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

86 CHAPTER 4. (BIO-)MEDICAL APPLICATIONS
γ (l)
i (x) =
θ (l)
i
=
⎧
1
⎪⎨
PK �x−θ(l−1) �
k=1 ( i
⎪⎩
2
�x−θ (l−1)
if Ix is empty,
1
) m−1
�2 k �
r∈Ix γ(l)
(4.3.12)
r (x) = 1 if Ix is not empty, i ∈ Ix,
0 if Ix is not empty, i �∈ Ix,
� |X|
t=1 γ(l)
i (xt) · xt
� |X|
t=1 γ(l)
i (xt)
. (4.3.13)
where Ix = {p ∈ 1, . . . , K | � x − θ (l−1)
p
� 2 = 0} (Höppner et al., 1999). As it
follows from (4.3.12), for any fixed fuzzifier m, the cluster affiliations γ (l)
i (x)
can take values between 0 and 1, for m → ∞ γ (l)
1
i(x) → K
. This feature allows
clustering of overlapping data. However, the results are quite dependent on
the choice of the fuzzifier m and there is no mathematically sound strategy
on how to choose this parameter. Moreover, it is not clear a-priori how many
clusters exist in the data and thus which value K should be initialized with.
Another problem not solved by this extensions is that the data is assumed
being (locally) stationary, that is, that the conditional expectation values θi
calculated for the respective clusters i are assumed to be time independent.
This can result in misinterpretation of the clustering results, if the data has
indeed a temporal trend, which would be the case when time series data is
analyzed.
Problems Clustering High Dimensional Data
Most real-world data sets (such as a spectrum) are very high-dimensional.
However, the performance of clustering algorithms tends to scale poorly as
the dimension of the data grows. This is often referred to as the curse of
dimensionality which seems to be a major obstacle in the development of data
mining techniques in several ways.
In (Beyer et al., 1999; Hinneburg et al., 2000) the authors show that under
certain reasonable assumptions on the data distribution in high dimensional
space all pairs of points are almost equidistant from one another for a wide
range of distributions and distance functions. In this paper the authors proved
that the difference between the nearest and the farthest data point to a given
query point (e.g. a cluster center) does not increase as fast as the distance
from the query point to the nearest points when the dimensionality goes to
infinity. In other words, the ratio of the distances of the nearest and farthest
neighbors to a given target in high dimensional space is almost 1.
In such a case, the evaluation of the distance functional becomes ill defined,
since the contrast between the distances to different data points does not exist.
Thus, even the concept of proximity may not be meaningful from a qualitative
perspective: a problem which is even more fundamental than the performance
degradation of high dimensional algorithms.
Parameterizing Distance Metrics
As indicated in the previous section the main problem of clustering highdimensional
data is the number of dimensions used in the cluster distance
functional (Equation 4.3.3). This distance metric d(x, y) is a function over