org.apache.commons.statistics.distribution

Class TrapezoidalDistribution

java.lang.Object

org.apache.commons.statistics.distribution.TrapezoidalDistribution

All Implemented Interfaces:

ContinuousDistribution

public abstract class TrapezoidalDistribution
extends Object

Implementation of the trapezoidal distribution.

The probability density function of $X$ is:

$$f(x;a,b,c,d)=\{\begin{array}{ll}\frac{2}{d+c-a-b}\frac{x-a}{b-a}& \text{for}a\le x<b\\ \frac{2}{d+c-a-b}& \text{for}b\le x<c\\ \frac{2}{d+c-a-b}\frac{d-x}{d-c}& \text{for}c\le x\le d\end{array}$$

for $-\mathrm{\infty }<a\le b\le c\le d<\mathrm{\infty }$ and $x\in [a,d]$.

Note the special cases:

• $b=c$ is the triangular distribution
• $a=b$ and $c=d$ is the uniform distribution

See Also:

Trapezoidal distribution (Wikipedia)

Nested Class Summary

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

ContinuousDistribution.Sampler

Field Summary

Fields

Modifier and Type	Field and Description
protected double	a Lower limit of this distribution (inclusive).
protected double	b Start of the trapezoid constant density.
protected double	c End of the trapezoid constant density.
protected double	d Upper limit of this distribution (inclusive).

Method Summary

Modifier and Type	Method and Description
ContinuousDistribution.Sampler	createSampler(UniformRandomProvider rng) Creates a sampler.
double	getB() Gets the start of the constant region of the density function.
double	getC() Gets the end of the constant region of the density function.
abstract double	getMean() Gets the mean 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 TrapezoidalDistribution	of(double a, double b, double c, double d) Creates a trapezoidal 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

Field Detail

a

protected final double a

Lower limit of this distribution (inclusive).

b

protected final double b

Start of the trapezoid constant density.

c

protected final double c

End of the trapezoid constant density.

d

protected final double d

Upper limit of this distribution (inclusive).

Method Detail

of

public static TrapezoidalDistribution of(double a,
                                         double b,
                                         double c,
                                         double d)

Creates a trapezoidal distribution.

The distribution density is represented as an up sloping line from a to b, constant from b to c, and then a down sloping line from c to d.

Parameters:

a - Lower limit of this distribution (inclusive).

b - Start of the trapezoid constant density (first shape parameter).

c - End of the trapezoid constant density (second shape parameter).

d - Upper limit of this distribution (inclusive).

Returns:

the distribution

Throws:

IllegalArgumentException - if a >= d, if b < a, if c < b or if c > d.

getMean

public abstract double getMean()

Gets the mean of this distribution.

For lower limit $a$, start of the density constant region $b$, end of the density constant region $c$ and upper limit $d$, the mean is:

$$\frac{1}{3(d+c-b-a)}(\frac{d^3-c^3}{d-c}-\frac{b^3-a^3}{b-a})$$

Returns:

the mean.

getVariance

public abstract double getVariance()

Gets the variance of this distribution.

For lower limit $a$, start of the density constant region $b$, end of the density constant region $c$ and upper limit $d$, the variance is:

$$\frac{1}{6(d+c-b-a)}(\frac{d^4-c^4}{d-c}-\frac{b^4-a^4}{b-a})-{\mu }^2$$

where $\mu$ is the mean.

Returns:

the variance.

getB

public double getB()

Gets the start of the constant region of the density function.

This is the first shape parameter b of the distribution.

Returns:

the first shape parameter b

getC

public double getC()

Gets the end of the constant region of the density function.

This is the second shape parameter c of the distribution.

Returns:

the second shape parameter c

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 equal to the lower limit parameter a of the distribution.

Returns:

the lower bound of the support.

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 equal to the upper limit parameter d of the distribution.

Returns:

the upper bound of the support.

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:

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

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(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.