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GenHVL[Yp2] (Overload Shapes Only)
GenHVL[Yp2] - [Yp2] Generalized Error 2-width ZDD
The GenHVL[Yp2] model is a modification of the GenHVL[Y] model where there is both a left and right side width parameter. This model is used only for preparative (overload) shapes. It is the best model we have thus found for fitting overload shapes. The starting estimates for the [Yp2] density assume such shapes are present.
The GenHVL[Yp2] a1 center parameter is the mode of the underlying error model peak.
The a5 parameter in the ZDD nomenclature below adjusts the power of the decay, and thus the fourth moment or kurtosis of the peak. This value is often close to 1, exponential or Laplace tails, for overload shapes. The a5 parameter in the ZDD nomenclature adjusts the asymmetry, and thus the third moment or skew of the overall peak. In this model, the primary asymmetry is determined the by difference between the right and left side widths, set by a4, the decay width divided by the rise width.
The [Yp2] ZDD allows both the skew, the third moment, and the kurtosis, the fourth moment or 'fatness' of the tails to vary.
By inserting the [Yp2] Generalized Error 2-width Model ZDD for the PDF, CDF, and CDFc in the GenHVL template, we produce the GenHVL[Yp2] 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 = Width Asymmetry (right side width)/(left side width)
a5 = Power n in exp(-zn) decay ( .25 > a5 > 4 ) adjusts kurtosis (fourth moment)
a6 = ZDD asymmetry ( -1 > a6 > 1 ), adjusts skew (third moment)
Built in model: GenHVL[Yp2]
User-defined peaks and view functions: GenHVL[Yp2](x,a0,a1,a2,a3,a4,a5,a6)
The GenHVL[Yp2] model with a4=1
(left and right widths equal) produces GenHVL[Y]
The GenHVL[Yp2] model with a5=1 (the specialization for the exp(-z) exponential tailing) reduces to the GenHVL[Yp2E] model.
Using the GenHVL[Yp2] for Preparative Peaks
When the a4 power is close to 1, the ZDD approaches a double sided exponential or Laplace density and it can replicate a host of high overload preparatory chromatography peak shapes. When using the GenHVL[Yp] to approximate the extreme overload shapes, the a5 asymmetry mainly controls the shape of the envelope as in the plot below.
The plots above use an an a5=1, an exponential power of decay. The first two sets of peaks use -a3 chromatographic distortion (fronted) with a constant a2 width and a4 varying. The slope at the plateau is determined by the a6 ZDD asymmetry, the first set of peaks with -a6 and the second set with +a6. The last two sets use a +a3 chromatographic distortion (tailed) with a constant a2 width and again with varying a4.
In the graph above, the power is varied from 1 to 1.2 for [-a3,-a6] [-a3,+a6] [+a3,-a6] [+a3,+a6]. It is this power of decay that makes the modeling of partial overload envelopes possible. The power=1 GenHVL[Yp2E] can be used for extreme overload shapes where the power would fit to unity, but for all other cases, the GenHVL[Yp2] should be fitted.
When a4=2 and a5=0, the ZDD becomes a Gaussian and the model reduces to the HVL.
When a4=2, the ZDD becomes a [Z] generalized normal and the model reduces to the GenHVL[Z].
As should be apparent from the plots above, two very different estimation algorithms are needed for the gradient and high overload assumptions. One assumes a more compact decay and the other an extremely elongated one. You should use the GenHVL[Y] for the HPLC gradient fits, and the GenHVL[Yp] model for fitting high overload shapes. The GenHVL[Yp] is mathematically identical to the GenHVL[Y] but generates starting estimates that assume a high overload state.
The GenHVL[YpE] model directly uses a Laplace ZDD (there is no power of decay adjustment-a power=1 exponential decay is used). In high overload experiments, we found the [Y] power converged asymptotically to 1.0. This [YpE] model is faster and may be more robust when fitting very high overload shapes. For overload fits, there is also a GenHVL[Yp2] and its power of 1 simplification, GenHVL[Yp2E], where separate left and right side widths are fitted.
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).
In gradient peaks, the instrumental distortions may be masked entirely by the gradient. If the IRF is not removed or included in a gradient fit (neither may be possible), the a5 parameter will reflect whatever measure of this IRF this skew adjustment is able to capture. Please note that the a5 adjustment occurs to the skew of the ZDD to which the chromatographic distortion operator is subsequently applied, a very different matter than an IRF convolution integral applied to the end result of the intrinsic chromatographic distortion. The net effect of ignoring the IRF in a gradient fit is to have the a4 understate the compression that is actually occurring in the gradient since it will be diminished to the extent it also accounts the IRF tailing.
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 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).
We have often observed a small modeling power improvement when using the GenHVL[Y] model with non-gradient analytic peaks. The power is typically between 1.96-1.98, and as such the benefit of adding the kurtosis to the modeling will be small. You should use the GenHVL[Y] model cautiously for fitting analytic peaks. Use the F-statistic of the fit of the GenHVL[Y] model against the F-statistic for the GenHVL or GenHVL[Z] models to ensure there is an actual improvement in the modeling. The GenHVL[Y] 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[Y] 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[Y] 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[Y]<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[Y]<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). Especially for this model, 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[Y] model is part of the unique content in the product covered by its copyright.