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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 a_{4} and a_{5} 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:

a_{0} = Area

a_{1} = Center (as mean of underlying normal ZDD)

a_{2} = Width (SD of underlying normal ZDD)

a_{3} = HVL Chromatographic distortion ( -1 > a_{3}
> 1 )

a_{4} = The GMG half-Gaussian convolution width, adjusts skew (third moment)

a_{5} = ZDD asymmetry ( -1 > a_{5} > 1 ), adjusts skew (third moment)

Built in model: GenHVL[V]

User-defined peaks and view functions: GenHVL[V](x,a_{0},a_{1},a_{2},a_{3},a_{4},a_{5})

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 a_{4} value must be positive.

As with the GenHVL[G], a convolution width, a right-sifted (positive)
a_{4} in the same direction as the a_{3}
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 a_{3},
the differences are more significant for the same magnitude of a_{4}.
This secondary skew adjustment should probably be very small to share the a_{4}
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 a_{4} approaches 0 and a_{5}=0, the ZDD becomes a Gaussian and the model reduces
to the HVL.

When a_{4} approaches 0, ZDD becomes a [Z]
generalized normal and the model reduces to the GenHVL[Z].

When a_{5} =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 a_{4}
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 a_{5} 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 a_{5}
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 a_{4}
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 a_{4}
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 a_{4} is very small, can it be assumed constant (shared) across all peaks in the chromatogram.

Both the a_{4} and a_{5} can be seen as indicators of the deviation from this Gaussian
ideality, and thus indicative of column health.

Note that the a_{4} and a_{5} 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 a_{2} will likely be varied (independently
fitted) for each peak.

Since peaks often evidence increased tailing with retention time, the a_{3}
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, a_{2} and/or a_{3} 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.