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GenHVL[G]

GenHVL[G] - [G] Half-Gaussian Modified Gaussian (GMG)

By inserting the [G] Half-Gaussian Modified Gaussian (GMG) ZDD for the PDF, CDF, and CDFc in GenHVL template, we produce the GenHVL[G] model:

a_{0} = Area

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

a_{2} = Width (SD of underlying Gaussian)

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

a_{4} = The GMG half-Gaussian convolution width

Built in model: GenHVL[G]

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

The GenHVL[G] model uses a GMG (half-Gaussian modified Gaussian) ZDD to produce the asymmetry. Note that the GMG and the Skew Normal density used in statistics produce identical shapes. The GenHVL and GenHVL[Z] models thus use one form of an asymmetric generalized normal, and the GenHVL[G] a different one, each producing its own unique type of third moment skew.

The above plots vary the a_{4}
GMG convolution width from 0.01 to 0.25. Because the GenHVL[G] uses 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 far greater for the same magnitude of a_{4}.
Note that the GenHVL[G] model allows only positive a_{4}
values.

Although the disparity of response in a_{4}
with chromatographic a_{3} distortion could be corrected
by using the statistical asymmetry parameter in the Skew Normal, this GMG parameterization acknowledges
the possibility of a theoretical underpinning for a half-Gaussian distortion in the ZDD itself, a measurable
real-world probabilistic delay in the ZDD. As suggested by the plots above, the GenHVL[G] a_{4}
parameter will not be easily shared across a chromatogram with both tailed and fronted peaks.

GenHVL[G] Considerations

When a_{4 }approaches 0, the ZDD becomes a Gaussian and the model reduces to the HVL. Using discrete
Fourier methods, it is not possible to have a zero a_{4}. There will be a built-in limiting point
which assures information in a minimum count of Fourier frequency channels.

This a_{4} GMG convolution width in the ZDD manages the deviations from the Gaussian ideality
assumed in the theoretical infinite dilution HVL. Again, note that small widths in an opposite direction
to the chromatographic distortion produce large deviations in shapes. For a retention scale, the a_{4}
may be in the vicinity of 0.01-0.05 for analytic peaks . At these values, this positive a_{4}
will produce an increased tailing on tailed peaks and mostly a positional shift on the fronted peaks.

This may be an instance there a_{4} cannot be assumed constant (shared) across all peaks in the
chromatogram.

The a_{4} can be an indicator of not only the deviation from this Gaussian ideality, but of the
actual probabilistic width of this additional distortion in the ZDD. Changes in the a_{4}, in
fitting a given standard, may be especially indicative of column health, assuming this model is able to
accurately represent the peak. The greater the a_{4} value, the greater this unwanted intrinsic
distortion.

Note that the a_{4} 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 need the GenHVL[Y] model where the fourth moment of the peak is also adjusted.

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