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-based clustering Granular representation of computing with fuzzy sets
Human Centricity and information granules Information granules as conceptual entities inherently associated with human pursuits (decision-making, perception control, 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 (HC 2 ) 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 interpretation Cast in the framework of users/designer Qualitative, user-centric interpretation of results of computing with fuzzy sets
Clustering as a conceptual and algorithmic framework of information granulation Data 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-centricity Guidance mechanisms
Fuzzy Clustering: Fuzzy C-Means (FCM) Given data x 1, x 2, …, x N, determine its structure by forming a collection of information granules – fuzzy sets Objective function Minimize Q; structure in data (partition matrix and prototypes)
Fuzzy Clustering: Fuzzy C-Means (FCM) V i – prototypes U- partition matrix
FCM – optimization Minimize subject to (a) prototypes (b) partition matrix
Domain Knowledge: Category of knowledge-oriented guidance Context-based guidance: clustering realized in a certain context specified 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 in terms of their proximity (resemblance, closeness)
Clustering with domain knowledge (Knowledge-based 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 (compound)
Context-oriented FCM Data (x k, target k ), k=1,2,…,N Contexts: fuzzy sets W 1, W 2, …, W p w jk = W i (target k ) membership of j-th context for k-th data Context-driven partition matrix
Context-oriented FCM: Optimization flow Objective function Iterative adjustment of partition matrix and prototypes Subject to constraint U in U(W j )
Viewpoints: definition Description of entity (concept) which is deemed essential in describing phenomenon (system) and helpful in casting an overall analysis in a required setting “external”, “reinforced” clusters
Viewpoints: definition viewpoint (a,b)viewpoint (a,?)
Viewpoints: definition Description of entity (concept) which is deemed essential in describing phenomenon (system) and helpful in casting an overall analysis in a required setting “external”, “reinforced” clusters
Viewpoints in fuzzy clustering B- Boolean matrix characterizing structure: viewpoints prototypes (induced by data)
Viewpoints in localization of “extreme” information granules specification of viewpoints through evolutionary/population-based optimization
Viewpoints in fuzzy clustering
Labelled data and their description Characterization in terms of membership degrees: F = [f ik ] i=12,…,c, k=1,2, …., N supervision indicator b = [b k ], k=1,2,…, N
Augmented objective function > 0
Proximity hints Characterization in terms of proximity degrees: Prox(k, l), k, l=1,2, …., N and supervision indicator matrix B = [b kl ], 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: Linkages with kernel functions K(x k, x l )
Augmented objective function > 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 granules Type -1 Information granules Type -2
Two general development strategies (2) HIERARCHICAL DEVELOPMENT OF INFORMATION GRANULES AND THE USE OF VIEWPOINTS Information granules Type -1 Information granules Type -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= [ …]
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 membership grades A= [ …] A= [L L L M M L L M…]
Granulation of membership grades: optimization A= [L L L M M L L M…] Entropy minimization G= {G 1, G 2, …, G c }
Granulation of fuzzy sets A= [L M L M…]
Granulation of fuzzy sets: optimization G1G1 GiGi GcGc 11 ii cc
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 level interpretability: 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 interpretability capabilities