ECE471-571 Pattern Recognition Lecture 3 Discriminant Function and Normal Density Hairong Qi, Gonzalez Family Professor Electrical Engineering and Computer Science University of Tennessee, Knoxville http://www.eecs.utk.edu/faculty/qi Email: [email protected] Pattern Classification Statistical Approach Supervised Basic concepts: Baysian decision rule (MPP, LR, Discri.) Non-Statistical Approach Unsupervised Basic concepts: Distance Agglomerative method Parameter estimate (ML, BL) k-means Non-Parametric learning (kNN) Winner-takes-all LDF (Perceptron)
Kohonen maps NN (BP) Mean-shift Decision-tree Syntactic approach Support Vector Machine Deep Learning (DL) Dimensionality Reduction FLD, PCA Performance Evaluation ROC curve (TP, TN, FN, FP) cross validation Stochastic Methods local opt (GD) global opt (SA, GA) Classifier Fusion majority voting NB, BKS Bayes Decision Rule P (w j | x)=
Maximum Posterior Probability Likelihood Ratio p(x | w j )P (w j ) p(x) For a given x, if P (w1 | x)> P (w 2 | x), then x belongs to class 1, otherwise, 2. If , then x belongs to class 1, otherwise, 2. 3 Discrimimant Function One way to represent pattern classifier- use discriminant functions gi(x) The classifier will assign a feature vector x to class w i if gi (x)> g j (x) For two-class cases, g( x) =g1 (x) - g2 (x) =P ( w1 | x) - P ( w 2 | x) 4 Multivariate Normal Density
p( x) = 1 1 T -1 exp - ( x - m) S ( x - m) d/2 1/ 2 2 ( 2p ) S x : d - component column vector m : d - component mean vector S : d - by - d covariance matrix S : determinant S -1 : inverse x1 m1 x = , m = xd m d 2 s 11 s 1d s 1 s 1d
S = = s d1 s dd s d1 s d 2 2 1 1 (x - m ) When d =1, p(x)= exp 2 2p s 2 s 5 Discriminant Function for Normal Density p(x | w)= 1 (2p )d / 2 S 1/ 2
1 T exp - (x - m ) S - 1 (x - m ) 2 gi (x)=ln p(x | wi )+ ln P (wi ) 1 =- (x 2 1 =- (x 2 T -1 d 1 m i ) S i (x - m i )- ln (2p )- ln S i + ln P (wi ) 2 2 T -1 1 m i ) S i (x - m i )- ln S i + ln P (wi ) 2 6 Case 1: Si=s2I The features are statistically independent, and have the same variance Geometrically, the samples fall in equal-size hyperspherical clusters
Decision boundary: hyperplane of d-1 dimension 1 s 2 0 0 s 2 S = , S =s 2d ,S- 1 = 1 0 s 2 0 2 s 7 Linear Discriminant Function and Linear Machine x x- gi (x)=-
x - mi 2 m i : the Euclidean norm (distance) 2 T m i =(x - m i ) (x - m i ) 2 + ln P (wi ) 2s T T T x x - 2m i x + m i m i =+ ln P (wi ) 2 2s m i T m iT m i gi (x)= 2 x + ln P (wi ) 2 s 2s 8 Minimum-Distance Classifier
When P(wi) are the same for all c classes, the discriminant function is actually measuring the minimum distance from each x to each of the c mean vectors gi (x)=- x - mi 2 2s 2 9 Case 2: Si = S The covariance matrices for all the classes are identical but not a scalar of identity matrix. Geometrically, the samples fall in hyperellipsoidal Decision boundary: hyperplane of d-1 dimension gi ( x) =ln p( x | w i ) + ln P ( w i ) 1 T -1 =- ( x - mi ) S i ( x - mi ) + ln P ( w i ) 2
Squared T - 1 T 1 T - 1 Mahalanobis =mi ( S ) x - mi S mi + ln P ( w i ) distance 2 10 Case 3: Si = arbitrary The covariance matrices are different from each category Quadratic classifier Decision boundary: hyperquadratic for 2-D Gaussian gi (x)=ln p(x | wi )+ ln P (wi ) 1 T -1 1 =- (x - m i ) S i (x - m i )- ln S i + ln P (w i ) 2 2 1 T - 1 T - 1 T 1 T - 1 1 =- x S i x + m i S i x - m i S i m i - ln S i + ln P (wi ) 2 2 2 ( ) 11 Case Study
a1 3.1 11.70 a b1 8.3 1.00 b c1 10.2 6.40 c u1 5.1 0.4 b a2 3.0 1.30 a b2 3.8 0.20 b c2 9.2 7.90 c u2 12.9 5.0 c a3 1.9 0.10 a b3 3.9 0.60 b c3 9.6 3.10 c u3 13.0 0.8 b a4 3.8 0.04 a b4 7.8 1.20 b c4 53.8 2.50 c u4 2.6 0.1 a
a5 4.1 1.10 a b5 9.1 0.60 b c5 15.8 7.60 c u5 30.0 0.1 o a6 1.9 0.40 a b6 15.4 3.60 b u6 20.5 0.8 o b7 7.7 1.60 b b8 6.5 0.40 b b9 5.7 0.40 b b10 13.6 1.60 b Calculate m Calculate S Derive the discriminant function gi(x) 12
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