Page 73 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

62 PARAMETER ESTIMATION

Substitution of (3.32), (3.28) and (3.29) in (3.26) gives rise to the
following estimator:

1
T
T

^ x x ulMMSE ðzÞ¼ m þ Kðz Hm Þ with K ¼ C x H
x x HC x H þ C v
ð3:33Þ
This version of the unbiased linear MMSE estimator is the so-called
Kalman form of the estimator.
Examination of (3.20) reveals that the MMSE estimator in the Gaussian
case with linear sensors is also expressed as a linear combination of m and
x
z. Thus, in this special case (that is, Gaussian densities þ linear sensors)
x
x
^ x x MMSE (z) is a linear estimator. Since ^ x ulMMSE (z)and ^ x MMSE (z)are based on
the same optimization criterion, the two solutions must be identical here:
^ x x ulMMSE (z) ¼ ^ x MMSE (z). We conclude that (3.20) is an alternative form of
x
(3.33). The forms are mathematically equivalent. See exercise 5.
The interpretation of ^ x ulMMSE (z) is as follows. The term m represents
x
x
the prior knowledge. The term Hm is the prior knowledge that we have
x
about the measurements. Therefore, the factor z Hm is the informa-
x
tive part of the measurements (called the innovation). The so-called
Kalman gain matrix K transforms the innovation into a correction term
K(z Hm ) that represents the knowledge that we have gained from the
x
measurements.

3.2 PERFORMANCE OF ESTIMATORS

No matter which precautions are taken, there will always be a difference
between the estimate of a parameter and its true (but unknown) value.
The difference is the estimation error. An estimate is useless without an
indication of the magnitude of that error. Usually, such an indication is
quantified in terms of the so-called bias of the estimator, and the vari-
ance. The main purpose of this section is to introduce these concepts.
Suppose that the true, but unknown value of a parameter is x.An
x
estimator ^ x(:) provides us with an estimate ^ x ¼ ^ x(z) based on measure-
x
x
ments z. The estimation error e is the difference between the estimate
and the true value:
e ¼ ^ x x ð3:34Þ
x
Since x is unknown, e is unknown as well.