InverseFromDensityGen (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.randvar.RandomVariateGen
- - umontreal.iro.lecuyer.randvar.InverseFromDensityGen

public class InverseFromDensityGen
extends RandomVariateGen

Implements a method for generating random variates by numerical inversion of an arbitrary continuous distribution when only the probability density is known. 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 α < 1) if one requires too high a precision (a too small eps, for ex. ε∼10^-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 10⁸ 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 10⁸ times; for this class, we initialized 10⁴ times.

Distribution	Inversion	Gauss-Lobatto	time ratio
	10⁸ init	10⁴ 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 Summary

Constructors
Constructor and Description
`InverseFromDensityGen(RandomStream s, ContinuousDistribution dis, double xc, double eps, int order)` Creates a new generator for the continuous distribution `dis`, using stream `s`.
`InverseFromDensityGen(RandomStream s, MathFunction dens, double xc, double eps, int order, double xleft, double xright)` Creates a new generator from the continuous probability density `dens`.

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`getEpsilon()` Returns the u-resolution `eps` associated with this object.
`int`	`getOrder()` Returns the order associated with this object.
`double`	`getXc()` Returns the `xc` given in the constructor.
`double`	`nextDouble()` Generates a new random variate.

Methods inherited from class umontreal.iro.lecuyer.randvar.RandomVariateGen
getDistribution, getStream, nextArrayOfDouble, setStream

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - InverseFromDensityGen
```
public InverseFromDensityGen(RandomStream s,
                     ContinuousDistribution dis,
                     double xc,
                     double eps,
                     int order)
```
    Creates a new generator for the continuous distribution dis, using stream s. 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 setXinf and 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 <= order <= 12.
  - InverseFromDensityGen
```
public InverseFromDensityGen(RandomStream s,
                     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.
- Method Detail
  - nextDouble
```
public double nextDouble()
```
    Generates a new random variate.
    
    Overrides:
    
    nextDouble in class RandomVariateGen
    
    Returns:
    the generated value
  - getXc
```
public double getXc()
```
    Returns the xc given in the constructor.
  - getEpsilon
```
public double getEpsilon()
```
    Returns the u-resolution eps associated with this object.
  - getOrder
```
public int getOrder()
```
    Returns the order associated with this object.

Class InverseFromDensityGen

Constructor Summary

Method Summary

Methods inherited from class umontreal.iro.lecuyer.randvar.RandomVariateGen

Methods inherited from class java.lang.Object

Constructor Detail

InverseFromDensityGen

InverseFromDensityGen

Method Detail

nextDouble

getXc

getEpsilon

getOrder