R. L. Coldwell
Department of Physics, University of Florida, Gainesville, FL 32611 and
Constellation Technology Corporation, 7887 Brian Dairy Road, Largo, FL 33777
Spectra consist of continuum features that vary over
many channels, and peaks that vary over few channels. In a fit to the continuum the peaks appear as outliers. Robust methods in which the weights associated
with the peaks are reduced allow the continuum to be fitted almost independently
of the peaks. This requires a very
smooth background. Cubic splines, which
are continuous with continuous first and second derivatives, are a good choice
for this task when the locations of the discontinuities in the third
derivatives, the knots, are included in the parameters optimized in the fitting
process. An extension to the Marquardt
method allows a Newton-Raphson method to be used in this optimization.
.
A spline can be defined as
(3) Where
(4)
With
c1=1 and x1 and all others 0, this is
FIGURE
1 A single knot spline and its
derivatives
The spline is the function representing the data
with the smoothest possible second derivatives. (7) (8)
Consider
a peak like bump which is 1 at x=0, extending from -1 to 1 and is zero
elsewhere, which we want to fit to a spline.
This bspline is
(5)
FIGURE 2 The
bspline of equation 5
Five knots from -1 to +1, approximately twice the
fwhm of the peak are required to produce this. It is interesting to attempt to produce this result with only
three splines.
FIGURE 3. Attempt to produce a bspline with
3 knots
The
practical value of this is that a cubic spline will be able to reproduce a peak
only if it has approximately 5 knots within a few half widths of the peak. In addition, the natural tendency of the
splines defined here is to become large for small values of x. This is also the natural tendency for spectral
data. The only coding needed to take
advantage of this reluctance of a spline to produce a peak is the requirement
that new knots be introduced at a distance from the old knots.
A
positive definite continuum is best fitted to the exponential of this
spline. The hard part of fitting
splines to data is the fitting of the knots.
Allowing these to vary, however, significantly decreases the number of
constants required to fit the continuum and more importantly eliminates the
usual curve fit problem of oscillations in the fitting function.
The final detail needed to keep the background below the peaks is the outlier regression, which increases the error estimates for data above the continuum and decreases it for those below.