Extends the class ContinuousDistribution for the triangular distribution (see [100] (page 297) and [118] (page 317)) with domain \([a,b]\) and mode (or shape parameter) \(m\), where \(a\le m\le b\). More...

Inheritance diagram for TriangularDist:

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Collaboration diagram for TriangularDist:

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Public Member Functions
	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 \(m\).

double	density (double x)

double	cdf (double x)
	Returns the distribution function \(F(x)\). More...

double	barF (double x)
	Returns \(\bar{F}(x) = 1 - F(x)\). More...

double	inverseF (double u)
	Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More...

double	getMean ()
	Returns the mean of the distribution function.

double	getVariance ()
	Returns the variance of the distribution function.

double	getStandardDeviation ()
	Returns the standard deviation of the distribution function.

double	getA ()
	Returns the value of \(a\) for this object.

double	getB ()
	Returns the value of \(b\) for this object.

double	getM ()
	Returns the value of \(m\) for this object.

void	setParams (double a, double b, double m)
	Sets the value of the parameters \(a\), \(b\) and \(m\) for this object.

double []	getParams ()
	Return a table containing the parameters of the current distribution. More...

String	toString ()
	Returns a `String` containing information about the current distribution.

Public Member Functions inherited from ContinuousDistribution
abstract double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\). More...

double	barF (double x)
	Returns the complementary distribution function. More...

double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More...

double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More...

double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\). More...

double	getMean ()
	Returns the mean. More...

double	getVariance ()
	Returns the variance. More...

double	getStandardDeviation ()
	Returns the standard deviation. More...

double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXinf (double xa)
	Sets the value \(x_a=\) `xa`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

void	setXsup (double xb)
	Sets the value \(x_b=\) `xb`, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More...

Static Public Member Functions
static double	density (double a, double b, double m, double x)
	Computes the density function.

static double	cdf (double a, double b, double m, double x)
	Computes the distribution function.

static double	barF (double a, double b, double m, double x)
	Computes the complementary distribution function.

static double	inverseF (double a, double b, double m, double u)
	Computes the inverse distribution function.

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 \(x[i]\), \(i = 0, 1,…, n-1\). More...

static TriangularDist	getInstanceFromMLE (double[] x, int n, double a, double b)
	Creates a new instance of a triangular distribution with parameters `a` and `b`. More...

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 \(a\), \(b\), \(m\). More...

static double	getVariance (double a, double b, double m)
	Computes and returns the variance \(\mbox{Var}[X] = (a^2 + b^2 + m^2 - ab - am - bm)/18\) of the triangular distribution with parameters \(a\), \(b\), \(m\). More...

static double	getStandardDeviation (double a, double b, double m)
	Computes and returns the standard deviation of the triangular distribution with parameters \(a\), \(b\), \(m\). More...

Additional Inherited Members
Public Attributes inherited from ContinuousDistribution
int	decPrec = 15

Protected Attributes inherited from ContinuousDistribution
double	supportA = Double.NEGATIVE_INFINITY

double	supportB = Double.POSITIVE_INFINITY

Static Protected Attributes inherited from ContinuousDistribution
static final double	XBIG = 100.0

static final double	XBIGM = 1000.0

static final double []	EPSARRAY

Detailed Description

Extends the class ContinuousDistribution for the triangular distribution (see [100] (page 297) and [118] (page 317)) with domain \([a,b]\) and mode (or shape parameter) \(m\), where \(a\le m\le b\).

The density function is

\[ f(x) = \left\{\begin{array}{ll} \frac{2(x-a)}{(b-a)(m-a)} & \mbox{ if } a\le x\le m, \\ \frac{2(b-x)}{(b-a)(b-m)} & \mbox{ if } m\le x\le b, \\ 0 & \mbox{ elsewhere, } \end{array}\right. \tag{ftrian} \]

the distribution function is

\[ F (x) = \left\{\begin{array}{ll} 0 & \mbox{ for } x < a, \\ \frac{(x - a)^2}{(b - a)(m - a)} & \mbox{ if } a\le x\le m, \\ 1 - \frac{(b - x)^2}{(b - a)(b - m)} & \mbox{ if } m\le x\le b, \\ 1 & \mbox{ for } x > b, \end{array}\right. \]

and the inverse distribution function is given by

\[ F^{-1}(u) = \left\{\begin{array}{ll} a + \sqrt{(b - a)(m - a)u} & \mbox{ if } 0\le u\le\frac{m-a}{b-a}, \\ b - \sqrt{(b - a)(b - m)(1 - u)} & \mbox{ if } \frac{m-a}{b-a}\le u \le1. \end{array}\right. \]

Member Function Documentation

◆ barF()

double barF ( double x )

Returns \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Implements Distribution.

◆ cdf()

double cdf ( double x )

Returns the distribution function \(F(x)\).

Parameters

x	value at which the distribution function is evaluated

Returns: distribution function evaluated at x

Implements Distribution.

◆ getInstanceFromMLE()

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

static

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()

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

static

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

◆ getMLE()

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

static

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 [189], [92], [114] .

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 [ \(m\)]

◆ getParams()

double [] getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(a\), \(b\), \(m\)].

Implements Distribution.

◆ getStandardDeviation()

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

static

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

Returns: the standard deviation of the triangular distribution

◆ getVariance()

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

static

Computes and returns the variance \(\mbox{Var}[X] = (a^2 + b^2 + m^2 - ab - am - bm)/18\) of the triangular distribution with parameters \(a\), \(b\), \(m\).

Returns: the variance of the triangular distribution

◆ inverseF()

double inverseF ( double u )

Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).

Parameters

u	value in the interval \((0,1)\) for which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Implements Distribution.

The documentation for this class was generated from the following file:

TriangularDist.java

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Detailed Description

Member Function Documentation

◆ barF()

◆ cdf()

◆ getInstanceFromMLE()

◆ getMean()

◆ getMLE()

◆ getParams()

◆ getStandardDeviation()

◆ getVariance()

◆ inverseF()