Noninvasive Cardiac Signal Analysis Using Data Decomposition Techniques - Modeling in Computational Biology and Biomedicine - page 99

2. Any linear combination of independent sources should decrease the objective.
Put in mathematical terms, given a trivial filter t , then for any spatial filter g we
have:

T T

Ψ [ g s ] ≤ Ψ [ t s ] .

This property is called domination .
3. Finally, the maximum of Ψ should be reached only for the expected sources. In
other words, the equality should occur in the inequality above only for trivial
mixtures. This can be written under the form of the discrimination property
below:

T T

If ∃ g : Ψ [ g s ]=max

t

Ψ [ t s ] ⇒ g trivial .

The discrimination property avoids the existence of spurious maxima. Optimization
criteria satisfying these properties are referred to as contrast criteria . In particular,
the above properties define the so-called multiple-input single-output contrasts,
since the extracting system generates a single output signal (an estimate of one of
the sources) from several mixtures of the sources acting as multiple inputs to the
system. These quite natural properties have already been put forward in [ 7 ]forthe
multiple-input multiple-output BSS problem, and in [ 10 ] for the single-input single-
output blind channel equalization problem.

3.3.3.3 Kurtosis Contrast Criterion

A variety of optimization criteria can be devised depending on the assumptions
available on the sources [ 8 , 9 ]. In the remaining of this chapter, we will solely
assume that the M sources are mutually statistically independent and that at least
( M −

1) are non Gaussian. Moreover, contrary to [ 7 , 27 ], we will concentrate here
on the extraction of sources one by one, a separation procedure known as deflation .
Contrast criteria are in fact not the same in both cases [ 9 , Chap. 3].
The first idea is to search for extremal values of γ , the fourth-order cumulant
of s , linked to the cumulants of the observations through Eq. ( 3.23 ). According to
Sect. 3.3.3.1 , this search will indeed maximize the gap to Gaussianity. But one can
notice that is unbounded, which is undesirable. This

problem can be fixed by normalizing s by its standard deviation, leading to the
kurtosis maximization (KM) criterion:

|γ| is unbounded above if w

γ
var 2

= E

{ s

4

}− 3E 2 { s

2

}

Ψ KM [ s ]= . (3.24)

{ s} E 2 { s

2

}

This criterion can be expressed in terms of vector w by exploiting the multilinearity
property ( 3.23 ):

Ψ KM ( w )= mnpq w m w n w p w q C mnpq

( ij w i w j R ij ) 2

, (3.25)