Page 341 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

330 WORKED OUT EXAMPLES

n (t) and u n (t) are eigenvalues and eigenvectors of C zjt . Using (9.16) the
expression for (zjt) can be moulded into the following equivalent form:

N 1 u n ðtÞu ðtÞ ! N 1 ðz u n ðtÞÞ 2
T
T
X
X
ðzjtÞ¼ z T n z ¼ ð9:17Þ
n ðtÞ l n ðtÞ
n¼0 n¼0
The computational savings are obtained by discarding all terms in (9.17) that
do not capture much information about the true value of t. Suppose that n
and u n are arranged according to their importance with respect to the
estimation, and that above some value of n,say J, the importance is negli-
gible. With that, the number of terms in (9.17) reduces from N to J. Experi-
ments show that J is in the order of 10. A speed up by a factor of 1000 is
feasible.

Selection of good components
The problem addressed now is how to order the eigenvectors in (9.17) such
that the most useful components come first, and thus will be selected. The
eigenvectors u n (t) are orthonormal and span the whole space. Therefore:

N 1
T 2 2
X
ðz u n ðtÞÞ ¼kzk ð9:18Þ
n¼0
Substitution in (9.17) yields:

N 1 T 2 N 1 T 2 2
X ðz u n ðtÞÞ X ðz u n ðtÞÞ kzk
ðzjtÞ¼ þ 2 2
n ðtÞ
n¼0 n¼0 v v
ð9:19Þ
N 1 2 2
X n ðtÞ v T 2 kzk
¼ ðz u n ðtÞÞ
n ðtÞ 2 2
n¼0 v v
The term containing kzk does not depend on t and can be omitted. The
maximum likelihood estimate for t appears to be equivalent to the one
that maximizes:

N 1 2
X n ðtÞ
T 2 v

n ðtÞðz u n ðtÞÞ with
n ðtÞ¼ ð9:20Þ
n ðtÞ 2
n¼0 v
The weight
n (t) is a good criterion to measure the importance of an
eigenvector. Hence, a plot of the
n versus n is helpful to find a reasonable