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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 a₃ parameter controls the width of the half-Gaussian convolution in the GMG density while the the a₄ 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, a₀, a₁, and a₃ will be identical. The a₂ width or second moment term will reflect a kinetic time constant, and a₄ 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 a₄ 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

a₀ = Area

a₁ = Center

a₂ = Width

a₃= Half-Gaussian convolution width

a₄ = 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.