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Automatic Speech Recognition II Hidden Markov Models Neural Network

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Hidden Markov Model DTW, VQ => recognize pattern, use distance measurement. HMM: statistical method for characterizing the properties of the frame of pattern,

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Discrete-time Markov Processes Consider a system with: N distinct states A set of probabilities associated with the state. => probabilities to change from one state to another state Time instants

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Discrete-time Markov Processes First order Markov chain: the probability depends on just the preceding state. The set of state-transition probabilities a ij :

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Discrete-time Markov Processes Ex. Consider a simple three-state Markov model of the weather. What is the probability that the weather for the next seven consecutive days is “sun-sun-snow-snow-sun-cloudy-sun”. Given the weather for today is “sun” and the weather condition for each day depends on the condition on a previous day. O=(sun, sun, sun, snow, snow, sun, cloudy, sun) O=(3, 3, 3, 1, 1, 3, 2, 3) State1: snow State2: cloudy State3: sunny Prob. for initial state

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Discrete-time Markov Processes Ex. Given a single fair coin, i.e., P(Heads)=P(Tails)=0.5 What is the probability that the next 10 tosses will provide the sequence (HHTHTTHTTH) What is the probability that 5 of the next 10 tosses will be tails?

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Coin-Toss Models You are in a room with a barrier which you cannot see what is happening. On the other side of the barrier is another person who is performing a coin- tossing experiment (using one or more coins). The person will not tell you which coin he selects at any time; he will only tell you the result of each coin flip. How do we build an HMM to explain the observe sequence of heads and tails? What the states in the model correspond to How many states should be in the model

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Coin-Toss Models Single coin Two states: heads or tails Observable Markov Model => not hidden headstails P(H) 1-P(H) P(H)

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Coin-Toss Models Two coins (Hidden Markov Model) Two state: coin 1, coin 2 Each state(coin) is characterized by a probability distribution of heads and tails There are probabilities of state transition (state transition matrix) Coin 1Coin 2 a11 a22 1-a11 1-a22 P(H)=P1 P(T)=1-P1) P(H)=P2 P(T)=1-P2)

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Coin-Toss Models Three coins (Hidden Markov Model) Three state: coin 1, coin 2, coin 3 Each state(coin) is characterized by a probability distribution of heads and tails There are probabilities of state transition (state transition matrix) a11 a22 a12 a21 a33 a31 a13 a23 a32 P(H)=P1 P(T)=1-P1) P(H)=P3 P(T)=1-P3) P(H)=P2 P(T)=1-P2)

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The Urn-and-Ball Model There are N-glass urns in the room. Each urn is a large quantity of colored balls : M distinct colors A genie is in the room and it chooses an initial urn. From this urn, a ball is chosen at random and its color is recorded as the observation. The ball is then replace to the same urn. A new urn is then selected according to the random selection of the current urn.

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Element of an HMM The number of states in the model (N) : S={1,2,…,N} The number of distinct observation symbols per state (M): V={v 1,v 2,…v M } The state transition probability distribution A={a ij } where a ij =P[q t+1 =j|q t =i] The observation symbol probability distribution, B={b j (k)}, in which b j (k)=P[o t =vk|q t =j] The initial state distribution Complete parameter set of the model

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HMM Generator of Observations Given appropriate values of N, M, A, B, and , the HMM can be used as a generator to give an observation sequence O=(o 1 o 2 …o T ) Choose an initial state q 1 =i Set t=1 Choose o t =v k according to the symbol probability distribution in state i, b j (k) Transit to the new state q t+1 = j according to the state-transition probability distribution for state i, a ij Set t=t+1; return to step 3 if t

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HMM Generator of Observations Ex. Consider an HMM representation of a coin-tossing problem. Assume a three-state model (three coins) with probabilities: All state transition probabilities = 1/3 State1State2State3 P(H)0.50.750.25 P(T)0.50.250.75

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HMM Generator of Observations 1. You observe the sequence O=(H H H H T H T T T T). What state sequence is most likely? What is the probability of the observation sequence and this most likely state sequence? Because all state transition probability are equal, the most likely state sequence is the one for which the probability of each individual observation is maximum. Thus for each H, the most likely state is 2 and for each T the most likely state is 3. The most likely state sequence is q=(2 2 2 2 3 2 3 3 3 3) with probability

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HMM Generator of Observations 2. What is the probability that the observation sequence came entirely from state 1? O=(H H H H T H T T T T), q=(1 1 1 1 1 1 1 1 1 1) The probability that the first H come from state 1 =0.5*1/3 The probability that the second H come from state 1 =0.5*1/3… The probability that the first T come from state 1 =0.5*1/3…

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HMM Generator of Observations If the state-transition probabilities were: What is the most likely state sequence for O=(H H H H T H T T T T). a 11 =0.9a 21 =0.45a 31 =0.45 a 12 =0.05a 22 =0.1a 32 =0.45 a 13 =0.05a 23 =0.45a 33 =0.1

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The three basic problems for HMM Problem 1: How do we compute P(O| ) Problem 2: How do we choose the state sequence q=(q 1, q 2,…q T ) that is optimal? (most likely) Problem 3: How do we adjust the model to maximize P(O| ) Speech recognition sense Training Model Samples of W word vocab W1 Model Wn Model Problem3

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The three basic problems for HMM To study the physical meaning of model states. Initial VowelFinal Problem2

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The three basic problems for HMM Unknown word Recognize an unknown word. Calculate P (O| 1 ) Calculate P (O| n ) compare Prediction Problem1

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Artificial Neural Network An artificial neural network (ANN), usually called "neural network" (NN), is a mathematical model or computational model that tries to simulate the structure and/or functional aspects of biological neural networks.

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Composition of NN Input nodes: each node is the feature vector of each sample. Hidden nodes: can be more than 1 layer. Output nodes: the output of the correspond input sample. The connections of input nodes, hidden nodes, and output nodes are specified by weight values. Input nodes Hidden nodes Output nodes Connections

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Feedforward operation and classification A simple three-layer NN x1x1 x2x2 y1y1 y2y2 zkzk bias Output k Hidden j Input i w ji w kj

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Feedforward operation and classification Net activation: the inner product of the inputs with the weights at the hidden unit. Where i = index of input layer, j =index of hidden layer node Each hidden unit emits an output that is a nonlinear function of its activation, f(net) that is: Simple of sign function:

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Feedforward operation and classification Each output unit computes its net activation based on the hidden unit signals as The output unit computes the nonlinear function of its net:

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Back propagation Backpropagation is the simplest and the most general methods for supervised training of multilayer NN. The basic approach in learning starts with an untrained network and follows these steps: Present a training pattern to the input layer. Pass the signals through the net and determine the output. Compare the output with the target values => difference (error) The weights are adjusted to reduce the error.

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Exercise Implement the vowel classifier by using Neural Network. Use the same speech samples that you use in VQ exercise. What is the important feature to classify vowels? Separate your samples into 2 groups: training and testing Label the class for the training sample. Train Multilayer perceptron from the training samples and perform testing on the testing data.

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Hidden Markov Models (HMMs) Chapter 3 (Duda et al.) – Section 3.10 (Warning: this section has lots of typos) CS479/679 Pattern Recognition Spring 2013.

Hidden Markov Models (HMMs) Chapter 3 (Duda et al.) – Section 3.10 (Warning: this section has lots of typos) CS479/679 Pattern Recognition Spring 2013.

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