Respiratory Motion Estimation with Hybrid Implementation - Prediction and Classification of Respiratory Motion - page 73

another. For i = 1, 2, …, c, let define w CP
i

k jðÞ as filtered weight vector, P CP

i k jðÞ
and G C i ðÞ are subset of the filtering-error covariance matrix and Kalman gain
matrix for the channel number i, respectively. In light of these new notations, we
can write the HEKF as follows:

ðÞ¼ d i ðÞ½ X

p

a CP
i

l ij y j ; ð 4 : 6 Þ

j ¼ 1

ðÞ¼½ X

p

X

p

Þ T þ R ðÞ 1 ;

C CP l ij B i ðÞ P CP

ð k j k 1 Þ B i ðÞ ð ð 4 : 7 Þ

i

i ¼ 1 j ¼ 1

Þ T C CP G CP

i

ðÞ¼ P CP
i

ð k j k 1 Þ B i ðÞ ð ðÞ; ð 4 : 8 Þ

w CP
i

Þ w CP
i

Þ G CP
i

ðÞ a CP
i

ð k þ 1 j k ð k j k 1 ðÞ; ð 4 : 9 Þ

P CP
i

Þ P CP
i

ð k þ 1 j k k jðÞþ Q i ðÞ; ð 4 : 10 Þ

P CP
i

k jðÞ¼ P CP
i

Þ G CP
i

ðÞ B i ðÞ P CP ð k j k 1 ð k j k 1 Þ; ð 4 : 11 Þ

i

where a CP
i

ðÞ; C CP ðÞ and P CP

i k þ 1 jð Þ denote the difference between the desired
response d i (k) for the linearized system and coupled estimations for the channel
number i, the global conversion factor for the entire-coupled network, and the
prediction-error covariance matrix for the coupled, respectively. In the case of
HEKF, we have c identical networks for c input channels. Each input sequence is
inserted into individual neural network process for each channel prediction, as
shown in Fig. 4.5 .

4.3.3 Optimized Group Number for Recurrent
Multilayer Perceptron

In DEKF algorithm the weights connecting inputs to a node are grouped together,
whereas each group in HEKF algorithm corresponds to the individual channel that
is composed of position vector sequence with respect to time k. In order to analyze
the group number, we can incorporate Fisher Linear Discriminant on the dis-
criminant analysis, which employs the within-class scatter value (S W ) and the
between-class scatter value (S B ) in the given samples [ 56 ].
We have a set of nD-dimensional samples, which correspond to the filtering-
error covariance matrices (P i &P C i ) defined in Eqs. ( 4.19 ) and ( 4.26 ) for each
group i. Let m i denote the D-dimensional sample mean for group i, and then define
m i as follows [ 56 ]: