Page 94 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

A GENERAL FRAMEWORK FOR ONLINE ESTIMATION 83

time-continuous to time-discrete models are described in many text-
books, for instance, on control engineering.

4.1.1 Models

We assume that the continuous time t is equidistantly sampled with
period . The discrete time index is denoted by an integer variable i.
Hence, the moments of sampling are t i ¼ i . Furthermore, we assume
that the estimation problem starts at t ¼ 0. Thus, i is a non-negative
integer denoting the discrete time.

The state space model
The state at time i is denoted by x(i) 2 X where X is the state space. For
discrete states, X ¼
¼f! 1 , ... , ! K g where ! k is the k-th symbol (label,
or class) out of K possible classes. For real-valued vectors with dimen-
M
sion M, we have X ¼ R .
Suppose for a moment that we have observed the state of a process
during its whole history, i.e. from the beginning of time up to the
present. In other words, the sequence of states x(0), x(1), .. . , x(i) are
observed and as such fully known. i denotes the present time. In add-
ition, suppose that – using this sequence – we want to estimate (predict)
the next state x(i þ 1). Assuming that the states can be modelled as
random variables, we need to evaluate the conditional probability dens-
ity 1 p(x(i þ 1)jx(0), x(1), ... , x(i)). Once this probability density is
known, the application of the theory in Chapters 2 and 3 will provide
the optimal estimate of x(i þ 1). For instance, if X is a real-valued vector
space, the Bayes estimator from Chapter 3 provides the best prediction
of the next state (the density p(x(i þ 1)jx(0), x(1), ... , x(i)) must be used
instead of the posterior density).
Unfortunately, the evaluation of p(x(i þ 1)jx(0), x(1), .. . , x(i)) is a
nasty task because it is not clear how to establish such a density in
real-world problems. Things become much easier if we succeed to define
the state such that the so-called Markov condition applies:

pðxði þ 1Þjxð0Þ; xð1Þ; .. . ; xðiÞÞ ¼ pðxði þ 1ÞjxðiÞÞ ð4:1Þ

1
For the finite-state case, the probability densities transform into probabilities, and appropriate
summations replace the integrals.