Geoscience Reference
In-Depth Information
this property is characteristic of white noise. Although the separated signals
are uncorrelated, they can still be interdependent, i.e., they may retain a
nonlinear correlation. h is phenomenon arises when, for example, the data
are not Gaussian distributed and the PCA consequently does not yield good
results. h e
independent component analysis
(ICA) was developed for this
type of task; it separates the variables
X
into independent variables
S
, which
are then nonlinearly uncorrelated. h e basis of an ICA, according to the
central limit theorem, is that the mixture of standardized random numbers
is Gaussian distributed. h e ICA algorithms therefore use a criterion that
estimates how Gaussian the combined distribution of the independent
components is (Hyvärinen 1999). h e less Gaussian this distribution, the
more independent the individual components.
According to the linear mixing model,
p
independent variables
X
are
linearly mixed in
n
measurements,
in which we are interested in the source variables
S
and the mixing matrix
A
(see Section 9.2). For example we can imagine that we are at a party in
which a lot of people are carrying on independent conversations. We can
hear a mixture of these conversations but perhaps cannot distinguish them
individually. We could install some microphones and use these to separate
out the individual conversations: hence, this dilemma is sometimes known
as the
cocktail party problem
. Its correct term is
blind source separation
,
which is dei ned by
where
W
T
is the separation matrix required to reverse the mixing and obtain
the original signals. In earth sciences we encounter similar problems, for
example, if we want to determine the relative contributions of dif erent
source rocks to basin sediments, as we did with the PCA but this time with
the possibility that there are nonlinear dependencies in the data and that
these are not Gaussian distributed (Section 9.2).
We again create a synthetic data set consisting of thirty measurements (the
proportions of each of the three minerals) from each of the thirty sediment
samples. In contrast to the PCA example, however, the temporal variation
in the source rocks is not Gaussian distributed but is uniformly distributed,
since we use
rand
instead of
randn
to create the pseudorandom numbers.
clear
