ARGUS DISTRIBUTION

Phitter implementation

Distribution Definition

import phitter

distribution = phitter.continuous.Argus({"chi": *, "loc": *, "scale": *})

💡 The distribution's parameters are defined equation section below

Distribution Methods and Attributes

## CDF, PDF, PPF receive float or numpy.ndarray.
distribution.cdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.pdf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.ppf(float | numpy.ndarray) # -> float | numpy.ndarray
distribution.sample(int) # -> numpy.ndarray

## STATS
distribution.mean # -> float
distribution.variance # -> float
distribution.standard_deviation # -> float
distribution.skewness # -> float
distribution.kurtosis # -> float
distribution.median # -> float
distribution.mode # -> float

Equations

Distribution Definition

$$X\sim \mathrm{Argus}(\chi ,\text{Loc},\text{Sc})$$

Distribution Domain

$$x\in (\text{Loc},\text{Loc}+\text{Sc})$$

Parameters Domain and Constraints

$$\chi \in {\mathbb{R}}^{+},\text{Loc}\in \mathbb{R},\text{Sc}\in {\mathbb{R}}^{+}$$

Cumulative Distribution Function

$$F_X\left(x\right)=1-\frac{\mathrm{\Psi }\left(\chi \sqrt{1-z(x{)}^2}\right)}{\mathrm{\Psi }(\chi )}$$

Probability Density Function

$$f_X\left(x\right)=\frac{1}{\text{Sc}}\cdot \frac{{\chi }^3}{\sqrt{2\pi }\mathrm{\Psi }(\chi )}\cdot z(x)\sqrt{1-z(x{)}^2}\exp (-\frac{1}{2}{\chi }^2(1-z(x{)}^2))$$

Percent Point Function / Sample

$$F_X^{-1}\left(u\right)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2{\text{P}}^{-1}(\frac{3}{2},(1-u)\text{P}(\frac{3}{2},\frac{{\chi }^2}{2}))}{{\chi }^2}}$$

Parametric Centered Moments

$${\mu }_k^{\prime }=E[X^k]={\int }_{\text{Loc}}^{\text{Loc}+\text{Sc}}{x}^k{f}_X\left(x\right)dx$$

Parametric Mean

$$\mathrm{Mean}(X)={\mu }_1^{\prime }=\text{Loc}+\text{Sc}\sqrt{\pi /8}\frac{\chi e^{-\frac{{\chi }^2}{4}}{I}_1(\frac{{\chi }^2}{4})}{\mathrm{\Psi }(\chi )}$$

Parametric Variance

$$\mathrm{Variance}(X)={\mu }_2^{\prime }-{\mu }_1^{\prime 2}={\text{Sc}}^2\cdot (1-\frac{3}{{\chi }^2}+\frac{\chi \varphi (\chi )}{\mathrm{\Psi }(\chi )})-(\mu -\text{Loc}{)}^2$$

Parametric Skewness

$$\mathrm{Skewness}(X)=\frac{{\mu }_3^{\prime }-3{\mu }_2^{\prime }{\mu }_1^{\prime }+2{\mu }_1^{\prime 3}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^{1.5}}$$

Parametric Kurtosis

$$\mathrm{Kurtosis}(X)=\frac{{\mu }_4^{\prime }-4{\mu }_1^{\prime }{\mu }_3^{\prime }+6{\mu }_1^{\prime 2}{\mu }_2^{\prime }-3{\mu }_1^{\prime 4}}{({\mu }_2^{\prime }-{\mu }_1^{\prime 2}{)}^2}$$

Parametric Median

$$\mathrm{Median}(X)=\text{Loc}+\text{Sc}\sqrt{1-\frac{2{\text{P}}^{-1}(\frac{3}{2},\frac{1}{2}\text{P}(\frac{3}{2},\frac{{\chi }^2}{2}))}{{\chi }^2}}$$

Parametric Mode

$$\mathrm{Mode}(X)=\text{Loc}+\frac{\text{Sc}}{\sqrt{2}\chi }\sqrt{({\chi }^2-2)+\sqrt{{\chi }^4+4}}$$

Additional Information and Definitions

• $\text{Loc}:\text{Location parameter}$
• $\text{Sc}:\text{Scale parameter}$
• $z(x)=(x-\text{Loc})/\text{Sc}$
• $u:\text{Uniform[0,1] random varible}$
• $\mathrm{\Psi }(\chi )=\mathrm{\Phi }(\chi )-\chi \varphi (\chi )-\frac{1}{2}$
• $\mathrm{\Phi }(x):\text{CDF normal standard distribution}$
• $\varphi (x):\text{PDF normal standard distribution}$
• $I_{\alpha }\left(x\right):\text{Modified Bessel function of the first kind of order }\alpha \in \mathbb{N}$
• $\text{P}(a,x)=\frac{\gamma (a,x)}{\mathrm{\Gamma }(a)}:\text{Regularized lower incomplete gamma function}$
• ${\text{P}}^{-1}(a,y):\text{Inverse of regularized lower incomplete gamma function}$
• $\gamma (a,x):\text{Lower incomplete gamma function}$
• $\mathrm{\Gamma }(x):\text{Gamma function}$

Spreadsheet Documents

• Phitter playground
• Download Excel Spreadsheet
• Excel file from GitHub repository
• Google spreadsheet document