algorithmic facets of human centricity in computing with fuzzy sets
DESCRIPTION
Algorithmic Facets of Human Centricity in Computing with Fuzzy Sets. Witold Pedrycz Department of Electrical & Computer Engineering University of Alberta, Edmonton, Canada and Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland. [email protected]. - PowerPoint PPT PresentationTRANSCRIPT
Algorithmic Facets of Human Centricity in Computing with Fuzzy
Sets
ISDA-2009, Pisa, Italy, November 30-December 2, 2009
Agenda
Human centricity and information granules
Design of information granules – approaches of knowledge-basedclustering
Granular representation of computing with fuzzy sets
Human Centricity and information granules
Information granules as conceptual entities inherently associated with human pursuits (decision-making, perceptioncontrol, prediction)
Interaction with and processing in intelligent systems realized at the level of information granules (fuzzy sets, rough sets, intervals…)
Emergence of Human-Centric computing (HC2)
Knowledge sharing and collaboration in distributed systems
Human Centricity and fuzzy sets
Two fundamental quests:
Construction of information granules (fuzzy sets);use of existing experimental evidence and its interpretationCast in the framework of users/designer
Qualitative, user-centric interpretation of results of computing with fuzzy sets
Clustering as aconceptual and algorithmic framework of information
granulationData information granules (clusters) abstraction of data
Formalism of: set theory (K-Means) fuzzy sets (FCM) rough sets
shadowed sets
Main categories of clustering
Graph-oriented and hierarchical (single linkage, complete linkage, average linkage..)
Objective function-based clustering
Diversity of formalisms and optimization tools(e.g., methods of Evolutionary Computing)
Key challenges of clustering
Data-driven methods
Selection of distance function (geometry of clusters)
Number of clusters
Quality of clustering results
The dichotomy and the shift of paradigm
Human-centricityGuidance mechanisms
Fuzzy Clustering: Fuzzy C-Means (FCM)
Given data x1, x2, …, xN, determine its structure byforming a collection of information granules – fuzzy sets
Objective function
2ik
N
1k
mik
c
1i||||uQ vx
Minimize Q; structure in data (partition matrix and prototypes)
Fuzzy Clustering: Fuzzy C-Means (FCM)
Vi – prototypes
U- partition matrix
FCM – optimization
2ik
N
1k
mik
c
1i||||uQ vx
Minimize
subject to
(a) prototypes
(b) partition matrix
Domain Knowledge:Category of knowledge-
oriented guidance
Context-based guidance: clustering realized in a certain contextspecified with regard to some attribute
Viewpoints: some structural information is provided
Partially labeled data: some data are provided with labels (classes)
Proximity knowledge: some pairs of data are quantified interms of their proximity (resemblance, closeness)
Clustering with domain knowledge
(Knowledge-based clustering)
Data
Information granules (structure)
CLUSTERING
Domain knowledge
Data-driven Data- and knowledge-driven
Data
Information granules (structure)
CLUSTERING
Context-based clustering
Clustering : construct clusters in input space X
Context-based Clustering : construct clusters in input space X given some context expressed in output space Y
Active role of the designer [customization of processing]
Context-based clustering:Conmputational considerations
•computationally more efficient,•well-focused, •designer-guided clustering process
Data
structure
Data
structure
context
Context-based clustering:focus mechanism
Determine structure in input space given the output is high
Determine structure in input space given the output is medium
Determine structure in input space given the output is low
Input space (data)
Context-based clustering:examples
Find a structure of customer data [clustering]
Find a structure of customer data considering customers making weekly purchases in the range [$1,000 $3,000]
Find a structure of customer data considering customers making weekly purchases at the level of
around $ 2,500
Find a structure of customer data considering customers making significant weekly purchases who
are young
no context
context
context
context(compound)
Context-oriented FCM
Data (xk, targetk), k=1,2,…,N
Contexts: fuzzy sets W1, W2, …, Wp
wjk = Wi(targetk) membership of j-th context for k-th data
c
1i
N
1kikjkikikj iNu0andk wu|0,1u)(WU
Context-driven partition matrix
Context-oriented FCM:Optimization flow
Objective function
Iterative adjustment of partition matrix and prototypes
2ik
c
1i
N
1k
mik ||||uQ vx
c
1j
1m
2
jk
ik
jkik
wu
vx
vx
N
1k
mik
N
1kk
mik
i
u
u xv
Subject to constraint U in U(Wj)
Viewpoints: definition
Description of entity (concept) which is deemed essential in describing phenomenon (system) and helpful in castingan overall analysis in a required setting
“external” , “reinforced” clusters
Viewpoints: definition
-150
-100
-50
0
50
100
150
200
0 100 200 300 400 500
x1
x2
a
b
x1
x2
a
viewpoint (a,b) viewpoint (a,?)
Viewpoints: definition
Description of entity (concept) which is deemed essential in describing phenomenon (system) and helpful in castingan overall analysis in a required setting
“external” , “reinforced” clusters
Viewpoints in fuzzy clustering
x1
x2
a
b
otherwise 0,
viewpointby the determined is B of rowth -i theof featureth -j theif 1,b ij
0
0
1
0
0
1
B
0
0
b
0
0
a
F
B- Boolean matrix characterizing structure: viewpoints prototypes (induced by data)
Viewpoints in localization of “extreme” information granules
specification of viewpoints through evolutionary/population-basedoptimization
Viewpoints in fuzzy clustering
Q = 2ijkj
n
1:bji,1j
mik
c
1i
N
1k
2ijkj
n
0:bji,1j
mik
c
1i
N
1k
)f(xu)v(xu
ijij
1b if f
0bif vg
ijij
ijijij
2ijkj
n
1j
mik
c
1i
N
1k
)g(xuQ
Labelled data and their description
Characterization in terms of membership degrees:
F = [fik] i=12,…,c , k=1,2, …., N
supervision indicator b = [bk], k=1,2,…, N
Augmented objective function
€
Q =i=1
c
∑ uik2
k=1
N
∑ || xk − vi ||2 +β∑ (uik − fik )2bk || xk − vi ||2∑
> 0
Proximity hints
Characterization in terms of proximity degrees:
Prox(k, l), k, l=1,2, …., N
and supervision indicator matrix B = [bkl], k, l=1,2,…, N
Prox(k,l)
Prox(s,t)
Proximity measure
Properties of proximity:
(a)Prox(k, k) =1
(b)Prox(k,l) = Prox(l,k)
Proximity induced by partition matrix U:
€
Prox(k,l) = min(uik
i=1
c
∑ ,uil )
Linkages with kernel functions K(xk, xl)
Augmented objective function
€
Q =i=1
c
∑ uik2
k=1
N
∑ || xk − vi ||2 +βi=1
c
∑k1=1
N
∑ [Prox(k1,k2) − Prox(U)(k1,k2)]2b(k1, k2) || xk1 − xk2 ||2
k2=1
N
∑
> 0
Two general development strategies
SELECTION OF A “MEANINGFUL” SUBSET OF INFORMATION GRANULES
Two general development strategies
(1) HIERARCHICAL DEVELOPMENT OF INFORMATION GRANULES (INFORMMATION GRANULES OF HIGHER TYPE)
Information granulesType -1
Information granulesType -2
Two general development strategies
(2) HIERARCHICAL DEVELOPMENT OF INFORMATION GRANULES AND THE USE OF VIEWPOINTS
Information granulesType -1
Information granulesType -2
viewpoints
Two general development strategies
(3) HIERARCHICAL DEVELOPMENT OF INFORMATION GRANULES – A MODE OF SUCCESSIVE CONSTRUCTION
Fuzzy Computing:Interpretability
Interpretation of fuzzy sets - departure from pure numeric quantification of membership grades
A= [0.11 0.19 0.34 0.45 1.00 0.98 0.821 0.447…]
Granulation of fuzzy sets
Granulation of membership grades
low, high, medium membership of alternative x
Granulation of membership grades and universe of discourse
low membership for a collection of alternatives….
Granulation of membershipgrades
A= [0.11 0.19 0.34 0.45 1.00 0.98 0.821 0.447…]
A= [L L L M M L L M…]
Granulation of membershipgrades: optimization
A= [L L L M M L L M…]
Entropy minimization
G= {G1, G2, …, Gc}
€
x
∑ H(G i
i=1
c
∑ (x))⇒ MinG
Granulation of fuzzy sets
A= [L M L M…]
Granulation of fuzzy sets:optimization
G1
Gi
Gc
1
i c
€
Vol = Vol(G i
i=1
c
∑ ,Wi) ⇒ MinG
Interpretability of fuzzy set computing
Fuzzy set computing
Interpretability layer
Granulation of fuzzy sets
Interpretability of fuzzy set computing
Fuzzy set computing
Interpretability layer
Granulation of fuzzy sets
Interpretability of fuzzy set computing
Equivalence sought with respect with assumed levelinterpretability:•stability•Equivalence of models
distinguishability
Non-distinguishability
Fuzzy set computing: a retrospective
interpretability
accuracy
~1970
after ~1990
neurofuzzy
evolutionary
Rule-based
Conclusions
Leitmotiv of human-centricity and its underlying reliance on information granules
Design of information granules – shift from data to knowledge-enhanced clustering
Revisiting the practice of fuzzy computing and its interpretabilitycapabilities