New Statistical Algorithms for the Analysis of Mass - FU Berlin, FB MI ...

50 CHAPTER 3. MATHEMATICAL MODELING AND ALGORITHMS
3.7 Identifying Potential Features
This step identifies masterpeak (potential features) that can be used to discriminate
two groups based on their respective properties (e.g. differences in
average height, see Fig. 3.6.13). That is, we consider a masterpeak a feature if
this masterpeak can be used to discriminate two sets of spectra. For example,
if a masterpeak at position x does only occur in one of these spectra sets it is
a feature since the detection or absence of this peak would clearly assign it to
one of the groups.
3.7.1 Our Approach
After the preprocessing steps we now have information about masterpeaks
of two patient (spectra) groups under scrutiny. To enable the creation of
fingerprints (see next section 3.8) we first need to create a set of potential
differences between these two groups of spectra. We define two spectra to be
different (with respect to one particular property) if
a) a masterpeak existent in one group does not occur in the other group
b) a masterpeak exists in both groups but differs significantly in some property
between the two groups.
In other words, the feature detection step identifies a set of masterpeaks that
differ significantly in particular properties (e.g. height, width) between two
groups of spectra with respect to some metric. With these information we can
subsequently analyze for sub-sets / patterns (fingerprints) by detection and
subsequent selection of the most significant combination of features.
Choosing the Metric
A metric or distance function defines a distance between two elements of a
set. The elements of our set are masterpeaks that are defined by property
distributions of their assigned single peaks, such as m/z values, height or
area. What we want is a distance function that equals to some very large
number (or infinity) if it does not make sense to compare them (that is, their
respective m/z values are too different) or incorporates the (dis-)similarity of
their property distributions otherwise.
Therefore, we need some (symmetric) method of measuring the similarity
between two probability distributions which we found in the Jensen-Shannon
(JS) divergence (see e.g. (Gómez-Lopera et al., 2000) and references therein)
because it can be computed quickly, has shown good results in similar applications
and does not assume strong properties of the data, such as being
normally distributed: For probability distributions ”P” and ”Q” of a discrete
variable the JS-divergence of ”Q” from ”P” is defined as:
Definition 3.7.1. Kullback-Leibler (KL) divergence (S. Kullback, 1951):
DKL(P �Q) = �
i
DJS = 1
2
P (i) log P (i)
Q(i) .
Definition � 3.7.2. � �Jensen-Shannon
� � (JS) � divergence �� (Lin, 1991):
�
�
DKL P � + DKL Q�
.
P +Q
2
P +Q
2
Of course there are other probability distance measures, for example histogram
intersection (Jia et al., 2006), Kolmogorov-Smirnov distance (Fasano
and Franceschini, 1987) or the earth mover’s distance (Rubner et al., 2000),
but these usually have quite strong requirements to the data.