Page 72 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN ESTIMATION 61

Example 3.5 Unbiased linear MMSE estimation of the
backscattering coefficient
In the scalar case, the linear MMSE estimator takes the form:

cov½x; zŠ
^ x x ulMMSE ðzÞ¼ ðz E½zŠÞ þ E½xŠ ð3:30Þ
Var½zŠ
where cov[x,z] is the covariance of x and z. In the backscattering
problem, the required moments are difficult to obtain analytically.
However, they are easily estimated from the population of the 500
realizations shown in Figure 3.6 using techniques from Chapter 5.
The resulting estimator is shown in Figure 3.6. MATLAB code to plot
the ML and unbiased linear MMSE estimates of the backscattering
coefficient on a data set is given in Listing 3.2.

Listing 3.2
MATLAB code for unbiased linear MMSE estimation.

load scatter; % Load dataset (zset,xset)
z ¼ 0.005:0.005:1.5; % Interesting range of z
x_ml ¼ z; % Maximum likelihood
mu_x ¼ mean(xset); mu_z ¼ mean(zset);
K ¼ ((xset-mu_x)’*(zset-mu_z))*inv((zset-mu_z)’*(zset-mu_z));
a ¼ mu_x K mu_z;

x_ulmse ¼ K z þ a; % Unbiased linear MMSE
figure; clf; plot(zset,xset,’.’); hold on;
plot(z,x_ml,’k-’); plot(z,x_ulmse,’k--’);
Linear sensors
The linear MMSE estimator takes a particular form if the sensory system
is linear and the sensor noise is additive:
z ¼ Hx þ v ð3:31Þ

This case is of special interest because of its crucial role in the Kalman filter
(to be discussed in Chapter 4). Suppose that the noise has zero mean with
covariance matrix C v . In addition, suppose that x and v are uncorrelated,
i.e. C xv ¼ 0. Under these assumptions the moments of z are as follows:
m ¼ H m x
z
T
C z ¼ HC x H þ C v
ð3:32Þ
C xz ¼ C x H T
C zx ¼ HC x
The proof is left as an exercise for the reader.