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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.