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

4.3. STUDY RESULTS 85
d(x, θi) = � x − θi � 2 . (4.3.7)
Since we have only discrete observation xt, t = 1, . . . , |X|, the functional
(4.3.4) gets the form
K� |X| �
γi(xt) � xt − θi � 2 → min . (4.3.8)
Γ(X),Θ
i=1 t=1
K-means algorithm iteratively minimizes the functional (4.3.8) subject to
constraints (4.3.5 - 4.3.6) assigning the new cluster affiliations γ (l) (x) and
updating the cluster centers θ (l)
i in iteration (l) according to the following
formulas
γ (l)
i (x) =
θ (l)
i
=
�
1 i = arg min � x − θ (l−1)
i �2 0
,
otherwise,
(4.3.9)
� |X|
t=1 γ(l)
i (xt) · x
� .
|X|
(xt)
(4.3.10)
t=1 γ(l)
i
Iterations (4.3.9-4.3.10) are repeated until the change of the averaged clustering
functional value does not exceed a certain predefined threshold value.
The complexity of the k-means algorithm is O(K · |X| · L), where L is
the number of iterations. Note that it is possible that a good cluster solution
will be missed due to the algorithm converging to a local rather than global
minimum of the scoring function. One way to alleviate this problem is to carry
out multiple searches from different randomly chosen starting points for the
initial cluster centers. This can even be taken further to adopt a simulated
annealing strategy to try to avoid getting trapped in local minima of the score
function.
Fuzzy c-Means Clustering
Experiments have shown that a spectrum is very unlikely to be assigned to
exactly one (sub-)cluster. In most cases a spectra reflects a transient disease
status between two or more extrema (e.g. some stage between healthy and
fully diseased). As it can be seen from (4.3.9), this can not be represented
by the k-means algorithm and thus geometrically overlapping clusters can not
be resolved. This issue was addressed by (Bezdek, 1981) who proposed the
following modification of the averaged clustering functional (4.3.8):
K� |X| �
i=1 t=1
γ m i (xt) � xt − θi � 2 → min , (4.3.11)
Γ(X),Θ
where m > 1 is a fixed parameter called the fuzzyfier (Bezdek, 1981; Bezdek
et al., 1987). Analogously to k-means, the fuzzy c-means algorithm is an
iterative procedure for the minimization of (4.3.11)