Supervised Learning: Linear Methods (1/2) Applied Multivariate Statistics Spring 2012
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1 Supervised Learning: Linear Methods (1/2) Applied Multivariate Statistics Spring 2012
2 Overview Review: Conditional Probability LDA / QDA: Theory Fisher s Discriminant Analysis LDA: Example Quality control: Testset and Crossvalidation Case study: Text recognition 1
3 Conditional Probability Sample space T: Med. Test positive T (Marginal) Probability: C: Patient has cancer C P(T), P(C) New sample space: People with cancer P(T C) large Conditional Probability: P(T C), P(C T) Bayes Theorem: New sample space: People with pos. test P(C T) small posterior P(CjT) = P (TjC)P (C) P (T ) Class conditional probability prior 2
4 One approach to supervised learning P(CjX) = P (C)P (XjC) P (X)» P(C)P(XjC) Find some estimate Prior / prevalence: Fraction of samples in that class Assume: XjC» N(¹ c ; c ) Bayes rule: Choose class where P(C X) is maximal (rule is optimal if all types of error are equally costly) Special case: Two classes (0/1) - choose c=1 if P(C=1 X) > 0.5 or - choose c=1 if posterior odds P(C=1 X)/P(C=0 X) > 1 In Practice: Estimate P C, μ C, Σ C 3
5 QDA: Doing the math P C X ~ P C P(X C) Use the fact: max P C X max(log P C X ) δ c x = log P C X = log P C + log P X C = p 1 exp 1 (2¼) d j C j 2 (x ¹ c) T 1 C (x ¹ c) = log P C 1 2 log Σ C 1 2 x μ C T Σ C 1 x μ C + c Prior Additional term Sq. Mahalanobis distance Choose class where δ c x is maximal Special case: Two classes Decision boundary: Values of x where δ 0 x = δ 1 (x) is quadratic in x Quadratic Discriminant Analysis (QDA) 4
6 Simplification Assume same covariance matrix in all classes, i.e. X C ~ N(μ c, Σ) δ c x = log P C 1 2 log Σ 1 2 x μ C T Σ 1 x μ C + c = Prior = log P C 1 x μ 2 C T Σ 1 x μ C + d = (= log P C + x T Σ 1 μ C 1 μ 2 C T Σ 1 μ C ) Fix for all classes Sq. Mahalanobis distance Decision boundary is linear in x Linear Discriminant Analysis (LDA) 1 Classify to which class (assume equal prior)? 0 Physical distance in space is equal Classify to class 0, since Mahal. Dist. is smaller 5
7 LDA vs. QDA + Only few parameters to estimate; accurate estimates - Inflexible (linear decision boundary) - Many parameters to estimate; less accurate + More flexible (quadratic decision boundary) 6
8 Fisher s Discriminant Analysis: Idea Find direction(s) in which groups are separated best 1. Principal Component 1. Linear Discriminant = 1. Canonical Variable Class Y, predictors X = X 1,, X d U = w T X Find w so that groups are separated along U best Measure of separation: Rayleigh coefficient D(U) J w = Var(U) where D U = E U Y = 0 E U Y = 1 2 E X Y = j = μ j, Var X Y = j = Σ E U Y = j = w T μ j, V U = w T Σw Concept extendable to many groups D(U) J w large D(U) J w small Var(U) Var(U) 7
9 LDA and Linear Discriminants - Direction with largest J(w): 1. Linear Discriminant (LD 1) - orthogonal to LD1, again largest J(w): LD 2 - etc. At most: min(nmb. dimensions, Nmb. Groups -1) LD s e.g.: 3 groups in 10 dimensions need 2 LD s Computed using Eigenvalue Decomposition or Singular Value Decomposition Proportion of trace: Captured % of variance between group means for each LD R: Function «lda» in package MASS does LDA and computes linear discriminants (also «qda» available) 8
10 Example: Classification of Iris flowers Iris setosa Iris versicolor Iris virginica Classify according to sepal/petal length/width 9
11 Quality of classification Use training data also as test data: Overfitting Too optimistic for error on new data Separate test data Training Test Cross validation (CV; e.g. leave-one-out cross validation): Every row is the test case once, the rest in the training data 10
12 Measures for prediction error Confusion matrix (e.g. 100 samples) Truth = 0 Truth = 1 Truth = 2 Estimate = Estimate = Estimate = Error rate: 1 sum(diagonal entries) / (number of samples) = = 1 76/100 = 0.24 We expect that our classifier predicts 24% of new observations incorrectly (this is just a rough estimate) 11
13 Example: Digit recognition 7129 hand-written digits Each (centered) digit was put in a 16*16 grid Measure grey value in each part of the grid, i.e. 256 grey values Sample of digits Example with 8*8 grid 12
14 Concepts to know Idea of LDA / QDA Meaning of Linear Discriminants Cross Validation Confusion matrix, error rate 13
15 R functions to know lda 14
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