Blending Independent Components and
Principal Components Analysis
2.3 The underlying rationale for ICA
[this page | pdf | references | back links | custom searches]
Return to
Abstract and Contents
Next page
2.3 The underlying
rationale for ICA
ICA is based on the generic yet often physically realistic
assumption that if different input signals are coming from different underlying
physical processes then these input signals will be largely independent of each
other. ICA aims to identify how to decompose output signals into (linear
combination) mixtures of different input signals that are as independent as
possible of each other. Several variants exist, which we also describe below, where
‘independent’ is replaced by an alternative statistical property that we might
also expect might differentiate between input signals.
ICA is related to more traditional methods of analysing
large data sets believed to involve linear combinations of underlying factors,
such as principal components analysis (PCA) and factor analysis
(FA). However, it arguably differs from them in important ways. ICA seeks to
find a set of independent source signals. In contrast, PCA and FA seek
to find a set of signals which are merely uncorrelated with each other.
By uncorrelated we mean that the correlation coefficients between the different
supposed input signals are zero. Lack of correlation is a potentially much
weaker property than independence. Independence implies a lack of correlation,
but lack of correlation does not imply independence. The correlation
coefficient in effect ‘averages’ the correlation across the entire
distributional form. For example, two signals might be strongly positively
correlated in one tail, strongly negatively correlated in another tail, and
show little correspondence in the middle of the distribution. The correlation
between them, as measured by their correlation coefficient, might thus be zero,
but it would be wrong then to conclude that the behaviour of the two signals
were independent of each other (particularly, in this instance, in the tails of
the distributional form).
How this works in practice can perhaps best be introduced,
as in Stone (2004), by using the example of two people speaking into two
different microphones, the aim of the exercise being to differentiate, as far
as possible, between the two voices. The microphones give different weights to
the different voices (e.g. there might be a muffler between one of the speakers
and one of the microphones). To simplify matters the microphones are assumed to
be equidistant from each source, so that phase differentials are not relevant
to the problem at hand. ICA and related techniques rely on the following
observations:
(a) The two input
signals, i.e. the two individual voices, are likely to be largely independent
of each other, when examined at fine time intervals. However, the two output
signals, i.e. the signals coming from the microphones will not be as
independent, since they involve mixtures (albeit differently weighted) of the
same underlying input signals.
(b) If histograms of the
amplitudes of each voice (when examined at these fine time intervals) are
plotted then they will most probably differ from the traditional bell-shaped
histogram corresponding to random noise. Conversely, the signal mixtures are
likely to be more normal in nature.
(c) The temporal
complexity of any mixture is typically greater than (or equal to) that of its
simplest, i.e. least complex, constituent source signal.
These observations lead to the following algorithm for
source signal extraction:
If source signals have some property X and signal
mixtures do not (or have less of it) then given a set of signal mixtures we
should attempt to extract signals with as much X as possible, since these
extracted signals are then likely to correspond as closely as possible to the
original source signals.
Different variants of ICA and its related techniques ‘unmix’
output signals, thus aiming to recover the original input signals, by
substituting ‘independence’, ‘non-normality’ and ‘lack of complexity’ for X
in the above prescription.
NAVIGATION LINKS
Contents | Prev | Next