Page 97 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

86 STATE ESTIMATION

The joint probability density of the sequence x(0), x(1), ... , x(i) follows
readily

pðxð0Þ; ... ; xðiÞÞ ¼ pðxðiÞjxði 1ÞjÞpðxð0Þ; .. . ; xði 1ÞÞ
i
ð4:5Þ
Y
¼ pðxð0ÞÞ pðxðjÞjxðj 1ÞÞ
j¼1

The measurement model
In addition to the state space model, we also need a measurement model
that describes the data from the sensor in relation with the state. Suppose
that at moment i the measurement data is z(i) 2 Z where Z is the
measurement space. For the real-valued state variables, the measurement
N
space is often a real-valued vector space, i.e. Z ¼ R . For the discrete
case, one often assumes that the measurement space is also finite, i.e.
Z ¼f# 1 , ... , # N g.
The probabilistic model of the sensory system is fully defined by the
conditional probability density p(z(i)jx(0), ... , x(i), z(0), .. . , z(i 1)).
We assume that the sequence of measurements starts at time i ¼ 0.
In order to shorten the notation, the sequence of all measurements up to
the present will be denoted by:

ZðiÞ¼fzð0Þ; ... ; zðiÞg ð4:6Þ

We restrict ourselves to memoryless sensory systems, i.e. systems where
z(i) depends on the value of x(i), but not on previous states nor on
previous measurements. In other words:

pðzðiÞjxð0Þ; ... ; xðiÞ; Zði 1ÞÞ ¼ pðzðiÞjxðiÞÞ ð4:7Þ

4.1.2 Optimal online estimation

Figure 4.2 presents an overview of the scheme for the online estimation
of the state. The connotation of the phrase online is that for each time
x
index i an estimate ^ x(i)of x(i) is produced based on Z(i), i.e. based on
all measurements that are available at that time. The crux of optimal
online estimation is to maintain the posterior density p(x(i)jZ(i)) for