Page 61 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

50 PARAMETER ESTIMATION

N probes N probes z
pðzjxÞ¼ UðzÞ gamma pdf ; N probes ð3:5Þ
x x
Figure 3.4(b) shows the conditional density.

Cost functions
The optimization criterion of Bayes, minimum risk, applies to statistical
parameter estimation provided that two conditions are met. First, it
must be possible to quantify the cost involved when the estimates differ
from the true parameters. Second, the expectation of the cost, the risk,
should be acceptable as an optimization criterion.
Suppose that the damage is quantified by a cost function
x
C(^ xjx): R M R M ! R. Ideally, this function represents the true cost.
In most applications, however, it is difficult to quantify the cost accur-
ately. Therefore, it is common practice to choose a cost function whose
mathematical treatment is not too complex. Often, the assumption is
that the cost function only depends on the difference between estimated
x
and true parameters: the estimation error e ¼ ^ x x. With this assump-
tion, the following cost functions are well known (see Table 3.1):

. quadratic cost function:
M 1
2
X
x
x
Cð^ xjxÞ¼ ^ x xk ¼ ð ^ x x m x m Þ 2 ð3:6Þ
k
2
m¼0
. absolute value cost function:
M 1
X
x
x
Cð^ xjxÞ¼ ^ x xk ¼ j ^ x x m x mj ð3:7Þ
k
1
m¼0
Table 3.1 Three different Bayes estimators worked out for the scalar case
MMSE estimation MMAE estimation MAP estimation
Quadratic cost function Absolute cost function Uniform cost function
ˆ (x–x) 2 ˆ |x–x| C(x ˆ |x)
∆
ˆ x–x ˆ x–x ˆ x–x

^ x x MMSE (z) ¼ E[xjz] ^ x x MMAE (z) ¼ ^ x ^ x x MAP (z) ¼ argmaxfp(xjz)g
x
R R ^ x x 1 x
¼ xp(xjz)dx with p(xjz)dx ¼
x 1 2