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
First order Markov chain: the probability depends on just the preceding state.
The set of state

transition probabilities
a
ij
:
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
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
a22
1

a11
1

a22
P(H)=P1
P(T)=1

P1
)
P(H)=P2
P(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
a22
a12
a21
a33
a31
P(H)=P1
P(T)=1

P1
)
P(H)=P3
P(T)=1

P3
)
P(H)=P2
P(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={v
1
,v
2
,…
v
M
}
The state transition probability distribution
A={
a
ij
}
where
a
ij
=P[q
t+1
=
jq
t
=
i
]
The observation symbol probability distribution,
B={
b
j
(k)}
, in which
b
j
(k)=P[
o
t
=
vkq
t
=j]
The initial state distribution
Complete parameter set of the model
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<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
State3
P(H)
0.5
0.75
0.25
P(T)
0.5
0.25
0.75
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
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…
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
=0.9
a
21
=0.45
a
31
=0.45
a
12
=0.05
a
22
=0.1
a
32
=0.45
a
13
=0.05
a
23
=0.45
a
33
=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=(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
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 nodes
Hidden nodes
Output nodes
Connections
Feedforward
operation and
classification
A simple three

layer NN
x
1
x
2
y
1
y
2
z
k
bias
Output k
Hidden j
Input
i
w
ji
w
kj
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:
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:
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.
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