TriangularDistribution.java
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.statistics.distribution;
/**
* Implementation of the triangular distribution.
*
* <p>The probability density function of \( X \) is:
*
* <p>\[ f(x; a, b, c) = \begin{cases}
* \frac{2(x-a)}{(b-a)(c-a)} & \text{for } a \le x \lt c \\
* \frac{2}{b-a} & \text{for } x = c \\
* \frac{2(b-x)}{(b-a)(b-c)} & \text{for } c \lt x \le b \\
* \end{cases} \]
*
* <p>for \( -\infty \lt a \le c \le b \lt \infty \) and
* \( x \in [a, b] \).
*
* @see <a href="https://en.wikipedia.org/wiki/Triangular_distribution">Triangular distribution (Wikipedia)</a>
* @see <a href="https://mathworld.wolfram.com/TriangularDistribution.html">Triangular distribution (MathWorld)</a>
*/
public final class TriangularDistribution extends AbstractContinuousDistribution {
/** Lower limit of this distribution (inclusive). */
private final double a;
/** Upper limit of this distribution (inclusive). */
private final double b;
/** Mode of this distribution. */
private final double c;
/** Cached value ((b - a) * (c - a). */
private final double divisor1;
/** Cached value ((b - a) * (b - c)). */
private final double divisor2;
/** Cumulative probability at the mode. */
private final double cdfMode;
/** Survival probability at the mode. */
private final double sfMode;
/**
* @param a Lower limit of this distribution (inclusive).
* @param c Mode of this distribution.
* @param b Upper limit of this distribution (inclusive).
*/
private TriangularDistribution(double a,
double c,
double b) {
this.a = a;
this.c = c;
this.b = b;
divisor1 = (b - a) * (c - a);
divisor2 = (b - a) * (b - c);
cdfMode = (c - a) / (b - a);
sfMode = (b - c) / (b - a);
}
/**
* Creates a triangular distribution.
*
* @param a Lower limit of this distribution (inclusive).
* @param c Mode of this distribution.
* @param b Upper limit of this distribution (inclusive).
* @return the distribution
* @throws IllegalArgumentException if {@code a >= b}, if {@code c > b} or if
* {@code c < a}.
*/
public static TriangularDistribution of(double a,
double c,
double b) {
if (a >= b) {
throw new DistributionException(DistributionException.INVALID_RANGE_LOW_GTE_HIGH,
a, b);
}
if (c < a) {
throw new DistributionException(DistributionException.TOO_SMALL,
c, a);
}
if (c > b) {
throw new DistributionException(DistributionException.TOO_LARGE,
c, b);
}
return new TriangularDistribution(a, c, b);
}
/**
* Gets the mode parameter of this distribution.
*
* @return the mode.
*/
public double getMode() {
return c;
}
/** {@inheritDoc} */
@Override
public double density(double x) {
if (x < a) {
return 0;
}
if (x < c) {
final double divident = 2 * (x - a);
return divident / divisor1;
}
if (x == c) {
return 2 / (b - a);
}
if (x <= b) {
final double divident = 2 * (b - x);
return divident / divisor2;
}
return 0;
}
/** {@inheritDoc} */
@Override
public double cumulativeProbability(double x) {
if (x <= a) {
return 0;
}
if (x < c) {
final double divident = (x - a) * (x - a);
return divident / divisor1;
}
if (x == c) {
return cdfMode;
}
if (x < b) {
final double divident = (b - x) * (b - x);
return 1 - (divident / divisor2);
}
return 1;
}
/** {@inheritDoc} */
@Override
public double survivalProbability(double x) {
// By symmetry:
if (x <= a) {
return 1;
}
if (x < c) {
final double divident = (x - a) * (x - a);
return 1 - (divident / divisor1);
}
if (x == c) {
return sfMode;
}
if (x < b) {
final double divident = (b - x) * (b - x);
return divident / divisor2;
}
return 0;
}
/** {@inheritDoc} */
@Override
public double inverseCumulativeProbability(double p) {
ArgumentUtils.checkProbability(p);
if (p == 0) {
return a;
}
if (p == 1) {
return b;
}
if (p < cdfMode) {
return a + Math.sqrt(p * divisor1);
}
return b - Math.sqrt((1 - p) * divisor2);
}
/** {@inheritDoc} */
@Override
public double inverseSurvivalProbability(double p) {
// By symmetry:
ArgumentUtils.checkProbability(p);
if (p == 1) {
return a;
}
if (p == 0) {
return b;
}
if (p >= sfMode) {
return a + Math.sqrt((1 - p) * divisor1);
}
return b - Math.sqrt(p * divisor2);
}
/**
* {@inheritDoc}
*
* <p>For lower limit \( a \), upper limit \( b \), and mode \( c \),
* the mean is \( (a + b + c) / 3 \).
*/
@Override
public double getMean() {
return (a + b + c) / 3;
}
/**
* {@inheritDoc}
*
* <p>For lower limit \( a \), upper limit \( b \), and mode \( c \),
* the variance is \( (a^2 + b^2 + c^2 - ab - ac - bc) / 18 \).
*/
@Override
public double getVariance() {
return (a * a + b * b + c * c - a * b - a * c - b * c) / 18;
}
/**
* {@inheritDoc}
*
* <p>The lower bound of the support is equal to the lower limit parameter
* {@code a} of the distribution.
*/
@Override
public double getSupportLowerBound() {
return a;
}
/**
* {@inheritDoc}
*
* <p>The upper bound of the support is equal to the upper limit parameter
* {@code b} of the distribution.
*/
@Override
public double getSupportUpperBound() {
return b;
}
}