Page 69 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

58 PARAMETER ESTIMATION

Example 3.4 Maximum likelihood estimation of the backscattering
coefficient
The maximum likelihood estimator for the backscattering coefficient
(see previous examples) is found by maximizing (3.5):

dpðzjxÞ
¼ 0 ) ^ x x ML ðzÞ¼ z ð3:23Þ
dx

The estimator is depicted in Figure 3.6 together with the MAP esti-
mator. The figure confirms the statement above that in areas of flat
prior probability density the MAP estimator and the ML estimator
coincide. However, the figure also reveals that the ML estimator can
produce an estimate of the backscattering coefficient that is larger
than one; a physical impossibility. This is the price that we have to
pay for not using prior information about the physical process.

1.5
realizations
MAP estimator
x ML estimator
ulMMSE estimator

backscattering coefficient 0.5
1

0
0 0.5 1 1.5
measurement z
Figure 3.6 MAP estimation, ML estimation and linear MMSE estimation