Generalized Normal

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Generalized Normal (Asymmetric)

Generalized Normal ([Z] ZDD)

Generalized Normal1.png

a₀ = Area

a₁ = Median (of asymmetric peak)

a₂ = Width (SD of underlying normal)

a₃ = Asymmetry ( -1 > a₃ > 1 )

v5_ZDD_[Z].png

Built-In Model User-Defined Peak/View Fn

GenNorm GenNorm(x,area,median,width,shape) Asym Generalized Normal Area

GenNorm_C GenNorm_C(amplitude,center,width,z-asym) Gen. Normal cumulative

GenNorm_CR GenNorm_CR(amplitude,center,width,z-asym) Gen. Normal reverse cumulative

GenHVL[Z] GenHVL[Z](x,area,center,width,distortn,z-asym) ZDD=Asym Gen. Normal

This is the [Z] ZDD (zero-distortion density) in the GenHVL[Z] and GenNLC[Z] models.

If the x variable scaled for time and location is designated z = (x-a1)/a2, a normal is represented as Exp[-z²/2]. The transform of the x variable is simply:

Although the parametrizations vary, this a₃ based variable logarithmic transformation of the x scale is the premise of all models in this primary class of generalized asymmetric normal models.

Generalized Normal [m] (Default GenHVL, GenHLC ZDD)

Generalized Normal2.png

a₀ = Area

a₁ = Mean (of asymmetric peak)

a₂ = Width (SD of underlying normal)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[m] GenNorm[m](x,area,mean,width,shape) Asym Generalized Normal Area

GenNorm[m]_C GenNorm[m]_C(x,amplitude,mean,width,shape) Asym Generalized Normal cumulative

GenNorm[m]_CR GenNorm[m]_CR(x,amplitude,mean,width,shape) Asym Generalized Normal reverse cumulative

GenHVL GenHVL(x,area,mean,width,distortn,z-asym) Generalized Haarhoff-Van der Linde a1=mean

This is the Default ZDD (zero-distortion density) in the GenHVL and GenNLC models. Here a₁ is the mean of the peak. In the standard asymmetric generalized normal (our [Z] ZDD in the chromatographic models), a₁ is the median.

Generalized Normal [c]

Generalized Normal4.png

a₀ = Area

a₁ = Mode (of asymmetric peak)

a₂ = Width (SD of underlying normal)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[c] GenNorm[c](x,area,mode,width,shape) Asym Generalized Normal Area

GenNorm[c]_C GenNorm[c]_C(x,amplitude,mode,width,shape) Asym Generalized Normal cumulative

GenNorm[c]_CR GenNorm[c]_CR(x,amplitude,mode,width,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zc](x,area,mode,width,distortn,z-asym) ZDD=Asym Gen. Normal [c]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ fits the mode, the x value at the apex of the peak.

Generalized Normal [d]

Generalized Normal5.png

a₀ = Area

a₁ = Median (of asymmetric peak)

a₂ = SD (SD of asymmetric peak)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[d] GenNorm[d](x,area,median,sd,shape) Asym Generalized Normal Area

GenNorm[d]_C GenNorm[d]_C(x,amplitude,median,sd,shape) Asym Generalized Normal cumulative

GenNorm[d]_CR GenNorm[d]_CR(x,amplitude,median,sd,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zd](x,area,median,SD,distortn,z-asym) ZDD=Asym Gen. Normal [d]

This parameterization is not used in the built-in generalized chromatographic models. Here a₂ directly fits the standard deviation of the density.

Generalized Normal [a]

Generalized Normal6.png

a₀ = Area

a₁ = Median (of asymmetric peak)

a₂ = Width (SD of underlying normal)

a₃ = Half-Height Asymmetry (Asym50 ratio)

Built-In Model User-Defined Peak/View Fn

GenNorm[a] GenNorm[a](x,area,median,width,asym) Asym Generalized Normal Area

GenNorm[a]_C GenNorm[a]_C(x,amplitude,median,sd,asym) Asym Generalized Normal cumulative

GenNorm[a]_CR GenNorm[a]_CR(x,amplitude,median,sd,asym) Asym Generalized Normal reverse cumulative

GenHVL[Za](x,area,median,width,distortn,Asym50) ZDD=Asym Gen. Normal [a]

This parameterization is not used in the built-in generalized chromatographic models. Here a₃ directly fits the half-height asymmetry ratio, Asym50.

Generalized Normal [cw]

Generalized Normal7.png

a₀ = Area

a₁ = Mode (of asymmetric peak)

a₂ = Half-Height Width (FWHM of asymmetric peak)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[cw] GenNorm[cw](x,area,mode,fwhm,shape) Asym Generalized Normal Area

GenNorm[cw]_C GenNorm[cw]_C(x,amplitude,median,sd,shape) Asym Generalized Normal cumulative

GenNorm[cw]_CR GenNorm[cw]_CR(x,amplitude,median,sd,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zcw](x,area,mode,FWHM,distortn,z-asym) ZDD=Asym Gen. Normal [cw]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ fits the mode, the x value at the apex of the peak, and a₂ directly fits the half-height width, the FWHM.

Generalized Normal [md]

Generalized Normal8.png

a₀ = Area

a₁ = Mean (of asymmetric peak)

a₂ = SD (SD of asymmetric peak)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[md] GenNorm[md](x,area,mean,sd,shape) Asym Generalized Normal Area

GenNorm[md]_C GenNorm[md]_C(x,amplitude,mean,sd,shape) Asym Generalized Normal cumulative

GenNorm[md]_CR GenNorm[md]_CR(x,amplitude,mean,sd,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zmd](x,area,mean,SD,distortn,z-asym) ZDD=Asym Gen. Normal [md]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ directly fits the mean of the density, and a₂ directly fits the standard deviation.

Generalized Normal [cd]

Generalized Normal9.png

a₀ = Area

a₁ = Mode (of asymmetric peak)

a₂ = SD (SD of asymmetric peak)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[cd] GenNorm[cd](x,area,mode,sd,shape) Asym Generalized Normal Area

GenNorm[cd]_C GenNorm[cd]_C(x,amplitude,mode,sd,shape) Asym Generalized Normal cumulative

GenNorm[cd]_CR GenNorm[cd]_CR(x,amplitude,mode,sd,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zcd](x,area,mode,SD,distortn,z-asym) ZDD=Asym Gen. Normal [cd]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ directly fits the mode, the x value at the apex of the peak, and a₂ directly fits the standard deviation.

Generalized Normal [mv]

Generalized Normal10.png

a₀ = Area

a₁ = Mean (of asymmetric peak)

a₂ = Variance (Variance of asymmetric peak)

a₃ = Asymmetry ( -1 > a₃ > 1 )

Built-In Model User-Defined Peak/View Fn

GenNorm[mv] GenNorm[mv](x,area,mean,variance,shape) Asym Generalized Normal Area

GenNorm[mv]_C GenNorm[mv]_C(x,amplitude,mean,variance,shape) Asym Generalized Normal cumulative

GenNorm[mv]_CR GenNorm[mv]_CR(x,amplitude,mean,variance,shape) Asym Generalized Normal reverse cumulative

GenHVL[Zmv](x,area,mean,variance,distortn,z-asym) ZDD=Asym Gen. Normal [mv]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ directly fits the mean (or first moment) of the density, and a₂ directly fits the variance (the second moment, the square of the standard deviation).

Generalized Normal [mvs]

Generalized Normal20.png

a₀ = Area (0th moment)

a₁ = Mean (of asymmetric peak, 1st moment)

a₂ = Variance (of asymmetric peak, 2nd moment)

a₃ = Skew (of asymmetric peak, 3rd moment)

Built-In Model User-Defined Peak/View Fn

GenNorm[mvs] GenNorm[mvs](x,area,mean,variance,skewness) Asym Generalized Normal Area

GenNorm[mvs]_C GenNorm[mvs]_C(x,amplitude,mean,variance,skewness) Asym Generalized Normal cumulative

GenNorm[mvs]_CR GenNorm[mvs]_CR(x,amplitude,mean,variance,skewness) Asym Generalized Normal reverse cumulative

GenHVL[Zmvs](x,area,mean,variance,distortn,skew) ZDD=Asym Gen. Normal [mvs]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ directly fits the mean (or first moment) of the density, a₂ directly fits the variance (the second moment, the square of the standard deviation), and a₃ directly fits the skew (the third statistical moment).

Generalized Normal [cwa] (LN4 Model)

Generalized Normal13.png

a₀ = Area

a₁ = Mode (of asymmetric peak)

a₂ = Half-Height Width (FWHM of asymmetric peak)

a₃ = Half-Height Asymmetry (Asym50 ratio)

Built-In Model User-Defined Peak/View Fn

GenNorm[cwa] GenNorm[cwa](x,area,mode,fwhm,asym) Asym Generalized Normal Area

GenNorm[cwa]_C GenNorm[cwa]_C(amplitude,mode,fwhm,l-asym) Gen. Normal LN4 FWHM cumulative

GenNorm[cwa]_CR GenNorm[cwa]_CR(amplitude,mode,fwhm,l-asym) Gen. Normal LN4 FWHM reverse cumulative

GenHVL[Zcwa](x,area,mode,FWHM,distortn,Asym50) ZDD=Asym Gen. Normal [cwa]

This parameterization is not used in the built-in generalized chromatographic models, is the 4-parameter LogNormal within the product. Here a₁ directly fits the mode , the x value at the apex of the peak, a₂ directly fits the half-height width, the FWHM, and a₃ directly fits the half-height asymmetry ratio, Asym50.

Generalized Normal [ca]

Generalized Normal14.png

a₀ = Area

a₁ = Mode (of asymmetric peak)

a₂ = Width (SD of Gaussian)

a₃ = Half-Height Asymmetry (Asym50 ratio)

Built-In Model User-Defined Peak/View Fn

GenNorm[ca] GenNorm[ca](x,area,mode,width,asym) Asym Generalized Normal Area

GenNorm[ca]_C GenNorm[ca]_C(amplitude,mode,width,l-asym) Gen. Normal LN4 Gauss width cumulative

GenNorm[ca]_CR GenNorm[ca]_CR(amplitude,mode,width,l-asym) Gen. Normal LN4 Gauss width reverse cumulative

GenHVL[Zca](x,area,mode,width,distortn,Asym50) ZDD=Asym Gen. Normal [ca]

This parameterization is not used in the built-in generalized chromatographic models. Here a₁ directly fits the mode , the x value at the apex of the peak, and a₃ directly fits the half-height asymmetry ratio, Asym50.

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