Implements a method for computing the inverse of an arbitrary continuous distribution function when only the probability density is known. More...

Inheritance diagram for umontreal.ssj.probdist.InverseDistFromDensity:

Public Member Functions
	InverseDistFromDensity (ContinuousDistribution dist, double xc, double eps, int order)
	Given a continuous distribution dist with a well-defined density method, this class will compute tables for the numerical inverse of the distribution.
	InverseDistFromDensity (MathFunction dens, double xc, double eps, int order, double xleft, double xright)
	Given a continuous probability density dens, this class will compute tables for the numerical inverse of the distribution.
double	density (double x)
	Computes the probability density at \(x\).
double	cdf (double x)
	Computes the distribution function at \(x\).
double	inverseF (double u)
	Computes the inverse distribution function at \(u\).
double	getXc ()
	Returns the xc given in the constructor.
double	getEpsilon ()
	Returns the \(u\)-resolution eps associated with this object.
int	getOrder ()
	Returns the order associated with 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	barF (double x)
	Returns the complementary distribution function.
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	getMean ()
	Returns the mean.
double	getVariance ()
	Returns the variance.
double	getStandardDeviation ()
	Returns the standard deviation.
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 \([x_a, x_b]\).

Detailed Description

Implements a method for computing the inverse of an arbitrary continuous distribution function when only the probability density is known.

[42] . The cumulative probabilities (cdf) are pre-computed by numerical quadrature of the density using Gauss-Lobatto integration over suitably small intervals to satisfy the required precision, and these values are kept in tables. Then the algorithm uses polynomial interpolation over the tabulated values to get the inverse cdf. The user can select the desired precision and the degree of the interpolating polynomials.

The algorithm may fail for some distributions for which the density becomes infinite at a point (for ex. the Gamma and the Beta distributions with \(\alpha< 1\)) if one chooses too high a precision (a too small eps, for ex. \(\epsilon\sim10^{-15}\)). However, it should work also for continuous densities with finite discontinuities.

While the setup time for this class is relatively slow, the numerical inversion is extremely fast and practically independent of the required precision and of the specific distribution. For comparisons between the times of standard inversion and inversion from this class as well as comparisons between setup times, see the introduction in class umontreal.ssj.randvar.InverseFromDensityGen from package randvar.

 Thus if only a few inverses are needed, then using this class is not
 efficient because of the slow set-up. But if one wants to call
 `inverseF` thousands of times or more, then using this class will be
 very efficient.

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Definition at line 65 of file InverseDistFromDensity.java.

Constructor & Destructor Documentation

◆ InverseDistFromDensity() [1/2]

umontreal.ssj.probdist.InverseDistFromDensity.InverseDistFromDensity	(	ContinuousDistribution	dist,
		double	xc,
		double	eps,
		int	order )

Given a continuous distribution dist with a well-defined density method, this class will compute tables for the numerical inverse of the distribution.

The user may wish to set the left and the right boundaries between which the density is non-zero by calling methods umontreal.ssj.probdist.ContinuousDistribution.setXinf and umontreal.ssj.probdist.ContinuousDistribution.setXsup of dist, for better efficiency. Argument xc can be the mean, the mode or any other \(x\) for which the density is relatively large. The

\(u\)-resolution eps is the required absolute error in the cdf, and order is the degree of the Newton interpolating polynomial over each interval. An order of 3 or 5, and an eps of \(10^{-6}\) to \(10^{-12}\) are usually good choices. Restrictions: \(3 \le\mathtt{order} \le12\).

Definition at line 237 of file InverseDistFromDensity.java.

◆ InverseDistFromDensity() [2/2]

umontreal.ssj.probdist.InverseDistFromDensity.InverseDistFromDensity	(	MathFunction	dens,
		double	xc,
		double	eps,
		int	order,
		double	xleft,
		double	xright )

Given a continuous probability density dens, this class will compute tables for the numerical inverse of the distribution.

The left and the right boundaries of the density are xleft and xright (the density is 0 outside the interval [xleft, xright]). See the description of the other constructor.

Definition at line 249 of file InverseDistFromDensity.java.