Implements a method for generating random variates by numerical inversion of an arbitrary continuous distribution when only the probability density is known [44] . More...
Public Member Functions | |
| InverseFromDensityGen (RandomStream stream, ContinuousDistribution dis, double xc, double eps, int order) | |
Creates a new generator for the continuous distribution dis, using stream stream. More... | |
| InverseFromDensityGen (RandomStream stream, MathFunction dens, double xc, double eps, int order, double xleft, double xright) | |
Creates a new generator from the continuous probability density dens. More... | |
| double | nextDouble () |
| Generates a new random variate. | |
| double | getXc () |
Returns the xc given in the constructor. | |
| double | getEpsilon () |
Returns the \(u\)-resolution eps. | |
| int | getOrder () |
| Returns the order of the interpolating polynomial. | |
 Public Member Functions inherited from RandomVariateGen | |
| RandomVariateGen (RandomStream s, Distribution dist) | |
Creates a new random variate generator from the distribution dist, using stream s. More... | |
| double | nextDouble () |
| Generates a random number from the continuous distribution contained in this object. More... | |
| void | nextArrayOfDouble (double[] v, int start, int n) |
Generates n random numbers from the continuous distribution contained in this object. More... | |
| double [] | nextArrayOfDouble (int n) |
Generates n random numbers from the continuous distribution contained in this object, and returns them in a new array of size n. More... | |
| RandomStream | getStream () |
| Returns the umontreal.ssj.rng.RandomStream used by this generator. More... | |
| void | setStream (RandomStream stream) |
Sets the umontreal.ssj.rng.RandomStream used by this generator to stream. | |
| Distribution | getDistribution () |
| Returns the umontreal.ssj.probdist.Distribution used by this generator. More... | |
| String | toString () |
Returns a String containing information about the current generator. | |
Additional Inherited Members | |
| Protected Attributes inherited from RandomVariateGen | |
| RandomStream | stream |
| Distribution | dist |
Detailed Description
Implements a method for generating random variates by numerical inversion of an arbitrary continuous distribution when only the probability density is known [44] .
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 requires 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 is relatively slow, the generation of random variables is extremely fast and practically independent of the required precision and of the specific distribution. The following table shows the time needed (in seconds) to generate \(10^8\) random numbers using inversion from a given class, then the numerical inversion with Gauss-Lobatto integration implemented here, and finally the speed ratios between the two methods. The speed ratio is the speed of the latter over the former. Thus for the beta distribution with parameters (5, 500), generating random variables with the Gauss-Lobatto integration implemented in this class is more than 1700 times faster than using inversion from the BetaDist class. These tests were made on a machine with processor AMD Athlon 4000, running Red Hat Linux, with clock speed at 2403 MHz.
| Distribution | Inversion | Gauss-Lobatto | speed ratio |
NormalDist(10.5, 5) | 9.19 | 8.89 | 1.03 |
ExponentialDist(5) | 17.72 | 8.82 | 2.0 |
CauchyDist(10.5, 5) | 18.30 | 8.81 | 2.1 |
BetaSymmetricalDist(10.5) | 242.80 | 8.85 | 27.4 |
GammaDist(55) | 899.50 | 8.89 | 101 |
ChiSquareNoncentralDist(10.5, 5) | 5326.90 | 8.85 | 602 |
BetaDist(5, 500) | 15469.10 | 8.86 | 1746 |
The following table gives the time (in sec.) needed to create an object (setup time) and to generate one random variable for this class compared to the same for the inversion method specific to each class, and the ratios of the times (init + one random variable) of the two methods. For inversion, we initialized \(10^8\) times; for this class, we initialized \(10^4\) times.
| Distribution | Inversion | Gauss-Lobatto | time ratio |
| \(10^8\) init | \(10^4\) init | for 1 init | |
NormalDist(10.5, 5) | 5.30 | 38.29 | 26426 |
ExponentialDist(5) | 3.98 | 27.05 | 12466 |
CauchyDist(10.5, 5) | 5.05 | 58.39 | 25007 |
BetaSymmetricalDist(10.5) | 90.66 | 68.33 | 2049 |
GammaDist(55) | 13.15 | 58.34 | 639 |
ChiSquareNoncentralDist(10.5, 5) | 190.48 | 248.98 | 451 |
BetaDist(5, 500) | 63.60 | 116.57 | 75 |
If only a few random variables are needed, then using this class is not efficient because of the slow set-up. But if one wants to generate large samples from the same distribution with fixed parameters, then this class will be very efficient. The following table gives the number of random variables generated beyond which, using this class will be worthwhile.
| Distribution | number of generated variables |
NormalDist(10.5, 5) | 41665 |
ExponentialDist(5) | 15266 |
CauchyDist(10.5, 5) | 31907 |
BetaSymmetricalDist(10.5) | 2814 |
GammaDist(55) | 649 |
ChiSquareNoncentralDist(10.5, 5) | 467 |
BetaDist(5, 500) | 75 |
Thus, for example, if one needs to generate less than 15266 exponential random variables, then using the InverseFromDensityGen class is not wortwhile: it will be faster to use inversion from the ExponentialGen class.
Constructor & Destructor Documentation
◆ InverseFromDensityGen() [1/2]
| InverseFromDensityGen | ( | RandomStream | stream, |
| ContinuousDistribution | dis, | ||
| double | xc, | ||
| double | eps, | ||
| int | order | ||
| ) |
Creates a new generator for the continuous distribution dis, using stream stream.
dis must have a well-defined density method; its other methods are unused. For a non-standard distribution dis, 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 dis, 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 desired 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: \(3 \le\mathtt{order} \le12\).
◆ InverseFromDensityGen() [2/2]
| InverseFromDensityGen | ( | RandomStream | stream, |
| MathFunction | dens, | ||
| double | xc, | ||
| double | eps, | ||
| int | order, | ||
| double | xleft, | ||
| double | xright | ||
| ) |
Creates a new generator from the continuous probability density dens.
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.
The documentation for this class was generated from the following file:
- InverseFromDensityGen.java
