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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 a2 is transformed to a Giddings kinetic time constant, the a5 is transformed to a Gidding's indexed asymmetry.

a0 = Area

a1 = Center (as mean of generalized normal ZDD)

a2 = Kinetic Width (Giddings time constant of ZDD)

a3 = NLC/HVL Chromatographic distortion ( -1 > a3 > 1 )

a4 = Power n in exp(-zn) decay ( .25 > a4 > 4 ), adjusts kurtosis (fourth moment)

a5 = NLC indexed asymmetry ( -10 > a5 > 10 ) a5=0.5 NLC (Giddings), adjusts skew (third moment)


Built in model: Gen2NLC

User-defined peaks and view functions: Gen2NLC(x,a0,a1,a2,a3,a4,a5)


In this plot, various a4 values which adjust the kurtosis or fourth moment are shown for a series of fronted and tailed Gen2NLC shapes which vary only in this a4 fat tail adjustment. The a5 asymmetry is set at 1.5, a typical value for analytic peaks, an asymmetry greater than the theoretical NLC (a5=0.5). An a4>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 a1 is the mean of the deconvolved asymmetric generalized normal within the ZDD. For an a4 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 a1 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 a4=2, the Gen2 ZDD becomes a Generalized Normal and the model reduces to the GenNLC.

When a4=2 and a5=0, the ZDD becomes a Gaussian and the model reduces to the HVL.

When a4=2 and a5=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 a4 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 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 tailing of the IRF.

We have found this a4 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 a5 asymmetry parameter is indexed to the NLC and thus the absolute peak asymmetry is not independent of the peak's a1 location. Use the Gen2HVL if you wish to fit an absolute statistical asymmetry.

This a5 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 a5=0.5. For most IC and non-gradient HPLC peaks, you should expect an a5 between 1.1 and 2.0 (the deviation from non-ideality is right skewed or further tailed from the Giddings).

In most instances, both 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 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 a4 and a5 were very close to constant if the S/N was strong.

The addition of a shared a4 and shared a5 parameter to an overall fit can result in orders of magnitude improvement in the goodness of fit.

The a4 and a5 are indicators of deviation from this ideality. Changes in the a4 and/or a5, in fitting a given standard, may well be indicative of column health. The greater the a5 value varies from 0.5, the greater the deviation from this Giddings ZDD assumption of the NLC. The more a4 drops in value from 2.0, the more the tailing is increasing, possibly from a slow 'drizzling' off the column.

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 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 a5 intrinsic skew to the chromatographic distortion, the a4 compression or dilation of the tailing, 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 more complex peak data.

Since peaks often slow in kinetic rates 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 segment, a2 and/or a3 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.

C:\1pf2022\Help\home.gif Gen2HVL GenHVL[Z]