cluster analysis & hybrid models business application & conceptual issues oct 4, 2005

William B. Hakes, Ph.D.-V

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Cluster Analysis & Hybrid ModelsBusiness Application & Conceptual

Issues

Oct 4, 2005


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• Introduction & Intro to Clustering

• Applied Problem- Dissertation– Conceptual/Practical Issues– Research Ideas– Good Clusters/Bad Clusters & Interpretation

• Applied Problem II/Binary Clustering

• Applied Problem III- Interpretation- Clustering from a Survey

• Trees (RI)- Intro

• Dissertation RI

• For Further Research

Today’s Outline


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1. A financial analyst of an investment firm is interested in identifying groups of mutual funds that are look alike in a “true” context, not simply based on the way Morningstar rates them.

2. A marketing manager is interested in identifying similar cities (across multiple dimensions) that can be used for a test marketing campaign in which a new product might be introduced.

3. The Director of Marketing at a telecom firm wants to understand the types of people that he already knows are candidates for the firm’s new internet data service

4. A Golf Club General Manager wants to understand the “natural” segments of his members so that he can better utilize his clubs assets and understand how he might ideally want the club to look in the future.

Introduction


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1. Cluster Analysis- its easy when:

a. You have a relatively small sample

b. You have nice, neat data

c. Your variables are continuous

2. Cluster Analysis- The Real World

a. Sometimes sample are small, but in business they’re large

b. We’d like our data to be free from error, containing no outliers, but that is rarely the case.

c. Variables are often a mix of continuous and categorical data

Cluster Overview


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A) Cluster entire customer base (General Purpose Clusters)

• Build predictive models across products

• See how your “targeted” customers fall into the clusters, if they provide separability

-or-

B) Build predictive models on base

• Determine the “targets” for a specific campaign

• Cluster those “targets” based only on actionable information (Specific Purpose Clusters)

-or-

C) Cluster Analysis as a primary end-analysis

Correct option depends on how you’ll use it!

Clustering- Some Competing Macro Views


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Quantitative Problem Domain

• Why Cluster Analysis?

– Commonly Applied:• Targeted Army recruitment- Faulds and Gohmann (2001)

• Identify “natural” segments of Euro tourists- Yuksel and Yuksel (2002)

• Uncover “natural” groups of common business goals across 15 countries- Hofstede et al. (2002)

• Prostate cancer treatment on various types of cells- Li & Sarkar (2002)

• Cluster analysis…

– Identifies subgroups within a larger group

– Makes each object (customer, product etc.)within each group as similar as possible while making the subgroups as different as possible from one another.

Cluster Analysis


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Quantitative Problem Domain• How Does Cluster Analysis Work?

– Variable Selection• Generally must be on similar/identical scales (standardized)• Metric- Ordinal/Interval/Ratio• Non-metric data• Correlation & outliers distort results• Principle Components & Factor Analysis as Inputs

• Construct similarity/proximity matrix to view relationship between all observations across all variables:– Euclidean Distance (most commonly used) – Other distances measures include Euclidean, Squared

Euclidean, City Block, Mahalonobis – Correlation- but consider (1, 2, 1, 2) and (9, 10, 9, 10) vs.

(1, 2, 1, 2) and (1, 1, 2, 2)– Association (Jaccard coefficient for binary variables)

Cluster Analysis cont’d


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Quantitative Problem Domain• How Does Cluster Analysis Work?

– Choose clustering algorithm • Hierarchical- n clusters of size 1 until one cluster remains (EDA &

Research)– Choose algorithm to determine how distance is to be computed

between clusters.– Iteratively, we continue to “relax" our criterion as to what is and is

not unique….lower our threshold regarding the decision when to declare two or more objects to be members of the same cluster (see Dendogram)

– Ward’s Method*, single Linkage, centroid, etc.• Non-Hierarchical (K-Means)- assign objects to clusters based on pre-

specified number of clusters (Practical Implementation)– Choose seeds for n clusters (often pre-determined)– Clusters are formed, new centroids computed, new clusters formed

• A Dual approach is recommended (Hartigan,1975 ; Milligan, 1980 ; Punj & Stewart, 1983)

– Use hierarchical clustering to compute estimated cluster centroids– Use centroids as the Cluster seeds for a K-Means analysis



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Quantitative Problem Domain• Choosing the Number of Clusters

• Hierarchical- n clusters are formed (use dendogram)

• K-Means– Form pre-specified k number based

on theory (Milligan, 1980; Hair et al., 1998)

– Form pre-specified k number based on application


– Consult the “Pseudo-F” in either case to assess solution (Lattin, et al., 2003; Punj & Stewart, 1983)

• Interpretation of Clusters– Which variables are important? How important?

• Univariate F-Tests on Cluster Centroids• Perspicuity via “art” in initial steps• Perspicuity via a different technique (a hybrid)


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Toward a Hybrid Approach

After a predictive model is built, how can variables best be pre-processed for cluster analysis so that rule induction on the resulting clusters provides maximum perspicuity while minimizing the art involved??

…. perhaps a hybrid model so as to minimize the “art” involved while maximizing perspicuity and applicability.


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Take the following example: You are a firm trying to generate clusters about the Atlanta area with the objective of understanding zip codes to which you want to “mass”market your products.

a. Many different races exist. How do you cluster them? Typically, its:

• 1) White

• 2) African American

• 3) Asian

• 4) Hispanic

• 5) Native American

• 6) Non-white other

b. What will clustering do with this variable as it groups people?

Clustering- A Problematic Example (I)


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A Problematic Example cont’d

1. Can you cluster this simple example?

2. How will you interpret it (e.g., what’s a common way to look at the “answer” to see if you agree with the differentiation)?

Person Zip Race101 30066 1102 30068 2103 30066 3104 30062 4105 30066 6106 30067 2107 30068 5108 30062 4109 30067 3110 30068 4111 30062 2112 30062 1113 30066 1114 30066 2115 30068 3116 30062 5117 30067 6118 30068 1119 30062 2120 30066 6


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1. Cluster Means- What do they tell us?

2. Assume we have three clusters, and along the “race” dimension, they are as follows:

• Cluster 1- Mean=2



How do you:

Use this data to assign people into clusters?

Interpret means?

A Problematic Example cont’dPerson Zip Race

101 30066 1102 30068 2103 30066 3104 30062 4105 30066 6106 30067 2107 30068 5108 30062 4109 30067 3110 30068 4111 30062 2112 30062 1113 30066 1114 30066 2115 30068 3116 30062 5117 30067 6118 30068 1119 30062 2120 30066 6


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Binary Variables- One Possible Solution?

Person Zip Race Race1 Race2 Race3 Race4 Race5 Race6101 30066 1 1 0 0 0 0 0102 30068 2 0 1 0 0 0 0103 30066 NA 0 0 0 0 0 0104 30062 4 0 0 0 1 0 0105 30066 6 0 0 0 0 0 1106 30067 2 0 1 0 0 0 0107 30068 5 0 0 0 0 1 0108 30062 4 0 0 0 1 0 0109 30067 3 0 0 1 0 0 0110 30068 4 0 0 0 1 0 0111 30062 2 0 1 0 0 0 0112 30062 1 1 0 0 0 0 0113 30066 1 1 0 0 0 0 0114 30066 2 0 1 0 0 0 0115 30068 3 0 0 1 0 0 0116 30062 5 0 0 0 0 1 0117 30067 6 0 0 0 0 0 1118 30068 1 1 0 0 0 0 0119 30062 NA 0 0 0 0 0 0120 30066 6 0 0 0 0 0 1


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Binary Variables- A Closer Look

Person Zip Race Race1 Race2 Race3 Race4 Race5 Race6101 30066 1 1 0 0 0 0 0102 30068 2 0 1 0 0 0 0103 30066 NA 0 0 0 0 0 0

1. How will these cases cluster?

2. What can we do about it?

How similar are persons 101, 102 & 103 to one another….are they more alike or more different?


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Applied Problem II

1. A Golf Club General Manager wants to understand the “natural” segments of his members so that he can better utilize his clubs assets and understand how he might ideally want the club to look in the future.

2. How can cluster analysis help?

3. We took a look at the following

• Demographic Information

• Usage Information

• Cost Information

• Some data was measured and some was survey data

Note that in clustering you may use N dummy variables (rather than N-1 in dependent techniques like regression)


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Application of Binary Clustering

Golf Use: Never .5/mo 1/mo 4/mo 10/mo 15+/mo

Cluster AClub01 AClub02 AClub03 AClub04 AClub05 AClub06 SUM N =

1 2% 23% 36% 33% 6% 0% 100% 10002 2% 29% 31% 22% 15% 1% 100% 5003 10% 27% 21% 19% 23% 1% 100% 3804 0% 35% 30% 26% 9% 0% 100% 2205 6% 21% 23% 33% 17% 0% 100% 1700

Total N 161 926 1046 1127 534 6

Cluster BClub01 BClub02 BClub03 BClub04 BClub05 BClub06 SUM N =

1 1% 7% 29% 40% 23% 0% 100% 10002 1% 15% 38% 28% 18% 0% 100% 5003 9% 20% 31% 20% 19% 0% 100% 3804 2% 13% 48% 31% 4% 2% 100% 2205 4% 16% 31% 30% 16% 3% 100% 1700

Total N 128 517 1236 1199 662 58

Cluster Means

- Above data taken from one question off of 30Q customer survey. 5 Clusters were formed.

- Note that a dummy separated n ways will sum to 100% only if there are no missing responses.


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Many different uses , but its works great for clustering (see SPSS)

Jaccard Coefficient

a___ Sj = a + b + cwhere a is the sum of agreement (+ +) and b, c represent the sums of absent/present combinations (i.e. + - , and - +, respectively). The table below shows this convention of lettering for counts when calculating the similarity between two objects. Values of d are not considered because they represent complete disagreement.

OBJECT 1

+ -

OBJECT2

+ a(1,1)

b(1,0)

- c(0,1)

d(0,0)

Var --> a b c d e f g h

Person 1 1 0 0 1 0 0 1 0Person 2 1 0 0 0 0 0 1 0Person 3 0 0 1 0 1 0 1 0

Matrix Person 1 Person 2 Person 3Person 1 1Person 2 67% 1Person 3 20% 25% 1

i.e. Person 1 vs. Person 2 = (2 / (2+1))

Jaccard Process Sample


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Summary of Binary Clustering1. Assists when we want to understand the “natural” segments

2. How can binary cluster analysis help (using Jaccard or otherwise)?

• Allows us to use categorical data.

• Gives us unique summary insight into the true percentages of each cluster along various dimensions.

• Not tricked by the zero problem- if zero’s are “true” zero’s, then clustering can be VERY interpretable

• No program as of yet that integrates Jaccard algorithm with traditional algorithms.

• Cluster different sets of variables and then cluster the clusters using Jaccard (dummy the cluster membership)

• Invent your own technique!!!! (i.e. K-Modes)

3. All clustering should be “checked” with domain experts for validation.


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Age By Years of Membership

0

5

10

15

20

25

30

35

40

30 40 50 60 70 80

Age

Ye

ars

Me

mb

erS

hip

Applied Problem III- F&B Survey AnalysisApplied Problem III- F&B Survey Analysis

For Illustration Purposes Only


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Can Clustering Help……Can Clustering Help……

For Illustration Purposes Only

Age By Years of Membershipwith Cluster Means

0

5

10

15

20

25

30

35

40

30 40 50 60 70 80Age

Yea

rs M

emb

erS

hip

36%

32%

21%

11%


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F&B Survey Analysis – Clusters vs. Member InfoF&B Survey Analysis – Clusters vs. Member Info

Can we look at member information we know to be true in order to measure the accuracy of member responses & therefore the clusters?

Survey Average Age Actual Member Average Age

Cluster 1 52.95 52.86

Cluster 2 47.23 47.77

Cluster 3 69.25 70.5

Cluster 4 56.06 56.11

Cluster 5 52.5 53.04


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F&B Survey Analysis – Clustering OverviewF&B Survey Analysis – Clustering Overview

What are the primary Clusters that exist at the Club?

Big Spenders (8% of Member Base)Age = 53 1 Child Golf/Tennis/Fitness = 4x per month

Opportunity Knocks w/Kids (58% of Member Base)Age = 47 1.5 Children, under 14 Golf=2x/mo SwimPool=8x/mo

Seniors (15% of Member Base)Age = 69 No Children Golf /Tennis= 3x/mo

Opportunity Knocks no Kids (17% of Member Base)Age = 56 If Kids, most are 18+ Heaviest Fitness 3x/mo Golf 6x/mo Pool

Heavy All-Around Users (4% of Member Base)Age = 52 1 Child, Age 11+ Golf/Swim/Fitness = 15x per month


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Q4 and Q5 plotted together…What is the relationship between the factors that members find important when selecting a restaurant and their level of satisfaction?

Sati

sfa

cti

on

Importance3 5

5

Quality of Food

Quality of Wine Menu Variety

ServiceAtmospherePrice

Speed

F&B Survey Analysis – Results by Question F&B Survey Analysis – Results by Question

4

High Importance +High Satisfaction = Increased Loyalty

High Importance +High Satisfaction = Increased Loyalty

4

Q4- What factors are important to you in selecting a restaurant?Q5- How satisfied are you with the same factors at the Club?


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F&B Survey Analysis – Importance vs. SatisfactionF&B Survey Analysis – Importance vs. Satisfaction

Note that the actual scale begins at “1” but there were no responses measured below “3”

Cluster 1

3.003.203.403.603.804.004.204.404.604.805.00

3.00 3.50 4.00 4.50 5.00

Importance

Sati

sfa

cti

on

Variety

SpeedQuality Food

Quality Wine

Price

Service

Atmosphere

“Big Spenders”


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Cluster 2

3.003.203.403.603.804.004.204.404.604.805.00

3.00 3.50 4.00 4.50 5.00

Importance

Sati

sfa

cti

on

VarietySpeed

Quality FoodQuality Wine

Price ServiceAtmosphere


“Opp Knocks/kids”


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Cluster 3

3.003.203.403.603.804.004.204.404.604.805.00

3.00 3.50 4.00 4.50 5.00

Importance

Sati

sfa

cti

on

Variety

Speed

Quality Food

Quality Wine below 3

PriceService

Atmosphere


“Seniors”


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Cluster 4

3.003.203.403.603.804.004.204.404.604.805.00

3.00 3.50 4.00 4.50 5.00

Importance

Sati

sfa

cti

on

Variety

SpeedQuality Food

Quality Wine

Price ServiceAtmosphere


“Opp Knocks no kids”


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Cluster 5

3.003.203.403.603.804.004.204.404.604.805.00

3.00 3.50 4.00 4.50 5.00

Importance

Sati

sfa

cti

on

Variety

Speed

Quality Food

Quality Wine

Price

Service

Atmosphere


“Heavy Users”


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Points to Ponder- Clustering

Pros-

1) Good for exploratory analysis

2) Helps discover previous unsuspected relationships

3) One of very few techniques that focuses on the groups it creates, not the variate that creates them

Cons-

1) Difficult to interpret/often not actionable

2) Deemed as too “soft” by some statisticians and businesses

3) Out of sample customer assignment is very tough

Solution- Hierarchical Non-Hierarchical Consult Expert -or-

Hierarchical Non-Hierarchical RuleInduction Consult Expert

Logistic Regression

1- Original Vars

2- Stdized Version of 1

4- Principal Component Scores

3- X1 B1 – XnBn

(from Logit)

Rule Induction

(CART)

Expert Panel Review

Each of 4 RI Solutions tested for Usefulness/Perspicuity

using ANOVA/Tukey’s HSD)

K-means Cluster Analysis

(refined solutions)

Hierarchical Cluster Analysis

(seeds developed)

Determine “Target “Group

Original Vars

Std Vars Logit Vars PCA Vars

Orig Var RI Solution

Std Var RI Solution

Logit Var RI Solution

PCA Var RI Solution

Variable pre-processing as inputs

Extract the pre-specified number of clusters as seeds for next stage

Generate a pre-specified number of clusters using seeds from prior CA

Input each cluster solution into RI program and create RI solutions

Transform rules into text descriptions and submit to Expert Panel for review

Extracted Group of Targeted Customers

Original Vars

Std Vars Logit Vars PCA Vars

Hybrid Test


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Dissertation- Testing a Hybrid Method• Credit Data: real-world data from financial services (auto loans)

– Predictive Model differentiating “goods” vs. “bads”• Given that we think you’re “good”, what else is there? •Cross selling opportunities

– You’re a good risk, but certainly there is more to offer you – Consider GE Capital

• Purchasing Data: real-world motor-home data from overseas company

– Predictive Model differentiating buyers vs. non-buyers• Given that we think you’re a “buyer”, what else is

there? •Compelling qualitative messages

– You’re likely a buyer, but certainly all buyers are not the same – Consider XYZ Telecom


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A Hybrid Approach• Why is a Hybrid model necessary for perspicuity?

– Predictive Models- Most methods are 1st Order• Differentiate target group from non-target group (binary) • Determine model which best differentiates pre-defined groups (3 or

more categories).

– Cluster Analysis- • Simply identifies similar items/customers within a population (Fasulo,

1999).

– Rule Induction- • Requires a pre-defined dependent variable so that key independent

variables may be distinguished

• The Hybrid model:– Converts a numerical model into a second order model

expressed in natural language If-Then rules, exposing the behavior of the system to view (Whalen & Gim, 1999).

– “Explains group membership” rather than simply “proportioning variance”.


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RI- Rules are induced based upon a set of inputs and a criterion (dependent) variable

Although many different techniques exist such as CART, ID3, CHAID, and many others, they all tend to utilize the following procedures, even though they each have different splitting rules (Whalen & Gim, 1999):

1) Identify a dependent variable of interest along with a set of independent (predictor) variables.

2) Compare all cutpoints of predictor variables to find the one that best predicts the dependent variable (using some statistical rule, though these rules differ among methods).

3) Identify the next best rule (a predictor variable along a certain cutpoint) in each of the sub-samples already defined by (2).

4) Continue to split until all remaining subsamples are homogenous with respect to the dependent variable.

5) The set of if-then rules from the analysis are applied to a validation set to determine performance.

Overview of Decision Trees/Rule Induction


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PATIENTS = 215

SURVIVE 178 82.8%DEAD 37 17.2%

Is BP<=91?

Terminal Node A

SURVIVE 6 30.0%DEAD 14 70.0%

NODE = DEAD

Terminal Node B

SURVIVE 102 98.1%DEAD 2 1.9%

NODE = SURVIVE

PATIENTS = 195

SURVIVE 172 88.2%DEAD 23 11.8%

Is AGE<=62.5?

Terminal Node C

SURVIVE 14 50.0%DEAD 14 50.0%

NODE = DEAD

PATIENTS = 91

SURVIVE 70 76.9%DEAD 21 23.1%

Is SINUS<=.5?

Terminal Node D

SURVIVE 56 88.9%DEAD 7 11.1%

NODE = SURVIVE

<= 91 > 91

<= 62.5 > 62.5

>.5<=.5

Trees (Binary) Are Fundamentally Simple


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Credit:

Hierarchical

K-Means

Input Set Cluster 1 Cluster 2 Cluster 3 Total

Original Variables 1780 424 980 3184

55.90% 13.32% 30.78% 100.00%

Standardized Variables

1868 290 1026 3184

58.67% 9.11% 32.22% 100.00%

Logistic Variables 2463 482 239 3184

77.36% 15.14% 7.51% 100.00%

PCA Variables 2635 292 257 3184

82.76% 9.17% 8.07% 100.00%

Input Set Cluster 1 Cluster 2 Cluster 3 Total

Original Variables 1980 328 876 3184

62.19% 10.30% 27.51% 100.00%

Standardized Variables 1511 295 1378 3184

47.46% 9.27% 43.28% 100.00%

Logistic Variables 2463 482 239 3184

77.36% 15.14% 7.51% 100.00%

PCA Variables 1583 295 1306 3184

49.72% 9.27% 41.02% 100.00%

Credit Cluster Size Comparison

Category % n1 49.05 7973 41.48 6742 9.48 154Total (100.00) 1625

Node 0

Category % n1 22.97 1643 77.03 5502 0.00 0Total (43.94) 714

Node 2

Category % n1 11.74 663 88.26 4962 0.00 0Total (34.58) 562

Node 6Category % n1 64.47 983 35.53 542 0.00 0Total (9.35) 152

Node 5

Category % n1 69.48 6333 13.61 1242 16.90 154Total (56.06) 911

Node 1

Category % n1 83.62 6333 16.38 1242 0.00 0Total (46.58) 757

Node 4

Category % n1 33.33 323 66.67 642 0.00 0Total (5.91) 96

Node 8Category % n1 90.92 6013 9.08 602 0.00 0Total (40.68) 661

Node 7

Category % n1 0.00 03 0.00 02 100.00 154Total (9.48) 154

Node 3

PCCLUS10 (Training Sample)

T2924XImprovement=0.3963

>1.5

TOTBALImprovement=0.0819

>3836.5<=3836.5

<=1.5

AGEAVGImprovement=0.3150

>-1.5

RVTRDSImprovement=0.0684

>5.5<=5.5

<=-1.5

Variable Definitions:

T2924X= <=1 Trd 30dpd in 24 mo

AGEAVG = Age Avg Open Trd

TOTBAL = TotBal of all trades

RVTRDS = # Rev Trades

2a 1b

3a

3b

1a

Appendix 6a Credit RI Tree: PCA Vars

back


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Issues for Further Research

1. Predictive Models differentiate one group from another, but what about types of groups within a target group?…How Many?

2. Cluster Analysis

a. Which variables are important in clustering?

b. What about out-of-sample assignment?

3. Clustering followed by 2nd Order Rule Induction (a.k.a. Decision Trees)

a. Develop clusters (2-stage)

b. Use as inputs into algorithm (“Best”Algorithm??)

c. Take simple rules and use to assess cases across a database

4. Cluster Analysis vs. Unsupervised NN’s


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Some Parting Thoughts….Q- How much time should you spend properly defining the

quantitative issue and designing the test??

A- A lot more than you think (up to 3x more than the actual “analysis”)

Q- Are there opportunities for Analytics in the marketplace??

A- Yes- tremendous opportunities for people who can do more than Pivot Tables and Regression in Excel.

Q- How do I “get my foot in the door” of analytics?

A- Continue formal education

Continue “informal” education as well

Make “networking” part of your daily/weekly to-do list

Join a firm that has years of experience in applied problem-solving


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Appendix

Following are some slides that we may not have time to talk thru. Feel free to contact me with any questions.


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• Classification and Regression Trees

• Origins in research conducted at Berkeley & Stanford

– Leo Breiman, University of California, Berkeley

– Jerry Friedman, Stanford University

– Charles J. Stone, University of California, Berkeley

– Richard Olshen, Stanford University

• Solved a number of problems plaguing other decision tree methods (CHAID, ID3)

• Very well known in biomedical and engineering arenas

• Only recently becoming known in IT, DM, and AI circles

Overview of CART


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• A binary splitting procedure can always reproduce a multi-way split

• A binary splitting procedure will only partially partition on a database field if another sequence is better

FYI-• CART (and trees in general) handle missing data very

well• Tests show that when data are missing at random even

25% missing rates have minimal effect on CART accuracy

• Costs of misclassification: allow for certain errors to be more serious than others

• Fundamentally detects non-linear relationships• Rules can be automatically detected, or modified by user

Why CART Works Well


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If the multi-way split is best, binary split method will find itIf it is not best, binary method will move to another variable

No

No

Yes No

Age <= 65

Age > 65Age <= 45

45 < Age <= 65Age <= 25

Age < 25 Age <= 55

Binary Split Detects MultiWay Splits


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• Split made all at once could be too hasty• Even if age group isis different, other variables might be even

more valuable after the Age > 65, Age < 65 split• The database is fragmented rapidly• Even with 500,000 records, 5 consecutive 4-way splits leave

about 2,000 records per partition• Binary splits are more patient, giving a better chance to

find important structure

Age

< 25 25 - 45 45 - 65 > 65

MultiWay Splits


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Credit RI Translation (NOTE: this dataset is tough to translate in this context)• Credit RI Tree PCA Vars- Translated Rules:Cluster 1- (51% of Customers)- (1a) 81% of the customers in this cluster have:

• Over the last 24 months, 1 or less trades rated 30 days past due.• Some information available regarding the average age of their open trades.• 5 or less “revolving” accounts.

- (1b) The other 19% of the customers in this cluster have:• Over the last 24 months, 2 or more trades rated 30 days past due.• A total balance of all trades less than $3837.00.

Cluster 2- (9% of Customers)- (2) Over the last 24 months, these customers have 1 or less trades rated 30 days past due.- (2) These customers have either no record of the age of their current accounts, or they only

have “inquiries” into their credit history.Cluster 3- (40% of Customers)- (3a) 15% of the customers in this cluster have:

• Over the last 24 months, 1 or less trades rated 30 days past due.• Some information available regarding the average age of their open accounts.• 6 or more “revolving” trades.

- (3b) 85% of the customers in this cluster have:• Over the last 24 months, 2 or more trades rated 30 days past due.• A total balance of all trades is greater than or equal to $3837.00.

cluster analysis & hybrid models business application & conceptual issues oct 4, 2005

Documents

cluster analysis work

continuouscluster analysis

introductioncluster

marketing manager

euclidean distance

new product

director of marketing

neat datayour variables