umontreal.iro.lecuyer.probdist

Class TriangularDist

java.lang.Object

umontreal.iro.lecuyer.probdist.ContinuousDistribution

umontreal.iro.lecuyer.probdist.TriangularDist

All Implemented Interfaces:

Distribution

public class TriangularDist
extends ContinuousDistribution

Extends the class ContinuousDistribution for the triangular distribution with domain [a, b] and mode (or shape parameter) m, where a <= m <= b. The density function is

f (x) =	2(x - a)/[(b - a)(m - a)]	for a <= x <= m,
f (x) =	2(b - x)/[(b - a)(b - m)]	for m <= x <= b,
f (x) =	0	elsewhere,

the distribution function is

F(x) =	0	for x < a,
F(x) =	(x - a)2/[(b - a)(m - a)]	if a <= x <= m,
F(x) =	1 - (b - x)2/[(b - a)(b - m)]	if m <= x <= b,
F(x) =	1	for x > b,

and the inverse distribution function is given by

F-1(u) =	a + ((b - a)(m - a)u)1/2	if 0 <= u <= (m - a)/(b - a),
F-1(u) =	b - ((b - a)(b - m)(1 - u))1/2	if (m - a)/(b - a <= u <= 1.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

Constructors

Constructor and Description
TriangularDist() Constructs a TriangularDist object with default parameters a = 0, b = 1, and m = 0.5.
TriangularDist(double m) Constructs a TriangularDist object with parameters a = 0 , b = 1 and m = m.
TriangularDist(double a, double b, double m) Constructs a TriangularDist object with parameters a, b and m.

Method Summary

Modifier and Type	Method and Description
double	barF(double x) Returns the complementary distribution function.
static double	barF(double a, double b, double m, double x) Computes the complementary distribution function.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double a, double b, double m, double x) Computes the distribution function.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double a, double b, double m, double x) Computes the density function.
double	getA() Returns the value of a for this object.
double	getB() Returns the value of b for this object.
static TriangularDist	getInstanceFromMLE(double[] x, int n, double a, double b) Creates a new instance of a triangular distribution with parameters a and b.
double	getM() Returns the value of m for this object.
double	getMean() Returns the mean.
static double	getMean(double a, double b, double m) Computes and returns the mean E[X] = (a + b + m)/3 of the triangular distribution with parameters a, b, m.
static double[]	getMLE(double[] x, int n, double a, double b) Estimates the parameter m of the triangular distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1.
double[]	getParams() Return a table containing the parameters of the current distribution.
double	getStandardDeviation() Returns the standard deviation.
static double	getStandardDeviation(double a, double b, double m) Computes and returns the standard deviation of the triangular distribution with parameters a, b, m.
double	getVariance() Returns the variance.
static double	getVariance(double a, double b, double m) Computes and returns the variance Var[X] = (a2 + b2 + m2 - ab - am - bm)/18 of the triangular distribution with parameters a, b, m.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double a, double b, double m, double u) Computes the inverse distribution function.
void	setParams(double a, double b, double m) Sets the value of the parameters a, b and m for this object.
java.lang.String	toString() Returns a String containing information about the current distribution.

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

TriangularDist

public TriangularDist()

Constructs a TriangularDist object with default parameters a = 0, b = 1, and m = 0.5.

TriangularDist

public TriangularDist(double m)

Constructs a TriangularDist object with parameters a = 0 , b = 1 and m = m.

TriangularDist

public TriangularDist(double a,
                      double b,
                      double m)

Constructs a TriangularDist object with parameters a, b and m.

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Specified by:

density in class ContinuousDistribution

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class ContinuousDistribution

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class ContinuousDistribution

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

getMean

public double getMean()

Description copied from class: ContinuousDistribution

Returns the mean.

Specified by:

getMean in interface Distribution

Overrides:

getMean in class ContinuousDistribution

Returns:

the mean

getVariance

public double getVariance()

Description copied from class: ContinuousDistribution

Returns the variance.

Specified by:

getVariance in interface Distribution

Overrides:

getVariance in class ContinuousDistribution

Returns:

the variance

getStandardDeviation

public double getStandardDeviation()

Description copied from class: ContinuousDistribution

Returns the standard deviation.

Specified by:

getStandardDeviation in interface Distribution

Overrides:

getStandardDeviation in class ContinuousDistribution

Returns:

the standard deviation

density

public static double density(double a,
                             double b,
                             double m,
                             double x)

Computes the density function.

cdf

public static double cdf(double a,
                         double b,
                         double m,
                         double x)

Computes the distribution function.

barF

public static double barF(double a,
                          double b,
                          double m,
                          double x)

Computes the complementary distribution function.

inverseF

public static double inverseF(double a,
                              double b,
                              double m,
                              double u)

Computes the inverse distribution function.

getMLE

public static double[] getMLE(double[] x,
                              int n,
                              double a,
                              double b)

Estimates the parameter m of the triangular distribution using the maximum likelihood method, from the n observations x[i], i = 0, 1,…, n - 1. The estimated parameter is returned in a one-element array: [hat(m)]. See.

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

a - lower limit of range

b - upper limit of range

Returns:

returns the parameter [m]

getInstanceFromMLE

public static TriangularDist getInstanceFromMLE(double[] x,
                                                int n,
                                                double a,
                                                double b)

Creates a new instance of a triangular distribution with parameters a and b. m is estimated using the maximum likelihood method based on the n observations x[i], i = 0, 1,…, n - 1.

Parameters:

x - the list of observations used to evaluate parameters

n - the number of observations used to evaluate parameters

a - lower limit of range

b - upper limit of range

getMean

public static double getMean(double a,
                             double b,
                             double m)

Computes and returns the mean E[X] = (a + b + m)/3 of the triangular distribution with parameters a, b, m.

Returns:

the mean of the triangular distribution

getVariance

public static double getVariance(double a,
                                 double b,
                                 double m)

Computes and returns the variance Var[X] = (a² + b² + m² - ab - am - bm)/18 of the triangular distribution with parameters a, b, m.

Returns:

the variance of the triangular distribution

getStandardDeviation

public static double getStandardDeviation(double a,
                                          double b,
                                          double m)

Computes and returns the standard deviation of the triangular distribution with parameters a, b, m.

Returns:

the standard deviation of the triangular distribution

getA

public double getA()

Returns the value of a for this object.

getB

public double getB()

Returns the value of b for this object.

getM

public double getM()

Returns the value of m for this object.

setParams

public void setParams(double a,
                      double b,
                      double m)

Sets the value of the parameters a, b and m for this object.

getParams

public double[] getParams()

Return a table containing the parameters of the current distribution. This table is put in regular order: [a, b, m].

toString

public java.lang.String toString()

Returns a String containing information about the current distribution.

Overrides:

toString in class java.lang.Object