann and Thomas Hofmann, (1997), Pairwise DataClustering by Deterministic Annealing , IEEE Transactions on PatternAnalysis and Machine Intelligence, Vol. 19, No. 1, January 1997.

http://www-dbv.informatik.uni-bonn.de/papers.html

Describes a maximum entropy approach to clustering which is similar to the

previous article. This approach uses the deterministic annealing optimization technique.

The following two papers are related to graph partitioning.

[KK98a] George Karypis and Vipin Kumar, (1998), hMETIS 1.5: A hypergraph partitioning package, Technical report, Department of Computer Science,

University of Minnesota, 1998.http://www-users.cs.umn.edu/~karypis/metis/

[KK98b] George Karypis and Vipin. Kumar, (1998), METIS 4.0: Unstructured graph partitioning and sparse matrix ordering system, Technical report,

Department of Computer Science, University of Minnesota, 1998.

http://www-users.cs.umn.edu/~karypis/metis/

The following papers are related to clustering in information retrieval.

[BD95] Michael W. Berry and Susan T. Dumais, (1995), Using Linear Algebra for Intelligent Information Retrieval , SIAM Review, 37:573-595.http://research.microsoft.com/~sdumais/

Describes the technique of Latent Semantic Indexing. This is a dimensionalityreduction technique that uses the data matrix, i.e., the matrix of the original data vectors.

This technique is used extensively in information retrieval.



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[FO95] Christos Faloutsos and Douglas W. Oard, (1995), A Survey of Information Retrieval and Filtering Methods, Technical Report CS-TR-3514, University

of Maryland, College Park, Maryland.http://www.cs.cmu.edu/~christos/

Describes a variety of techniques used in information retrieval. Among the topicscovered are clustering and Latent Semantic Indexing.

[SKK00b] Michael Steinbach, George Karypis, and Vipin Kumar, A Comparison of Document Clustering Techniques, University of Minnesota, Technical Report

#00-034, 2000.

This paper presents the results of an experimental study of some commondocument clustering techniques. In particular, it compares the two main approaches to

document clustering, agglomerative hierarchical clustering and K-means. (For K-means

we used a “standard” K-means algorithm and a variant of K-means, “bisecting” K-

means.) Hierarchical clustering is often portrayed as the better quality clusteringapproach, but is limited because of its quadratic time complexity. In contrast, K-means

and its variants have a time complexity which is linear in the number of documents, butare thought to produce inferior clusters. Sometimes K-means and agglomerative

hierarchical approaches are combined so as to “get the best of both worlds.” However,

the paper’s results indicate that the bisecting K-means technique is better than thestandard K-means approach and as good or better than the hierarchical approaches that

were tested.

The following papers are related to nearest neighbor search and nearest

neighbor trees. In the literature such techniques are found in the related area of

multi-dimensional access methods.

[BBK98] Stefan Berchtold, Christian Bohm, Hans-Peter Kriegel, (1998), The Pyramid Technique: Towards Breaking the Curse of Dimensionality, Proceedings of 1998 International Conference on the Management of Data (SIGMOD '98),

Seattle, Washington.


Describes a new indexing method for high dimensional spaces that attempts to

overcome some of the problems of previous methods, i.e., for high dimensional spaceslinear search is often better. The technique bounds the data by a hyper-cube, splits the

hyper-cube into pyramids whose tips meet at the center of the cube, and then slices the

pyramids into sections. By doing this, the technique is able to map each point in a highdimensional space into a linear space. B+ trees can then be used to manage the data. So

far this technique only works well for hyper-cube shaped queries.

[BGRS99] Kevin Beyer, Jonathan Goldstein, Raghu Ramakrishnan, and Uri Shaft,

(1998), When is “Nearest Neighbor” Meaningful?, International Conferenceon Database Theory, 1999, pp. 217-235.

http://www.cs.wisc.edu/~raghu/



73

An article on the problems encountered by nearest neighbor search in high

dimensions. This paper proposes the ε-Radius Nearest Neighbor problem, which is to

find all points within a radius of DMIN * (1+ε) of a query point, where DMIN is theminimum distance from the query point to its closest neighbor. Such a query is only

meaningful if the number of points within the radius is a small fraction of all the points.The paper then shows that for many situations, particularly if the number of dimensionsis greater than 10 or 15, such queries are meaningless. A major contribution of this paper

is to show that any new nearest neighbor search techniques should always be compared to

a linear scan.

[BKK96] Stefan Berchtold, Daniel A. Keim, Hans-Peter Kriegel, (1996), The X-tree: An Index Structure for High-Dimensional Data, Proceedings of 1996

International Conference on Very Large Databases (VLDB'96), Bombay,India.


Describes the X-tree, a high dimensional indexing structure that is based on theR* tree. This paper contains an analysis of the number of nodes of the R* tree that have

to be searched for a particular query. For dimensions more than 5, typically 90% of thetree must be searched. The X-tree tries to be more efficient, by allowing larger nodes,

i.e., by going more to a linear search, as the dimensionality increases.

[Bri95] Sergey Brin, (1995), Near Neighbor Search in Large Metric Spaces,

Proceedings of the 21st International Conference on Very Large DataBases,

pp. 574-584, Zurich Switzerland.http://www-db.stanford.edu/~sergey/

Describes the GNAT (Geometric Nearest Neighbor) tree. This tree provides adata structure for high dimensional data that attempts to exploit (model) the geometry of

the data. This technique is oriented towards cases where distance calculations are

relatively expensive and initial build time is not as important. The technique is notintended for cases where data is added or deleted. Finally, this technique is intended for

data whose distance function is a metric.

[CPZ97] Paolo Ciaccia, Marco Patella, Pavel Zezula, (1997), M-tree: An Efficient Access Method for Similarity Search in Metric Spaces, Proceedings of 1997

International Conference on Very Large Databases (VLDB'97), pp. 426-435,

Athens, Greece.

http://www.cs.unibo.it/~ciaccia/

Describes the M-tree, a high dimensional access technique for data whose

distance function is a metric. This technique generates balanced trees and can be used asa secondary storage based access method (like the R*-tree, the X-tree, and the pyramid

technique). This technique relies on a careful use of the triangle inequality to minimize

potentially expensive distance calculations.



74

[GG98] Volker Gaede and Oliver Gunther, (1998), Multidimensional AccessMethods, ACM Computing Surveys, Vol. 30, No. 2,

http://www.wiwi.hu-berlin.de/~gaede/

A very understandable and comprehensive survey of multi-dimensional access

methods.

[JW96] Ramesh Jain and David A. White, (1996), Similarity Indexing with the SS-tree, Proceedings of the 12

thInternational Conference on Data Engineering,

pp. 516-523, New Orleans, USA.

http://vision.ucsd.edu/~jain/

Describes the SS-tree and demonstrates that it is somewhat better than the R* tree.Basically, the SS tree uses hyper-spherical clusters of points, while the R*-tree uses

hyper-rectangular clusters. Like the R* tree, the SS tree can be used as a secondary

storage access method.

[KS97] Norio Katayama and Shin’ichi Satoh, (1997), The SR-tree: An IndexStructure for High-Dimensional Nearest Neighbor Queries, Proceedings of

the 1997 ACM SIGMOD International Conference on Management of Data,Tucson, Arizona.

http://www.rd.nacsis.ac.jp/~katayama/homepage/research/srtree/English.html

Describes an improvement of the SS tree that uses the intersection of a hyper-

sphere and a hyper-rectangle.

The following are papers on miscellaneous topics.[Fi96] Doug Fisher, (1996), Iterative Optimization and Simplification of

Hierarchical Clusterings, Journal of Artificial Intelligence 4 (1996) 147-179.http://www.vuse.vanderbilt.edu/~dfisher/

Describes techniques for improving hierarchical clusterings. A number of

iterative techniques are introduced and evaluated. The viewpoints taken in this article is

a that of machine learning.

[Ko97] T. Kohonen, (1997), Self-Organizing Maps, Second Extended Edition,

Springer Series in Information Sciences, Vol. 30, Springer, Berlin,Heidelberg, New York, 1995, 1997.

http://www.cis.hut.fi/teuvo/

[FL95] Christos Floutsos and King-Ip Lin, (1995), Fastmap: A fast algorithm for indexing of traditional and multimedia databases. Proceedings of SIGMOD

1995.

http://www.cs.cmu.edu/~christos/

Describes the Fastmap algorithm. This algorithm takes maps a set of m points

into Euclidean space in time linear in m and (approximately) preserves the distances



Introduction toDataflow Analysis

15-745 Optimizing CompilersSpring 2006

Peter Lee

On to global optimizations!

Most of the important optimizationopportunitie s (e.g., in loops) are not local

We’ll start wi th a simple globa loptimizat ion: simple constant propagation

Then develop a general framework —dataflow analysis — that supports this andother key optimizations

Classic reference: Matthe w S. Hecht, “Flow Analysisof Computer Programs, Elsevier Science, NY, 1977

Simple constant propagation

a = 5;

b = 3;

...

n = a + b;

for (i=0; i<n; ++i)

..

a = 5;

b = 3;

...

n = 5 + 3;

for (i=0; i<n; ++i)

..

If a and b can be determined to be constants, then replace them

Must analyze the code, to determine if it islegal to perform this program transformation

A sample programint fib10 () {

var n, old, older,

result, i : int;

n = 10;

older = 0;

old = 1;

result = 0;

if (n <= 1) return n;

for (i=2; i<n; i+=1) {

result = old + older;

older = old;

old = result;

}

return result;

}

MOVE(TEMP(15), CONST(10))




CJUMP(LE, TEMP(15), CONST(1), .L8, .L7)

LABEL(.L8)

MOVE(TEMP(1), TEMP(15))

JUMP(.L1)

LABEL(.L7)


JUMP(.L3)

LABEL(.L3)

CJUMP(LT, TEMP(11), TEMP(15), .L4, .L5)

LABEL(.L4)

MOVE(TEMP(12), BINOP(PLUS, TEMP(14), TEMP(13)))



MOVE(TEMP(11), BINOP(PLUS, TEMP(11), CONST(1)))

JUMP(.L3)

LABEL(.L5)


JUMP(.L1)

source code linearized IR




var n, old, older,

result, i : int;

n = 10;

older = 0;

old = 1;

result = 0;


for (i=2; i<n; i+=1) {


older = old;

old = result;

}

return result;

}

n = 10

older = 0

old = 1

result = 0

if n > 1 then goto 7

return n

i = 2

if i > n goto 14

result = old + olderolder = old

old = result

i = i + 1

goto 8

return result

source code easier-to-read version

of linearized IR


var n, old, older,

result, i : int;

n = 10;

older = 0;

old = 1;

result = 0;


for (i=2; i<n; i+=1) {


older = old;

old = result;

}

return result;

}

1: n = 10

2: older = 0

3: old = 1

4: result = 0

5: if n > 1 then goto 7

6: return n

7: i = 2

8: if i > n goto 14

9: result = old + older10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

source code linearized IR, with

program point labels

Terminology alert!1: n = 102: older = 0

3: old = 1

4: result = 0


6: return n

7: i = 2

8: if i > n goto 14

9: result = old + older

10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

13

12

11

10

914

8

76

5

4

3

2

1

pred(n): immediate predecessors of node nsucc(n): immediate successors of node n

E.g .: pred(8) = {7, 13}

a control-flow graph

More terminology

A definiti on is a statement th at defines a value for some variable v

e.g., d: v = x y

defs(v) = the set of definitions that define v



Yet more terminology...

For each statement, in isolation, it is easy to see if it is definition

Consider a statement at point d

d: v = x op y

This statement generates a definition...

...and kills all other definitions that define a value for v

GENs and KILLsGEN? KILLs

1: n = 10 Y {}

2: older = 0 Y {10}

3: old = 1 Y {11}

4: result = 0 Y {9}

5: if n > 1 then goto 7 N {}

6: return n N {}

7: i = 2 Y {12}

8: if i > n goto 14 N {}

9: result = old + older Y {4}

10: older = old Y {2}11: old = result Y {3}

12: i = i + 1 Y {7}

13: goto 8 N {}

14: return result N {}

for a statement d that defines v,KILL(d) = defs (v) - {d}

Reaching definitionsA definit ion at programpoint d reaches programpoint u if there is acontrol-flow path from d to u that does not cont aina definit ion of the same variable as d

1: n = 102: older = 0

3: old = 1

4: result = 0


6: return n

7: i = 2

8: if i > n goto 14


10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

Does 1 reach 5?What use s of n are reached by 1?What definitions reach 8?

Reaching definitions, formally

rd(n) = the set of definitions that reachstatement n

rd(n) = ( gen(p) (rd(p) - kill(p)) )ppred(n)



Alternatively: In and Out sets

in(n): the se t of defs that reach the beginning of node n

out (n): the set of defs that reach the endof node n

in(n) = out(p)

out (n) = gen(n) (in(n) - kill(n))

ppred(n)

Solving reaching definitions

In general, we won’t be able tocompute an exact solution toreaching definitions

We want, the refore, aconservative approximation

if d reaches n for some execut ion of P, then drd(n)

so, rd(n) = {1, 2, ..., 14} would work for our current example (though obviously

we want to do better)

...

if (...)

x = 1;

...

a = x

...

Fixed point solut ions

Notice, informally:

in() and out () are monotonic (increasing)

finite number of definitions

So, can init ialize in()=out()={}, for all n,and the find a fixed point iterat ively

in(n) = out(p)

out (n) = gen(n) (in(n) - kill(n))

ppred(n)

Warning!

We are being dangerously informal here

what is meant by “conservative approximation”

why is a fixed poin t solu tion to rd(n) areasonable solut ion, and is it reallyconservative?

We will definite ly need to address these and other related quest ions (next time)



GEN? KILLs IN0 OUT0

0: entry N {} {} {}

1: n = 10 Y {} {} {}

2: older = 0 Y {10} {} {}

3: old = 1 Y {11} {} {}

4: result = 0 Y {9} {} {}

5: if n > 1 then goto 7 N {} {} {}

6: return n N {} {} {}

7: i = 2 Y {12} {} {}8: if i > n goto 14 N {} {} {}

9: result = old + older Y {4} {} {}

10: older = old Y {2} {} {}

11: old = result Y {3} {} {}

12: i = i + 1 Y {7} {} {}

13: goto 8 N {} {} {}

14: return result N {} {} {}

in(n) = out(p)

out(n) = gen(n) (in(n) - kill(n))

ppred(n)

GEN? KILLs IN1 OUT1

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {1} {1,2}

3: old = 1 Y {11} {1,2} {1,2,3}

4: result = 0 Y {9} {1,2,3} {1,2,3,4}

5: if n > 1 then goto 7 N {} {1,2,3,4} {1,2,3,4}

6: return n N {} {1,2,3,4} {1,2,3,4}

7: i = 2 Y {12} {1,2,3,4} {1,2,3,4,7}8: if i > n goto 14 N {} {1,2,3,4,7} {1,2,3,4,7}

9: result = old + older Y {4} {1,2,3,4,7} {1,2,3,7,9}

10: older = old Y {2} {1,2,3,7,9} {1,3,7,9,10}

11: old = result Y {3} {1,3,7,9,10} {1,7,9,10,11}

12: i = i + 1 Y {7} {1,7,9,10,11} {1,9,10,11,12}

13: goto 8 N {} {1,9,10,11,12}{1,9,10,11,12}

14: return result N {} {1,2,3,4,7} {1,2,3,4,7}

in(n) = out(p)


ppred(n)

GEN? KILLs IN2 OUT2

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {1} {1,2}

3: old = 1 Y {11} {1,2} {1,2,3}

4: result = 0 Y {9} {1,2,3} {1,2,3,4}

5: if n > 1 then goto 7 N {} {1,2,3,4} {1,2,3,4}

6: return n N {} {1,2,3,4} {1,2,3,4}

7: i = 2 Y {12} {1,2,3,4} {1,2,3,4,7}

8: if i > n goto 14 N {} {1-4,7,9-12} {1-4,7,9-12}

9: result = old + older Y {4} {1-4,7,9-12} {1-3,7,9-12}

10: older = old Y {2} {1-3,7,9-12} {1,3,7,9-12}

11: old = result Y {3} {1,3,7,9-12} {1,7,9-12}

12: i = i + 1 Y {7} {1,7,9-12} {1,9-12}

13: goto 8 N {} {1,9-12} {1,9-12}

14: return result N {} {1-4,7,9-12} {1-4,7,9-12}

in(n) = out(p)


ppred(n)

GEN? KILLs IN3 OUT3

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {1} {1,2}

3: old = 1 Y {11} {1,2} {1,2,3}

4: result = 0 Y {9} {1,2,3} {1,2,3,4}

5: if n > 1 then goto 7 N {} {1,2,3,4} {1,2,3,4}

6: return n N {} {1,2,3,4} {1,2,3,4}

7: i = 2 Y {12} {1,2,3,4} {1,2,3,4,7}

8: if i > n goto 14 N {} {1-4,7,9-12} {1-4,7,9-12}

9: result = old + older Y {4} {1-4,7,9-12} {1-3,7,9-12}

10: older = old Y {2} {1-3,7,9-12} {1,3,7,9-12}

11: old = result Y {3} {1,3,7,9-12} {1,7,9-12}

12: i = i + 1 Y {7} {1,7,9-12} {1,9-12}

13: goto 8 N {} {1,9-12} {1,9-12}

14: return result N {} {1-4,7,9-12} {1-4,7,9-12}

in(n) = out(p)


ppred(n)

done!



GEN? KILLs IN0 OUT0

0: entry N {} {} {}

1: n = 10 Y {} {} {}

2: older = 0 Y {10} {} {}

3: old = 1 Y {11} {} {}

4: result = 0 Y {9} {} {}

5: if n > 1 then goto 7 N {} {} {}

6: return n N {} {} {}

7: i = 2 Y {12} {} {}8: if i > n goto 14 N {} {} {}

9: result = old + older Y {4} {} {}

10: older = old Y {2} {} {}

11: old = result Y {3} {} {}

12: i = i + 1 Y {7} {} {}

13: goto 8 N {} {} {}


in(n) = out(p)


ppred(n)Visiting inreverse order,

14...0GEN? KILLs IN1 OUT1

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {} {2}

3: old = 1 Y {11} {} {3}

4: result = 0 Y {9} {} {4}

5: if n > 1 then goto 7 N {} {} {}

6: return n N {} {} {}

7: i = 2 Y {12} {} {7}8: if i > n goto 14 N {} {} {}

9: result = old + older Y {4} {} {9}

10: older = old Y {2} {} {10}

11: old = result Y {3} {} {11}

12: i = i + 1 Y {7} {} {12}

13: goto 8 N {} {} {}


in(n) = out(p)



14...0

GEN? KILLs IN2 OUT2

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {1} {1,2}

3: old = 1 Y {11} {2} {2,3}

4: result = 0 Y {9} {3} {3,4}

5: if n > 1 then goto 7 N {} {4} {4}

6: return n N {} {} {}

7: i = 2 Y {12} {} {7}

8: if i > n goto 14 N {} {7} {7,12}

9: result = old + older Y {4} {} {9}

10: older = old Y {2} {9} {9,10}

11: old = result Y {3} {10} {10,11}

12: i = i + 1 Y {7} {11} {11,12}

13: goto 8 N {} {12} {12}


in(n) = out(p)



14...0GEN? KILLs IN3 OUT3

0: entry N {} {} {}

1: n = 10 Y {} {} {1}

2: older = 0 Y {10} {1} {1,2}

3: old = 1 Y {11} {1,2} {1,2,3}

4: result = 0 Y {9} {2,3} {2,3,4}

5: if n > 1 then goto 7 N {} {3,4} {3,4}

6: return n N {} {4} {4}

7: i = 2 Y {12} {4} {4,7}

8: if i > n goto 14 N {} {7,11,12} {7,11,12}

9: result = old + older Y {4} {7,12} {7,9,12}

10: older = old Y {2} {9} {9,10,12}

11: old = result Y {3} {9,10} {9,10,11}

12: i = i + 1 Y {7} {10,11} {10,11,12}

13: goto 8 N {} {11,12} {11,12}

14: return result N {} {7,12} {7,12}

in(n) = out(p)


ppred(n)

st ill not done!

Visiting inreverse order,

14...0



Practical matters

What order to visit the nodes?

Iteration over basic blocks, not s tatements?

Efficient representation of sets?Representing reaching definitions info?

!

Basic block nodes

For straight-line code:in[s1; s2] = in[s1]out[s1; s2] = out[s2]

So, for each basic block we can compute (in

a linear pass) the Ge n and Kill sets, and then perform the iterati ve analysis on the blocks instead of individual statements

Finally perform a linear analysis for eachstatement in each block

Basic blocks1: n = 102: older = 0

3: old = 1

4: result = 0


6: return n

7: i = 2

8: if i > n goto 14


10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

B1

B2B3

B5

B6

B4

B5

B6

B2 B3

B1

B4

GEN KILL

{1,2,3,4} {9,10,11}

{} {}

{7} {12}

{} {}

{9,10,11,12} {2,3,4,7}

{} {}

Basic blocks, cont’d1: n = 102: older = 0

3: old = 1

4: result = 0


6: return n

7: i = 2

8: if i > n goto 14


10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

B1

B2B3

B5

B6

B4

Typicall y, far fe wer nodes in th e control-flow graph,hence much faster convergence in practice



Practical matters




!

!

Bit vectors

Bit vectors are commonly used torepresent the gen, kill, in, and out se ts

each definition is a bit posi tion

gen: 1 in each position generated, else 0

kill: 0 in each position kille d, else 1

out (n) = in(n) (gen(n) kill(n))

Practical matters



Efficient representation of sets?

Representing reaching definitions info?

!

!

!

Use-def chains1: n = 102: older = 0

3: old = 1

4: result = 0


6: return n

7: i = 2

8: if i > n goto 14


10: older = old

11: old = result

12: i = i + 1

13: goto 8

14: return result

For each use of var v ina statement s, a list of definit ions of v that reach s

Note that def-use chainsare also useful, b ut NOTgiven by reachingdefinitions analysis



Practical matters




!

!

!!

Simple constant propagation

Calculate reaching definitions

For a use of v in statement n, if the onlydefinit ion of v that reache s n is of the form

v = c

for a constant c, then replace v with c

Uninitialized variables?...if (...)

x = 1;

...

a = x

...

This def reaches this use

...but the def might not get executed!

Language s like C typicall y do not define the behavior of programs with uninitialized variables, so simple constant propagation is OK

Q: How would you use RD to detect possi ble uninitialized variables?

Better constant propagation

In genera l, however, we will want to dea l with uninitia lized variables in a more principled way

In genera l, this is done by defining a latt ice and then binding each dataflow value witha lattice element



Constant propagation lattice All variables initialized to

If defs d1: v=c1 and d2: v=c2reach n, then v is join(c1,c2)

For gen d: v = c, bind d with c

For gen d: v = x

y, bind d with T

or calculate the lattice values of x and y and then join them

.. . -2 -1 0 1 2 ...

Next time

Dead-code eliminat ion

based on liveness analysis

Common-subexpression elimination

based on available-expressions analysis

The general dataflow analysis f ramework



Performance Analysis of Complex Systems by Integration of Dataflow Graphs and

Compositional Performance Analysis

Simon Schliecker, Steffen Stein, Rolf ErnstInstitute of Computer and Communication Network Engineering

Technical University of Braunschweig, Germany[ schliecker | stein | ernst ] @ida.ing.tu-bs.de

Abstract

In this paper we integrate two established approaches to

formal multiprocessor performance analysis, namely Synchro-

nous Dataflow Graphs and Compositional Performance Analy-

sis. Both make different trade-offs between precision and ap-

plicability. We show how the strengths of both can be combined

to achieve a very precise and adaptive model. We couple these

models of completely different paradigms by relying on load

descriptions of event streams. The results show a superior performance analysis quality.

1 Introduction

Formal approaches to system performance analysis are

gaining industrial importance with increasing system complex-

ity. Research in this area has produced a variety of models of

computation to describe and reason about the system’s behav-

ior. Each of these established models represents a focus on dif-

ferent system properties or is adapted to specific multiproces-

sor design styles. Some models may be very well suited to give

performance information for a specific setup, and be inexact or

inapplicable for another.The timing of real-time systems remains a large applica-

tion area for the use of such formal methods. Key metrics are

latency and throughput of the system. In this paper, we fo-

cus on two analysis methods based on Synchronous Dataflow

(SDF) graphs and event model driven compositional perfor-

mance analysis.

SDF graphs are well suited to describe signal processing

applications such as filters, data-driven hardware components,

and also software functions. The data-dependencies and par-

allelism can be well expressed with SDF, but some dynamic

paradigms such as priority driven scheduling can not directly

be modeled. Determining optimal schedules and computing

the average achievable throughput of the described system hasbeen widely discussed in literature e.g. [10, 4, 3]. Less at-

tention has been given to the worst-case traffic and latency of

events in such a graph, as required for our mixed setup.

In the case of more dynamic multiprocessor behavior, com-

positional methods based on local schedulability analysis yield

InIn AA BB OutOut

DSPDSP

uCuC ARMARM

T1 T2

T3

C1 C2

SPP

In1

In2

Event Stream

SDF

Out1

Out2

Figure 1. Integrated Setup Overview

good results. System properties can be derived by composing

local analysis instances using event streams describing key in-

tegration issues [5, 12]. This approach allows to integrate a

wide range of local details but can not evade overestimations

between component boundaries.

Modern embedded systems integrate applications from di-

verse domains. Take for example the system depicted in fig-

ure 1. Here, a DSP, a microcontroller and an ARM processor

core are interconnected by a bus. The bus as well as the ARMcore are assumed to be arbitrated in a static priority preemptive

fashion. A filtering application implemented on the DSP runsa fixed order schedule. The DSP application is best modeled

using SDF graphs, whereas schedulability analysis is required

for the bus timing as well as for the ARM core.

Integrating both analysis approaches in a single framework

as presented in this paper will increase compositional analysis

accuracy and extend the applicability of SDF analysis into the

domain of systems that contain priority driven components. It

allows the designer to choose the most suitable model.

This paper presents the involved models for performance

analysis (Secs. 2.1, 2.2), our integrated setup (Sec. 3), and algo-

rithms to derive the required data from an SDF graph (Sec. 4).

Experiments are shown in Sec. 5, and we conclude in Sec. 6.

2 Related Work

2.1 Compositional System Level Performance Analysis

The compositional performance analysis methodology pro-

posed in [5] (available as the tool SymTA/S), solves the global

system-level performance verification problem by decompos-

1

978-3-9810801-2-4/DATE07 © 2007 EDAA



ing the system into independently investigated components.

Each resource (as seen in in Fig. 1) is a component and the

possible I/O timing between the tasks (event streams) is cap-

tured with event models. Input event models capture event

patterns leading to task activations. These are used to per-

form a local scheduling analysis of a resource to derive the

local response times as well as output event models. Theseoutput event models are propagated to subsequent resources

where they are used, in turn, as input event models. In setups

with cyclic dependencies the assumed event streams become

increasingly more generic. This procedure either converges

(and provides a conservative estimation of system properties

such as jitter and latencies which can be checked against given

constraints), or the system’s schedulability can not be guaran-

teed.

Fundamental properties of the event models used in [12] and

[5] are the maximum and minimum number of events that may

occur during a time interval of given size. Correspondingly, the

minimum δ−(i) and maximum distance δ+(i) between succes-

sive events can be used. To receive a more compact description,these models are expressed in [9] with a set of key parameters:

the period P , the signal jitter J , and the minimum distance be-

tween any two events dmin. The distance for i ≥ 2 events in a

bursty event stream ES can then be expressed as follows:

δ+(n) = (i − 1)P + J (1)

δ−(n) = max((i − 1)dmin, (i − 1)P − J ) (2)

Event models only capture key integration aspects such as

send/receive message jitters, etc., and ignore details of the con-

crete computation or communication resources. They there-

fore represent a suitable abstraction for the integration of other

analysis methodologies or application models.

2.2 Synchronous Dataflow Graphs

A Synchronous Dataflow graph is a directed graph. The

nodes of the graph model tasks and are referred to as actors.

The edges model communication channels or other dependen-

cies between actors. Tokens on the edges represent data sam-

ples that are consumed or produced by the execution ( firing)

of an actor. An actor may consume or produce more than one

token upon each firing. The actual number of consumed or

produced token is called consumption or production rate re-

spectively. If the consumption and production rates of all ac-

tors equal one, the graph is referred to as a Homogeneous SDF

(HSDF) graph.The classical SDF model [7] is untimed. However, fortim-

ing analysis purposes, SDF Graphs have been annotated with

execution times for each actor [10, 4]. Many problems related

to SDF Graphs have been thoroughly discussed in literature.

This includes computation of the throughput of an SDF Graph

which involves calculating the Maximum Cycle Mean λ of the

graph [3]. In literature, given SDF graphs are often converted

to equivalent homogeneous SDF (HSDF) [10] graphs to sim-

plify reasoning about them. The conversion algorithm given

in [10] also yields a repetition vector r, indicating how many

firings of each node will return the graph into the initial state.

Also, r indicates for each node, how many copies of it will be

present in the equivalent HSDF graph. As the HSDF graph can

be considerably more complex, recent work [4] directly derives

throughput from the SDF representation.

SDF graphs can be used to express the (partial) order of the

application’s data dependencies. Likewise they can accommo-date additional constraints imposed by shared resources and

scheduling. This is done to reason about and derive an optimal

static order schedule e.g. in [10]. A much larger application

area is described in [8], where the self-timed execution of an

SDF model of the actual system is used to also model the de-

lay of computation, communication, and arbitration of suitable

hardware components. Under self-timed execution each actor

fires as soon as all input tokens are available. Reasoning about

this model delivers conservative estimates of the actual hard-

ware behavior. In this paper we assume that an SDF model of

the application exists for which the previous is true.

SDF graphs under self-timed execution have been shown

to behave monotonically, which means that the earlier arrivaltime of a token can not lead to the later activation of any ac-

tor. Furthermore, it is shown that consistent graphs that do

not deadlock , will after a transient phase settle into a periodic

regime [10]. During the periodic regime, each actor n will fire

N ∗ r[n] times until the graph returns into its initial state [1]. N can be derived from the number of tokens on the critical cycles

in the graph and is well bounded in most practical cases [10].

The state of the self-timed execution of an SDF graph is de-

fined by the current token placement and execution progress of

the involved actors. If we explicitly derive an initial state (as in

Secs. 4.3, 4.4), we reduce the state space by representing it by

a token placement alone as in [4]. Any state can be reduced to

such a simplified state, by either moving all unfinished tokensto the input or to the outputs of the processing actor.

2.3 Mixing Models

Previous work has investigated the composition of perfor-

mance analysis methodologies from different domains. In [2],

a mixed simulation setup is presented that embeds a dataflow

graph into a discrete event setup and vice-versa. Many other

domains have been coupled in the Ptolemy project. In [6]

a simulated component is integrated into a formal real-time

performance analysis method. This allows to regard features

which are difficult to capture with formal models on the one

hand, and speed up analysis time by abstracting detailed be-

havior on the other. Because they rely on simulation, both of the above approaches do however not derive worst-case guar-

antees for the integrated components.

3 Embedding SDF into Compositional Perfor-mance Analysis

With the tracking of specific event timing, SDF graphs are

a suitable model for tightly integrated systems with feedback

loops and many cyclic dependencies. On the other hand, the

compositional performance analysis can address the complex-

ity of larger systems with abstract event streams and decoupled

2



load on a given output edge p during self-timed execution

of the SDF graph. The observed token production times

directly translate to the minimum and maximum inter-event

times δ− p (i) and δ+ p (i) of an input event stream. To receive an

event model conform with the description in Sec. 1 and [9], we

derive the jitter from the lower and upper amount of observed

tokens.In this paper, we focus on the maximum load a SDF compo-

nent can impose on its environment. For completeness, we also

present a simple method to derive the minimum load imposed.

Maximum Output Load To construct a starting configura-

tion yielding the maximum output load for a given actor n, we

evaluate a set of equations describing the activation timepoints

of all actors in an HSDF Graph. Let πn be the set of prede-

cessors of n, d(e( j,n)) denote the initial number of tokens on

the edge from j to n, and an(k) denote the kth activation of n.

Then

an(k) = maxj∈πn

{aj(k − d(e( j,n)) + E ( j)} (6)

with

an(k) = −E (n) ∀k < 0 ∨ E ( j) ∈ [E min, E max] (7)

captures the initial state of the graph and all future activations

(k ≥ 0) of an actor n (corresponding to the evolution equations

of [1]). Eq. 6 states that an actor becomes activated as soon as

there are tokens on all its input edges and models the depen-

dencies between successive actor activations. The first part of

Eq. 7 provides that the initial tokens have been produced at

time zero. This setup reduces the set of possible initial states to

those with no ongoing computation. Any diverging initial state

can conservatively be mapped to this set as in Sec. 2.2.The above equations describe the evolution of the states of

the SDF graph during self-timed execution from a given start

configuration. Maximizing the load induced by a given out-

put actor n implies maximizing the density of produced tokens

over time. Due to monotonicity, a maximum density will bereached for minimal execution times of all actors. Assuming

constant execution time for all actors, production of a token on

an output edge can be related to activation of the output nodeby a static offset (its minimum execution time). Thus, the prob-

lem of finding a maximum density for produced tokens on an

output edge is equivalent to maximizing the activation density

of the output actor. Then, the problem of finding the maximum

load at an output can be formulated as

minimize an(k) − an(0) ∀k ≥ 0 (8)

This way we reformulated the problem of maximizing the out-

put load of a given node to minimizing equation 6 for all k ≥ 0and a given n by altering the initial token placement. It is hard

to find the reachable token placement that minimizes equation

8 for all k. Thus, we introduce a method to derive a token

placement that will overestimate all reachable placements in

minimizing equation 8.

It is easy to see that any actor n will instantly fire most of-

ten, if all input edges initially bear the maximum amount of

tokens possible. For more tokens to be produced on the incom-

ing edges of n as soon as possible, the same argumentation

applies to the predecessors of n and their input edges. Due to

the cyclic nature of the system of equations [1], this chain of

argumentation can be continued until all firings of nodes aredependant on the first firings of actor n. Thus minimizing the

timepoints of these will minimize all activation timepoints of

n. A solution that will yield minimum activation times for nis to put the maximum number of tokens on the edges with

increasing distance from n.

The maximum number of tokens placeable on any edge is

well bounded in a strongly connected HSDF graph, as the num-

ber of tokens in a simple cycle must remain constant [10].

This constraint limits the number of initial token placements

to a superset of the actually reachable token placements. Fur-

ther constraints taking into account i.e. actor execution times

and properties of the connected event streams will tighten this

approximation and improve the results of the algorithms to cal-culate the upper envelope described below.

It can be shown that the method outlined above yields a

unique token placement for all HSDF graphs [11]. Algorithm 1

fires all nodes except the output as often as possible, resulting

in the token placement yielding the highest output load during

self-timed execution.

Algorithm 1 Construct Token PlacementINPUT: HSDF graph G, Output node n ∈ GOUTPUT: Token placement will produce maximum output load

oneFired = true

while oneFired dooneFired = false

for all nodes v ∈ G\{n} do

if v can fire thenfire voneFired = true

To find the actual activation pattern of the output node given

the above initial token placement, we assume all input event

streams to produce tokens at a rate corresponding to their max-

imal load δ−(i) and simulate the resulting graph assuming min-

imal execution times for all actors. The simulation can be

stopped once the graph has entered the periodic regime [10].

From the observed timepoints tkout at which the k-th to-

ken was produced on the output edge, we construct a function

δ(i) = tiout − t1out, describing the distances between the ob-

served tokens. δ−(2) equals the dmin parameter used in the

event model of [9]. Since dmin and P are known, the jitter can

be easily discovered from the distances δ−(i).

Minimum Output Load The minimal load an SDF compo-

nent can impose on the connected SymTA/S model can be de-

rived from the maximum length of time during which no to-

ken has to occur at all. This time span is conservatiely ap-

proximated by the difference between maximum and minimum

latency (as introduced in the Sec. 4.4). From this, a jitter

as necessary for the event model of [9] can be deduced from

Lmax − Lmin = P + J .

4



This approach is trivial. For improved considerations, it can

be replaced by a similiar procedure as undertaken for the max-

imum load.

4.4 Path Latency

End-to-end latency is a fundamental metric to quantify sys-

tem performance. As we model a subsystem with SDF, we re-

quire the local latency for further system wide considerations.

The path latency as the time between the arrival of an event at

an input and the first causally dependant reaction at the out-

put. A token x is causally dependant on all tokens that are

consumed with the actor activation that leads to the production

of x.

The maximum latency is difficult to predict from an SDF

representation directly. We therefore assume to have an HSDF

representation which duplicates the actors according to the rep-

etitions vector (see Sec. 2.2). We assume for the remainder of

this paper that the input and output actors of the path, whose

latency we are interested in, are not duplicated. This is the case

for many applications. In other cases, the worst case latency is

given by the maximum over all latencies from any correspond-

ing input actor to any corresponding output actor.

We are now interested in the latency of an arbitrary token,

that arrives at an input edge ein at time t0in. The latency to an

output edge eout can be derived by simulation of the self-timed

schedule [10]. The first causally dependent reaction at the out-

put can easily be tracked with a colored token at the input that

progressively causes produced tokens to be colored as well. It

will be observed at the output after it has propagated over a path

from ein to eout. Let ρ(ein, eout) be the minimum number of

tokens on any such path at time t0in, then the 1 + ρ(ein, eout)-th

firing of the output actor after the arrival of the token at the in-

put must be causally dependant. Thus, the latency for the tokenarriving at the input is given by

L = t1+ρ(ein,eout)out − t0in (9)

To derive the maximum latency, we assume the graph in a

worst case state at t0in and let it behave as slow as possible

thereafter. As the arrival time of the token t0in is fixed, we

maximize its latency by maximizing t1+ρ(ein,eout)out .

From Eq.6 it can immediately be seen that for any actor jdecreasing actor execution times (E j) or activation times (aj)

can not lead to a later activation (and therefore finishing) of

another (an). This is true throughout the graph. Therefore,

t1+ρ(ein,eout)out is maximal under the following conditions: (i)

throughout the graph every actor always executes with its worst

case execution time (see also [4]), and (ii) events at the graph’s

inputs arrive as late as possible.

Also, we again do not need to precisely model the state of

the HSDF graph at t0in. We assume the initial state to be de-

fined only by the token placement. Any differing state can be

expressed by assuming that tokens currently processed by an

actor are placed on the respective incoming edge. This overes-

timates the resulting latency, but greatly reduces state space.

Assume for now that the only external input event stream is

ES in. At the input edge ein, the latest possible arrival of to-

kens previous to t0in are given by the minimum distances inher-ent to the event stream: t−1in = t0in−d−in(2), t−2in = t0in−d−in(3),

a.s.o. Subsequent tokens arriving after t0in do not effect the la-

tency due to the FIFO ordering throughout the model.

To produce the worst case state at t0in, we do however not

need to begin the simulation at a time tsim = t−∞in in the past.

For any event stream which can be modeled as in Sec. 2.1,

we can assume the additional distance to previous events to

be constant after a transient phase. This is e.g. the case for the

burst event model of Eq. 1 when the P term dominates the dmin

term [9]. In other models this may not be so easy to determine,

but it is always conservative to assume constant distances to

previous events starting with an arbitrary event arrival t−Sin . As

the distance increment is non-decreasing, this means that any

prior events will be assumed later than is actually possible (and

thus lead to a higher latency). Thus, after a number of events S in the past the additional minimum distance to any prior events

can be assumed to be constant.

The effect is that the graph that produces the maximum la-tency at time t0in behaved purely periodic at some time in the

past. The self-timed execution of an HSDF graph with constant

execution times is in the same state after N firings of every ac-

tor (once it has entered its periodic regime) [10] . The same is

true for our setup before t−Sin , as all graph elements have a con-

stant behavior (at their bounds). Thus, the graph will be in the

same state at t−Sin as it was at t−(S+N )in , making it unnecessary

to investigate any prior states. Thus, we begin the simulation

at tsim = t−Sin .

The graph can be in any of the N states with the begin-

ning of the simulation. In general, each can lead to differ-

ent t1+ρ(ein,eout)

outand must therefore be tested with a different

simulation run. Alternatively, an approach similar to that in

Sec. 4.3 can be used to derive a conservative initial placement.

So far we have neglected other input event streams ES p ∈{Pin \ ES in} to the graph. On average these inputs will not

constrain the graph’s execution (see Sec. 4.1), but they may

introduce transient underflow situations that can increase the

searched latency. We create the most constraining scenario by

assuming events to arrive as late as possible after the beginning

of the simulation: t0 p = tsim+d+ p (2), t1 p = tsim+d+ p (3), a.s.o.

for all inputs. This setup puts transient underflow situations at

the beginning of the simulation.

Thus, we begin a simulation at a timepoint tsim sufficiently

in the past either in each of N graph states or a conservativeapproximation. We then let external events arrive as late as

possible, including the input token at t0in, and it will experience

the worst-case latency, which is determined by the first causally

dependant reaction observed at the output.

The minimum latency can be determined in a similar fash-

ion. However, an easy and appropriate simplification is that

the first causally dependant effect of an arriving event can not

observed before it has propagated at least through the shortest

path, where the shortest path is the path with the smallest sum

of execution times.

5