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

32 CHAPTER 3. MATHEMATICAL MODELING AND ALGORITHMS
in some of our test cases they alter the relative peak intensities where subsequent
analyses rely on (e.g. to obtain the relative compound concentrations
correctly). This has also been found by by others (e.g. (Li et al., 2007)).
Our preliminary experiments have shown that non-parametric techniques are
more appropriate. Out of these approaches a wavelet shrinkage approach (for
a review see (Taswell, 2000); a nice introduction can be found here (Louis
et al., 1998)) has performed best on artificial data and data from spiked experiments
where we knew the peak positions and intensities. This wavelet
shrinkage approach has been reported to be very well suited for denoising of
mass spectrometry data (see e.g. (Ojanen et al., 2004; Liu, Sera, Matsubara,
Otsuka and Terabe, 2003; Coombes et al., 2005)).
Opposed to other denoising algorithms, such as moving average or lowpass
filter (e.g. Savitzky-Golay), this approach utilizes the multi-scale nature
of the signal and therefore has better energy conservation properties, that is,
the amplitude of the signal decreases less through denoising.
The multi-scale method used here is based on a time-invariant discrete
orthogonal wavelet transformation (Nason and Silverman, 1995) because orthogonal
wavelets can give the most compact representation of a signal. During
the wavelet transform at each scale high- and low-pass filters are applied to
the signal. The output from a high-pass filter is recorded as the wavelet coefficients
and represents the details of the signal. The low-pass filter extracts
the low-frequency components which are used in the next stage where another
set of high- and low-pass filters is employed.
Wavelet shrinkage denoising does involve non-linear soft thresholding (shrinking)
in the wavelet transform domain (Donoho, 1995). It is based on the
assumption that wavelet coefficients of the true signal have high amplitude,
opposed to the lowest magnitude coefficients that represent the noise (see Figure
3.3.5 for an example). Thus, by eliminating coefficients that are smaller
than a predetermined threshold this noise can be removed. Summarized, it is a
three step process: a linear forward wavelet transform, a non-linear shrinkage
denoising of the resulting coefficients and a linear inverse wavelet transform.
None of these steps needs a-priori parameterization of a particular model.
We now give a more formal definition. We assume we are given a noisy
signal (the raw data) X consisting of m (noisy) samples
X(t) = S(t) + ɛ(t), t = 1 . . . m
This raw signal X is assumed to contain the real signal S with additive noise
ɛ. Let W(·) denote the forward and W −1 (·) the inverse wavelet transform
operators and Ds(·, λ) be the denoising operator with a data adaptive soft
threshold λ. Our aim is to denoise X to estimate ˆ S which should be as close
as possible to the original signal S. The three step process mentioned above
is then:
� Linear forward wavelet transform: Y = W(X). This results in m noisy
wavelet coefficients yj,k.
� Non-linear shrinkage denoising, that is thresholding the wavelet coefficients
which includes estimating the threshold λ = {λ1, . . . , λj} for each
level j (see below)
– λ = d(Y )
– Z = Ds(Y, λ)