PeakLab v1 Documentation Contents AIST Software Home AIST Software Support

GenNLC[V]

GenNLC[V] - [V] ZDD

The GenNLC[V] model is uses an asymmetric normal generalization of the
GMG
or Skew Normal density. The model reduces to the GenNLC[Z]
as the GMG becomes a Gaussian. By combining the two primary statistical generalized normals, there are
two third moment or skew adjustments. Because of the high likelihood of correlations between the
a_{4} and a_{5} parameters, this model should be treated as experimental and used with
caution.

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

To convert the GenHVL[V] to the GenNLC[V], 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} = The GMG half-Gaussian convolution width, adjusts skew (third moment)

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

Built in model: GenNLC[V]

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

The GenNLC[V] allows the skew in the ZDD to be additionally adjusted by a one-sided probabilisitic (Gaussian)
convolution. For this model to be theoretically valid, you must assume the zero distortion peak shape,
independent of instrumental effects, contains a one-sided Gaussian smearing, or delay, in the internal
chromatographic broadening. The a_{4} value must be positive.

A convolution width, a right-sifted (positive) a_{4}
in the same direction as the a_{3} chromatographic distortion,
produces only small differences with tailed shapes. On the other hand, on fronted shapes where this convolution
width is in the opposite direction of a_{3}, the differences
are more significant for the same magnitude of a_{4}.
This secondary skew adjustment should probably be very small to share the a_{4}
parameter.

If such a one-sided Gaussian spreading is present in the ZDD, as furnished by this model, the F-statistic of the GenNLC[V] fit should be higher than the F-statistic of the GenNLC[Z] fit.

GenHVL[V] Considerations

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

When a_{4} approaches 0 and a_{5}=0.5, the ZDD becomes a Giddings and the model reduces
to the NLC.

When a_{4} approaches 0, the ZDD becomes a [Z]
generalized normal and the model reduces to the GenNLC[Z].

The [V] ZDD model represents a generalization of the Asymmetric
Generalized Normal (the [Z]
density) and the Skew Normal or GMG (the [G]
density). If the logarithmic transform of the GenNLC
or GenNLC[Z]
is sufficient to statistically model the data, you will see the a_{4}
GMG convolution width iterate to values that approach zero, and there will be no statistical significance
for this parameter.

You will typically find the models which also adjust the kurtosis or fourth moment tailing, the GenNLC[Y] and GenNLC[T] densities, to be of greater utility than the GenNLC[V] density which combines two distinct third moment adjustments.

We have not found this model useful for fitting overload shapes.

The GenNLC[V] model should only be used if the simpler GenNLC or GenNLC[Z] models are unsuccessful in adjusting the skew of the fitted peaks. For most analytic peaks, GenNLC[V] fits will be statistically overspecified. Use cautiously.

The GenNLC[V]'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 GenHVL[V] 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).

We have at times observed a small modeling power improvement when using
the GenNLC[V] model with non-gradient analytic peaks. The a_{4}
width is typically very small, perhaps 0.01-0.02 on a retention x-scale. You should use the GenNLC[V]
model cautiously for fitting analytic peaks. Use the F-statistic of the fit of the GenNLC[V] model against
the F-statistic for the GenNLC
or GenNLC[Z]
models to ensure there is an actual improvement in the modeling. The GenNLC[V] F-statistic will increase
in contrast with the GenNLC or GenNLC[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. if the a_{4}
GMG convolution width in the ZDD is managing anything real, this should appear consistently in the F-statistic
of the GenNLC[V] model.

Only if a_{4} is very small, can it be assumed constant (shared) across all peaks in the chromatogram.

Both the a_{4} and a_{5} can be seen as indicators of the deviation from this Gaussian
ideality, and thus indicative of column health.

Note that the a_{4} and 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.

Since peaks often increase in width with retention time, the a_{2} will likely 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[V] model is part of the unique content in the product covered by its copyright.