【免费】补充材料-隐马尔科夫模型1资源-CSDN文库

需积分: 0 30 浏览量 2022-08-03 14:47:03 上传评论收藏 2.62MB PDF 举报

资源详情

资源评论

资源推荐

A Tutorial on Hidden Markov Models and

Selected Applications in Speech Recognition

LAWRENCE

RABINER,

FELLOW,

IEEE

Although initially introduced and studied in the late 1960s and

early 1970s, statistical methods of Markov source or hidden Markov

modeling have become increasingly popular in the last several

years. There are two strong reasons why this has occurred. First the

models are very rich in mathematical structure and hence can form

the theoretical basis for use in a wide range of applications. Sec-

ond the models, when applied properly, work very well in practice

for several important applications. In this paper we attempt

care-

fully and methodically review the theoretical aspects of this type

of statistical modeling and show how they have been applied to

selected problems in machine recognition of speech.

INTRODUCTION

Real-world processes generally produce observable out-

puts which can be characterized as signals. The signals can

bediscrete in

nature(e.g.,charactersfrom

afinitealphabet,

quantized vectors from a codebook, etc.), or continuous in

nature (e.g., speech samples, temperature measurements,

music, etc.). The signal source can be stationary (i.e., its sta-

tistical properties do not vary with time), or nonstationary

(i.e., the signal properties vary over time). The signals can

be pure (i.e., coming strictly from a single source), or can

be corrupted from other signal sources (e.g., noise) or by

transmission distortions, reverberation, etc.

problem of fundamental interest

characterizing such

real-world signals in terms of signal models. There are sev-

eral reasons why one

interested in applying signal models.

First of

all,

signal model can provide the basis for a the-

oretical description of a signal processing system which can

be used to process the signal

as to provide a desired out-

put. For example if we are interested in enhancing a speech

signal corrupted by noise and transmission distortion, we

can use the signal model to design a system which will opti-

mally remove the noise and undo the transmission distor-

tion.

second reason why signal models are important

that they are potentially capable of letting

learn a great

deal about the signal source (i.e., the real-world process

which produced the signal) without having to have the

sourceavailable. This property

especially important when

the cost of getting signals from the actual source

high.

Manuscript received January 15,1988; revised October 4,1988.

The author

with AT&T Bell Laboratories, Murray Hill,

07974-

IEEE

Log

Number 8825949.

2070, USA.

In this case, with a good signal model, we can simulate the

source and learn

much

possible via simulations.

Finally, the most important reason why signal models are

important

that they often workextremelywell in practice,

and enable

to realize important practical systems-e.g.,

prediction systems, recognition systems, identification sys-

tems, etc., in

very efficient manner.

These are several possible choices for what type of signal

model

used for characterizing the properties of a given

signal. Broadly one can dichotomize the types

signal

models into the class of deterministic models, and the class

of statistical models. Deterministic models generally exploit

some known specific properties of the signal, e.g., that the

signal

sine wave, or a sum

exponentials, etc. In these

cases, specification of the signal model

generally straight-

forward;all that

required

istodetermine(estimate)values

of the parameters

the signal model (e.g., amplitude, fre-

quency, phase of a sine wave, amplitudes and rates of expo-

nentials, etc.). The second broad class of signal models

the set

statistical models in which one tries to charac-

terize only the statistical properties of the signal. Examples

of such statistical models include Gaussian processes, Pois-

son processes, Markov processes, and hidden Markov pro-

cesses, among others. The underlying assumption of the

statistical model

that the signal can be well characterized

parametric random process, and that the parameters

of the stochastic process can be determined (estimated) in

a precise, well-defined manner.

For the applications of interest, namely speech process-

ing, both deterministic and stochastic signal models have

had good success. In this paper we will concern ourselves

strictlywith one typeof stochastic signal model, namelythe

hidden Markov model (HMM). (These models are referred

Markov sources or probabilistic functions of Markov

chains in the communications literature.) We will first

review the theory of Markov chains and then extend the

ideas to the class of hidden Markov models using several

simple examples. We will then focus our attention on the

three fundamental problems' for HMM design, namely: the

'The idea of characterizing the theoretical aspects of hidden

Markov modeling in terms of solving three fundamental problems

due to Jack Ferguson of IDA (Institute for Defense Analysis) who

introduced it in lectures and writing.

0018-9219/89/02000257$01.00

1989

IEEE

PROCEEDINGS OF THE IEEE,

VOL.

77,

NO.

FEBRUARY

1989

257

evaluation of the probability (or likelihood) of

sequence

of observations given a specific HMM; the determination

best sequence of model states; and the adjustment of

model parameters

as to best account for the observed

signal. We will show that once these three fundamental

problems are solved, we can apply HMMs to selected prob-

lems in speech recognition.

Neither the theory of hidden Markov models nor its

applications to speech recognition

new. The basic theory

was published in

series of classic papers by Baum and his

colleagues

[I]-[5]

in the late

1960s

and early

1970s

and was

implemented for speech processing applications by Baker

161

at CMU, and by Jelinek and his colleagues at IBM

[7-[13]

in the

1970s.

However, widespread understanding and

application of the theory of HMMs to speech processing

has occurred only within the past several years. There are

several reasons why this has been the case. First, the basic

theory of hidden Markov models was published in math-

ematical journals which were not generally read by engi-

neers working on problems in speech processing. The sec-

ond reason was that the original applications of the theory

to speech processing did not provide sufficient tutorial

material for most readers to understand the theory and to

be able to apply it to their own research.

result, several

tutorial papers were written which provided

sufficient

level of detail for

number of research labs to begin work

using HMMs in individual speech processing applications

[14]-[19].

This tutorial

intended to provide an overview

of the basic theory of HMMs (as originated by Baum and

his colleagues), provide practical details on methods of

implementation of the theory, and describe a couple

selected applications of the theory to distinct problems in

speech recognition. The paper combines results from a

number of original sources and hopefully provides

single

source for acquiring the background required to pursue

further this fascinating area of research.

The organization of this paper

follows. In Section I1

we review the theory of discrete Markov chains and show

how the concept of hidden states, where the observation

probabilistic function of the state, can be used effec-

tively. We illustrate the theory with two simple examples,

namely coin-tossing, and the classic balls-in-urns system.

In Section Ill we discuss the three fundamental problems

of HMMs, and give several practical techniques for solving

these problems. In Section IV we discuss the various types

of HMMs that have been studied including ergodic

well

left-right models. In this section we also discuss the var-

ious model features including the form of the observation

density function, the state duration density, and the opti-

mization criterion for choosing optimal HMM parameter

values. In Section Vwe discuss the issues that arise in imple-

menting HMMs including the topics of scaling, initial

parameter estimates, model size, model form, missingdata,

and multiple observation sequences. In Section

describean isolated word speech recognizer, implemented

with HMM ideas, and show how it performs

compared

to alternative implementations. In Section

VI1

we extend

the ideas presented in Section VI to the problem of recog-

nizing a string of spoken words based on concatenating

individual HMMsofeachword in thevocabulary. In Section

Vlll

we briefly outline how the ideas of HMM have been

applied to

largevocabulary speech recognizer, and in Sec-

tion

we summarize the ideas discussed throughout the

paper.

11.

DISCRETE

MARKOV

PROCESSES~

Consider a system which may be described at any time

being in one of

set of

distinct states,

S1,

SzI

. . .

SN,

as illustrated in Fig.

(where

for simplicity). At reg-

Fig.

Markov chain with

states (labeled

S,)

with

selected state transitions.

ularlyspaced discrete times, the system undergoesachange

of state (possibly back to the same state) according to

set

of probabilities associated with the state. We denote the

time instants associated with state changes as

. . .

and we denote the actual state at time

qr.

full

probabilistic description of the above system would, in gen-

eral, require specification of the current state (at time

t),

well

all

the predecessor states. For the special case of a

discrete, first order, Markov chain, this probabilistic

description

truncated to just the current and the pre-

decessor state, i.e.,

99,

qqt-1

SI,

qt-2

Skr

. .

9s:

S&:

SJ.

)

Furthermoreweonlyconsider

those processes in which the

right-hand side of

(1)

independent of time, thereby lead-

ing to the set of state transition probabilities

a,,

of the form

(2)

with the state transition coefficients having the properties

a,,

99,

S,(q,-,

S,],

i,j

a,,

/=1

(3a)

(3b)

since they obey standard stochastic constraints.

The above stochastic process could be called an observ-

able Markov model since the output of the process

the

set of states at each instant of time, where each state cor-

responds to

physical (observable) event.

set ideas, con-

sider

simple 3-state Markov model of the weather. We

assume that once

day (e.g., at noon), the weather

good overview

discrete Markov processes is in [20, ch.

51.

258

PROCEEDINGS

THE

IEEE,

VOL. 77,

NO.

FEBRUARY

1989

observed as being one of the following:

State

rain or (snow)

State 2: cloudy

State

sunny.

We postulate that the weather on day tis characterized by

a single one of the three states above, and that the matrix

of state transition probabilities

0.4

0.3 0.3

LO.1

0.1

0.81

Given that the weather on day

sunny (state

3),

we can ask the question: What

the probability (according

to the model) that the weather for the next 7 days will be

"sun-sun-rain-rain-sun-cloudy-sun

* *

a"?

Stated more for-

mally, we define the observation sequence

{S3,

S3,

S1, S1,

S3,

Sz,

S3} corresponding to

. .

and we wish to determine the probability of

given the

model. This probability can be expressed (and evaluated)

P(0IModel)

RS,,

S3,

S1, S1,

S3,

Sz,

S31Modell

SS31

RS3lS3l

SS3lS3l

RSlIS31

7r3

(0.8)(0.8)(0.1)(0.4)(0.3)(0.1)(0.2)

1.536

a33

a31

all

a13

a32

aZ3

where we use the notation

491

S;],

(4)

to denote the initial state probabilities.

Another interesting question we can ask (and answer

using the model)

is:

Given that the model

in a known state,

what

the probabilityit stays in that stateforexactlyddays?

This probability can be evaluated as the probability of the

observation sequence

{Si,

Si, Si,

S;},

123

dkl

given the model, which

P(OIMode1, ql

S;)

(aJd-'(l

a;;)

p,(d).

(5)

The quantityp;(d)

the (discrete) probability density func-

tion of duration din state

This exponential duration den-

sity

characteristic of the state duration in a Markovchain.

Based on pi(d), we can readily calculate the expected num-

ber of observations (duration) in

state, conditioned on

starting in that state as

dpi(d)

d=l

(6a)

(6b)

d(ajJd-'(1

a;,)

d=l

ai;

Thus the expected number of consecutive days of sunny

weather, according to the model,

140.2)

for cloudy

2.5; for rain it

1.67.

Extension to Hidden Markov Models

So far we have considered Markov models in which each

state corresponded to an observable (physical) event. This

model

too restrictive to be applicable to many problems

of interest. In this section we extend the concept of Markov

models to include the case where the observation

prob-

abilistic function of the state-i.e., the resulting model

(which iscalled a hidden Markovmodel) isadoublyembed-

ded stochastic process with an underlying stochastic pro-

cess that

not observable (it

hidden), but can only be

observed through another set of stochastic processes that

produce the sequence of observations.

fix ideas, con-

sider the following model of some simple coin tossing

experiments.

Coin

Toss

Models: Assume the following scenario. You

are in a room with

barrier (e.g.,

curtain) through which

you cannot see what

happening. On the other side

the

barrier

another person who

performing a coin (or mul-

tiplecoin) tossing experiment. Theother person will not tell

you anything about what he

doing exactly; he will only

tell you the result of each coin flip. Thus

sequence of

hid-

den coin tossing experiments

performed, with the obser-

vation sequence consisting of a series of heads and tails;

e.g., a typical observation sequence would be

. . .

x333x

...

where

stands for heads and

stands for tails.

Given the above scenario, the problem of interest

how

do we build an HMM to explain (model) the observed

sequence of heads and tails. The first problem one faces

deciding what the states in the model correspond to, and

then deciding how many states should be in the model. One

possiblechoicewould

betoassumethatonlyasingle

biased

coin was being tossed. In this case we could model the sit-

uation with a 2-state model where each state corresponds

to a side of the coin (i.e., heads or tails). This model

depicted in Fig. 2(a).3 In this case the Markov model

observable, and the only issue for complete specification

of the model would be to decide on the best value for the

bias (i.e., the probability of, say, heads). Interestingly, an

equivalent HMM to that of Fig. 2(a) would be

degenerate

I-state model, where the state corresponds to the single

biased coin, and the unknown parameter

the bias of the

coin.

A second form of HMM for explaining the observed

sequence of coin toss outcome

given in Fig. 2(b). In this

case there are 2 states in the model and each state corre-

sponds to a different, biased, coin being tossed. Each state

characterized by a probability distribution of heads and

tails, and transitions between states are characterized by a

state transition matrix. The physical mechanism which

accounts for how state transitions are selected could itself

set of independent coin tosses, or some other prob-

abilistic event.

third form of HMM for explaining the observed

sequence of coin toss outcomes

given in Fig. 2(c). This

model corresponds to using

biased coins, and choosing

from among the three, based on some probabilistic event.

3The model

Fig.

2(a)

a memoryless process and thus

degenerate case

a Markov model.

RABINER: HIDDEN MARKOV MODELS

259

P(H1

P(HI

HEADS TAILS

STATE

t23

P(H1 PI Pp P3

P(T) I-P, i-Pp I-P3

---

0-HHTTHTHHTTH

...

S=l122421122i

...

=HHTTHTHHTTH

...

S=21I22212212

...

HHTTHTHHTTH

...

s=3123314

231

3...

Fig.

Three possible Markov models which can account

for the resultsof hidden coin tossing experiments.

(a)

I-coin

model. (b) 2-coins model.

(c)

3-coins model.

Given the choice among the three models shown in Fig.

2 for explaining the observed sequence of heads and tails,

a natural question would bewhich model best matches the

actual observations. It should beclearthat the simple I-coin

model of Fig. 2(a) has only

unknown parameter; the 2-coin

model of Fig. 2(b) has4 un known parameters; and the 3-coin

model of Fig. 2(c) has

unknown parameters. Thus, with

the greater degrees of freedom, the larger

HMMs

would

seem to inherently be more capable of modeling

series

of coin tossing experiments than would equivalently smaller

models. Although this

theoretically true, we will see later

in this paper that practical considerations impose some

strong limitations on the size of models that we can con-

sider. Furthermore, it might just be the case that only

sin-

glecoin

being tossed. Then using the 3-coin model of Fig.

2(c) would be inappropriate, since the actual physical event

would not correspond to the model being used-i.e., we

would be using an underspecified system.

The

Urn

and

BallMode14:To extend the ideas of the HMM

to a somewhat more complicated situation, consider the

urn and ball system of Fig.

We assume that there are

(1arge)glassurnsin aroom. Withineach urntherearealarge

number of colored balls. We assume there are

distinct

colorsofthe balls. The physical

processforobtainingobser-

vations

as follows. A genie

in the room, and according

to some random process, he (or she) chooses an initial urn.

From this urn, a ball

chosen at random, and

its

color

recorded as theobservation.The ball

then replaced in the

urn from which it was selected. A new urn

then selected

4The urn and ball model

was

introduced by

Jack

Ferguson, and

his colleagues,

lectures

on HMM theory.

{GREEN,

GREEN,

BLUE,

RED,

YELLOW,

RED,

. . .

...

BLUE}

Fig.

N-state urn and ball model which illustrates the

general

case

discrete symbol HMM.

according to the random selection process associated with

the current urn, and the ball selection process

repeated.

This entire process generates afinite observation sequence

of colors, which we would like to model as the observable

output of an

HMM.

It should be obvious that the simplest

HMM

that cor-

responds to the urn and ball process

one in which each

state corresponds to a specific urn, and for which a (ball)

color probability

defined for each state. The choice of

urns

dictated by the state transition matrix of the

HMM.

Elements

HMM

The above examples give us a pretty good idea of what

HMM

and how

can be applied to some simple sce-

narios. We now formally define the elements of an

HMM,

and explain how the model generates observation

sequences.

HMM

characterized by the following:

the number of states in the model. Although the

states are hidden, for many practical applications there

often some physical significance attached to the states or

to sets of states of the model. Hence,

the coin tossing

experiments, each state corresponded to a distinct biased

coin. In the urn and ball model, the states corresponded

to the urns. Generally the states are interconnected in such

a way that any state can be reached from any other state

(e.g., an ergodic model); however, we will see later in this

paper that other possible interconnections of states are

often of interest. We denote the individual states as

{Sl,

S2,

. . .

SN},

and the state at time

g,.

the number of distinct observation symbols per

state, i.e., the discrete alphabet size. The observation sym-

bols

correspond to the physical output

the system being

modeled. For the coin toss experiments the observation

symbols were simply heads or tails; for the ball and urn

model they were the colors of the balls selected from the

urns. We denote the individual symbols as

{vl,

v,,

The state transition probability distribution

{a,}

(7)

For the special case where any state can reach any other

state in a single step, we have

for all

For other

types of

HMMs,

we would have

a,]

for one or more

(i,

pairs.

VM).

where

a,,

p[q,+l

S,lq,

S,],

260

PROCEEDINGS

THE

IEEE,

VOL.

77,

NO.

FEBRUARY

1989

The observation symbol probability distribution in

statej,

{b,(k)}, where

b,(k)

p[vk

t)qt

S,],

IikiM.

(8)

The initial state distribution

{T~}

where

p[ql

SI],

(9)

Given appropriate values of

and

ir,

the HMM

can be used as a generator to give an observation sequence

0=01O~~~~oJ

(10)

(where each observation

one of the symbols from

and Tis the number of observations

the sequence) as

follows:

Choose an initial state

according to the initial

state distribution

Set

3) Choose

according to the symbol probability

distribution in state

SI,

i.e., b,(k).

Transit to a new state

q,,,

according to the state

transition probability distribution for state

S,,

i.e.,

a,.

Set

return to step 3)if

otherwise ter-

minate the procedure.

The above procedure can be used as both a generator

observations, and as a model for how a given observation

sequence was generated by an appropriate HMM.

It can be seen from the above discussion that a complete

specification of an HMM requires specification of two

model parameters

and

M),

specification of observation

symbols, and the specification of the three probability mea-

sures

and

For convenience, we use the compact

notation

(A,

(11)

to indicate the complete parameter set of the model.

The Three Basic Problems

for

HMMs5

Given the form of HMM of the previous section, there are

three basic problems of interest that must be solved for the

model to be useful in real-world applications. These prob-

lems are the following:

Problem

Given the observation sequence

. .

Or,

and a model

(A,

ir),

how do

we efficiently compute P(OIA), the proba-

bilityof theobservation sequence,given the

model?

Problem

Given the observation sequence

. . .

Or,

and the model

how do we choose

a corresponding state sequence

. . .

qJwhich

optimal in some meaningful

sense (i.e., best “explains” the observa-

ion

s)?

Problem

How do we adjust the model parameters

(A,

to maximize P(OJA)?

5The material in this section and in Section

Ill

based on the

ideas presented by Jack Ferguson

IDA

in lectures at

Bell

Lab-

oratories.

Problem

the evaluation problem, namely given a

model and asequenceof observations, how dowecompute

the probability that the observed sequence was produced

by the model. We can also view the problem as one of scor-

ing how well a given model matches a given observation

sequence. The latter viewpoint

extremely useful. For

example, if we consider the case in which we are trying to

choose among several competing models, the solution to

Problem

allows

to choose the model which best

matches the observations.

Problem

the one

which we attempt to uncover the

hidden part of the model, i.e., to find the “correct” state

sequence. It should be clear that for all but the case of

degenerate models, there

no “correct” state sequence

to be found. Hence for practical situations, we usually use

an optimality criterion to solve this problem as best as pos-

sible. Unfortunately, as we will see, there are several rea-

sonable optimality criteria that can be imposed, and hence

the choice of criterion

a strong function of the intended

use for the uncovered state sequence. Typical uses might

be to learn about the structure of the model, to find optimal

state sequences for continuous speech recognition, or to

get average statistics of individual states, etc.

Problem

the one in which we attempt to optimize the

model parameters

as to best describe how a given obser-

vation sequence comes about. The observation sequence

used to adjust the model parameters

called a training

sequence since

used to “train” the HMM. The training

problem

the crucial one for most applications of HMMs,

since

allows us to optimally adapt model parameters to

observed training data-i.e., to create best models for real

phenomena.

To fix ideas, consider the following simple isolated word

speech recognizer. For each word of a Wword vocabulary,

we want to design a separate N-state HMM. We represent

the speech signal of

given word as a time sequence of

coded spectral vectors. We assume that the coding

done

using

spectral codebook with

unique spectral vectors;

hence each observation

the index of the spectral vector

closest (in some spectral sense) to the original speech sig-

nal. Thus, for each vocabulary word, we have a training

sequence consisting of a number of repetitions of

sequencesofcodebook indicesoftheword (byoneor more

talkers). The first task

to build individual word models.

This task

done by using the solution to Problem

to opti-

mally estimate model parameters for each word model. To

develop an understanding of the physical meaning of the

model states, we use the solution to Problem

to segment

each of the word training sequences into states, and then

study the properties of the spectral vectors that lead to the

observations occurring in each state. The goal here would

be to make refinements on the model (e.g., more states,

different codebook size, etc.)

as to improve its capability

of modeling the spoken word sequences. Finally, once the

set of

HMMs has been designed and optimized and thor-

oughly studied, recognition of an unknown word

per-

formed using the solution to Problem

to score each word

model based upon the given test observation sequence,

and select the word

whose

modelscore

highe5t

[k,?

the

highest

kel

hood).

In the next section we present formal mathematical solu-

tionstoeachofthethreefundamental

problemsfor HMMs.

RABINER:

HIDDEN

MARKOV MODELS

261

剩余29页未读，继续阅读

评论收藏

内容反馈

本本纲目

粉丝: 27
资源: 293

补充材料-隐马尔科夫模型1

评论0

最新资源

补充材料-隐马尔科夫模型1

评论0

“使用高频数据估计跳跃扩散模型：一种基于过滤的方法”的补充材料-研究论文

论文研究-基于隐马尔科夫模型的中国股票信息探测.pdf

高斯--马尔科夫过程的matlab代码以及仿真

基于隐马尔科夫模型

HMM （隐马尔科夫模型）详细资料（含语音识别介绍）

机器学习C++源码解析-隐马尔科夫中文分词HMM_CWS算法-源码+数据

隐马尔科夫模型程序范例

第二章_隐马尔科夫模型

隐马尔科夫模型Python实现

HMM隐马尔科夫模型

隐马尔科夫分词源代码

基于BP神经网络-加权马尔科夫模型的泄水闸水平位移预测.pdf

隐马尔科夫模型教材ppt

毕业设计93基于连续隐马尔科夫模型的语音识别 (2).docx

隐马尔科夫模型.zip_python_机器学习_隐马尔科夫_隐马尔科夫模型_马尔科夫

Python基于隐马尔科夫模型的笔势识别源码(课程设计).zip

隐马尔科夫模型及其在语音处理中的应用

隐马尔科夫模型

BurpLoaderKeygen.jar.zip

最新版ISO/IEC 27001:2022、ISO 27002:2022中英文合集

Goby红队版-win-x64-2.4.7版本

Chrome Header Editor 插件

ISO SAE 21434-2021 中文版.pdf

OpenVAS GVM 中文翻译补丁

安全认证cisp教材全套

STM32F103C8T6核心板-电路原理图1.PDF

软件工程导论(第六版)课后习题答案1

最新资源