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Statistical Functions


Normal (Gaussian)

 

Statistical Functions1.png

a0 = Area

a1 = Center (mean)

a2 = Width (SD)

 

Built in model: Normal

User-defined peaks and view functions: Normal(x,a0,a1,a2)

 

Normal (Gaussian) Amplitude

 

Statistical Functions2.png

a0 = Amplitude

a1 = Center (mean)

a2 = Width (SD)

 

Built in model: Normal[amp]

User-defined peaks and view functions: Normal[amp](x,a0,a1,a2)

 

v5_Statistical_Normal.png

Cauchy (Lorentzian)

 

Statistical Functions3.png

a0 = Area

a1 = Center

a2 = Width

 

Built in model: Cauchy

User-defined peaks and view functions: Cauchy(x,a0,a1,a2)

 

Cauchy (Lorentzian) Amplitude

 

Statistical Functions4.png

a0 = Amplitude

a1 = Center

a2 = Width

 

Built in model: Cauchy[amp]

User-defined peaks and view functions: Cauchy[amp](x,a0,a1,a2)

 

v5_Statistical_Cauchy.png

Log Normal

 

Statistical Functions5.png

a0 = Area

a1 = Center (0)

a2 = Width (>0)

 

Built in model: LogNormal

User-defined peaks and view functions: LogNormal(x,a0,a1,a2)

 

Log Normal Amplitude

 

Statistical Functions6.png

a0 = Amplitude

a1 = Center (0)

a2 = Width (>0)

 

Built in model: LogNormal[amp]

User-defined peaks and view functions: LogNormal[amp](x,a0,a1,a2)

 

v5_Statistical_LogNormal.png

 

Logistic

 

Statistical Functions7.png

a0 = Area

a1 = Center

a2 = Width

 

Built in model: Logistic

User-defined peaks and view functions: Logistic(x,a0,a1,a2)

 

Logistic Amplitude

 

Statistical Functions8.png

a0 = Amplitude

a1 = Center

a2 = Width

 

Built in model: Logistic[amp]

User-defined peaks and view functions: Logistic[amp](x,a0,a1,a2)

 

v5_Statistical_Logistic.png

 

Laplace (DblSided Exp)

 

Statistical Functions9.png

a0 = Area

a1 = Center

a2 = Width

 

Built in model: Laplace

User-defined peaks and view functions: Laplace(x,a0,a1,a2)

 

Laplace (DblSided Exp) Amplitude

 

Statistical Functions10.png

a0 = Amplitude

a1 = Center

a2 = Width

 

Built in model: Laplace[amp]

User-defined peaks and view functions: Laplace[amp](x,a0,a1,a2)

 

v5_Statistical_Laplace.png

Extreme Value

 

Statistical Functions11.png

a0 = Area

a1 = Center

a2 = Width (>0)

 

Built in model: ExtrValue

User-defined peaks and view functions: ExtrValue(x,a0,a1,a2)

 

Extreme Value Amplitude

 

Statistical Functions12.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

 

Built in model: ExtrValue[amp]

User-defined peaks and view functions: ExtrValue[amp](x,a0,a1,a2)

 

v5_Statistical_ExtrValue.png

 

Gamma Peak

 

Statistical Functions13.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (1.01<a3<167.92)

 

Built in model: GammaPeak

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

 

Gamma Peak Amplitude

 

Statistical Functions14.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (1.01<a3<167.92)

 

Built in model: GammaPeak[amp]

User-defined peaks and view functions: GammaPeak[amp](x,a0,a1,a2,a3)

 

v5_Statistical_GammaPeak.png

 

Inverted Gamma Peak

 

Statistical Functions15.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (0<a3<167.92)

 

Built in model: InvGamma

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

 

Inverted Gamma Peak Amplitude

 

Statistical Functions16.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (0<a3<167.92)

 

Built in model: InvGamma[amp]

User-defined peaks and view functions: InvGamma[amp](x,a0,a1,a2,a3)

 

v5_Statistical_InvGamma.png

 

Weibull

 

Statistical Functions74.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (>1.01)

 

Built in model: Weibull

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

 

Weibull Amplitude

 

Statistical Functions80.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (>1.01)

 

Built in model: Weibull[amp]

User-defined peaks and view functions: Weibull[amp](x,a0,a1,a2,a3)

 

v5_Statistical_Weibull.png

Error

 

Statistical Functions21.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: Error

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

 

Error Amplitude

 

Statistical Functions22.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: Error[amp]

User-defined peaks and view functions: Error[amp](x,a0,a1,a2,a3)

 

v5_Statistical_Error.png

Chi Squared

 

Statistical Functions23.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (2.01<a3<335.84)

 

Built in model: ChiSq

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

 

Chi Squared Amplitude

 

Statistical Functions24.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (2.01<a3<335.84)

 

Built in model: ChiSq[amp]

User-defined peaks and view functions: ChiSq[amp](x,a0,a1,a2,a3)

 

v5_Statistical_ChiSq.png

Student's t

 

Statistical Functions25.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: Student

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

 

Student's t Amplitude

 

Statistical Functions26.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: Student[amp]

User-defined peaks and view functions: Student[amp](x,a0,a1,a2,a3)

 

v5_Statistical_Student.png

Beta Peak

 

Statistical Functions27.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape1 (1.01)

a4 = Shape2 (1.01)

 

Built in model: BetaPeak

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

 

Beta Peak Amplitude

 

Statistical Functions28.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape1 (1.01)

a4 = Shape2 (1.01)

 

Built in model: BetaPeak[amp]

User-defined peaks and view functions: BetaPeak[amp](x,a0,a1,a2,a3,a4)

 

v5_Statistical_BetaPeak.png

 

F Variance

 

Statistical Functions29.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape1 (2<a3<167.92)

a4 = Shape2 (2<a4<167.92)

 

Built in model: FVariance

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

 

FVariance Amplitude

 

Statistical Functions30.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape1 (2<a3<167.92)

a4 = Shape2 (2<a4<167.92)

 

Built in model: FVariance[amp]

User-defined peaks and view functions: FVariance[amp](x,a0,a1,a2,a3,a4)

v5_Statistical_FVariance.png

Pearson VII

 

Pearson VII1.png

a0 = Area

a1 = Center (mode)

a2 = FWHM

a3 = Shape (1=Lorentzian, infinity=Gaussian)

 

Built in model: PearsonVII

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

 

Pearson VII (Amplitude)

 

Pearson VII2.png

a0 = Amplitude

a1 = Center (mode)

a2 = FWHM

a3 = Shape (1=Lorentzian, infinity=Gaussian)

 

Built in model:PearsonVII[amp]

User-defined peaks and view functions: PearsonVII[amp](x,a0,a1,a2,a3)

 

v5_Statistical_PearsonVII.png

Pearson IV

 

Statistical Functions31.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape1 (>0)

a4 = Shape2

 

Built in model: PearsonIV

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

 

Pearson IV Amplitude

 

Statistical Functions32.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape1 (>0)

a4 = Shape2

 

Built in model: PearsonIV[amp]

User-defined peaks and view functions: PearsonIV[amp](x,a0,a1,a2,a3,a4)

 

v5_Statistical_PearsonIV.png

 

Generalized Normal

 

Statistical Functions33.png

a0 = Area

a1 = Center (median)

a2 = Width (>0)

a3 = Asymmetry

 

Built in model: GenNorm

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

 

v5_Statistical_GenNorm.png

Generalized Normal Amplitude

 

Statistical Functions34.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Asymmetry

 

Built in model: GenNorm[amp]

User-defined peaks and view functions: GenNorm[amp](x,a0,a1,a2,a3)

 

Generalized Normal Alternate Parametrizations

The Asymmetric Generalized Normal is available in a number of different parametrizations.

a1=mean (Default ZDD)

 

Statistical Functions35.png

a0 = Area

a1 = Center (mean)

a2 = Width (>0)

a3 = Asymmetry

 

Built in model: GenNorm[m]

User-defined peaks and view functions: GenNorm[m](x,a0,a1,a2,a3)

 

a1=mode

 

Statistical Functions36.png

a0 = Area

a1 = Center (mode)

a2 = Width (>0)

a3 = Asymmetry

 

Built in model: GenNorm[c]

User-defined peaks and view functions: GenNorm[c](x,a0,a1,a2,a3)

 

a2=SD

 

Statistical Functions37.png

a0 = Area

a1 = Center (mean)

a2 = Width (SD)

a3 = Asymmetry

 

Built in model: GenNorm[d]

User-defined peaks and view functions: GenNorm[d](x,a0,a1,a2,a3)

 

a3=asym50

 

Statistical Functions38.png

a0 = Area

a1 = Center (mean)

a2 = Width (>0)

a3 = Half-Height Asymmetry Asym50

 

Built in model: GenNorm[a]

User-defined peaks and view functions: GenNorm[a](x,a0,a1,a2,a3)

 

a1=mode, a2=FWHM

 

Statistical Functions39.png

a0 = Area

a1 = Center (mode)

a2 = Width (FWHM)

a3 = Asymmetry

 

Built in model: GenNorm[cw]

User-defined peaks and view functions: GenNorm[cw](x,a0,a1,a2,a3)

 

a1=mean, a2=SD

 

Statistical Functions40.png

a0 = Area

a1 = Center (mean)

a2 = Width (SD)

a3 = Asymmetry

 

Built in model: GenNorm[md]

User-defined peaks and view functions: GenNorm[md](x,a0,a1,a2,a3)

 

a1=mode, a2=SD

 

Statistical Functions41.png

a0 = Area

a1 = Center (mode)

a2 = Width (SD)

a3 = Asymmetry

 

Built in model: GenNorm[cd]

User-defined peaks and view functions: GenNorm[cd](x,a0,a1,a2,a3)

 

a1=mean, a2=variance

 

Statistical Functions42.png

a0 = Area

a1 = Center (mean)

a2 = Width (variance)

a3 = Asymmetry

 

Built in model: GenNorm[mv]

User-defined peaks and view functions: GenNorm[mv](x,a0,a1,a2,a3)

 

a1=mean, a2=variance, a3=skew

 

Statistical Functions45.png

a0 = Area

a1 = Center (mean)

a2 = Width (variance)

a3 = Asymmetry (skew)

 

Built in model: GenNorm[mvs]

User-defined peaks and view functions: GenNorm[mvs](x,a0,a1,a2,a3)

 

a1=mode, a2=FWHM, a3=Asym50

 

Statistical Functions46.png

a0 = Area

a1 = Center (mode)

a2 = Half-Height Width (FWHM)

a3 = Half-Height Asymmetry (Asym50)

 

Built in model: GenNorm[cwa]

User-defined peaks and view functions: GenNorm[cwa](x,a0,a1,a2,a3)

 

a1=mode, a3=Asym50

 

Statistical Functions47.png

a0 = Area

a1 = Center (mode)

a2 = Width

a3 = Half-Height Asymmetry (Asym50)

 

Built in model: GenNorm[ca]

User-defined peaks and view functions: GenNorm[ca](x,a0,a1,a2,a3)

 

Skew Normal

 

Statistical Functions48.png

a0 = Area

a1 = Center

a2 = Width

a3 = Asymmetry

 

Built in model: SkewNormal

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

 

v5_Statistical_SkewNormal.png

GMG (Half-Gaussian Gaussian)

 

Statistical Functions49.png

a0 = Area

a1 = Center

a2 = Full-Gaussian Width

a3 = Half-Gaussian Width

 

Built in model: GMG

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

 

v5_Statistical_GMG.png

 

EMG (Exponential Gaussian)

 

Statistical Functions50.png

a0 = Area

a1 = Center

a2 = Gaussian Width

a3 = Exponential Decay Width (time constant)

 

Built in model: EMG

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

 

v5_Statistical_EMG.png

EMG Alternate Parametrizations

a2=SD

 

Statistical Functions51.png

a0 = Area

a1 = Center

a2 = SD of Convolution

a3 = Exponential Decay Width (time constant)

 

Built in model: EMG[d]

User-defined peaks and view functions: EMG[d](x,a0,a1,a2,a3)

 

a1=mean, a2=SD

 

Statistical Functions52.png

a0 = Area

a1 = Center (mean)

a2 = Width (SD)

a3 = Exponential Decay Width (time constant)

 

Built in model: EMG[md]

User-defined peaks and view functions: EMG[md](x,a0,a1,a2,a3)

 

NError (Alternate Error Model-Q ZDD)

 

ZDD - [Q]5.png

a0 = Area

a1 = Center

a2 = Width

a3= Error model power (.25 > a3 > 4 , Laplace=1, Gaussian=2)

 

Built in model: NError

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

 

v5_Statistical_NError.png

 

Extreme Value 4-Parameter

 

Statistical Functions55.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: ExtrVal4

User-defined peaks and view functions: ExtrVal4(x,a0,a1,a2,a3)

v5_Statistical_ExtrVal4.png

Extreme Value 4-Parameter Amplitude

 

Statistical Functions56.png

a0 = Amplitude

a1 = Center

a2 = Width (>0)

a3 = Shape (>0)

 

Built in model: ExtrVal4[amp]

User-defined peaks and view functions: ExtrVal4[amp](x,a0,a1,a2,a3)

 

Generalized Extreme Value

 

Statistical Functions53.png

a0 = Area

a1 = Center

a2 = Width

a3 = Shape

Built in model: GenExtrVal

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

v5_Statistical_GenExtrVal.png

Generalized Extreme Value Amplitude

 

Statistical Functions54.png

a0 = Amplitude

a1 = Center

a2 = Width

a3 = Shape

 

Built in model: GenExtrVal[amp]

User-defined peaks and view functions: GenExtrVal[amp](x,a0,a1,a2,a3)

 

Generalized Logistic

 

Statistical Functions60.png

a0 = Area

a1 = Center

a2 = Width

a3 = Shape

 

Built in model: GenLogistic

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

 

v5_Statistical_GenLogistic.png

Generalized Logistic Amplitude

 

Statistical Functions61.png

a0 = Amplitude

a1 = Center

a2 = Width

a3 = Shape

 

Built in model: GenLogistic[amp]

User-defined peaks and view functions: GenLogistic[amp](x,a0,a1,a2,a3)

 

Generalized Error

 

Statistical Functions83.png

a0 = Area

a1 = Center

a2 = Width

a3 = Power of Tail Decay

a4 = Asymmetry

 

Built in model: GenError

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

 

v5_Statistical_GenError.png

Generalized Error, Two-Width

 

 

ZDD - [Yp2]2.png

a0 = Area

a1 = Center

a2 = Width, right
a3 = Fraction of a2 for Width, left
a4 = Error model power (.25 > a4 > 4 , Laplace=1, Gaussian=2)

a5 = Asymmetry ( fronted -1 > a5 > 1 tailed)

 

Built in model: GenError2

User-defined peaks and view functions: GenError2(x,a0,a1,a2,a3,a4)

 

v5_Statistical_GenError2.png

Generalized Student's t

 

Statistical Functions84.png

a0 = Area

a1 = Center

a2 = Width

a3 = Tailing (Student's t nu or DOF)

a4 = Asymmetry

 

Built in model: GenStudent

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

 

v5_Statistical_GenStudent.png

 

SDS (Symmetric Double Sigmoid)

 

Statistical Functions89.png

a0 = Area

a1 = Center

a2 = Width

a3 = Sigmoid Width

 

Built in model: SDS

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

 

v5_Statistical_SDS.png

SDS (Symmetric Double Sigmoid) Amplitude

 

Statistical Functions88.png

a0 = Amplitude

a1 = Center

a2 = Width

a3 = Sigmoid Width

 

Built in model: SDS[amp]

User-defined peaks and view functions: SDS[amp](x,a0,a1,a2,a3)

 

Generalized SDS (Symmetric Double Sigmoid)

 

Statistical Functions87.png

a0 = Area

a1 = Center

a2 = Width

a3 = Adjusted Sigmoid Width (approximates Gaussian width in Generalized Models)

a4 = Asymmetry

 

Built in model: GenSDS

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

 

v5_Statistical_GenSDS.png

 

Generalized GMG (Half-Gaussian Modified Gaussian)

 

ZDD - [V]4.png

a0 = Area

a1 = Center

a2 = Width (>0)

a3= Half-Gaussian convolution width (>0)

a4 = Asymmetry ( fronted -1 > a4 > 1 tailed)

 

Built in model: GenGMG

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

v5_Statistical_GenGMG.png



c:\1pf\v5 help\home.gif GMG Generalized Normal