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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] a_{1} center parameter is the mode of the underlying error model peak.

The a_{5} 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 a_{5}
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 a_{4},
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:

a_{0} = Area

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

a_{2} = Width (SD of underlying normal ZDD)

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

a_{4} = Width Asymmetry
(right side width)/(left side width)

a_{5} = Power
n in exp(-zn) decay ( .25 > a_{5}
> 4 ) adjusts kurtosis (fourth moment)

a_{6}
= ZDD asymmetry ( -1 > a_{6}
> 1 ), adjusts skew (third moment)

Built in model: GenHVL[Yp2]

User-defined peaks and view functions: GenHVL[Yp2](x,a_{0},a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})

The GenHVL[Yp2] model with a_{4}=1
(left and right widths equal) produces GenHVL[Y]
shapes.

The GenHVL[Yp2] model with a_{5}=1
(the specialization for the exp(-z) exponential tailing) reduces to the GenHVL[Yp2E]
model.

Using the GenHVL[Yp2] for Preparative Peaks

When the a_{4}
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 a_{5}
asymmetry mainly controls the shape of the envelope as in the plot below.

The plots above use an an a_{5}=1,
an exponential power of decay. The first two sets of peaks use -a_{3}
chromatographic distortion (fronted) with a constant a_{2}
width and a_{4} varying. The slope at the plateau is determined
by the a_{6} ZDD asymmetry, the first set of peaks with
-a_{6} and the second set with +a_{6}.
The last two sets use a +a3 chromatographic distortion (tailed) with a constant a_{2}
width and again with varying a_{4}.

In the graph above, the power is varied from 1 to 1.2 for [-a_{3},-a_{6}]
[-a_{3},+a_{6}]
[+a_{3},-a_{6}]
[+a_{3},+a_{6}].
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.

GenHVL[Yp2] Considerations

When a_{4}=2 and a_{5}=0, the ZDD becomes a Gaussian and the model reduces to the HVL.

When a_{4}=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 a_{5} 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 a_{5}
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
a_{5} parameter
will reflect whatever measure of this IRF this skew adjustment is able to capture. Please note that the
a_{5} 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 a_{4}
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 a_{5} 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).

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, a_{4}
and a_{5} can be assumed constant (shared) across all
peaks in the chromatogram. It is strongly recommended that a_{4}
and a_{5} be shared across all peaks.

The addition of a shared a_{4}
and a_{5} parameter to an overall fit can result in orders
of magnitude improvement in the goodness of fit.

Both a_{4} and a_{5}
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 a_{5}
value, the more the skew is deviating from this Gaussian ZDD assumption of the HVL. For the GenHVL[Y]
model, the a_{4}
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 a_{4}
or a_{5}.

Note that the a_{5}
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 a_{5}
intrinsic skew to the chromatographic distortion, the a_{4}
intrinsic tailing in the ZDD, and the primary a_{3}
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 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.

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