Probability Distributions

Introduction

This component provides subroutines for computing probability distribution functions (i.e., pdf, cdf, etc.). For any continuous probability distribution represented by a class D, we have the following static methods to compute different function values:

D::pdf(x, ...) - Probability density function at x.
D::cdf_lower(x, ...) - Cumulative probability function at x.
D::cdf_upper(x, ...) - Cumulative (from upper bound) probability function at x.
D::invcdf_lower(P, ...) - The inverse of cumulative probability function at P.
D::invcdf_lower(Q, ...) - The inverse of cumulative (from upper bound) probability function at Q.

Here ... represents the additional arguments specific to the target distribution.

Note that these methods are optional, i.e., for some distributions, not all of these methods are available.

The following variables, functions and classes are all defined within the namespace HIPP::NUMERICAL.

Example: we compute the probability density function and cumulative probability function of the Gaussian distribution by the static methods of class GaussianDist:

using namespace HIPP;
using namespace HIPP::NUMERICAL;

double sigma = 2.0;
pout << "Gaussian distribution p(x) and cumulative P(x):\n";
for( auto x : {0., 1., 2., 3., 4., 5.} )
    pout << "p(", x, ") = ",  GaussianDist::pdf(x, sigma),
        ", P(", x, ") = ", GaussianDist::cdf_lower(x, sigma), endl;

The output is:

Gaussian distribution p(x) and cumulative P(x):
p(0) = 0.199471, P(0) = 0.5
p(1) = 0.176033, P(1) = 0.691462
p(2) = 0.120985, P(2) = 0.841345
p(3) = 0.0647588, P(3) = 0.933193
p(4) = 0.0269955, P(4) = 0.97725
p(5) = 0.00876415, P(5) = 0.99379

Table of Available Distributions

We summary the available distribution class and its methods in the following table:

Class	Probability Distribution	Methods
`UniformRealDist`	Uniform real distribution in the range \([a,\,b]\)	`pdf(x,a,b)`, `cdf_lower(x,a,b)`, `cdf_upper(x,a,b)`, `invcdf_lower(P,a,b)`, `invcdf_upper(Q,a,b)`
`GaussianDist`	Normal/Gaussian distribution with standard deviation \(\sigma\)	`pdf(x,sigma)`, `cdf_lower(x,sigma)`, `cdf_upper(x,sigma)`, `invcdf_lower(P,sigma)`, `invcdf_upper(Q,sigma)`
	Specific to \(\sigma=1\)	`pdf_unit(x)`, `cdf_lower_unit(x)`, `cdf_upper_unit(x)`, `invcdf_lower_unit(P)`, `invcdf_upper_unit(Q)`
`GaussianTailDist`	The gaussian tail distribution for \(x>a\) with standard deviation \(\sigma\)	`pdf(x,a,sigma)`
	Specific to \(\sigma\)	`pdf_unit(x,a)`
`BivariateGaussianDist`	Bivariate Gaussian distribution with standard deviations \((\sigma_x, \sigma_y)\) and correlation coefficient \(\rho\)	`pdf(x,y,sigma_x,sigma_y, rho)`
`LognormalDist`	Lognormal distribution for \(x > 0\)	`pdf(x,zeta,sigma)`, `cdf_lower(x,zeta,sigma)`, `cdf_upper(x,zeta,sigma)`, `invcdf_lower(P,zeta,sigma)`, `invcdf_upper(Q,zeta,sigma)`
`ExpPowDist`	Exponential power distribution functions with scale parameter \(a\) and exponential \(b\)	`pdf(x,a,b)`, `cdf_lower(x,a,b)`, `cdf_upper(x,a,b)`
`ExponentialDist`	Exponential distribution functions with mean \(\mu\)	`pdf(x,mu)`, `cdf_lower(x,mu)`, `cdf_upper(x,mu)`, `invcdf_lower(P,mu)`, `invcdf_upper(Q,mu)`
`LaplaceDist`	Laplace distribution functions with with \(a\)	`pdf(x,a)`, `cdf_lower(x,a)`, `cdf_upper(x,a)`, `invcdf_lower(P,a)`, `invcdf_upper(Q,a)`
`CauchyDist`	Cauchy distribution (also known as Lorentz distribution) with scale parameter \(a\)	`pdf(x,a)`, `cdf_lower(x,a)`, `cdf_upper(x,a)`, `invcdf_lower(P,a)`, `invcdf_upper(Q,a)`
`RayleighDist`	Rayleigh distribution with scale parameter \(\sigma\)	`pdf(x,sigma)`, `cdf_lower(x,sigma)`, `cdf_upper(x,sigma)`, `invcdf_lower(P,sigma)`, `invcdf_upper(Q,sigma)`
`RayleighTailDist`	Tail of Rayleigh distribution with scale parameter \(\sigma\) and a lower limit \(a\)	`pdf(x,a,sigma)`

Uniform Real Distribution

class UniformRealDist

UniformRealDist - the distribution function is uniform in \([a, b]\), i.e.,

\[p(x) = \frac{1}{b-a} .\]

static double pdf(double x, double a, double b) noexcept
static double cdf_lower(double x, double a, double b) noexcept
static double cdf_upper(double x, double a, double b) noexcept
static double invcdf_lower(double P, double a, double b) noexcept
static double invcdf_upper(double Q, double a, double b) noexcept

Gaussian Distributions

Gaussian Distribution

class GaussianDist

The Normal (Gaussian) distribution with standard deviation \(\sigma\).

static double pdf(double x, double sigma) noexcept
static double cdf_lower(double x, double sigma) noexcept
static double cdf_upper(double x, double sigma) noexcept
static double invcdf_lower(double P, double sigma) noexcept
static double invcdf_upper(double Q, double sigma) noexcept: Calls for the Gaussian dist with standard deviation \(\sigma\).

static double pdf_unit(double x) noexcept
static double cdf_lower_unit(double x) noexcept
static double cdf_upper_unit(double x) noexcept
static double invcdf_lower_unit(double P) noexcept
static double invcdf_upper_unit(double Q) noexcept: Calls for the unit gaussian dist., i.e., \(\sigma = 1\).

Gaussian Tail Distribution

class GaussianTailDist

GaussianTailDist - for the gaussian tail distribution, i.e., its pdf is the upper tail of a full normal distribution \(\mathcal{N}(x|0, \sigma)\), with \(x > a\).

static double pdf(double x, double a, double sigma) noexcept
static double pdf_unit(double x, double a) noexcept

Bivariate Gaussian Distribution

class BivariateGaussianDist

static double pdf(double x, double y, double sigma_x, double sigma_y, double rho) noexcept

Lognormal Distribution

class LognormalDist

LognormalDist - the lognormal distribution for \(x > 0\), i.e.

\[p(x) = \frac{1}{ x \sqrt{ 2 \pi \sigma^2} } \exp( - \frac{( {\rm ln}(x) - \zeta )^2}{2 \sigma^2} ) .\]

static double pdf(double x, double zeta, double sigma) noexcept
static double cdf_lower(double x, double zeta, double sigma) noexcept
static double cdf_upper(double x, double zeta, double sigma) noexcept
static double invcdf_lower(double P, double zeta, double sigma) noexcept
static double invcdf_upper(double Q, double zeta, double sigma) noexcept

Exponential Distributions

Exponential Power Distribution

class ExpPowDist

ExpPowDist - exponential power distribution functions with scale parameter a and exponential b, i.e.

\[p(x) = \frac{1}{ 2 a \Gamma(1+1/b) } \exp( -| x / a |^b ),\]

for \(x >= 0\).

Special cases:

\(b = 1\): Laplace distribution.
\(b = 2\): Gaussian distribution with \(a = \sqrt{2} \sigma\).

static double pdf(double x, double a, double b) noexcept
static double cdf_lower(double x, double a, double b) noexcept
static double cdf_upper(double x, double a, double b) noexcept

Exponential Distribution

class ExponentialDist

ExponentialDist - exponential distribution functions with mean \(\mu\), i.e.,

\[p(x) = \frac{1}{\mu} \exp( - \frac{x}{\mu} ),\]

for \(x \geqslant 0\).

static double pdf(double x, double mu) noexcept
static double cdf_lower(double x, double mu) noexcept
static double cdf_upper(double x, double mu) noexcept
static double invcdf_lower(double P, double mu) noexcept
static double invcdf_upper(double Q, double mu) noexcept

Laplace Distribution

class LaplaceDist

LaplaceDist - laplace distribution functions with with \(a\), i.e.,

\[p(x) = \frac{1}{2a} \exp( -|x/a| ),\]

for \(x \in (-\infty, \infty)\).

static double pdf(double x, double a) noexcept
static double cdf_lower(double x, double a) noexcept
static double cdf_upper(double x, double a) noexcept
static double invcdf_lower(double P, double a) noexcept
static double invcdf_upper(double Q, double a) noexcept

Cauchy Distribution

class CauchyDist

Cauchy distribution (also known as Lorentz distribution) with scale parameter \(a\), i.e.,

\[p(x) = \frac{1}{a \pi ( 1 + (x/a)^2 )},\]

for \(x \in (-\infty, \infty)\).

double pdf(double x, double a) noexcept
double cdf_lower(double x, double a) noexcept
double cdf_upper(double x, double a) noexcept
double invcdf_lower(double P, double a) noexcept
double invcdf_upper(double Q, double a) noexcept

Rayleigh Distributions

Rayleigh Distribution

class RayleighDist

Rayleigh distribution with scale parameter \(\sigma\), i.e.,

\[p(x) = \frac{x}{\sigma^2} \exp( - \frac{x^2}{2 \sigma^2} ),\]

for \(x > 0\).

double pdf(double x, double sigma) noexcept
double cdf_lower(double x, double sigma) noexcept
double cdf_upper(double x, double sigma) noexcept
double invcdf_lower(double P, double sigma) noexcept
double invcdf_upper(double Q, double sigma) noexcept

Rayleigh Tail Distribution

class RayleighTailDist

Tail of Rayleigh distribution with scale parameter \(\sigma\) and a lower limit \(a\), i.e.,

\[p(x) = \frac{x}{\sigma^2} \exp( \frac{a^2-x^2}{2 \sigma^2} ),\]

for \(x > a\).

double pdf(double x, double a, double sigma) noexcept

HIPP 0.0.1 documentation

Probability Distributions

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