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ZDD - [V] Generalized GMG

This is the Zero
Distortion Density (ZDD) used in the GenHVL[V] models. The [V] model is a further generalization of
the [Z] ZDD. Instead of a generalized Gaussian, the [V] uses a generalized GMG (half-Gaussian modified
Gaussian). The a_{3} parameter controls the width of the half-Gaussian convolution in the GMG
density while the the a_{4}
parameter in the ZDD nomenclature below adjusts the asymmetry as it does in the generalized normal. The
net effect is that there are two parameters that adjust the third moment or skew of the peak. There is
no specific adjustment of the kurtosis or fourth moment.

This is also the ZDD used in the GenNLC[V] models. For the GenNLC[V]
models, a_{0}, a_{1},
and a_{3} will be identical. The a_{2}
width or second moment term will reflect a kinetic time constant, and a_{4}
will reflect an NLC asymmetry normalized to 0.5 for the Giddings ZDD. Please see the GenHVL
and GenNLC equivalence topic for further details.

The [V] ZDD model represents a generalization of the Asymmetric
Generalized Normal (the [Z]
density) and the Skew
Normal or GMG
(the [G]
density). If the logarithmic transform of the [Z] or default
densities is sufficient to statistically model the data, you will see the GMG convolution width (this
will be a_{4} in the GenHVL[V] and GenNLC[V] models) iterate
to values that approach zero, and there will be no statistical significance for this convolution width
parameter.

You will typically find the models which also adjust the kurtosis or fourth moment tailing, the [Y] and [T] densities, to be of greater utility than the [V] density which combines two distinct third moment adjustments.

The [V] models should only be used if the simpler [Z] and default densities are unsuccessful in adjusting the skew of the fitted peaks. For most analytic peaks, GenHVL[V] or GenHVL[V] fits will be statistically overspecified. Use cautiously.

[V] ZDD

a_{0} = Area

a_{1} = Center

a_{2} = Width

a_{3}= Half-Gaussian convolution width

a_{4} = Asymmetry ( fronted -1 > a4 > 1 tailed)

Built-In Peak Model

GenGMG (Statistical family)

User-Defined Peaks and View Functions

GenGMG(x,area,center,width,g-sd,shape) Generalized
HalfGauss Modified Gaussian Area

GenGMG_C(x,area,center,width,g-sd,shape) Generalized
HalfGauss Modified Gaussian cumulative

GenGMG_CR(x,area,center,width,g-sd,shape) Generalized
HalfGauss Modified Gaussian reverse cumulative

This generalized GMG is part of the unique content in the product covered by its copyright.