Implements a method for computing the inverse of an arbitrary continuous distribution function when only the probability density is known [43] . More...
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. More... | |
| 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. More... | |
| 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. 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... | |
Protected Member Functions | |
| void | printArray (double[] U) |
| void | setParams (ContinuousDistribution dist, MathFunction dens, double xc, double eps, int order) |
Static Protected Attributes | |
| static final double | HALF_PI = Math.PI/2.0 |
| Static Protected Attributes inherited from ContinuousDistribution | |
| static final double | XBIG = 100.0 |
| static final double | XBIGM = 1000.0 |
| static final double [] | EPSARRAY |
Package Attributes | |
| MathFunction | m_dens |
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 |
Detailed Description
Implements a method for computing the inverse of an arbitrary continuous distribution function when only the probability density is known [43] .
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.
Constructor & Destructor Documentation
◆ InverseDistFromDensity() [1/2]
| 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: \(3 \le\mathtt{order} \le12\).
◆ InverseDistFromDensity() [2/2]
| 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.
Member Function Documentation
◆ getParams()
| double [] getParams | ( | ) |
Return a table containing the parameters of the current distribution.
This table is returned as: [xc, eps, order].
Implements Distribution.
The documentation for this class was generated from the following file:
- InverseDistFromDensity.java
