Principles of Pattern Recognition. C. A. Murthy Machine Intelligence Unit Indian Statistical Institute Kolkata

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1 Principles of Pattern Recognition C. A. Murthy Machine Intelligence Unit Indian Statistical Institute Kolkata

2 Pattern Recognition Measurement Space > Feature Space >Decision space Main Tasks : Feature Selection and Supervised / Unsupervised Classification

3 Supervised Classification Classification Two cases 1. Conditional Probability density functions and prior probabilities are known 2. Training sample points are given

4 Bayes decision rule M classes class conditional density functions p1( x), p2( x),..., pm ( x) N x R prior probabilities 0 P < 1 < i M P i = 1 1 P 1, P2,... i =1,2,... M P M Put x in class i if Pi pi ( x) Pj p j ( x); j i Best decision rule Minimizes the prob. of misclassification

5 We may not know p i ' s. (Estimation of density functions), Normal distribution We may not know P i ' s Error prob. may be difficult to obtain Other decision rules are needed

6 Normal distribution case 1 1 ' 1 p i ( x) = 1 exp ( x µ i ) i ( x µ i ) ; 2 N ( 2π ) 2 i i =1,2,...M If M = 2 and 1 = 2 = then the decision boundary is linear. In general the decision boundary is nonlinear

7 K-Nearest Neighbour decision rule Suppose we are given n points x 1,x 2, x n. Let there be M classes. Let n i of these sample points belong to class i; i=1,2, M. M i.e., n i = n. i= 1 Let x be the point to be classified. Let k be a positive integer. Find k nearest neighbors of x among x 1,..,x M n Let k i of these nearest k neighbors i = k. belong to i= 1 i th class; i=1,2, M;

8 Minimum distance classifier Let µ 1,µ 2, µ M be the means of the M classes. Let d(a,b) denote the distance between a & b. (Examples : Euclidean, Minkowski) Put x in class i if d(x, µ i ) < d(x, µ j ) j i

9 Some remarks Standardization & normalization Choosing the appropriate distance function Probability of misclassification Cost of misclassification

10 Clustering Problem: Finding natural groups in data set Example 1: Example 2:

11 Clustering (contd..) Let us assume that the given data set S = x, x,..., } { 1 2 x n R M No. of clusters K may not be known Choice of similarity/dissimilarity measure Algorithms

12 Dissimilarity Measures Metrics ' a = ( a1, a2,..., a M ) ' b = ( b1, b2,..., b M ) d p ( a, b) M = i= 1 a i b i p 1 p ; p 1. p = 2 Euclidean distance

13 Similarity Measures s( a, b) = M a i i= 1 2 ai b i b 2 i Other such measures are also available

14 K-Means Algorithm Several versions of K-Means algorithm are available. One version is given below Number of clusters = K S = x, x,..., } R { 1 2 x n d Euclidean distance 1. A11, A12,... A1 k Partitions of S into K subsets 2. A21 = A22 =... = A2k = Φ 3. y i = mean of A 1i i = 1,2,... k. j = 1,2,...,n 4. For d( x put x j in A 2i if j, yi ) < ( x j, yi ), i i If A 1i =A 2i for all i=1,2, k then stop o.w. i = 1,2,...,k Rename A 2i as A 1i and goto step 2. M

15 K-Means Algorithm (contd..) Number of iterations is usually decided by the user provides basically convex clusters Non convex clusters may not be obtained Two different initial partitions may give rise to two different clusterings

16 Hierarchical Clustering Techniques Agglomerative Divisive

17 Agglomerative Techniques S = x, x,..., } { 1 2 x n d dissimilarity measure R M 1. N clusters c 1 = { x1}, c2 = { x2},... c n = { xnlevel } 1 2. Clusters at the level i c1, c2,..., c n i + 1 Merge two clusters c i, cif j D( ci, c j ) < D( ci, c ), 1 j i1, j 1 1 (one cluster is reduced) Rename the clusters as 3. Repeat step 2 till the required c1, no. c2,..., of clusters n i is obtained

18 Agglomerative Techniques (contd..) D( A, B) = Min( d( x, y)) single linkage. x y B A D( A, B) = Max( d( x, y)) complete linkage. x y B A several other such D s can be considered. single linkage provides non-convex clustering generally.

19 Feature selection Feature X 1, X 2,,X N b no. of features to be selected. b < N Uses : Reduction in computational complexity Redundant features act as noise. Noise removal Insight into the classification problem.

20 Steps of feature selection Objective function J which attaches a value to every subset of features is to be defined. Algorithms for feature selection are to be formulated.

21 Objective functions for feature selection (Devijver & Kittler) Probabilistic separability (Chernoff, Bhattacharyya, Matusita, Divergence) Inter class distance

22 Feature Selection Criteria: Supervised Criterion: (notations) ω i i = 1,, M : classes n i, i = 1,, M : number of points in class i P i : a priori probability of class i x ik : k th point of i th class 1. Interclass Distance Measures: J = 1 2 c i= 1 P i c j= 1 P 1 j ninj n i k = 1 n j l= 1 δ ( x δ : Euclidean, Minkowski, Manhattan Reference: Devijver & Kittler, Pattern Recognition: A Statistical Approach, Englewood Cliffs, 1982 ik, x jl )

23 2. Probabilistic Separability Measures: Bhattacharyya Distance: 1 2 [ p( x ω x ] 1)p( ω dx JB = - ln 2) 3. Information Theoretic Measures: Mutual Information: J I = M i= 1 P i p( x ω )ln i p( x ωi ) p( x) dx Difficulty: Computing the probabilities. Empirical estimates are used.

24 Unsupervised Criterion: Entropy (E): Similarity between points x i and x j : S ij = e αδ(xi,xj) i,j = 1,, l l l E = Sijlog(Sij ) + (1 Sij )log(1- Sij ) i= 1 j= 1 Other unsupervised indices: Fuzzy Feature Evaluation Index Neuro-fuzzy Feature Evaluation Index

25 Search Algorithms: If total number of features = D Number of features to be selected = d Computational complexities: Exhaustive search [( D C d )] D =100 and d =10 the no. of computations is greater than Branch and Bound (gives optimal set for a class of evaluation criteria) Worst case: ( D C d )

26 Algorithms for feature selection Sequential forward selection Sequential backward selection (l,r) algorithm Branch and bound algorithm

27 Sequential forward selection A o = φ. A k denotes the k features already selected Let a 1 {X 1, X N } A k be such that J(A k U{a 1 }) J (A k U{a}) a {X 1, X N } A k then A k+1 =A k U{a 1 }. Run the program b times

28 Sequential Forward/Backward Search (greedy algorithms, very approximate, gives poor result on most real life data) Sequential Floating F/B Search (l-r algorithm) (relatively better than SFS/SBS) Non-Monotonicity Property: Two best features are not always the best two

29 KDD Process Raw Data Data Preparation Data Condensation Dimensionality Reduction Reduced Clean Data Machine Learning Pattern Recognition Knowledge Extraction/ Evaluation Knowledge Noise Removal Data Mining

30 Feature Selection Selection Criteria Supervised Unsupervised Search Algorithms (for the best subset according to the criterion) Exhaustive search not feasible Heuristic Search

31 Difficulties: For data mining applications involving datasets large both in dimension (D) and number of samples (l): Search is time consuming/does not obtain the optimal feature subset. Computation of feature evaluation criterion is time consuming. (most often of polynomial in l) The criteria are specific to classification/ discrimination (supervised) and convex clustering (unsupervised) not suited for general data mining tasks.

32 Thank You!!

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