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GenHVL[V]

GenHVL[V] - [V] Generalized Error Model ZDD

The GenHVL[V] model is a further generalization of the GenHVL[G] and GenHVL[Z] models. By combining two different asymmetric generalized normals, the [V] ZDD allows two separate adjustments to the skew, the third moment, of the peak. Only indirectly is the kurtosis altered. Because of the high likelihood of correlations between the a₄ and a₅ parameters, this model should be treated as experimental and used with caution.

By inserting the [V] Generalized GMG ZDD for the PDF, CDF, and CDFc in GenHVL template, we produce the GenHVL[V] model:

a₀ = Area

a₁ = Center (as mean of underlying normal ZDD)

a₂ = Width (SD of underlying normal ZDD)

a₃ = HVL Chromatographic distortion ( -1 > a₃ > 1 )

a₄ = The GMG half-Gaussian convolution width, adjusts skew (third moment)

a₅ = ZDD asymmetry ( -1 > a₅ > 1 ), adjusts skew (third moment)

Built in model: GenHVL[V]

User-defined peaks and view functions: GenHVL[V](x,a₀,a₁,a₂,a₃,a₄,a₅)

The GenHVL[V] allows the skew in the ZDD to be additionally adjusted by a one-sided probabilisitic (Gaussian) convolution. For this model to be theoretically valid, you must assume the zero distortion peak shape, independent of instrumental effects, contains a one-sided Gaussian smearing, or delay, in the internal chromatographic broadening. The a₄ value must be positive.

As with the GenHVL[G], a convolution width, a right-sifted (positive) a₄ in the same direction as the a₃ chromatographic distortion, produces only small differences with tailed shapes. On the other hand, on fronted shapes where this convolution width is in the opposite direction of a₃, the differences are more significant for the same magnitude of a₄. This secondary skew adjustment should probably be very small to share the a₄ parameter.

If such a one-sided Gaussian spreading is present in the ZDD, as furnished by this model, the F-statistic of the GenHVL[V] fit should be higher than the F-statistic of the GenHVL[Z] fit.

GenHVL[V] Considerations

When a₄ approaches 0 and a₅=0, the ZDD becomes a Gaussian and the model reduces to the HVL.

When a₄ approaches 0, ZDD becomes a [Z] generalized normal and the model reduces to the GenHVL[Z].

When a₅ =0, the ZDD becomes a [G] Half-Gaussian Modified Gaussian (GMG) ZDD, and the model reduces to the GenHVL[G].

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 GenHVL or GenHVL[Z] is sufficient to statistically model the data, you will see the a₄ GMG convolution width iterate to values that approach zero, and there will be no statistical significance for this parameter.

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

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

This a₅ skew adjustment in the ZDD manages the deviations from the Gaussian ideality assumed in the theoretical infinite dilution HVL. This is the statistical asymmetry parameter; small differences in values produce large deviations in analytic shapes. For most IC and non-gradient HPLC peaks, you should expect an a₅ between +0.01 and +0.03 (the deviation from non-ideality is a right skewed or tailed).

We have often observed a small modeling power improvement when using the GenHVL[V] model with non-gradient analytic peaks. The a₄ width is typically very small, perhaps 0.01-0.02 on a retention x-scale. You should use the GenHVL[V] model cautiously for fitting analytic peaks. Use the F-statistic of the fit of the GenHVL[V] model against the F-statistic for the GenHVL or GenHVL[Z] models to ensure there is an actual improvement in the modeling. The GenHVL[V] F-statistic will increase in contrast with the GenHVL or GenHVL[Z] model when this adjustment to the fourth moment is statistically beneficial. A high S/N will definitely be needed to even see this benefit. if the a₄ GMG convolution width in the ZDD is managing anything real, this should appear consistently in the F-statistic of the GenHVL[V] model.

Only if a₄ is very small, can it be assumed constant (shared) across all peaks in the chromatogram.

Both the a₄ and a₅ can be seen as indicators of the deviation from this Gaussian ideality, and thus indicative of column health.

Note that the a₄ and a₅ will be most effectively estimated and fitted when the peaks are skewed with some measure of fronting or tailing. Higher concentrations are very good for this model, assuming that one does not enter into a condition of overload that impacts the quality of the fit.

This model will be least effective in highly dilute samples with a poor S/N ratio since such peaks will generally have much less intrinsic skew.

Since peaks often increase in width with retention time, the a₂ will likely be varied (independently fitted) for each peak.

Since peaks often evidence increased tailing with retention time, the a₃ will probably be varied (independently fitted) for each peak.

If you are dealing with a small range of time, however, or of you are dealing with overlapping or hidden peaks in a narrow band, a₂ and/or a₃ can be held constant across the peaks in this band.

If you are addressing gradient peaks, or the overload shapes of preparative chromatography, you will probably need the GenHVL[Y] model where the fourth moment of the peak is also adjusted.

The GenHVL[V] model is part of the unique content in the product covered by its copyright.