Extends the class ContinuousDistribution for the logistic distribution (e.g., [96] (page 115)). More...

Inheritance diagram for umontreal.ssj.probdist.LogisticDist:

Public Member Functions
	LogisticDist ()
	Constructs a LogisticDist object with default parameters.
	LogisticDist (double alpha, double lambda)
	Constructs a LogisticDist object with parameters \(\alpha\) = alpha and \(\lambda\) = lambda.
double	density (double x)
	Returns \(f(x)\), the density evaluated at \(x\).
double	cdf (double x)
	Returns the distribution function \(F(x)\).
double	barF (double x)
	Returns the complementary distribution function.
double	inverseF (double u)
	Returns the inverse distribution function \(x = F^{-1}(u)\).
double	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
double	getAlpha ()
	Return the parameter \(\alpha\) of this object.
double	getLambda ()
	Returns the parameter \(\lambda\) of this object.
void	setParams (double alpha, double lambda)
	Sets the parameters \(\alpha\) and \(\lambda\) of this object.
double[]	getParams ()
	Return a table containing the parameters of the current distribution.
String	toString ()
	Returns a String containing information about the current distribution.
Public Member Functions inherited from umontreal.ssj.probdist.ContinuousDistribution
double	inverseBrent (double a, double b, double u, double tol)
	Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method.
double	inverseBisection (double u)
	Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection.
double	getXinf ()
	Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
double	getXsup ()
	Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXinf (double xa)
	Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).
void	setXsup (double xb)
	Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\).

Static Public Member Functions
static double	density (double alpha, double lambda, double x)
	Computes the density function \(f(x)\).
static double	cdf (double alpha, double lambda, double x)
	Computes the distribution function \(F(x)\).
static double	barF (double alpha, double lambda, double x)
	Computes the complementary distribution function \(1-F(x)\).
static double	inverseF (double alpha, double lambda, double u)
	Computes the inverse distribution function \(F^{-1}(u)\).
static double[]	getMLE (double[] x, int n)
	Estimates the parameters \((\alpha, \lambda)\) of the logistic distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).
static LogisticDist	getInstanceFromMLE (double[] x, int n)
	Creates a new instance of a logistic distribution with parameters.
static double	getMean (double alpha, double lambda)
	Computes and returns the mean \(E[X] = \alpha\) of the logistic distribution with parameters \(\alpha\) and \(\lambda\).
static double	getVariance (double alpha, double lambda)
	Computes and returns the variance \(\mbox{Var}[X] = \pi^2 /(3\lambda^2)\) of the logistic distribution with parameters.
static double	getStandardDeviation (double alpha, double lambda)
	Computes and returns the standard deviation of the logistic distribution with parameters \(\alpha\) and \(\lambda\).

Detailed Description

Extends the class ContinuousDistribution for the logistic distribution (e.g., [96] (page 115)).

It has location parameter \(\alpha\) and scale parameter \(\lambda> 0\). The density is

\[ f (x) = \frac{\lambda e^{-\lambda(x - \alpha)}}{(1 + e^{-\lambda(x - \alpha)})^2} \qquad\qquad\mbox{for } -\infty< x < \infty, \tag{flogistic} \]

and the distribution function is

\[ F(x) = \frac{1}{1 + e^{-\lambda(x - \alpha)}} \qquad\qquad\mbox{for } -\infty< x < \infty. \tag{Flogistic} \]

For \(\lambda=1\) and \(\alpha=0\), one can write

\[ F(x) = \frac{1 + \tanh({x/2})}{2}. \]

The inverse distribution function is given by

\[ F^{-1}(u) = \ln(u/(1-u))/\lambda+ \alpha\qquad\mbox{for } 0 \le u < 1. \]

Definition at line 48 of file LogisticDist.java.

Constructor & Destructor Documentation

◆ LogisticDist() [1/2]

umontreal.ssj.probdist.LogisticDist.LogisticDist ( )

Constructs a LogisticDist object with default parameters.

\(\alpha= 0\) and \(\lambda=1\).

Definition at line 128 of file LogisticDist.java.

◆ LogisticDist() [2/2]

umontreal.ssj.probdist.LogisticDist.LogisticDist	(	double	alpha,
		double	lambda )

Constructs a LogisticDist object with parameters \(\alpha\) = alpha and \(\lambda\) = lambda.

Definition at line 136 of file LogisticDist.java.

Member Function Documentation

◆ barF() [1/2]

double umontreal.ssj.probdist.LogisticDist.barF	(	double	alpha,
		double	lambda,
		double	x )

static

Computes the complementary distribution function \(1-F(x)\).

Definition at line 197 of file LogisticDist.java.

◆ barF() [2/2]

double umontreal.ssj.probdist.LogisticDist.barF ( double x )

Returns the complementary distribution function.

The default implementation computes \(\bar{F}(x) = 1 - F(x)\).

Parameters

x	value at which the complementary distribution function is evaluated

Returns: complementary distribution function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 148 of file LogisticDist.java.

◆ cdf() [1/2]

double umontreal.ssj.probdist.LogisticDist.cdf	(	double	alpha,
		double	lambda,
		double	x )

static

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

Definition at line 185 of file LogisticDist.java.

◆ cdf() [2/2]

double umontreal.ssj.probdist.LogisticDist.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 umontreal.ssj.probdist.Distribution.

Definition at line 144 of file LogisticDist.java.

◆ density() [1/2]

double umontreal.ssj.probdist.LogisticDist.density	(	double	alpha,
		double	lambda,
		double	x )

static

Computes the density function \(f(x)\).

Definition at line 171 of file LogisticDist.java.

◆ density() [2/2]

double umontreal.ssj.probdist.LogisticDist.density ( double x )

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

Parameters

x	value at which the density is evaluated

Returns: density function evaluated at x

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 140 of file LogisticDist.java.

◆ getAlpha()

double umontreal.ssj.probdist.LogisticDist.getAlpha ( )

Return the parameter \(\alpha\) of this object.

Definition at line 331 of file LogisticDist.java.

◆ getInstanceFromMLE()

LogisticDist umontreal.ssj.probdist.LogisticDist.getInstanceFromMLE	(	double[]	x,
		int	n )

static

Creates a new instance of a logistic distribution with parameters.

\(\alpha\) and \(\lambda\) estimated using the maximum likelihood method based on the \(n\) observations \(x[i]\), \(i = 0, 1, …, n-1\).

Parameters

x	the list of observations to use to evaluate parameters
n	the number of observations to use to evaluate parameters

Definition at line 282 of file LogisticDist.java.

◆ getLambda()

double umontreal.ssj.probdist.LogisticDist.getLambda ( )

Returns the parameter \(\lambda\) of this object.

Definition at line 338 of file LogisticDist.java.

◆ getMean() [1/2]

double umontreal.ssj.probdist.LogisticDist.getMean ( )

Returns the mean.

Returns: the mean

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 156 of file LogisticDist.java.

◆ getMean() [2/2]

double umontreal.ssj.probdist.LogisticDist.getMean	(	double	alpha,
		double	lambda )

static

Computes and returns the mean \(E[X] = \alpha\) of the logistic distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the mean of the logistic distribution \(E[X] = \alpha\)

Definition at line 293 of file LogisticDist.java.

◆ getMLE()

double[] umontreal.ssj.probdist.LogisticDist.getMLE	(	double[]	x,
		int	n )

static

Estimates the parameters \((\alpha, \lambda)\) of the logistic distribution using the maximum likelihood method, from the \(n\) observations \(x[i]\), \(i = 0, 1,…, n-1\).

The estimates are returned in a two-element array, in regular order: [ \(\alpha\), \(\lambda\)]. The maximum likelihood estimators are the values

\((\hat{\alpha}, \hat{\lambda})\) that satisfy the equations:

\begin{align*} \sum_{i=1}^n \frac{1}{1 + e^{\hat{\lambda} (x_i - \hat{\alpha})}} & = \frac{n}{2} \\ \sum_{i=1}^n \hat{\lambda} (x_i - \hat{\alpha}) \frac{1 - e^{\hat{\lambda} (x_i - \hat{\alpha})}}{1 + e^{\hat{\lambda} (x_i - \hat{\alpha})}} & = n. \end{align*}

See [55] (page 128).

Parameters

x	the list of observations used to evaluate parameters
n	the number of observations used to evaluate parameters

Returns: returns the parameter [ \(\hat{\alpha}\), \(\hat{\lambda}\)]

Definition at line 240 of file LogisticDist.java.

◆ getParams()

double[] umontreal.ssj.probdist.LogisticDist.getParams ( )

Return a table containing the parameters of the current distribution.

This table is put in regular order: [ \(\alpha\), \(\lambda\)].

Implements umontreal.ssj.probdist.Distribution.

Definition at line 356 of file LogisticDist.java.

◆ getStandardDeviation() [1/2]

double umontreal.ssj.probdist.LogisticDist.getStandardDeviation ( )

Returns the standard deviation.

Returns: the standard deviation

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 164 of file LogisticDist.java.

◆ getStandardDeviation() [2/2]

double umontreal.ssj.probdist.LogisticDist.getStandardDeviation	(	double	alpha,
		double	lambda )

static

Computes and returns the standard deviation of the logistic distribution with parameters \(\alpha\) and \(\lambda\).

Returns: the standard deviation of the logistic distribution

Definition at line 321 of file LogisticDist.java.

◆ getVariance() [1/2]

double umontreal.ssj.probdist.LogisticDist.getVariance ( )

Returns the variance.

Returns: the variance

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 160 of file LogisticDist.java.

◆ getVariance() [2/2]

double umontreal.ssj.probdist.LogisticDist.getVariance	(	double	alpha,
		double	lambda )

static

Computes and returns the variance \(\mbox{Var}[X] = \pi^2 /(3\lambda^2)\) of the logistic distribution with parameters.

\(\alpha\) and \(\lambda\).

Returns: the variance of the logistic distribution \(\mbox{Var}[X] = 1 / 3 \pi^2 * (1 / \lambda^2)\)

Definition at line 308 of file LogisticDist.java.

◆ inverseF() [1/2]

double umontreal.ssj.probdist.LogisticDist.inverseF	(	double	alpha,
		double	lambda,
		double	u )

static

Computes the inverse distribution function \(F^{-1}(u)\).

Definition at line 209 of file LogisticDist.java.

◆ inverseF() [2/2]

double umontreal.ssj.probdist.LogisticDist.inverseF ( double u )

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

Restrictions: \(u \in[0,1]\).

Parameters

u	value at which the inverse distribution function is evaluated

Returns: the inverse distribution function evaluated at u

Exceptions

IllegalArgumentException if \(u\) is not in the interval \([0,1]\)

Reimplemented from umontreal.ssj.probdist.ContinuousDistribution.

Definition at line 152 of file LogisticDist.java.

◆ setParams()

void umontreal.ssj.probdist.LogisticDist.setParams	(	double	alpha,
		double	lambda )

Sets the parameters \(\alpha\) and \(\lambda\) of this object.

Definition at line 345 of file LogisticDist.java.

◆ toString()

String umontreal.ssj.probdist.LogisticDist.toString ( )

Returns a String containing information about the current distribution.

Definition at line 364 of file LogisticDist.java.

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

src/main/java/umontreal/ssj/probdist/LogisticDist.java

SSJ API Documentation Stochastic Simulation in Java

Public Member Functions

Static Public Member Functions

Detailed Description

Constructor & Destructor Documentation

◆ LogisticDist() [1/2]

◆ LogisticDist() [2/2]

Member Function Documentation

◆ barF() [1/2]

◆ barF() [2/2]

◆ cdf() [1/2]

◆ cdf() [2/2]

◆ density() [1/2]

◆ density() [2/2]

◆ getAlpha()

◆ getInstanceFromMLE()

◆ getLambda()

◆ getMean() [1/2]

◆ getMean() [2/2]

◆ getMLE()

◆ getParams()

◆ getStandardDeviation() [1/2]

◆ getStandardDeviation() [2/2]

◆ getVariance() [1/2]

◆ getVariance() [2/2]

◆ inverseF() [1/2]

◆ inverseF() [2/2]

◆ setParams()

◆ toString()