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

60 CHAPTER 3. MATHEMATICAL MODELING AND ALGORITHMS
manifold M of smaller dimension than the original d-dimensional data space
X . Its primary goal is to find a representation of M, allowing to project the
data (Xn) N n=1 on it and thus obtain a low-dimensional, compact representation
of the noisy sample. The challenge is to find a linear or nonlinear mapping
f : Rd → Rm with m ≤ d such that reconstruction error of the sample,
Eλ[{Xn} N n=1] = 1
N
N�
λ(Xn, X ′ n)
is acceptably small. Here, X ′ n = f(Xn) is the reconstructed vector for Xn
and λ a suitable distance measure in X . At best one will get rid of some
computational obstacles:
n=1
i) efficiency and classification performance
ii) measurement, storage and computation costs
iii) ease of interpretation and modeling
The problem of DR technically decomposes into two tasks: First one has
to determine elements from the target space. Second, one has to construct a
basis of the target space from these elements. Hence DR is often achieved by
semi-parametric feature extraction.
In this work we use DR techniques to reduce the dimensionality of the
fingerprint feature vectors by projecting from the high dimensional feature
space to some low-dimensional embedding. This enables subsequent clustering
algorithms to work reliably in two or three dimensions, since they usually are
not designed to cluster points in high-dimensional space (see e.g. the curseof-dimensionality
and section 4.3.3).
As in the case of feature selection for example in the machine learning
community much effort has been put into the development of sophisticated
(general) DR algorithms. Therefore, we did not develop an own approach but
evaluated existing ones with the fingerprints gained from the feature selection
step. We can show that it is possible to reduce the complexity of these
fingerprints even more by only losing very little classification power. This results
in a classifier being a line or a plane and gains comprehensible clustering
possibilities in R 2 or R 3 which are easily visualizeable.
All of the dimensionality reduction (DR) algorithms project the n-dimensional
fingerprint data 18 to d = 2 dimensions 19 . Subsequently the resulting points
are clustered by k-means clustering 20 and a goodness of clusters (GOC) score
is calculated as follows:
GOC =
�n |Ci|
i=1 max(|Ci,G1|,|Ci,G2|)
n
(3.8.6)
where n: number of clusters, |Ci|: number of total points in cluster Ci, |Ci,Gx|:
number of points of group x in cluster i.
For the evaluation we used some widely used algorithms of three classes:
18 n being the number of features
19 The performance can be increased by projecting into d = 3 dimensions (data not shown)
but for easier visualization we took only two dimensions.
20 k = 5