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

4.3. STUDY RESULTS 87
pairs of objects x and y from some set X. It needs to have the following
properties for all x, y, z ∈ X ⊂ R n :
d(x, x) = 0, d(x, y) = d(y, x), d(x, y) ≥ 0, d(x, y) + d(y, z) ≥ d(x, z)
d(x, y) = 0 ⇒ x = y
Commonly, d has the following form:
�
dA,W (x, y) = (x − y) T AW AT (x − y) (4.3.14)
where x, y ∈ Rn , A can be any real matrix and W is a diagonal matrix with
non-negative entries - this corresponds to a metric that “weights” the axes
differently; more generally, W parameterizes a family of Mahalanobis distances
over Rn . Note that AW AT is semi-positive definite and thus dA,W (x, y) is a
valid distance metric.
The parameterization of A and W is very flexible. For example, setting
A = I (the identity matrix) would result in a weighted Euclidian distance
dI,W (x, y) = ��n i=1 Wii(xi − yi) 2 . Setting A �= I corresponds to applying a
linear transformation to the input data (AT x, AT y) 3 :
�
dA,W (x, y) = ((x − y) T A)W (AT (x − y)) (4.3.15)
Thus, we need to find a parameterization of A and W that allows rescaling
of the data such that only important dimensions are considered in the distance
calculation.
Clustering Spectra
Now we are coming back to our original problem: clustering spectra to detect
biologically interesting sub-clusters e.g. within the group of spectra of a particular
disease. The previous sections gave a good explanation why clustering
of whole spectra with tens of thousands of dimensions would not be very reasonable.
For the above reason the dimensionality of data sets is often reduced
by various techniques before it is clustered.
Figure 4.3.6 shows the strategy we are using to allow clustering of very
high-dimensional data. As described in sections 3.7 & 3.8.3, in this thesis we
have developed a new algorithm to learn a metric that allows low-dimensional
representations of high-dimensional mass spectra. This low-dimensional representation
can then be used in subsequent steps such as clustering. The studies
presented below (sections 4.4 & 4.5) show that our algorithms produce good
results. The following steps briefly summarize our algorithm:
1. We begin with the original spectrum after preprocessing has been performed
(see section 3.3). A raw spectrum typically has about 100.000
data points.
2. The peak detection step (see section 3.4) eliminates noise that “looks
like” peaks and identifies biologically relevant peaks. This steps detects
about 2000-4000 peaks per spectrum.
3 Note that by introducing a non-linear basis function φ and using
p (φ(x) − φ(y)) T W (φ(x) − φ(y)) a non-linear distance metrics could also be used.