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

3.5. PEAK DETECTION IN 2D MAPS 43
� Create 2D (orthogonal) range tree (de Berg et al., 2000) which needs
O(n log n) time and space for creation and storage 11 , respectively, and
can answer range queries in O((log n) 2 + k) 12 (k being the number of
results). Since in a typical (medium resolution) map n ∼ 2.000.000, a
query needs about (36 + k) comparisons in time and 12MB in space.
Of course, the analysis could also be performed directly by querying the
database but by using range trees this can be done on a remote worker
(see section 5.4) without the need for using the potentially slow database
connection.
� For each spectrum St (sorted increasingly by retention time t): If an yet
, t)
unprocessed peak is found at position ( m
z
– Use this peak as seed and extend a bounding box around it.
� The extension in m/z (x) direction is given by the length of
the isotope pattern + 10% (that is, if an isotope pattern spans
from 1000 to 1010da the box would have the x-dimension: 999
to 1011da).
� In retention time (y) direction (successively (t − i) and (t +
i)) the extension is done as follows: if Si is the current 1D
spectrum, get the peaks of the next spectrum (Sj = Si±1)
within the determined x range. If the peaks found are similar
(see below) to the peaks of Si this step is repeated until no
further extension is possible.
If the peaks found are not similar the next two spectra (Sk =
Si±2 and Sl = Si±3) are checked as well to account for missing
data. If the peaks of Sk or Sl are similar to the peak of Si
the respective spectrum is set as the current one and this step
started all over.
– Fit 2D isotope pattern (mixture of 2D Gaussians) on peaks found
within the bounding box. For each 2D Gaussian 13 we can compute
the center of mass along the retention time axis and the m/z spread
along the m/z axis for each gaussian from the 1D isotopic patterns
(consisting of a mixture of Gaussians).
– Mark used peaks as processed
This procedure results in a list of 2D peaks parameterized by a mixture of
2D Gaussians.
3.5.1 Similarity Measures for Curves
The similarity (or distance) d(A, B) of two curves A, B (e.g. isotope patterns
modeled as mixture of 1D Gaussians) is measured by the difference of their
curvature based Turning Function ΘA (Arkin et al., 1991) (see (Veltkamp and
Hagedoorn, 2000) for a discussion of differently similarity measures) that is
well suited to capture differences in isotope pattern shape but is not sensitive
to height differences.
11 n log n
Optimal storage of O( log log n
) is possible by using R-trees.
12
Or O((log n) + k) in an improved fractional cascading version.
13 (x−xo)
f(x, y) = A · exp(−( 2
2σ2 2
(y−yo)
) − (
x
2σ2 )) where A is the amplitude, x0, y0 the center
y
and σx, σy the x and y spreads.