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Generalized Normal (Asymmetric)

Generalized Normal ([Z] ZDD) a0 = Area

a1 = Median (of asymmetric peak)

a2 = Width (SD of underlying normal)

a3 = Asymmetry ( -1 > a3 > 1 ) 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) 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] 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] 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] 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] 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] 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] 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] 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] 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) 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] 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.   