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Constrained Gaussian

Event-related data that depend upon counting statistics, such as high energy spectra, will often have peak widths which increase with energy. This model allows a simplification of Gaussian fitting when fitting multiple peaks.

Constrained Gaussian (Area)

The constrained Gaussian with a0 as the peak area is defined as follows:

a0 = Area

a1 = Center (mode)

a2 = width 1 (frequency invariant)

a3 = width 2 (frequency dependent)

Built in model: GaussCnstr

User-defined peaks and view functions: GaussCnstr(x,a0,a1,a2,a3)

Constrained Gaussian (Amplitude)

The constrained Gaussian with a0 as the peak amplitude is defined as follows:

a0 = Amplitude

a1 = Center (mode)

a2 = width 1 (frequency invariant)

a3 = width 2 (frequency dependent)

Built in model:GaussCnstr[amp]

User-defined peaks and view functions: GaussCnstr[amp](x,a0,a1,a2,a3)

In this first example, wave numbers range from 10,000cm-1 to 70,000cm-1 with an a2 non-frequency dependent width varying from 0cm-1 to 300cm-1, and with a constant a3=0.1 across six points in the visible and UV bands.

In this example, a single 25,000cm-1 peak, 100% frequency dependent (a2=0), is varied in a3 from .02 to .05. The higher a3, the greater the frequency dependent spreading.

Multiple Peaks Only

Note that this model has no validity for fitting a single peak. In such a case the denominator is clearly overspecified, where a1a3+a2 is but a single parameter. What gives this model validity is sharing a2 and a3 across all peaks. The concept of this model is to fit many peaks with only two widths.

Single a2, a3

The a2 width represents a constant line spread function, the width of each peak due to effects which have no frequency or energy dependence. The a3 term simply creates a scaled width which is linearly proportional to energy. It is not a width per se, but is used to produce a unique frequency-dependent width component for each peak. When fitting constrained Gaussians, a single a2 and a3 is always fit. Widths and shapes cannot be varied.