Uncertainty Reasoning through Similarity in Context Claudia dAmato - PowerPoint PPT Presentation

Uncertainty Reasoning through Similarity in Context Claudia d’Amato Nicola Fanizzi Dipartimento di Informatica Universit` a degli studi di Bari, Italy 2nd ARCOE Workshop @ ECAI2010, Lisbon , PT

Introduction Motivations Motivation Reasoning with Web ontologies expressed in standard representations based on Description Logics difficult due to inherent incompleteness: OWA vs. CWA incoherence (+ noise ): heterogeneous and distributed sources Various solutions investigated in the URSW community, e.g modeling vague knowledge in terms of probability and fuzziness : support to evolution ? N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 2 / 26

Introduction Idea Idea try inductive methods : often efficient, noise-tolerant and incremental In particular, methods based on similarity (or a notion of distance ) proposed for many reasoning tasks, cast as inductive problems In the literature: most of the measures for concept-similarity inductive techniques borrowed from Machine Learning require a notion of similarity between individuals N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 3 / 26

Introduction Idea Outline Survey of applications of similarity in context Preliminaries 1 Contextual Metrics for Individuals 2 Similarity in Context Family of Metrics Inductive Instance Classification 3 Problem k -Nearest Neighbor Procedure Rough DLs 4 Rough Concept Approximations Induced Indiscernibility Relation Extensions 5 Conclusions and Outlook N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 4 / 26

Preliminaries Syntax & Semantics Preliminaries I Axioms in terms of a vocabulary of N C set of primitive concept names N R set of primitive role names N I set of individual names and language constructors Semantics defined by interpretations I = (∆ I , · I ) where ∆ I domain of the interpretation (non-empty) · I interpretation function that maps names to extensions each A ∈ N C to a set A I ⊆ ∆ I and each R ∈ N R to R I ⊆ ∆ I × ∆ I Then new concepts/roles defined using the language constructors N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 5 / 26

Preliminaries DL Knowledge Bases DL Knowledge Bases knowledge base K = �T , A� TBox T set of axioms C ⊑ D (resp. C ≡ D ) meaning C I ⊆ D I (resp. C I = D I ) where C is atomic and D is a concept description ABox A set of assertions — ground axioms e.g. C ( a ) and R ( a, b ) stating: a belongs to C and ( a, b ) belongs to R Ind ( A ) = set of individuals occurring in A Interpretations of interest (models) satisfy all the axioms in K N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 6 / 26

Contextual Metrics for Individuals Preliminaries 1 Contextual Metrics for Individuals 2 Similarity in Context Family of Metrics Inductive Instance Classification 3 Problem k -Nearest Neighbor Procedure Rough DLs 4 Rough Concept Approximations Induced Indiscernibility Relation Extensions 5 Conclusions and Outlook N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 7 / 26

Contextual Metrics for Individuals Similarity in Context Context & Similarity I A context of reference must express the essential features for comparing domain objects. similarity is not merely a relation between objects but rather between the two in a given context (which is subject to changes) [Goldstone et al.,1997] the task also matters ! N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 8 / 26

Contextual Metrics for Individuals Similarity in Context Context & Similarity II In the following. . . Context Given a knowledge base K , a context C is a finite set of concept descriptions ( features ) C = { F 1 , F 2 , . . . , F m } built on concepts and roles defined in K N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 9 / 26

Contextual Metrics for Individuals Similarity in Context Learning the Context given a fitness / criterion function J for the task methods for finding contexts based on distinguishability proposed; stochastic search using Genetic Programming Simulated Annealing Alternatively, since the metrics are based on weighted projections: consider as many features as possible (e.g. all defined concepts) find good choice for the weights � w based on information ( entropy ) based on variance N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 10 / 26

Contextual Metrics for Individuals Family of Metrics A Family of Metrics Given a context C and a weight vector � w , the family { d C p } p ∈ N of functions d C p : Ind ( A ) × Ind ( A ) �→ [0 , 1] is defined � m � 1 /p � d C w i | π i ( a ) − π i ( b ) | p p ( a, b ) = i =1 where ∀ i ∈ { 1 , . . . , m } the i -th projection function π i :  1 K | = F i ( a )  π i ( a ) = 0 K | = ¬ F i ( a )  u i (prior) otherwise Inspired by Minkowski ’s norms; can be proven to be semi-distances N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 11 / 26

Inductive Instance Classification Preliminaries 1 Contextual Metrics for Individuals 2 Similarity in Context Family of Metrics Inductive Instance Classification 3 Problem k -Nearest Neighbor Procedure Rough DLs 4 Rough Concept Approximations Induced Indiscernibility Relation Extensions 5 Conclusions and Outlook N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 12 / 26

Inductive Instance Classification Problem Inductive Classification Instance checking as a Learning Problem given a query concept Q and a query individual x q using S Q sample of prototype training instances with correct membership values h Q ( x i ) = v ∈ {− 1 , 0 , +1 } = V determine ˆ h Q ( x q ) i.e. estimate membership of x q w.r.t. Q We use well known non-parametric methods: k -NN, Parzen Windows no ind. model, only rel. distances RBF Nets, SVMs . . . build an inductive model N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 13 / 26

Inductive Instance Classification k -Nearest Neighbor Procedure k -Nearest Neighbor Procedure I A sort of analogical reasoning [d’Amato et al.,2008-URSW I] x 10 x 7 x 4 x 6 x q x 12 x 1 x 2 x 9 x 5 x 3 x 8 x 11 Selection of the k = 5 nearest neighbors. green=positive ex., red=negative ex. N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 14 / 26

Inductive Instance Classification k -Nearest Neighbor Procedure k -Nearest Neighbor Procedure II Weighted majority vote: given NN k ( x q ) = { x 1 , . . . , x k } of x q ’s nearest neighbors w.r.t. d C p , the estimate of the membership hypothesis is proximity weight vote k � �� � � �� � � ˆ γ ( d C h Q ( x q ) = argmax p ( x i , x q )) · δ ( v, h ( x i )) v ∈ V i =1 where: δ Kronecker indicator function γ decaying function e.g. γ ( x ) = (1 − x ) b or γ ( x ) = 1 /x b for some b > 0 N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 15 / 26

Inductive Instance Classification k -Nearest Neighbor Procedure Lessons Learned Applying this and similar methods based on density estimates (RBF Networks, SVMs, . . . ) build the inductive model once and classify efficiently many times may give an answer in case of uncertain class-membership (can be forced to do that) may provide an estimate of the likelihood of the answer experimentally: nearly sound and complete (few omission errors) measure used also in unsupervised tasks: e.g. clustering individual resources N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 16 / 26

Rough DLs Preliminaries 1 Contextual Metrics for Individuals 2 Similarity in Context Family of Metrics Inductive Instance Classification 3 Problem k -Nearest Neighbor Procedure Rough DLs 4 Rough Concept Approximations Induced Indiscernibility Relation Extensions 5 Conclusions and Outlook N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 17 / 26

Rough DLs Rough Concept Approximations Rough DL Recently Rough DLs introduced [Schlobach et al.,IJCAI2007] as a mechanism for modeling vague concepts by means of a crisp specification of its approximations Approximations Given an indiscernibility relation R , the upper approximation of a concept C is C = { a | ∃ b : R ( a, b ) ∧ b ∈ C } (typical instances) the lower approximation is C = { a | ∀ b : R ( a, b ) → b ∈ C } (prototypical instances) If R expressed in terms of the knowledge base then standard reasoners can be used N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 18 / 26

Rough DLs Rough Concept Approximations N. Fanizzi (University of Bari, IT) U. Reasoning through Similarity in Context Aug. 16th 2010 19 / 26