Respiratory Motion Estimation with Hybrid Implementation - Prediction and Classification of Respiratory Motion - page 70

by the previous weight vector w(k -1) and Kalman gain G(k) denoting an integral
part of the EKF algorithm. We show the hat symbol (^) for predicted or filtered
estimate vectors.
Note that in Fig. 4.2 the recurrent network produces the actual output vector
y(k) resulting from the input vector u(k) using the nonlinear measure model,
including the internal activation v(k) and measurement noise r(k). In summary,
RNN executes in the supervised-training part of the prediction, whereas EKF
executes in the part of the correction with predicted or filtered estimation. We may
use this supervised-training method for a particular purpose of training continuous
respiratory motion datasets with multiple patient interaction.

4.3 Multi-Channel Coupled EKF-RNN

4.3.1 Decoupled Extended Kalman Filter

The computational complexity of EKF for RNN is mainly determined by com-
puting the filtering-error covariance matrix P(k|k) at time-step k. For a recurrent
neural network containing p output nodes and s weights, the computational
complexity of the EKF is O(ps 2 ) and its storage requirement is O(s 2 )[ 15 ].
Puskorius et al. proposed the decoupled extended Kalman filter to improve the
computational complexity by ignoring the interactions between the estimates of
certain weights in the recurrent neural network [ 15 ]. If the weights in the network
are decoupled with ''mutually exclusive weight groups [ 55 ]'', we can reduce the
computational complexity of the covariance matrix P(k|k) as shown in the block-
diagonal form in the bottom left of Fig. 4.3 [ 15 , 55 ].
Haykin defined g as the designated number of mutually exclusive weight groups
[ 55 ]. Let definde w i k jðÞ as filtered weight vector, P i (k|k) as subset of the filtering-
error covariance matrix and G i (k) as Kalman gain matrix for the group i, for i = 1,
2, …, g, respectively [ 55 ]. DEKF can be expressed as shown in the Fig. 4.3 .
In Fig. 4.3 , we denote a i ðÞ as the difference between the desired response
d i (k) for the linearized system and its estimation for the ith weight group, C(k)as
the global conversion factor for the entire network, and P i k þ 1 j k

ð Þ as the pre-

diction-error covariance matrix, respectively [ 55 ].
DEKF can reduce the computational complexity and its storage requirement of
the EKF, but [ 15 ] restricts to a DEKF algorithm for which the weights are grouped
by node. That sacrifices the computational accuracy because of omitting the
interactions between the estimates of certain weights.
To verify the prediction accuracy, we used a certain marginal value that can be
explained in detail in Sect. 4.3.4 . Figure 4.4 shows the estimation of the respira-
tory motion with DEKF. As you can see in Fig. 4.4 , we can notice that the
percentage of prediction overshoot based on the marginal value is over 35 %. That
means we need a new approach to compensate the prediction accuracy with