automatic speech recognition ii
DESCRIPTION
Automatic Speech Recognition II. Hidden Markov Models Neural Network. Hidden Markov Model. DTW, VQ => recognize pattern, use distance measurement. HMM: statistical method for characterizing the properties of the frame of pattern,. Discrete-time Markov Processes. Consider a system with: - PowerPoint PPT PresentationTRANSCRIPT
Automatic Speech Recognition II Hidden Markov Models Neural Network
Hidden Markov Model DTW, VQ => recognize pattern, use distance
measurement. HMM: statistical method for characterizing the properties of
the frame of pattern,
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
Discrete-time Markov Processes
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First order Markov chain: the probability depends on just the preceding state.
The set of state-transition probabilities aij :
]|[ 1 iqjqPa ttij
N
jij
ij
ia
ija
1
,0
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: snowState2: cloudyState3: sunny
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)2.0)(1.0)(3.0)(4.0)(1.0)(8.0)(8.0)(1()2|3()3|2()1|3()1|1()3|1()3|3()3|3()3(
)3 2, 3, 1, 1, 3, 3, 3,()|(
PPPPPPPPPModelOP
Prob. for initial state
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?
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
Coin-Toss Models Single coin
Two states: heads or tails Observable Markov Model => not hidden
heads tails
P(H) 1-P(H)1-P(H)
P(H)
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 1 Coin 2
a11 a221-a11
1-a22P(H)=P1P(T)=1-
P1)
P(H)=P2P(T)=1-
P2)
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 a22a12
a21
a33
a31 a13
a23a3
2P(H)=P1P(T)=1-
P1)
P(H)=P3P(T)=1-
P3)
P(H)=P2P(T)=1-
P2)
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.
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={v1,v2,…vM} The state transition probability distribution A={aij} where
aij=P[qt+1=j|qt=i] The observation symbol probability distribution, B={bj(k)},
in which bj(k)=P[ot=vk|qt=j] The initial state distribution
Complete parameter set of the model
NiiqPi 1],[ 1
),,( BA
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=(o1o2…oT) Choose an initial state q1=i Set t=1 Choose ot=vk according to the symbol probability distribution in
state i , bj(k) Transit to the new state qt+1 =j according to the state-transition
probability distribution for state i, aij Set t=t+1; return to step 3 if t<T, otherwise, terminate the
procedure.
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
State1 State2 State3P(H) 0.5 0.75 0.25P(T) 0.5 0.25 0.75
HMM Generator of Observations1. 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 isq=(2 2 2 2 3 2 3 3 3 3) with probability
1010 )3/1()75.0()|,( qOP
HMM Generator of Observations2. 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*1/3The 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… 1010 )3/1()5.0()|,( qOP
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).
a11=0.9 a21=0.45 a31=0.45a12=0.05 a22=0.1 a32=0.45a13=0.05 a23=0.45 a33=0.1
The three basic problems for HMM Problem 1: How do we compute P(O|) Problem 2: How do we choose the state sequence q=(q1,
q2,…qT) that is optimal? (most likely) Problem 3: How do we adjust the model to
maximize P(O|) Speech recognition sense
),,( BA
Training ModelSamples
of W word vocab
W1Model
WnModel
),,( BA
),,( BAProblem3
The three basic problems for HMM
To study the physical meaning of model states.
Initial
Vowel Final
Problem2
The three basic problems for HMM
Unknown word
Recognize an unknown word.
Calculate P (O|1)
Calculate P (O|n)
compare Prediction
Problem1
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.
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 nodesHidden nodes
Output nodes
Connections
Feedforward operation and classification
A simple three-layer NN
x1 x2
y1 y2
zk
bias
Output k
Hidden j
Input i
wji
wkj
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:
01
jji
d
iij wwxnet
)( jj netfy
01
01)()(
netifnetif
netSgnnetf j
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:
01
kkj
n
jjk wwynet
H
)( kk netfz
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.
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.