Page 124 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

DISCRETE STATE VARIABLES 113

4.3 DISCRETE STATE VARIABLES

We consider physical processes that are described at any time as being in
one of a finite number of states. Examples of such processes are:

. The sequence of strokes of a tennis player during a game, e.g.
service, backhand-volley, smash, etc.
. The sequence of actions that a tennis player performs during a
particular stroke.
. The different types of manoeuvres of an airplane, e.g. a linear flight,
a turn, a nose dive, etc.
. The sequence of characters in a word, and the sequence of words in
a sentence.
. The sequence of tones of a melody as part of a musical piece.
. The emotional modes of a person: angry, happy, astonished, etc.

These situations are described by a state variable x(i) that can only take a
value from a finite set of states
¼f! 1 , .. . , ! K g.
The task is to determine the sequence of states that a particular process
goes through (or has gone through). For that purpose, at any time meas-
urements z(i) are available. Often, the output of the sensors is real-
valued. But nevertheless we will assume that the measurements take their
values from a finite set. Thus, some sort of discretization must take place
that maps the range of the sensor data onto a finite set Z ¼f# 1 , ... , # N g.
This section first introduces a state space model that is often used for
discrete state variables, i.e. the hidden Markov model. This model will be
used in the next subsections for online and offline estimation of the states.

4.3.1 Hidden Markov models

A hidden Markov model (HMM) is an instance of the state space model
discussed in Section 4.1.1. It describes a sequence x(i) of discrete states
starting at time i ¼ 0. The sequence is observed by means of measure-
ments z(i) that can only take values from a finite set. The model consists
of the following ingredients:

. The set
containing the K states ! k that x(i) can take.
. The set Z containing the N symbols # n that z(i) can take.
. The initial state probability P 0 (x(0)).