Pattern Classification

Transcription

1 Pattern Classification Introduction Parametric classifiers Semi-parametric classifiers Dimensionality reduction Significance testing 6345 Automatic Speech Recognition Semi-Parametric Classifiers 1

2 Semi-Parametric Classifiers Mixture densities ML parameter estimation Mixture implementations Expectation maximization (EM) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 2

3 Mixture Densities PDF is composed of a mixture of m component densities {ω 1,,ω m }: m p(x)= p(x ω j )P(ω j ) j=1 Component PDF parameters and mixture weights P(ω j ) are typically unknown, making parameter estimation a form of unsupervised learning Gaussian mixtures assume Normal components: p(x ω k ) N(µ k, Σ k ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 3

4 Gaussian Mixture Example: One Dimension x σ p(x) =06p 1 (x)+04p 2 (x) p 1 (x) N( σ,σ 2 ) p 2 (x) N(15σ,σ 2 ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 4

5 Gaussian Example First 9 MFCC s from[s]: Gaussian PDF s ( dimension 1 ) s ( dimension 2 ) s ( dimension 3 ) s ( dimension 4 ) s ( dimension 5 ) s ( dimension 6 ) s ( dimension 7 ) s ( dimension 8 ) s ( dimension 9 ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers

6 Independent Mixtures [s]: 2 Gaussian Mixture Components/Dimension s ( dimension 1 ) s ( dimension 2 ) s ( dimension 3 ) s ( dimension 4 ) s ( dimension 5 ) s ( dimension 6 ) s ( dimension 7 ) s ( dimension 8 ) s ( dimension 9 ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers

7 Mixture Components [s]: 2 Gaussian Mixture Components/Dimension s ( dimension 1 ) s ( dimension 2 ) s ( dimension 3 ) s ( dimension 4 ) s ( dimension 5 ) s ( dimension 6 ) s ( dimension 7 ) s ( dimension 8 ) s ( dimension 9 ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers

8 ML Parameter Estimation: 1D Gaussian Mixture Means log L(µ k ) = n log p(x i ) = n log m i=1 i=1 j=1 p(x i ω j )P(ω j ) log L(µ k ) µ k = i log p(x i ) = µ k i 1 p(x i ) µ k p(x i ω k )P(ω k ) p(x i ω k ) = (x i µ k ) 2 1 e 2σ 2 k = p(x i ω k ) (x i µ k ) µ k µ k 2πσk σ 2 k log L(µ k ) µ k = i since p(x i ω k )P(ω k ) p(x i ) P(ω k ) p(x i ) p(x i ω k ) (x i µ k ) σ 2 k = P(ω k x i ) ˆµ k = =0 P(ω k x i )x i i P(ω k x i ) i 6345 Automatic Speech Recognition Semi-Parametric Classifiers 8

9 Gaussian Mixtures: ML Parameter Estimation The maximum likelihood solutions are of the form: ˆµ k = 1 n 1 n i i ˆP(ω k x i )x i ˆP(ω k x i ) ˆΣ k = 1 n i ˆP(ω k x i )(x i ˆµ k )(x i ˆµ k ) t 1 n ˆP(ω k )= 1 n i ˆP(ω k x i ) i ˆP(ω k x i ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 9

10 Gaussian Mixtures: ML Parameter Estimation The ML solutions are typically solved iteratively: Select a set of initial estimates for ˆP(ω k ), ˆµ k, ˆΣ k Use a set of n samples to reestimate the mixture parameters until some kind of convergence is found Clustering procedures are often used to provide the initial parameter estimates Similar to K-means clustering procedure 6345 Automatic Speech Recognition Semi-Parametric Classifiers 10

11 Example: 4 Samples, 2 Densities 1 Data: X = {x 1,x 2,x 3,x 4 }={2,1, 1, 2} 2 Init: p(x ω 1 ) N(1, 1) p(x ω 2 ) N( 1, 1) P(ω i )=05 3 Estimate: P(ω 1 x) P(ω 2 x) x 1 x 2 x 3 x p(x) (e 05 + e 45 )(e 0 + e 2 )(e 0 + e 2 )(e 05 + e 45 ) Recompute mixture parameters (only shown for ω 1 ): ˆP(ω 1 )= = 05 ˆµ 1 = 98(2)+88(1)+12( 1)+02( 2) = 134 ˆσ 2 1 = 98(2 134)2 +88(1 134) 2 +12( 1 134) 2 +02( 2 134) = Repeat steps 3,4 until convergence 6345 Automatic Speech Recognition Semi-Parametric Classifiers 11

12 [s] Duration: 2 Densities Iter µ 1 µ 2 σ 1 σ Iter P(ω 1 ) P(ω 2 ) logp(x) Duration (sec) Duration (sec) Duration (sec) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 12

13 Gaussian Mixture Example: Two Dimensions 3-Dimensional PDF 4 PDF Contour Automatic Speech Recognition Semi-Parametric Classifiers 13

14 Two Dimensional Mixtures Diagonal Covariance Full Covariance s ( dimension 5 ) s ( dimension 6 ) s ( dimension 5 ) s ( dimension 6 ) Two Mixtures Three Mixtures s ( dimension 5 ) s ( dimension 6 ) s ( dimension 5 ) s ( dimension 6 ) Automatic Speech Recognition Semi-Parametric Classifiers 14

15 Two Dimensional Components Mixture Components s ( dimension 5 ) s ( dimension 5 ) s ( dimension 5 ) s ( dimension 5 ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 15 s ( dimension 6 ) s ( dimension 6 ) s ( dimension 6 ) s ( dimension 6 )

16 Mixture of Gaussians: Implementation Variations Diagonal Gaussians are often used instead of full-covariance Gaussians Can reduce the number of parameters Can potentially model the underlying PDF just as well if enough components are used Mixture parameters are often constrained to be the same in order to reduce the number of parameters which need to be estimated Richter Gaussians share the same mean in order to better model the PDF tails Tied-Mixtures share the same Gaussian parameters across all classes Only the mixture weights ˆP(ω i ) are class specific (Also known as semi-continuous) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 16

17 Richter Gaussian Mixtures [s] Log Duration: 2 Richter Gaussians log Duration (sec) log Duration (sec) Richter Density Richter Components 6345 Automatic Speech Recognition Semi-Parametric Classifiers 17

18 Expectation-Maximization (EM) Used for determining parameters, θ, forincomplete data, X = {x i } (ie, unsupervised learning problems) Introduces variable, Z = {z j }, to make data complete so θ can be solved using conventional ML techniques log L(θ) =logp(x,z θ) = i,j log p(x i,z j θ) In reality, z j can only be estimated by P(z j x i,θ), so we can only compute the expectation of log L(θ) E = E(log L(θ)) = P(z j x i,θ)logp(x i,z j θ) i j EM solutions are computed iteratively until convergence 1 Compute the expectation of log L(θ) 2 Compute the values θ, which maximize E 6345 Automatic Speech Recognition Semi-Parametric Classifiers 18

19 EM Parameter Estimation: 1D Gaussian Mixture Means Let z i be the component id, {ω j }, which x i belongs to E = E(log L(θ)) = P(z j x i,θ)logp(x i,z j θ) i j Convert to mixture component notation: E = E(log L(µ k )) = P(ω j x i )logp(x i,ω j ) i j Differentiate with respect to µ k : E µ k = i P(ω k x i ) log p(x i,ω k ) = µ k i P(ω k x i )x i i ˆµ k = P(ω k x i ) i P(ω k x i )( x i µ k σ 2 k ) = Automatic Speech Recognition Semi-Parametric Classifiers 19

20 EM Properties Each iteration of EM will increase the likelihood of X log p(x θ ) = log p(x i θ ) = p(x θ) p(x i i θ) i j = P(z j x i,θ)(log p(x i θ ) p(x i,z j θ) p(x i j i,z j θ ) p(x i θ) P(z j x i,θ)log p(x i θ ) p(x i θ) +log p(x i,z j θ ) p(x i,z j θ) ) Using Bayes rule and the Kullback-Liebler distance metric: p(x i,z j θ) p(x i θ) =P(z j x i,θ) j P(z j x i,θ)log P(z j x i,θ) P(z j x i,θ ) 0 Since θ was determined to maximize E(log L(θ)): i j P(z j x i,θ)log p(x i,z j θ ) p(x i,z j θ) 0 Combining these two properties: p(x θ ) p(x θ) 6345 Automatic Speech Recognition Semi-Parametric Classifiers 20

21 Dimensionality Reduction Given a training set, PDF parameter estimation becomes less robust as dimensionality increases Increasing dimensions can make it more difficult to obtain insights into any underlying structure Analytical techniques exist which can transform a sample space to a different set of dimensions If original dimensions are correlated, the same information may require fewer dimensions The transformed space will often have more Normal distribution than the original space If the new dimensions are orthogonal, it could be easier to model the transformed space 6345 Automatic Speech Recognition Dimensionality Reduction 1

22 Principal Components Analysis Linearly transforms d-dimensional vector, x, tod dimensional vector, y, via orthonormal vectors, W y = W t x W = {w 1,,w d } W t W =I If d <d,xcan be only partially reconstructed from y ˆx = Wy Principal components, W, minimize the distortion, D, between x, and ˆx, on training data, X = {x 1,,x n } D= n x i ˆx i 2 i=1 Also known as Karhunen-Loève (K-L) expansion (w i s are sinusoids for some stochastic processes) 6345 Automatic Speech Recognition Dimensionality Reduction 2

23 PCA Computation x 2 y 2 y 1 x 1 W corresponds to the first d eigenvectors, P, ofσ P={e 1,,e d } Σ=PΛP t w i =e i Full covariance structure of original space, Σ, is transformed to a diagonal covariance structure, Λ Eigenvalues, {λ 1,,λ d }, represent the variances in Λ Axes in d -space contain maximum amount of variance d D = i=d Automatic Speech Recognition Dimensionality Reduction 3 λ i

24 PCA Example Original feature vector mean rate response (d =40) Data obtained from 100 speakers from TIMIT corpus First 10 components explains 98% of total variance 100% Percent of Total Variance 90% 80% 70% 60% 50% Cosine PCA 40% Number of Components 6345 Automatic Speech Recognition Dimensionality Reduction 4

25 PCA Example 0 5 Value Value Value Value Value Frequency (Bark) Frequency (Bark) 6345 Automatic Speech Recognition Dimensionality Reduction 5

26 Left 8 Left 4 Left 2 Left 1 Right 1 Right 2 Right 4 Right 8 C13 C12 C11 C10 C9 C8 C7 C6 C5 C4 C3 C2 C1 C0 Left 8 Left 4 Left 2 Left 1 Right 1 Right 2 Right 4 Right C13 C12 C11 C10 C9 C8 C7 C6 C5 C4 C3 C2 C1 C PCA for Boundary Classification Eight non-uniform averages from 14 MFCCs First 50 dimensions used for classification Second Component Seventh Component 6345 Automatic Speech Recognition Dimensionality Reduction 6

27 PCA Issues PCA can be performed using Covariances Σ Correlation coefficients matrix P ρ ij = σ ij σii σ jj ρ ij 1 Pis usually preferred when the input dimensions have significantly different ranges PCA can be used to normalize or whiten original d-dimensional space to simplify subsequent processing Σ= P= Λ= I Whitening operation can be done in one step: z = V t x 6345 Automatic Speech Recognition Dimensionality Reduction 7

28 Significance Testing To properly compare results from different classifier algorithms, A 1,andA 2, it is necessary to perform significance tests Large differences can be insignificant for small test sets Small differences can be significant for large test sets General significance tests evaluate the hypothesis that the probability of being correct, p i, of both algorithms is the same The most powerful comparisons can be made using common train and test corpora, and common evaluation criterion Results reflect differences in algorithms rather than accidental differences in test sets Significance tests can be more precise when identical data are used since they can focus on tokens misclassified by only one algorithm, rather than on all tokens 6345 Automatic Speech Recognition Significance Testing 1

29 McNemar s Significance Test When algorithms A 1 and A 2 are tested on identical data we can collapse the results into a 2x2 matrix of counts A 1 /A 2 Correct Incorrect Correct n 00 n 01 Incorrect n 10 n 11 To compare algorithms, we test the null hypothesis H 0 that n 01 p 1 = p 2,orn 01 = n 10,orq= = 1 n 01 + n 10 2 Given H 0, the probability of observing k tokens asymmetrically classified out of n = n 01 + n 10 has a Binomial PMF ( ) (1 ) n n P(k) = k 2 McNemar s Test measures the probability, P, of all cases that meet or exceed the observed asymmetric distribution, and tests P<α 6345 Automatic Speech Recognition Significance Testing 2

30 McNemar s Significance Test (cont t) The probability, P, is computed by summing up the PMF tails l n P = P(k)+ P(k) l= min(n 01,n 10 ) m = max(n 01,n 10 ) k=0 k=m Probability Distribution 0 min max n For large n, a Normal distribution is often assumed 6345 Automatic Speech Recognition Significance Testing 3

31 Significance Test Example (Gillick and Cox, 1989) Common test set of 1400 tokens Algorithms A 1 and A 2 make 72 and 62 errors Are the differences significant? A 2 A A 2 A A 2 A n = 134 m =72 P=0437 n =16 m=13 P=00213 n =10 m=10 P= Automatic Speech Recognition Significance Testing 4

32 References Huang, Acero, and Hon, Spoken Language Processing, Prentice-Hall, 2001 Duda, Hart and Stork, Pattern Classification, John Wiley & Sons, 2001 Jelinek, Statistical Methods for Speech Recognition MIT Press, 1997 Bishop, Neural Networks for Pattern Recognition, Clarendon Press, 1995 Gillick and Cox, Some Statistical Issues in the Comparison of Speech Recognition Algorithms, Proc ICASSP, Automatic Speech Recognition Pattern Classification