Phantom: Prediction of Human Motion with Distributed Body Sensors - Prediction and Classification of Respiratory Motion - page 42

A model-based approach to deal with the clustering problem consists of certain
models, e.g., a Gaussian or a Poisson model for clusters and attempting to
optimize the fit between the data and the model. The most widely used clustering
method of this kind is a Gaussian mixture model (GMM) [ 30 - 32 ]. In GMM, the
joint

probability density that consists of the mixture of Gaussians

; where y ¼ 1...G ; should be solved [ 31 , 32 ]. Assume a training set
of independent and identically distributed points sampled from the mixture, and our
task is to estimate the parameters, i.e., prior probability

/ z; m y ; P y

a y , mean (m y ) and

covariance P y of the clustering components (G) that maximize the log-likelihood
function d ðÞ based on EM algorithm [ 32 , 33 ]. Given an initial estimation

, EM algorithm calculates the posterior probability p(y|z j ) in E-step.

a 0 ; m 0 ; P 0

Based on the estimated result we can calculate the prior probability a y , mean (m y )
and

covariance P y for

the next iteration, respectively, in the following

M-step [ 48 - 50 ].
The log-likelihood for the incomplete datasets in the EM algorithm can never
be decreased (see Sect 1.6 in [ 51 ]), because the EM algorithm iterates the com-
putations E-step and M-step until the convergence to a local maximum of the
likelihood function. That means that the consecutive log-likelihood functions
monotonically increase and could be very similar but not identical. We define the
discrepancy of consecutive values as the difference ðÞ . Now, we can define the
difference ðÞ as follows:

D ðÞ d H ; G ð Þ d H ; G 1 ð Þ: ð 3 : 3 Þ

n o G
y ¼ 1 we can find the optimal
cluster number (G*) with the conventional Brute-force search algorithm by
introducing D ðÞ that is a log-likelihood function after parameter learning with the
following equation: G ¼ argmin g D ðÞ . In practice, we can set a threshold D t ðÞ
that is almost closed to zero to save the redundant iteration step. We can start with
G = 2 for a first candidate solution for cluster number selection, estimate the set of
finite mixture model parameter H a y ; m y ; P y

Once we estimate the parameter H a y ; m y ; P y

n o G

using EM algorithms based

y ¼ 1

on the sample data, and calculate D ðÞ . After checking whether a candidate G is
an appropriate cluster number for L, we can use the cluster number G as an
appropriate cluster number as shown in Fig. 3.3 .
The search algorithm based on the log-likelihood function in Fig. 3.3 can only
work in the static data model, but cannot guarantee to work in the time sequential
data because of the lack of adaptive parameter for the time sequential data. Thus, it
has the following two limitations: (1) it can only work within limitation of the
initial dataset, and (2) it cannot guarantee the global optimal based on the time