Package org.apache.commons.statistics.distribution

Class LaplaceDistribution

java.lang.Object

org.apache.commons.statistics.distribution.LaplaceDistribution

All Implemented Interfaces:

ContinuousDistribution

public final class LaplaceDistribution
extends Object

Implementation of the Laplace distribution.

The probability density function of \( X \) is:

\[ f(x; \mu, b) = \frac{1}{2b} \exp \left( -\frac{|x-\mu|}{b} \right) \]

for \( \mu \) the location, \( b > 0 \) the scale, and \( x \in (-\infty, \infty) \).

See Also:

• Laplace distribution (Wikipedia)
• Laplace distribution (MathWorld)

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.
double	cumulativeProbability(double x)	For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x).
double	density(double x)	Returns the probability density function (PDF) of this distribution evaluated at the specified point x.
double	getLocation()	Gets the location parameter of this distribution.
double	getMean()	Gets the mean of this distribution.
double	getScale()	Gets the scale parameter of this distribution.
double	getSupportLowerBound()	Gets the lower bound of the support.
double	getSupportUpperBound()	Gets the upper bound of the support.
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.
double	logDensity(double x)	Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.
static LaplaceDistribution	of(double mu, double beta)	Creates a Laplace 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).
double	survivalProbability(double x)	For a random variable X whose values are distributed according to this distribution, this method returns P(X > x).

Methods inherited from class java.lang.Object

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

Method Details

of

public static LaplaceDistribution of(double mu,
 double beta)

Creates a Laplace distribution.

Parameters:

mu - Location parameter.

beta - Scale parameter (must be positive).

Returns:

the distribution

Throws:

IllegalArgumentException - if beta <= 0

getLocation

public double getLocation()

Gets the location parameter of this distribution.

Returns:

the location parameter.

getScale

public double getScale()

Gets the scale parameter of this distribution.

Returns:

the scale parameter.

density

public double density(double x)

Returns the probability density function (PDF) of this distribution evaluated at the specified point x. In general, the PDF is the derivative of the CDF. If the derivative does not exist at x, then an appropriate replacement should be returned, e.g. Double.POSITIVE_INFINITY, Double.NaN, or the limit inferior or limit superior of the difference quotient.

Parameters:

x - Point at which the PDF is evaluated.

Returns:

the value of the probability density function at x.

logDensity

public double logDensity(double x)

Returns the natural logarithm of the probability density function (PDF) of this distribution evaluated at the specified point x.

Parameters:

x - Point at which the PDF is evaluated.

Returns:

the logarithm of the value of the probability density function at x.

cumulativeProbability

public double cumulativeProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X <= x). In other words, this method represents the (cumulative) distribution function (CDF) for this distribution.

Parameters:

x - Point at which the CDF is evaluated.

Returns:

the probability that a random variable with this distribution takes a value less than or equal to x.

survivalProbability

public double survivalProbability(double x)

For a random variable X whose values are distributed according to this distribution, this method returns P(X > x). In other words, this method represents the complementary cumulative distribution function.

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

Parameters:

x - Point at which the survival function is evaluated.

Returns:

the probability that a random variable with this distribution takes a value greater than x.

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:

\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \le x) \ge p\} & \text{for } 0 \lt p \le 1 \\ \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} & \text{for } p = 0 \end{cases} \]

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).

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:

\[ x = \begin{cases} \inf \{ x \in \mathbb R : P(X \gt x) \le p\} & \text{for } 0 \le p \lt 1 \\ \inf \{ x \in \mathbb R : P(X \gt x) \lt 1 \} & \text{for } p = 1 \end{cases} \]

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).

getMean

public double getMean()

Gets the mean of this distribution.

The mean is equal to the location.

Returns:

the mean.

getVariance

public double getVariance()

Gets the variance of this distribution.

For scale parameter \( b \), the variance is \( 2 b^2 \).

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. \( \inf \{ x \in \mathbb R : P(X \le x) \gt 0 \} \).

The lower bound of the support is always negative infinity.

Returns:

negative infinity.

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.

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.