Page 59 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

48 PARAMETER ESTIMATION

p(x). Associated with x is a physical object, process or event (in short:
‘physical object’), of which x is a property. x is called a parameter
vector, and its density p(x) is called the prior probability density.
The object is sensed by a sensory system which produces an N-dimen-
sional measurement vector z. The task of the parameter estimator is to
recover the original parameter vector x given the measurement vector z.
M
x
This is done by means of the estimation function ^ x(z): R N ! R . The
conditional probability density p(zjx) gives the connection between the
parameter vector and measurements. With fixed x, the randomness of
the measurement vector z is due to physical noise sources in the sensor
system and other unpredictable phenomena. The randomness is charac-
terized by p(zjx). The overall probability density of z is found by averaging
the conditional density over the complete parameter space:

Z
pðzÞ¼ pðzjxÞpðxÞdx ð3:1Þ
x
M
The integral extends over the entire M-dimensional space R .
Finally, Bayes’ theorem for conditional probabilities gives us the
posterior probability density p(xjz):

pðzjxÞpðxÞ
pðxjzÞ¼ ð3:2Þ
pðzÞ

This density is most useful since z, being the output of the sensory
system, is at our disposal and thus fully known. Thus, p(xjz) represents
exactly the knowledge that we have on x after having observed z.

Example 3.2 Estimation of the backscattering coefficient
The backscattering coefficient x from Example 3.1 is within the
interval [0,1]. In most applications, however, lower values of the
coefficient occur more frequently than higher ones. Such a preference
can be taken into account by means of the prior probability density
p(x). We will assume that for a certain application x has a beta
distribution:

ða þ b þ 1Þ! a b
pðxÞ¼ x ð1 xÞ for 0 x 1 ð3:3Þ
a!b!
The parameters a and b are the shape parameters of the distribution.
In Figure 3.4(a) these parameters are set to a ¼ 1 and b ¼ 4. These