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Gen2NLC

Gen2NLC - Generalized Error Model ZDD

This is the probably the second most important of the GenNLC models. The Gen2NLC models use an additional generalization of the GenNLC where the fourth moment or kurtosis, the 'fatness' of the tails, is also adjustable.

By using the relationships of equivalence between the GenHVL and GenNLC models, we make the following simple substitution into the Gen2HVL model equation to derive the Gen2NLC model:

To convert the Gen2HVL to the Gen2NLC, the a_{2} is transformed to a Giddings kinetic time constant,
the a_{5} is transformed
to a Gidding's indexed asymmetry.

a_{0}
= Area

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

a_{2} = Kinetic Width (Giddings time constant of ZDD)

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

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

a_{5} = NLC indexed asymmetry
( -10 > a_{5}
> 10 ) a_{5}=0.5 NLC (Giddings), adjusts skew
(third moment)

Built in model: Gen2NLC

User-defined peaks and view functions: Gen2NLC(x,a_{0},a_{1},a_{2},a_{3},a_{4},a_{5})

In this plot, various a_{4} values which adjust the kurtosis or fourth moment are shown for a
series of fronted and tailed Gen2NLC shapes which vary only in this a_{4} fat tail adjustment.
The a_{5} asymmetry is set at 1.5, a typical value for analytic peaks, an asymmetry greater than
the theoretical NLC (a_{5}=0.5).
An a_{4}>2 is a decay that is more compact than a Gaussian, and is seen in gradient HPLC. When
used to fit non-gradient analytic peaks, the value is typically very slightly less than 2.0.

Gen2NLC vs. GenNLC[Y]

The Gen2NLC uses the generalized
error ZDD where a_{1} is the mean of the deconvolved asymmetric generalized normal within
the ZDD. For an a_{4} power=2.0 decay, this will be the mean of the deconvolved zero-distortion
peak. The GenNLC[Y]
model uses the [Y]
generalized error ZDD where the a_{1} center value is the fully deconvolved Gaussian mean.
Use the GenNLC[Y]
if you wish a full statistical deconvolution where the Gaussian mean is directly fitted and you want its
specific confidence bands. Both produce identical shapes.

Gen2NLC Considerations

When a_{4}=2, the Gen2
ZDD becomes a Generalized
Normal and the model reduces to the GenNLC.

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

When a_{4}=2 and a_{5}=0.5, the ZDD becomes a Giddings and the model reduces to the NLC.

The Gen2NLC model adjusts both the third and fourth moments and is the model of choice for gradient HPLC
peaks where a compression occurs from the gradient, and the decay of the peak will be more compact than
a Gaussian (a power of 2.0 in the a_{4} parameter).

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 tailing of the IRF.

We have found this a_{4} fourth moment adjustment in the Gen2NLC model to often be overspecified
(statistically insignificant) for non-gradient analytic peaks. The power
is typically between 1.96-1.99, and as such the benefit of adding the kurtosis to the modeling will be
small. You should use the Gen2NLC model cautiously for fitting analytic peaks. Use the F-statistic of
the fit of the GenNLC model against the F-statistic for the GenNLC
model to ensure there is an actual improvement in the modeling. The Gen2NLC F-statistic will increase
in contrast with the GenNLC model when this adjustment to the fourth moment is statistically beneficial.
A high S/N will definitely be needed to even see this benefit.

For fitting overload shapes, the Gen2NLC or GenNLC[Y] can be used, but the kinetics are not likely meaningful, and you will probably want to use the GenHVL[Yp] or GenHVL[YpE] specialization of the GenHVL[Y] model. These are designed specifically for fitting high overload shapes.

The Gen2NLC's a_{5} asymmetry parameter is indexed to the NLC and thus the absolute peak asymmetry
is not independent of the peak's a_{1} location. Use the Gen2HVL if you wish to fit an absolute
statistical asymmetry.

This a_{5} skew adjustment
in the ZDD manages the deviations from the Giddings ideality assumed in the theoretical infinite dilution
NLC. This is an asymmetry parameter indexed to the NLC at a_{5}=0.5.
For most IC and non-gradient HPLC peaks, you should expect an a_{5}
between 1.1 and 2.0 (the deviation from non-ideality is right skewed or further tailed from the Giddings).

In most instances, both 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 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}
and a_{5} were
very close to constant if the S/N was strong.

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

The a_{4}
and a_{5} are
indicators of deviation from this ideality. Changes in the a_{4}
and/or a_{5},
in fitting a given standard, may well be indicative of column health. The greater the a_{5}
value varies from 0.5, the greater the deviation from this Giddings ZDD assumption of the NLC. The more
a_{4} drops in
value from 2.0, the more the tailing is increasing, possibly from a slow 'drizzling' off the column.

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 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. Accurately fitting the tailing is even more demanding of data quality.

The Gen2NLC<irf> composite fits are available, the model with a convolution integral describing the instrumental distortions, in order to isolate the intrinsic chromatographic distortion from the IRF instrumental distortion, but this should be done with caution. The data must be 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 Gen2NLC<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}
compression or dilation of the tailing, 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 slow in kinetic rates 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 segment, a_{2} and/or a_{3} can be held constant across the peaks in
this band.

The Gen2NLC model is part of the unique content in the product covered by its copyright.