PROBABILISTIC DISTANCE MEASURES FOR PROTOTYPE-BASED RULES Marcin Blachnik, Tadeusz Wieczorek Department of Electrotechnology Faculty of Materials Engineering & Metallurgy, The Silesian University of Technology, Poland Włodzisław Duch Department of Informatics, Nicolaus Copernicus University, Poland, School of Computer Engineering, Nanyang Technological University Singapore.

Outline Type of rules What are prototype rules? Heterogeneous distance function Probability density function (PDF) estimation Results Conclusions ICONIP 2005 Taiwan

Types of rules Crisp logical rules. Rough sets and logic. Fuzzy rules (F-rules). Prototype rules (P-rules) – most general? P-rules with additive similarity functions may be converted into the neurofuzzy rules with “natural” membership functions, including nominal features. P-rules do not need the feature space. There are many neurofuzzy programs, but no P-rules so far. Krótko o typach reguł – cechy reguł twardych, zbiorów przybliżonych i rozmytych, na koniec nowe reguły prototypowe ICONIP 2005 Taiwan

Motivation Understanding data, situations, recognizing objects or making diagnosis people frequently use similarity to known cases, and rarely use logical reasoning, but soft computing experts use logic instead of similarity ... Relations between similarity and logic are not clear. Q1: How to obtain the same decision borders in Fuzzy Logic systems and Prototype Rule Based systems? Q2: What type of similarity measure corresponds to a typical fuzzy functions and vice versa? Q3: How to transform one type of a system into another type preserving their decision borders? Q4: Are there any advantages of such transformations? Q5: Can we understand data better using prototypes instead of logical rules? ICONIP 2005 Taiwan

Example Ograniczenia klasycznych reguł twardych - dwa Gausy – optymalna Bayesowska granica jest liniowa granica systemów regułowych twardych granica uzyskana dla systemów prototypowych, również możliwa do uzyskania dla systemów rozmytych ICONIP 2005 Taiwan

Prototype rules - advantages Inspired by cognitive psychology: understanding data, situations, recognizing objects or making diagnosis people frequently use similarity to known cases, and rarely use logical reasoning. With Heterogeneous Distance Functions P-rules supports all types of attributes: continues, discrete, symbolic and nominal, while F-rules require numerical inputs. Locally linear decision borders to avoid overfitting. Many algorithms for prototype selection and optimization exist but they have not been applied to understand data. Applications of P-rules to real datasets give excellent results generating small number of prototypes. Zalety reguł prototypowych ICONIP 2005 Taiwan

Prototype rules - learning Learning process involves: select similarity or dissimilarity (distance) functions model optimization: the number and positions of prototypes Decision making task consist of: calculating distance (similarity) to each prototype assigning P-rule to calculate the output class as a rule Nearest Neighbour rule: If P=argminp’(D(X,P’)) Then Class(X)=Class(P) Threshold rule: If D(X,P)≤dp Then Class(X)=Class(P) Taking D(X,P) - Chebychev distance crisp logic rules are obtained Krótko jak działają i przykłady różnych typów reguł możliwych do uzyskania z reguł prototypowych – Chebychev -> klasyczna logika ICONIP 2005 Taiwan

Applications to real data (ICONIP’2004) Gene expression data for 2 types of leukaemia (Golub et al, Science 286 (1999) 531-537 Description: 2 classes, 1100 features, 3 most relevant selected. Used methods: 1 prototype/class LVQ, DVDM similarity measure. Results (number of misclassified vectors): Data Set Golub et al P-rules Train 3 Test 5 Searching for Promoters in DNA strings Description: 2 classes, 57 features, all symbolic features. Used methods: 9 prototypes for promoters, 12 for nonpromoters, generated using C-means + LVQ, with VDM similarity measure. Results: 5 misclassified vectors in leave one out test. ICONIP 2005 Taiwan

Distance (similarity) functions N – number of attributes K – number of classes Input vectors X=[x1, x2, … , xN]T Y=[y1, y2, … , yN]T q – exponent value P(Ci|x) - posterior probab. for symbolic features, estimated as P(Ci|x)=ni /n Continuous attributes Probabilistic Metrics Typy miar odległości prawdopodobieństwa i klasycznej miary odległości Dwa typy funkcji heterogenicznych -> 1) mieszana Minkowski + Miary probabilistyczne 2) tylko miary probabilistyczne ICONIP 2005 Taiwan

Heterogeneous distance function Combine contributions from symbolic and real-valued features to get the distance. or use only probabilistic measures Typy miar odległości prawdopodobieństwa i klasycznej miary odległości Dwa typy funkcji heterogenicznych -> 1) mieszana Minkowski + Miary probabilistyczne 2) tylko miary probabilistyczne ICONIP 2005 Taiwan

Probability density function estimation Problem: how to combine influence of nominal/symbolic? 1. Normalization – continuous  symbolic 2. Estimation – continuous attributes => prob. If estimation, then several options to get probabilities: Discretization (DVDM) Discretization + Interpolation (IVDM) Gaussian kernel estimation (GVDM) Rectangular Parzen window (LVDM) Rectangular moving Parzen window (PVDM) Dla 1) problem z normalizacją odległości probabilistycznych i klasycznych typu Minkowskiego Dla 2) Konieczność Estymacji PDF dla danych ciągłych Porównane typu estymacji PDF ICONIP 2005 Taiwan

3 overlapping Gaussians in 4D, good parameters for estimation. Discretization Discretization & Interpolation Gaussian kernel Rect. Parzen window Moving Parzen windows. Porównanie jakości estymacji PDF dla różnych metod dla 3 nakładających się Gausów, dobrze dobrane parametry 3 overlapping Gaussians in 4D, good parameters for estimation. ICONIP 2005 Taiwan

3 overlapping Gaussians in 4D, bad parameters for estimation. Discretization Discretization & Interpolation Gaussian kernel Moving Parzen wind. Rect. Parzen window Porównanie dla tych samych danych przy złym doborze parametrów estymacji 3 overlapping Gaussians in 4D, bad parameters for estimation. ICONIP 2005 Taiwan

Testing and comparison procedure Two artificial datasets for testing, 2D 200 vectors/class uniform distribution 200 vectors/class normal distribution 6 real datasets with mixes symbolic/real features. Flags (UCI repository) Glass (UCI repository) Promoters (UCI repository) Wisconsin Brest Cancer, WBC (UCI repository) Pima Indians diabetes (UCI repository) Lancet (from A.J. Walker, S.S. Cross, R.F. Harrison, Visualization of biomedical datasets by use of growing cell structure networks: a novel diagnostic classification technique. Lancet Vol. 354, pp. 1518-1522, 1999.) For all tasks 10 fold CV test procedure is used. Opis danych użytych do porównania oraz omówienie zbiorów sztucznych – jeden to 3 gausy a drugi to trzy rozkłady jednaostajne ICONIP 2005 Taiwan

Classification results Results on artificial datasets. Left: Gaussian distributed. Right: uniform distributed. Similar results, except for convergence problems. Datasets with all symbolic or discrete values. leave-one-out results. Wyniki dla zbiorów sztucznych z zaznaczonymi najlepszymi wartościami oraz przekreślone wartości ze źle dobranymi parametrami estymacji -> np.. okno mniejsze od kroku Wyniki dla zbiorów z danymi symbolicznymi i dyskretnymi ICONIP 2005 Taiwan

Real datasets Wyniki dla pozostałych realnych zbiorów z zaznaczonymi najlepszymi wartościmi Jak widać nie ma najlepszej metody ICONIP 2005 Taiwan

Results & discussion Selection of appropriate parameters is very important. Incorrect values if one uses: too small sigma (Gaussian Estimation); too narrow window (Rectangular Parzen Window estimations) too many bins in discretization. Increased sensitivity of estimation methods => overfitting if too high sigma (Gaussian Estimation); too wide window (Rectangular Parzen Window estimations) Low number of bins in discretization. Decreased sensitivity of estimation methods leading to over-generalization. Middle values of parameters are best start points leading to good results (0.5, Parzen width0.5, Parzen step  0.01) Wnioski ICONIP 2005 Taiwan

Some conclusions First step in understanding relations between fuzzy and similarity-based systems. Prototype rules can be expressed using fuzzy rules and vice versa leading to new possibilities in both fields: new type of membership functions & new type of distance functions. Expert knowledge can be captured in any kind of rules, but sometimes it may be more natural to express knowledge as P-rules (similarity) or as F-rules (logical conditions). VDM measure used in P-rules leads to a natural shape of membership functions in fuzzy logic for symbolic data. There is no best choice of heterogeneous distance function type or PDF estimation method or probability metrics. Simplest methods may lead to good results. Selection of appropriate parameters is very important. P-systems should be as popular as neurofuzzy systems, although many open problems still remain, both theoretical and practical. Wnioski ICONIP 2005 Taiwan

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