Other Applications: Sequential Dependence Modelling and Data Mining - On Statistical Pattern Recognition in Independent Component Analysis Mixture Modelling - page 155

Table 7.1 SICAMM algorithm
Initialization n ¼ 0
X ð 0 Þ¼ x ð 0 Þ ½

s k ð 0 Þ¼ A 1
k

ð x ð 0 Þ b k Þ k ¼ 1...K;

p s k ð 0 Þ

det A 1
k

½

p s k 0 ð 0 Þ

pC k ð 0 Þ= X ð 0 Þ ½ ¼ P k 0 ¼ 1 det A 1

½

k 0

For n ¼ 1 to N
X ð n Þ¼ x ð 0 Þ x ð 1 Þ ...x ð n Þ ½

s k ð n Þ¼ A 1
k

ð x ð n Þ b k Þ k ¼ 1...K

¼ P

K

pC k ð n Þ= X ð n 1 Þ ½ pC k ð n Þ= C k 0 ð n 1 Þ ½ pC k 0 ð n 1 Þ= X ð n 1 Þ ½

k 0 ¼ 1

p s k ð n Þ

det A 1
k

½ pC k ð n Þ= X ð n 1 Þ ½

p s k 0 ð n Þ

pC k ð n Þ= X ð n Þ ½ ¼ P k 0 ¼ 1 det A 1

½ pC k 0 ð n Þ= X ð n 1 Þ ½

k 0

7.1.3 Simulations

We have considered a simple scenario that is similar to the first example included in
the classical ICAMM [ 1 ]. Observations are vectors of dimension 2, and the number of
classes is also 2. In class 1, the observation vectors are obtained by linearly trans-
forming independent component vectors where both components are obtained from
uniform distributions having zero mean and unit variance. In class 2, the observation
vectors are obtained in the same way as in class 1, but the distributions are zero mean
and unit variance Laplacian. The centroids were selected relatively close, thus b 1 ¼
½ 11 T and b 2 ¼½ 1 : 51 : 5 T . We have compared the error percentages in classifying
an observation as belonging to class 1 or to class 2 using ICAMM and SICAMM.
The parameters of the ICA mixtures were estimated using the Mixca procedure of
Chap. 3 for both simulations and for the application of analysis of hypnograms.
First of all, to simplify the comparison, we have considered that the probability
of staying in the same class is the same for both class 1 and class 2. Therefore, only
one parameter a is required to establish the degree of sequential dependence, i.e.,

pC 1 ð n Þ= C 1 ð n 1 Þ ½ ¼ pC 2 ð n Þ= C 2 ð n 1 Þ¼ a ½

ð 7 : 4 Þ

pC 1 ð n Þ= C 2 ð n 1 Þ

½ ¼ pC 2 ð n Þ= C 1 ð n 1 Þ ½ ¼ 1 a

In Fig. 7.2 , we represent the estimated error percentages for a varying from 0.5
(no sequential dependence at all) to 1 (total dependence).
The error percentage was estimated by the quotient between the number of
misclassified observations and the total number of generated observations. To
obtain reliable results we considered 300 pairs of transforming matrices, A 1 and
A 2 ; which were randomly generated: every element of these matrices was obtained
from a uniform distribution between 0 and 1. For every pair of matrices, a run of
200 observation vectors was produced taking the above defined centroids and
distributions into account. Hence, the total number of observation vectors for every
a value was 60,000.