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ZDD - Gen2 Default - Generalized Error

This is the Zero Distortion Density (ZDD) used in the default Gen2HVL models. This generalized normal's a1 center parameter is the mean of the asymmetric peak. The [Y] ZDD is a similar model with a parameterization where a1 is the center of the underlying normal. The a4 parameter in the ZDD nomenclature adjusts the third moment or skew. The a3 parameter in the ZDD nomenclature adjusts the fourth moment or kurtosis by varying the power of decay of the tails.

This is also the ZDD used in the default Gen2NLC models. For the Gen2NLC models, a0, a1, and a3 will be identical. The a2 width or second moment term will reflect a kinetic time constant, and a4 will reflect an NLC asymmetry normalized to 0.5 for the Giddings ZDD. Please see the GenHVL and GenNLC equivalence topic for further details.

Twice Generalized Default ZDD (Two Higher Moments)

a0 = Area

a1 = Center (as mean of asymmetric peak)

a2 = Width (SD of underlying Gaussian)

a3 = Power of Decay of Tails, controls kurtosis or fourth moment

a4 = Asymmetry ( fronted -1 > a3 > 1 tailed), controls skew or third moment

In this graph, the outer peaks vary the asymmetry a4 at positive and negative values. The peaks in the center of the plot vary the a3 tailing at a fixed positive and negative asymmetry.

Built-In Peak Model

GenError[m] (Statistical family)

User-Defined Peaks and View Functions

GenError[m](x,area,mean,width,n-power,z-asym) Asym Generalized Error
GenError[m]_C(x,area,mean,width,n-power,z-asym) Asym Generalized Error
GenError[m]_CR(x,area,mean,width,n-power,z-asym) Asym Generalized Error

This Asymmetric Error Model is part of the unique content in the product covered by its copyright.

Example Fit

Data generated from a Generalized Error peak are fitted. This is the shape of the Gen2HVL or Gen2NLC models when there is zero chromatographic distortion. In this example, the generated data use a power of decay of 2.2, a value higher than the 2.0 Gaussian decay, and what might be expected with gradient HPLC peaks.

"GenError(x,1,5,.25,2.2,0.1)"

Fitted Parameters

r^2 Coef Det  DF Adj r^2    Fit Std Err  F-value      ppm uVar

1.00000000   1.00000000   1.0065e-15   2.266e+32    0.00000000

Peak  Type               a0          a1          a2          a3          a4

1  GenError[m]  1.00000000  5.01253130  0.25000000  2.20000000  0.10000000

Measured Values

Peak  Type          Amplitude    Center       FWHM       Asym50     FW Base      Asym10

1  GenError[m]  1.40379500  4.95649171  0.67941140  1.24009278  1.29512868  1.32688030

Peak  Type            Area       % Area       Mean       StdDev     Skewness    Kurtosis

1  GenError[m]  1.00000000  100.000000  5.01415908  0.27628895  0.28335254  2.96345806

Analytic Moments

Peak  Type           FnArea     % FnArea

1  GenError[m]  1.00000000  100.000000

Deconvolved Moments

Peak  Type        Area        Mean       StdDev     Skewness    Kurtosis

1  GenNorm  1.00000000  5.01253130  0.25188257  0.30175910  3.16232386

Peak  Type        Area        Mean       StdDev     Skewness    Kurtosis

1  Gauss    1.00000000  5.00000000  0.25000000  5.6424e-8   2.99999967

The a1 of the GenError[m] ZDD is the mean of the Generalized Normal (the GenError with identical parameters except for an a3=2 Gaussian decay). When a3=2, the GenError[m] a1 will be the first moment or mean of the peak.