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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:
a0 = Area
a1 = Center (as mean of generalized normal ZDD)
a2 = Width (SD of underlying Gaussian)
a3 = HVL Chromatographic distortion ( -1 > a3 > 1 )
a4 = ZDD asymmetry ( -1 > a4 > 1 )
Built in model: GenHVL[Z]
User-defined peaks and view functions: GenHVL[Z](x,a0,a1,a2,a3,a4)
The only difference between the GenHVL and the GenHVL[Z] models is in a1. 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(-z2), power=2.0, Gaussian tailing.
When a4=0, the ZDD becomes a Gaussian and the model reduces to the HVL.
This a4 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 a4 between +0.01 and +0.03 (the deviation from non-ideality is a right skewed or tailed).
In most instances, a4 can be assumed constant (shared) across all peaks in the chromatogram. It is strongly recommended that a4 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 a4 was very close to constant.
The addition of this single a4 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 a4 is also an exacting indicator of the deviation from this ideality. Changes in the a4, in fitting a given standard, may well be indicative of column health. The greater the a4 value, the more the separation is deviating from this Gaussian ZDD assumption of the HVL.
Note that the a4 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 a4 intrinsic skew to the chromatographic distortion, and the primary a3 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 a2 will probably 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 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.