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3.6. PEAK REGISTRATION (ALIGNMENT) 47
PP being the set of peak properties. DA(Ci) is constructed as follows:
Kernel Density Estimation (KDE) 15 (Scheather, 2004) is performed on A(xk) ∀ xk ∈
Ci, interpolated and scaled to result in a probability density function.
We have used this particular clustering approach since preliminary experiments
have shown that this technique was able to best resolve the clusters in
the |PP|-dimensional space we work in (typically |PP| = 6 . . . 8). Analyses
have shown that the DA(Ci) are usually not unimodal thus favoring a clustering
schema that does not create spherical clusters, such as k-means clustering
(see e.g. (Deonia et al., 2007)).
Determine Cluster Properties
This step determines the properties (such as center, height, . . . ) of
the clusters found in the previous step. For each property a KDE k
is performed on the values of the single peaks the masterpeak consists
of. If k is not similar to a normal distribution (tested by: Kolmogorov-
Smirnov test, Lilliefors test, Anderson-Darling test, Ryan-Joiner test,
Shapiro-Wilk test, Normal probability plot (rankit plot), Normality
test, Jarque-Bera test) the masterpeak is flagged.
Merge Similar Clusters
Since the clustering in the previous step is not deterministic this step
repairs clusters that have been divided into two or more sub-clusters.
Two clusters are merged together if all properties or all properties except the
center are similar (measured by the Jensen-Shannon divergence, see section
3.7.1).
Outcome: List of Masterpeaks
The above procedure finally results in a list of masterpeaks. That is a list
of peaks clustered by positions and properties (such as position or height)
represented by the average values for these properties.
3.6.2 Other Approaches
An alternative approach to enable different spectra for comparison is to align
them and described below. This means, define some reference key peaks (e.g.
based on known house-keeping molecules), find these peaks in each spectrum
and reorientate each spectrum towards these peaks. Obvious problems are:
� How to detect the key peaks ?
� What if key peaks are missing ?
� What if there is a peak similar and next to a key peak ? Which is the
right one ?
The main issue here is that during the reorientation process a spectrum
gets partly distorted, because between two key peaks a linear adjustment takes
place but the parts are not directly tied together.
This is shown exemplarily in Figure 3.6.15: spectrum (b) is reorientated
on the basis of its detected key peaks towards the position of the reference key
15 Following the Parzen Window approach with Gaussian Kernel.
Figure 3.6.14: These
are the parameters being
determined for each peak.