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GenHVL[T] - [T] Generalized Student's t ZDD
The GenHVL[T] model is a further generalization of the GenHVL[S] model. The [T] ZDD allows both the skew, the third moment, and the kurtosis, the fourth moment or 'fatness' of the tails to vary. The GenHVL[T] model may be of value in fitting preparative peaks with their high overload shapes.
By inserting the [T] Generalized Student's t (Asymmetric) ZDD for the PDF, CDF, and CDFc in GenHVL template, we produce the GenHVL[Y] 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 = Student's t tailing, the nu or DOF ( 1 > a4 > 1,000,000 ) (fourth moment)
a5 = ZDD asymmetry ( -1 > a5 > 1 ), adjusts skew (third moment)
Built in model: GenHVL[T]
User-defined peaks and view functions: GenHVL[T](x,a0,a1,a2,a3,a4,a5)
The GenHVL[T] model with a5=0
reduces to the GenHVL[S]
The GenHVL[T] model with a5=0 and a4=infinity reduces to the HVL model.
Inapplicability of the GenHVL[T] for Gradient HPLC Peaks
In a gradient separation, the tailing of a peak is typically more compact than that of a Gaussian. Since the GenHVL[T]'s ZDD cannot produce such compaction, the GenHVL[T] model will not be useful for gradient peaks.
Using the GenHVL[T] for Preparative Peaks
Although the GenHVL[T] can approximate certain overload shapes, you should regard this as an empirical model only. If you can realize a strong fit of an overload shape using this model, the a4 and a5 will give you useful estimates of the higher moments of such peaks, absent the overload, which you may be able to tie to column health.
When a4 approaches infinity and a5=0, the ZDD becomes a Gaussian and the model reduces to the HVL.
This a4 kurtosis adjustment in the ZDD manages the deviations only from a Gaussian tail decay.
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[T] model with non-gradient analytic peaks. The nu will typically fit to 50-500, and as such the benefit of adding the kurtosis to the modeling will be small. You should use the GenHVL[T] model cautiously for fitting analytic peaks. Use the F-statistic of the fit of the GenHVL[T] model against the F-statistic for the GenHVL or GenHVL[Z] models to ensure there is an actual improvement in the modeling. The GenHVL[T] 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.
In most instances, a4 and a5 can be assumed constant (shared) across all peaks in the chromatogram. It is strongly recommended that a4 and a5 be shared across all peaks.
The addition of a shared a4 and a5 parameter to an overall fit can result in orders of magnitude improvement in the goodness of fit.
Both a4 and a5 are measures of the deviation from ideality. Changes in either, in fitting a given standard, may well be indicative of column health. The greater the a5 value, the more the skew is deviating from this Gaussian ZDD assumption of the HVL. For the GenHVL[T] model, the a4 may be of equally or even greater importance since additional tailing represents a drizzle of sorts that can impact adjacent peaks. You may wish to use the GenHVL[T] with a standard and watch for any unexpected changes in either a4 or a5.
Note that the 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 fitting analytic peaks with this model, assuming that one does not enter into a condition of overload that impacts the quality of the fit.
This model will probably not be effective at all in highly dilute samples with a poor S/N ratio since such peaks will generally have much less intrinsic skew and the tailing will be poorly defined due to inaccuracies in the baseline subtraction.
The GenHVL[T]<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 independent deconvolutions within the fitting. For very dilute and noisy analytic samples, you will probably have to remove the IRF prior using independent determinations of the IRF parameters.
The GenHVL[T]<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 a5 intrinsic skew to the chromatographic distortion, the a4 intrinsic tailing in the ZDD, 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 production 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.
The GenHVL[T] model is part of the unique content in the product covered by its copyright.