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## Package org.apache.commons.math.random

Random number and random data generators.

See:
Description

Interface Summary
EmpiricalDistribution | Represents an empirical probability distribution -- a probability distribution derived from observed data without making any assumptions about the functional form of the population distribution that the data come from.
NormalizedRandomGenerator | This interface represent a normalized random generator for scalars.
RandomData | Random data generation utilities.
RandomGenerator | Interface extracted from `java.util.Random`.
RandomVectorGenerator | This interface represents a random generator for whole vectors.

Class Summary
AbstractRandomGenerator | Abstract class implementing the `RandomGenerator` interface.
AbstractWell | This abstract class implements the WELL class of pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
BitsStreamGenerator | Base class for random number generators that generates bits streams.
CorrelatedRandomVectorGenerator | A `RandomVectorGenerator` that generates vectors with with correlated components.
EmpiricalDistributionImpl | Implements `EmpiricalDistribution` interface.
GaussianRandomGenerator | This class is a gaussian normalized random generator for scalars.
JDKRandomGenerator | Extension of `java.util.Random` to implement `RandomGenerator`.
MersenneTwister | This class implements a powerful pseudo-random number generator developed by Makoto Matsumoto and Takuji Nishimura during 1996-1997.
RandomAdaptor | Extension of `java.util.Random` wrapping a `RandomGenerator`.
RandomDataImpl | Implements the `RandomData` interface using a `RandomGenerator` instance to generate non-secure data and a `SecureRandom` instance to provide data for the `nextSecureXxx` methods.
UncorrelatedRandomVectorGenerator | A `RandomVectorGenerator` that generates vectors with uncorrelated components.
UniformRandomGenerator | This class implements a normalized uniform random generator.
UnitSphereRandomVectorGenerator | Generate random vectors isotropically located on the surface of a sphere.
ValueServer | Generates values for use in simulation applications.
Well1024a | This class implements the WELL1024a pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
Well19937a | This class implements the WELL19937a pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
Well19937c | This class implements the WELL19937c pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
Well44497a | This class implements the WELL44497a pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
Well44497b | This class implements the WELL44497b pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.
Well512a | This class implements the WELL512a pseudo-random number generator from François Panneton, Pierre L'Ecuyer and Makoto Matsumoto.

## Package org.apache.commons.math.random Description

Random number and random data generators.

Commons-math provides a few pseudo random number generators. The top level interface is RandomGenerator. It is implemented by three classes:

- `JDKRandomGenerator` that extends the JDK provided generator
- AbstractRandomGenerator as a helper for users generators
- BitStreamGenerator which is an abstract class for several generators and which in turn is extended by:
  - `MersenneTwister`
  - `Well512a`
  - `Well1024a`
  - `Well19937a`
  - `Well19937c`
  - `Well44497a`
  - `Well44497b`

The JDK provided generator is a simple one that can be used only for very simple needs. The Mersenne Twister is a fast generator with very good properties well suited for Monte-Carlo simulation. It is equidistributed for generating vectors up to dimension 623 and has a huge period: 219937 - 1 (which is a Mersenne prime). This generator is described in a paper by Makoto Matsumoto and Takuji Nishimura in 1998: Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator, ACM Transactions on Modeling and Computer Simulation, Vol. 8, No. 1, January 1998, pp 3--30. The WELL generators are a family of generators with period ranging from 2512 - 1 to 244497 - 1 (this last one is also a Mersenne prime) with even better properties than Mersenne Twister. These generators are described in a paper by François Panneton, Pierre L'Ecuyer and Makoto Matsumoto Improved Long-Period Generators Based on Linear Recurrences Modulo 2 ACM Transactions on Mathematical Software, 32, 1 (2006). The errata for the paper are in wellrng-errata.txt.

For simple sampling, any of these generators is sufficient. For Monte-Carlo simulations the JDK generator does not have any of the good mathematical properties of the other generators, so it should be avoided. The Mersenne twister and WELL generators have equidistribution properties proven according to their bits pool size which is directly linked to their period (all of them have maximal period, i.e. a generator with size n pool has a period 2n-1). They also have equidistribution properties for 32 bits blocks up to s/32 dimension where s is their pool size. So WELL19937c for exemple is equidistributed up to dimension 623 (19937/32). This means a Monte-Carlo simulation generating a vector of n variables at each iteration has some guarantees on the properties of the vector as long as its dimension does not exceed the limit. However, since we use bits from two successive 32 bits generated integers to create one double, this limit is smaller when the variables are of type double. so for Monte-Carlo simulation where less the 16 doubles are generated at each round, WELL1024 may be sufficient. If a larger number of doubles are needed a generator with a larger pool would be useful.

The WELL generators are more modern then MersenneTwister (the paper describing than has been published in 2006 instead of 1998) and fix some of its (few) drawbacks. If initialization array contains many zero bits, MersenneTwister may take a very long time (several hundreds of thousands of iterations to reach a steady state with a balanced number of zero and one in its bits pool). So the WELL generators are better to escape zeroland as explained by the WELL generators creators. The Well19937a and Well44497a generator are not maximally equidistributed (i.e. there are some dimensions or bits blocks size for which they are not equidistributed). The Well512a, Well1024a, Well19937c and Well44497b are maximally equidistributed for blocks size up to 32 bits (they should behave correctly also for double based on more than 32 bits blocks, but equidistribution is not proven at these blocks sizes).

The MersenneTwister generator uses a 624 elements integer array, so it consumes less than 2.5 kilobytes. The WELL generators use 6 integer arrays with a size equal to the pool size, so for example the WELL44497b generator uses about 33 kilobytes. This may be important if a very large number of generator instances were used at the same time.

All generators are quite fast. As an example, here are some comparisons, obtained on a 64 bits JVM on a linux computer with a 2008 processor (AMD phenom Quad 9550 at 2.2 GHz). The generation rate for MersenneTwister was about 27 millions doubles per second (remember we generate two 32 bits integers for each double). Generation rates for other PRNG, relative to MersenneTwister:

Example of performances
Name | generation rate (relative to MersenneTwister)
`MersenneTwister` | 1
`JDKRandomGenerator` | between 0.96 and 1.16
`Well512a` | between 0.85 and 0.88
`Well1024a` | between 0.63 and 0.73
`Well19937a` | between 0.70 and 0.71
`Well19937c` | between 0.57 and 0.71
`Well44497a` | between 0.69 and 0.71
`Well44497b` | between 0.65 and 0.71

So for most simulation problems, the better generators like `Well19937c` and `Well44497b` are probably very good choices.

Note that none of these generators are suitable for cryptography. They are devoted to simulation, and to generate very long series with strong properties on the series as a whole (equidistribution, no correlation ...). They do not attempt to create small series but with very strong properties of unpredictability as needed in cryptography.

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