Mathematics in Independent Component Analysis

Chapter 1
Statistical machine learning of
biomedical data
Biostatistics deals with the analysis of high-dimensional data sets originating from biological or
biomedical problems. An important challenge in this analysis is to identify underlying statistical
patterns that facilitate the interpretation of the data set using techniques from machine
learning. A possible approach is to learn a more meaningful representation of the data set,
which maximizes certain statistical features. Such often linear representations have several potential
applications including the decomposition of objects into ‘natural’ components (Lee and
Seung, 1999), redundancy and dimensionality reduction (Friedman and Tukey, 1975), biomedical
data analysis, microarray data mining or enhancement, feature extraction of images in nuclear
medicine, etc. (Alpaydin, 2004, Bishop, 2007, Cichocki and Amari, 2002, Hyvärinen et al., 2001c,
MacKay, 2003, Mitchell, 1997).
In the following, we will review some statistical representation models and discuss identifiability
conditions. The resulting separation algorithms will be applied to various biomedical
problems in the last part of this summary.
1.1 Introduction
Assume the data is given by a multivariate time series x(t) ∈ R m , where t indexes time, space
or some other quantity. Data analysis can be defined as finding a meaningful representation of
x(t) i.e. as x(t) = f(s(t)) with unknown features s(t) ∈ R m and mixing mapping f. Often, f is
assumed to be linear, so we are dealing with the situation
x(t) = As(t) (1.1)
with a mixing matrix A ∈ R m×n . Often, white noise n(t) is added to the model, yielding
x(t) = As(t) + n(t); this can be included in s(t) by increasing its dimension. In the situation
(1.1), the analysis problem is reformulated as the search for a (possibly overcomplete) basis, in
which the feature signal s(t) allows more insight into the data than x(t) itself. This of course
has to be specified within a statistical framework.
There are two general approaches to finding data representations or models as in (1.1):
3