Package org.apache.commons.statistics.distribution

Class FoldedNormalDistribution

java.lang.Object

org.apache.commons.statistics.distribution.FoldedNormalDistribution

All Implemented Interfaces:

ContinuousDistribution

public abstract class FoldedNormalDistribution
extends Object

Implementation of the folded normal distribution.

Given a normally distributed random variable $X$ with mean $\mu$ and variance ${\sigma }^2$, the random variable $Y=|X|$ has a folded normal distribution. This is equivalent to not recording the sign from a normally distributed random variable.

The probability density function of $X$ is:

$$f(x;\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2}+\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{x+\mu }{\sigma }\right)}^2}$$

for $\mu$ the location, $\sigma >0$ the scale, and $x\in [0,\mathrm{\infty })$.

If the location $\mu$ is 0 this reduces to the half-normal distribution.

Since:

1.1

See Also:

Folded normal distribution (Wikipedia), Half-normal distribution (Wikipedia)

Nested Class Summary

Nested classes/interfaces inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

ContinuousDistribution.Sampler

Method Summary

Modifier and Type	Method	Description
ContinuousDistribution.Sampler	createSampler(org.apache.commons.rng.UniformRandomProvider rng)	Creates a sampler.
abstract double	getMean()	Gets the mean of this distribution.
abstract double	getMu()	Gets the location parameter $\mu$ of this distribution.
double	getSigma()	Gets the scale parameter $\sigma$ of this distribution.
double	getSupportLowerBound()	Gets the lower bound of the support.
double	getSupportUpperBound()	Gets the upper bound of the support.
abstract double	getVariance()	Gets the variance of this distribution.
double	inverseCumulativeProbability(double p)	Computes the quantile function of this distribution.
double	inverseSurvivalProbability(double p)	Computes the inverse survival probability function of this distribution.
static FoldedNormalDistribution	of(double mu, double sigma)	Creates a folded normal distribution.
double	probability(double x0, double x1)	For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1).

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface org.apache.commons.statistics.distribution.ContinuousDistribution

cumulativeProbability, density, logDensity, survivalProbability

Method Detail

of

public static FoldedNormalDistribution of(double mu,
                                          double sigma)

Creates a folded normal distribution. If the location mu is zero this is the half-normal distribution.

Parameters:

mu - Location parameter.

sigma - Scale parameter.

Returns:

the distribution

Throws:

IllegalArgumentException - if sigma <= 0.

getMu

public abstract double getMu()

Gets the location parameter $\mu$ of this distribution.

Returns:

the mu parameter.

getSigma

public double getSigma()

Gets the scale parameter $\sigma$ of this distribution.

Returns:

the sigma parameter.

getMean

public abstract double getMean()

Gets the mean of this distribution.

For location parameter $\mu$ and scale parameter $\sigma$, the mean is:

$$\sigma \sqrt{\frac{2}{\pi }}\exp \left(\frac{-{\mu }^2}{2{\sigma }^2}\right)+\mu \mathrm{erf}\left(\frac{\mu }{\sqrt{2{\sigma }^2}}\right)$$

where $\mathrm{erf}$ is the error function.

Returns:

the mean.

getVariance

public abstract double getVariance()

Gets the variance of this distribution.

For location parameter $\mu$, scale parameter $\sigma$ and a distribution mean ${\mu }_Y$, the variance is:

$${\mu }^2+{\sigma }^2-{\mu }_Y^2$$

Returns:

the variance.

getSupportLowerBound

public double getSupportLowerBound()

Gets the lower bound of the support. It must return the same value as inverseCumulativeProbability(0), i.e. $inf\{x\in \mathbb{R}:P(X\le x)>0\}$.

The lower bound of the support is always 0.

Returns:

0.

getSupportUpperBound

public double getSupportUpperBound()

Gets the upper bound of the support. It must return the same value as inverseCumulativeProbability(1), i.e. $inf\{x\in \mathbb{R}:P(X\le x)=1\}$.

The upper bound of the support is always positive infinity.

Returns:

positive infinity.

probability

public double probability(double x0,
                          double x1)

For a random variable X whose values are distributed according to this distribution, this method returns P(x0 < X <= x1). The default implementation uses the identity P(x0 < X <= x1) = P(X <= x1) - P(X <= x0)

Specified by:

probability in interface ContinuousDistribution

Parameters:

x0 - Lower bound (exclusive).

x1 - Upper bound (inclusive).

Returns:

the probability that a random variable with this distribution takes a value between x0 and x1, excluding the lower and including the upper endpoint.

inverseCumulativeProbability

public double inverseCumulativeProbability(double p)

Computes the quantile function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X\le x)\ge p\}& \text{for}0<p\le 1\\ inf\{x\in \mathbb{R}:P(X\le x)>0\}& \text{for}p=0\end{array}$$

The default implementation returns:

• ContinuousDistribution.getSupportLowerBound() for p = 0,
• ContinuousDistribution.getSupportUpperBound() for p = 1, or
• the result of a search for a root between the lower and upper bound using cumulativeProbability(x) - p. The bounds may be bracketed for efficiency.

Specified by:

inverseCumulativeProbability in interface ContinuousDistribution

Parameters:

p - Cumulative probability.

Returns:

the smallest p-quantile of this distribution (largest 0-quantile for p = 0).

Throws:

IllegalArgumentException - if p < 0 or p > 1

inverseSurvivalProbability

public double inverseSurvivalProbability(double p)

Computes the inverse survival probability function of this distribution. For a random variable X distributed according to this distribution, the returned value is:

$$x=\{\begin{array}{ll}inf\{x\in \mathbb{R}:P(X>x)\le p\}& \text{for}0\le p<1\\ inf\{x\in \mathbb{R}:P(X>x)<1\}& \text{for}p=1\end{array}$$

By default, this is defined as inverseCumulativeProbability(1 - p), but the specific implementation may be more accurate.

The default implementation returns:

• ContinuousDistribution.getSupportLowerBound() for p = 1,
• ContinuousDistribution.getSupportUpperBound() for p = 0, or
• the result of a search for a root between the lower and upper bound using survivalProbability(x) - p. The bounds may be bracketed for efficiency.

Specified by:

inverseSurvivalProbability in interface ContinuousDistribution

Parameters:

p - Survival probability.

Returns:

the smallest (1-p)-quantile of this distribution (largest 0-quantile for p = 1).

Throws:

IllegalArgumentException - if p < 0 or p > 1

createSampler

public ContinuousDistribution.Sampler createSampler(org.apache.commons.rng.UniformRandomProvider rng)

Creates a sampler.

Specified by:

createSampler in interface ContinuousDistribution

Parameters:

rng - Generator of uniformly distributed numbers.

Returns:

a sampler that produces random numbers according this distribution.