Lecture 3: Machine learning, classification, and generative models

Transcription

1 EE E6820: Speech & Audio Processing & Recognition Lecture 3: Machine learning, classification, and generative models 1 Classification 2 Generative models 3 Gaussian models Michael Mandel <mim@ee.columbia.edu> 4 Hidden Markov models Columbia University Dept. of Electrical Engineering dpwe/e6820 February 7, 2008 Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

2 Outline 1 Classification 2 Generative models 3 Gaussian models 4 Hidden Markov models Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

3 Classification and generative models F2/Hz 4000 f/hz time/s x x F1/Hz ay ao Classification discriminative models discrete, categorical random variable of interest fixed set of categories Generative models descriptive models continuous or discrete random variable(s) of interest can estimate parameters Bayes rule makes them useful for classification Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

4 Building a classifier Define classes/attributes could state explicit rules better to define through training examples Define feature space Define decision algorithm set parameters from examples Measure performance calculated (weighted) error rate 1100 Pols vowel formants: "u" (x) vs. "o" (o) 1000 F2 / Hz F1 / Hz Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

5 Classification system parts Sensor signal segment feature vector class Pre-processing/ segmentation Feature extraction Classification Post-processing STFT Locate vowels Formant extraction Context constraints Costs/risk Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

6 Feature extraction Right features are critical waveform vs formants vs cepstra invariance under irrelevant modifications Theoretically equivalent features may act very differently in a particular classifier representations make important aspects explicit remove irrelevant information Feature design incorporates domain knowledge although more data less need for cleverness Smaller feature space (fewer dimensions) simpler models (fewer parameters) less training data needed faster training [inverting MFCCs] Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

7 Optimal classification Minimize probability of error with Bayes optimal decision ˆθ = argmax p(θ i x) θ i p(error) = p(error x)p(x) dx = (1 p(θ i x))p(x) dx i Λ i where Λ i is the region of x where θ i is chosen... but p(θ i x) is largest in that region so p(error) is minimized Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

8 Sources of error p(x ω 1 ) Pr(ω 1 ) X ω1 ^ X ω2 ^ p(x ω 2 ) Pr(ω 2 ) Pr(ω 2 X ω1 ^ ) = Pr(err X ω1 ^ ) x Pr(ω 1 X ω2 ^ ) = Pr(err X ω2 ^ ) Suboptimal threshold / regions (bias error) use a Bayes classifier Incorrect distributions (model error) better distribution models / more training data Misleading features ( Bayes error ) irreducible for given feature set regardless of classification scheme Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

9 Two roads to classification Optimal classifier is but we don t know p(θ i x) ˆθ = argmax θ i p(θ i x) Can model distribution directly e.g. Nearest neighbor, SVM, AdaBoost, neural net maps from inputs x to outputs θi a discriminative model Often easier to model data likelihood p(x θ i ) use Bayes rule to convert to p(θ i x) a generative (descriptive) model Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

10 Nearest neighbor classification 1800 Pols vowel formants: "u" (x), "o" (o), "a" (+) 1600 F2 / Hz x * o + new observation x F1 / Hz Find closest match (Nearest Neighbor) Naïve implementation takes O(N) time for N training points As N, error rate approaches twice the Bayes error rate With K summarized classes, takes O(K) time Locality sensitive hashing gives approximate nearest neighbors in O(dn 1/c2 ) time (Andoni and Indyk, 2006) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

11 Support vector machines (w. x) + b = 1 Large margin linear classifier for separable data x 2 regularization of margin avoids y i over-fitting = 1 w can be adapted to non-separable data (C parameter) made nonlinear using kernels k(x 1, x 2 ) = Φ(x 1 ) Φ(x 2 ). (w. x) + b = + 1 x 1 (w x) + b = 0 Depends only on training points near the decision boundary, the support vectors Unique, optimal solution for given Φ and C y i = +1 Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

12 Outline 1 Classification 2 Generative models 3 Gaussian models 4 Hidden Markov models Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

13 Generative models Describe the data using structured probabilistic models Observations are random variables whose distribution depends on model parameters Source distributions p(x θ i ) reflect variability in features reflect noise in observation generally have to be estimated from data (rather than known in advance) p(x ω i ) ω 1 ω 2 ω 3 ω 4 x Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

14 Generative models (2) Three things to do with generative models Evaluate the probability of an observation, possibly under multiple parameter settings p(x), p(x θ 1 ), p(x θ 2 ),... Estimate model parameters from observed data ˆθ = argmin C(θ, θ x) θ Run the model forward to generate new data x p(x ˆθ) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

15 Random variables review Random variables have joint distributions, p(x, y) y p(x,y) Marginal distribution of y p(y) = p(x, y) dx p(y) µy σ y p(x) µ x σ x σ xy x Knowing one value in a joint distribution constrains the remainder Conditional distribution of x given y p(x,y) y Y p(x y) p(x, y) p(y) = p(x, y) p(x, y) dy x p(x y = Y ) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

16 Bayes rule p(x y)p(y) = p(x, y) = p(y x)p(x) p(y x) = p(x y)p(y) p(x) can reverse conditioning given priors/marginals terms can be discrete or continuous generalizes to more variables p(x, y, z) = p(x y, z)p(y, z) = p(x y, z)p(y z)p(z) allows conversion between joint, marginals, conditionals Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

17 Bayes rule for generative models Run generative models backwards to compare them p(θ x) = Likelihood p(x θ) R p(x θ)p(θ) p(θ) Posterior prob Evidence = p(x) Prior prob Posterior is the classification we re looking for combination of prior belief in each class with likelihood under our model normalized by evidence (so posteriors = 1) Objection: priors are often unknown... but omitting them amounts to assuming they are all equal Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

18 Computing probabilities and estimating parameters Want probability of the observation under a model, p(x) regardless of parameter settings Full Bayesian integral p(x) = p(x θ)p(θ) dθ Difficult to compute in general, approximate as p(x ˆθ) Maximum likelihood (ML) ˆθ = argmax p(x θ) θ Maximum a posteriori (MAP): ML + prior ˆθ = argmax θ p(θ x) = argmax p(x θ)p(θ) θ Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

19 Model checking After estimating parameters, run the model forward Check that model is rich enough to capture variation in data parameters are estimated correctly there aren t any bugs in your code Generate data from the model and compare it to observations x p(x θ) are they similar under some statistics T (x) : R d R? can you find the real data set in a group of synthetic data sets? Then go back and update your model accordingly Gelman et al. (2003, ch. 6) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

20 Outline 1 Classification 2 Generative models 3 Gaussian models 4 Hidden Markov models Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

21 Gaussian models Easiest way to model distributions is via parametric model assume known form, estimate a few parameters Gaussian model is simple and useful. In 1D [ p(x θ i ) = 1 exp 1 ( ) ] x 2 µi σ i 2π 2 σ i Parameters mean µ i and variance σ i fit P M P M e 1/2 p(x ω i ) µ i σ i x Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

22 Gaussians in d dimensions p(x θ i ) = [ 1 (2π) d/2 exp 1 ] Σ i 1/2 2 (x µ i) T Σ 1 i (x µ i ) Described by a d-dimensional mean µ i and a d d covariance matrix Σ i Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

23 Gaussian mixture models Single Gaussians cannot model distributions with multiple modes distributions with nonlinear correlations What about a weighted sum? p(x) k c k p(x θ k ) where {ck } is a set of weights and {p(x θ k )} is a set of Gaussian components can fit anything given enough components Interpretation: each observation is generated by one of the Gaussians, chosen with probability c k = p(θ k ) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

24 Gaussian mixtures (2) e.g. nonlinear correlation resulting surface original data Gaussian components Problem: finding c k and θ k parameters easy if we knew which θ k generated each x Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

25 Expectation-maximization (EM) General procedure for estimating model parameters when some are unknown e.g. which GMM component generated a point Iteratively updated model parameters θ to maximize Q, the expected log-probability of observed data x and hidden data z Q(θ, θ t ) = p(z x, θ t ) log p(z, x θ) z E step: calculate p(z x, θt ) using θ t M step: find θ that maximizes Q using p(z x, θt ) can prove p(x θ) non-decreasing hence maximum likelihood model local optimum depends on initialization Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

26 Fitting GMMs with EM Want to find parameters of the Gaussians θk = {µ k, Σ k } weights/priors on Gaussians ck = p(θ k )... that maximize likelihood of training data x If we could assign each x to a particular θ k, estimation would be direct Hence treat mixture indices, z, as hidden form Q = E [p(x, z θ)] differentiate wrt model parameters equations for µ k, Σ k, c k to maximize Q Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

27 GMM EM updated equations Parameters that maximize Q ν nk p(z k x n, θ t ) µ k = n ν nkx n Σ k = c k = 1 N n ν nk n ν nk(x n µ k )(x n µ k ) T n ν nk n ν nk Each involves ν nk, fuzzy membership of x n in Gaussian k Updated parameter is just sample average, weighted by fuzzy membership Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

28 GMM examples Vowel data fit with different mixture counts 1 Gauss logp(x)= Gauss logp(x)= Gauss logp(x)= Gauss logp(x)= [Example... ] Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

29 Outline 1 Classification 2 Generative models 3 Gaussian models 4 Hidden Markov models Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

30 Markov models A (first order) Markov model is a finite-state system whose behavior depends only on the current state The future is independent of the past, conditioned on the present e.g. generative Markov model S A.1 B.1.1 C E p(q n+1 q n ) q n S A B C E q n+1 S A B C E S A A A A A A A A B B B B B B B B B C C C C B B B B B B C E Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

31 Hidden Markov models Markov model where state sequence Q = {q n } is not directly observable ( hidden ) But, observations X do depend on Q x n is RV that depends only on current state p(x n q n ) p(x q) p(x q) Emission distributions q = A q = B q = C q = A q = B q = C observation x xn xn State sequence AAAAAAAABBBBBBBBBBBCCCCBBBBBBBC Observation sequence time step n can still tell something about state sequence... Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

32 (Generative) Markov models HMM is specified by parameters Θ: - states q i k a t - transition probabilities a ij k a t k a t k a t emission distributions b i (x) p(x q) k a t x (+ initial state probabilities π i ) a ij p(q j n q i n 1) b i (x) p(x q i ) π i p(q i 1) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

33 Markov models for sequence recognition Independence of observations observation xn depends only on current state q n p(x Q) = p(x 1, x 2,... x N q 1, q 2,... q N ) = p(x 1 q 1 )p(x 2 q 2 ) p(x N q N ) N N = p(x n q n ) = b qn (x n ) n=1 n=1 Markov transitions transition to next state q i+1 depends only on q i p(q M) = p(q 1, q 2,... M) = p(q N q N 1... q 1 )p(q N 1 q N 2... q 1 )p(q 2 q 1 )p(q 1 ) = p(q N q N 1 )p(q N 1 q N 2 )p(q 2 q 1 )p(q 1 ) N N = p(q 1 ) p(q n q n 1 ) = π q1 a qn 1 q n n=2 n=2 Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

34 Model-fit calculations From state-based modeling : p(x Θ j ) = all Q p(x Q, Θ j )p(q Θ j ) For HMMs N p(x Q) = b qn (x n ) p(q M) = π q1 n=1 N a qn 1 q n n=2 Hence, solve for ˆΘ = argmax Θj p(θ j X ) Using Bayes rule to convert from p(x Θ j ) Sum over all Q??? Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

35 Summing over all paths 0.9 S S A B E 0.1 Model M A B E S A B E q 0 q 1 q 2 q 3 q 4 S A A A E S A A B E S A B B E S B B B E 0.2 x n p(x B) p(x A) States E B A Observations x 1, x 2, x Observation likelihoods p(x q) x 1 x 2 x 3 A q{ B S time n All possible 3-emission paths Q k from S to E Paths p(q M) = Πn p(q n q n-1 ) p(x Q,M) = Πn p(x n q n ) p(x,q M).9 x.7 x.7 x.1 = x.7 x.2 x.2 = x.2 x.8 x.2 = x.8 x.8 x.2 = x 0.2 x 0.1 = x 0.2 x 2.3 = x 2.2 x 2.3 = x 2.2 x 2.3 = Σ = Σ = p(x M) = n Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

36 The forward recursion Dynamic-programming-like technique to sum over all Q Define α n (i) as the probability of getting to state q i at time step n (by any path): α n (i) = p(x 1, x 2,... x n, q n = q i ) p(x n 1, q i n) α n+1 (j) can be calculated recursively: Model states q i α n (i+1) a i+1j a ij b j (x n+1 ) S α n+1 (j) = [Σ α n (i) a ij ] b j (x n+1 ) i=1 α n (i) Time steps n Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

37 Forward recursion (2) Initialize α 1 (i) = π i b i (x 1 ) Then total probability p(x1 N Θ) = S i=1 α N(i) Practical way to solve for p(x Θ j ) and hence select the most probable model (recognition) Observations X p(x M 1 ) p(m 1 ) p(x M 2 ) p(m 2 ) Choose best Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

38 Optimal path May be interested in actual q n assignments which state was active at each time frame e.g. phone labeling (for training?) Total probability is over all paths... but can also solve for single best path, Viterbi state sequence Probability along best path to state q j n+1 : [ ] ˆα n+1 (j) = max {ˆα n (i)a ij } b j (x n+1 ) i backtrack from final state to get best path final probability is product only (no sum) log-domain calculation is just summation Best path often dominates total probability p(x Θ) p(x, ˆQ Θ) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

39 Interpreting the Viterbi path Viterbi path assigns each x n to a state q i performing classification based on b i (x)... at the same time applying transition constraints a ij Inferred classification 3 x n 2 Viterbi labels: AAAAAAAABBBBBBBBBBBCCCCBBBBBBBC Can be used for segmentation train an HMM with garbage and target states decode on new data to find targets, boundaries Can use for (heuristic) training e.g. forced alignment to bootstrap speech recognizer e.g. train classifiers based on labels... Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

40 Aside: Training and test data A rich model can learn every training example (overtraining) But the goal is to classify new, unseen data sometimes use cross validation set to decide when to stop training For evaluation results to be meaningful: don t test with training data! don t train on test data (even indirectly... ) Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

41 Aside (2): Model complexity More training data allows the use of larger models More model parameters create a better fit to the training data more Gaussian mixture components more HMM states For fixed training set size, there will be some optimal model size that avoids overtraining Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

42 Summary Classification is making discrete (hard) decisions Basis is comparison with known examples explicitly or via a model Classification models discriminative models, like SVMs, neural nets, boosters, directly learn posteriors p(θ i x) generative models, like Gaussians, GMMs, HMMs, model likelihoods p(x θ) Bayes rule lets us use generative models for classification EM allows parameter estimation even with some data missing Parting thought Is it wise to use generative models for discrimination or vice versa? Michael Mandel (E6820 SAPR) Machine learning February 7, / 43

43 References Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In Proc. IEEE Symposium on Foundations of Computer Science, pages , Washington, DC, USA, IEEE Computer Society. Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin. Bayesian Data Analysis. Chapman & Hall/CRC, second edition, July ISBN X. Lawrence R. Rabiner. A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77(2): , Jeff A. Bilmes. A gentle tutorial on the em algorithm and its application to parameter estimation for gaussian mixture and hidden markov models, Christopher J. C. Burges. A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discov., 2(2): , June Michael Mandel (E6820 SAPR) Machine learning February 7, / 43