Phantom: Prediction of Human Motion with Distributed Body Sensors - Prediction and Classification of Respiratory Motion - page 45

iteration for the collaborative grouping for time sequential data, respectively, using
( 3.7 ). A local maximum at time k can be selected by iterating the above two steps.
We can select the collaborative cluster number (G* ADT ) by introducing D ADT ðÞ
that is a log-likelihood function after parameter learning with the following
equations:

G ADT ¼ arg min
G

D ADT ðÞ;
D ADT ðÞ¼ d ADT H ; G

ð 3 : 8 Þ

Þ d ADT H ; G 1 ð ð Þ

where d ADT is a log-likelihood function with the adaptive posterior probability.

Note that Eq. ( 3.3 ) is extended into Eq. ( 3.8 ) with the hyper-parameter b y .
Comparing with the previous algorithm, the collaborative grouping with time
sequential data can select local maxima at time k by iterating two steps: E-step and
M-step. We can select the global optimal from a series of local maxima of time k,
as shown in Fig. 3.4 .

3.3.2 Estimated Parameters Used for Interacting
Multiple Model Estimator

Collaborative grouping with time sequence data can select local maxima at time
k using the difference ðÞ of the consecutive log-likelihood functions. We can set
the difference ðÞ of the consecutive log-likelihood functions as D G ; ð Þ with
respect to time k. To reduce the notation complexity, D ðÞ is simply used for
D G ; ð Þ . Now we can use this D ðÞ for IMME to estimate the multi-channel
estimates and covariance. As mentioned, the log-likelihood function in each EM
step cannot decrease [ 51 ]. That means we can minimize the difference D ðð Þ of
the consecutive log-likelihood functions with respect to time k because D ðÞ

Fig. 3.4 Collaborative group
number selection with the
adaptive hyper-parameter

set β y ; // hyper-parameter

set

μ y ;

// switching probability

calculate β y ( k ) // adaptive hyper-parameter

while ( G ≠ L ))
{

E-Step : calculate p ADT ( yz j );
M-Step : calculate α y , m y ,and∑ y ;
calculate Δ ADT ( G );
temp = Δ ADT ( G );
if ( minDelta ≥ temp )
minDelta = temp ;

}