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GenHVL[Z]

GenHVL[Z] - [Z] Generalized Normal ZDD

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

a_{0} = Area

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

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

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

a_{4} = ZDD asymmetry ( -1 > a_{4} > 1 )

Built in model: GenHVL[Z]

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

The only difference between the GenHVL
and the GenHVL[Z] models is in a_{1}. For the GenHVL, a1 is the mean of the generalized normal
ZDD (the skewed zero-distortion peak). For the GenHVL[Z], a1 is the mean of the unskewed Gaussian in the
ZDD.

When the ZDD is further generalized to also adjust the kurtosis in the GenHVL[Y]
models, the GenHVL[Z] will be the specialization for the exp(-z^{2}), power=2.0, Gaussian tailing.

GenHVL[Z] Considerations

When a_{4}=0, the ZDD becomes a Gaussian and the model reduces to the HVL.

This a_{4} 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 shapes. For most IC and non-gradient HPLC peaks, you should expect
an a_{4} between +0.01 and +0.03 (the deviation from non-ideality is a right skewed or tailed).

In most instances, a_{4} can be assumed constant (shared) across all peaks in the chromatogram.
It is strongly recommended that a_{4} be shared across all peaks and only independently fitted
with each peak if the parameter significance allows and you find such necessary. In our experience, across
a wide range of concentrations, and across peaks ranging from highly fronted to highly tailed, the fitted
a_{4} was very close to constant.

The addition of this single a_{4} parameter to an overall fit can result in orders of magnitude
improvement in the goodness of fit. The impact of just this one additional parameter in a fit of perhaps
many dozens of parameters can be the difference between 5 ppm and 5000 ppm in the unaccounted variance
in the fit.

The a_{4} is also an exacting indicator of the deviation from this ideality. Changes in the a_{4},
in fitting a given standard, may well be indicative of column health. The greater the a_{4} value,
the more the separation is deviating from this Gaussian ZDD assumption of the HVL.

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.

The GenHVL[Z]<irf> composite fits, the model with a convolution integral describing the instrumental distortions, isolate the intrinsic chromatographic distortion from the IRF instrumental distortion only when the data are of a sufficient S/N and quality to realize two independent deconvolutions within the fitting. For very dilute and noisy samples, you will probably have to remove the IRF prior using independent determinations of the IRF parameters.

The GenHVL[Z]<ge> model uses the <ge>IRF, consistently the best of the convolution models
as it fits both kinetic and probabilistic instrument distortions. Bear in mind, however, that this fit
must extract the kinetic instrumental distortion, the probabilistic instrumental distortion, the a_{4}
intrinsic skew to the chromatographic distortion, and the primary a_{3} chromatographic distortion
(very possibly for for each peak). It is recommended the IRF parameters be determined by fits of a clean
standard, and the instrumental distortions removed by deconvolving the known IRF prior to fitting more
complex peak data.

Since peaks often increase in width with retention time, the a_{2} will probably 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[Z] model is part of the unique content in the product covered by its copyright.