Page 70 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

BAYESIAN ESTIMATION 59

If the measurement vector z is linear in x and corrupted by additive
Gaussian noise, as given in equation (3.18), the likelihood of x is given in
(3.21). Thus, in that case:

T 1 1 T 1
^ x x ML ðzÞ¼ H C H H C z ð3:24Þ
v v
A further simplification is obtained if we assume that the noise is white,
i.e. C v I:

1
T
T
^ x x ML ðzÞ¼ ðH HÞ H z ð3:25Þ
T
T
1
The operation (H H) H is the pseudo inverse of H. Of course, its
T
validity depends on the existence of the inverse of H H. Usually, such is
the case if the number of measurements exceeds the number of para-
meters, i.e. N > M. That is, if the system is overdetermined.
3.1.5 Unbiased linear MMSE estimation

The estimators discussed in the previous sections exploit full statistical
knowledge of the problem. Designing such an estimator is often difficult.
The first problem that arises is the adequate modelling of the probability
densities. Such modelling requires detailed knowledge of the physical
process and the sensors. Once we have the probability densities, the
second problem is how to minimize the conditional risk. Analytic
solutions are often hard to obtain. Numerical solutions are often
burdensome.
If we constrain the expression for the estimator to some mathematical
form, the problem of designing the estimator boils down to finding the
suitable parameters of that form. An example is the unbiased linear
MMSE estimator with the following form: 2
^ x x ulMMSE ðzÞ¼ Kz þ a ð3:26Þ

The matrix K and the vector a must be optimized during the design
phase so as to match the behaviour of the estimator to the problem at
hand. The estimator has the same optimization criterion as the MMSE

2
The connotation of the term ‘unbiased’ becomes clear in Section 3.2.1. The linear MMSE
x
(without the adjective ‘unbiased’) also exists. It has the form ^ x lMMSE (z) ¼ Kz. See exercise 1.