PeakLab v1 Documentation Contents AIST Software Home AIST Software Support

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 a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = width 1 (frequency invariant)

a_{3} = width
2 (frequency dependent)

Built in model: GaussCnstr

User-defined peaks and view functions: GaussCnstr(x,a_{0},a_{1},a_{2},a_{3})

Constrained Gaussian (Amplitude)

The constrained Gaussian with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = width 1 (frequency invariant)

a_{3} = width 2 (frequency
dependent)

Built in model:GaussCnstr[amp]

User-defined peaks and view functions: GaussCnstr[amp](x,a_{0},a_{1},a_{2},a_{3})

In this first example, wave numbers range from 10,000cm^{-1} to 70,000cm^{-1} with an
a_{2} non-frequency dependent width varying from 0cm^{-1} to 300cm^{-1}, and with
a constant a_{3}=0.1 across six points in the visible and UV bands.

In this example, a single 25,000cm^{-1} peak, 100% frequency dependent (a_{2}=0), is varied
in a_{3} from .02 to .05. The higher a_{3}, 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 a_{1}a_{3}+a_{2} is but a single parameter. What gives this
model validity is sharing a_{2} and a_{3} across all peaks. The concept of this model
is to fit many peaks with only two widths.

Single a_{2}, a_{3}

The a_{2} width represents a constant line spread function, the width of each peak due to effects
which have no frequency or energy dependence. The a_{3} 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 a_{2} and a_{3}
is always fit. Widths and shapes cannot be varied.