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| <!-- $Revision: 1054186 $ $Date: 2011-01-01 03:28:46 +0100 (sam. 01 janv. 2011) $ --> |
| <body> |
| <p>Random number and random data generators.</p> |
| <p>Commons-math provides a few pseudo random number generators. The top level interface is RandomGenerator. |
| It is implemented by three classes: |
| <ul> |
| <li>{@link org.apache.commons.math.random.JDKRandomGenerator JDKRandomGenerator} |
| that extends the JDK provided generator</li> |
| <li>AbstractRandomGenerator as a helper for users generators</li> |
| <li>BitStreamGenerator which is an abstract class for several generators and |
| which in turn is extended by: |
| <ul> |
| <li>{@link org.apache.commons.math.random.MersenneTwister MersenneTwister}</li> |
| <li>{@link org.apache.commons.math.random.Well512a Well512a}</li> |
| <li>{@link org.apache.commons.math.random.Well1024a Well1024a}</li> |
| <li>{@link org.apache.commons.math.random.Well19937a Well19937a}</li> |
| <li>{@link org.apache.commons.math.random.Well19937c Well19937c}</li> |
| <li>{@link org.apache.commons.math.random.Well44497a Well44497a}</li> |
| <li>{@link org.apache.commons.math.random.Well44497b Well44497b}</li> |
| </ul> |
| </li> |
| </ul> |
| </p> |
| |
| <p> |
| 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: 2<sup>19937</sup> - 1 (which is a Mersenne prime). This generator |
| is described in a paper by Makoto Matsumoto and Takuji Nishimura in 1998: <a |
| href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf">Mersenne Twister: |
| A 623-Dimensionally Equidistributed Uniform Pseudo-Random Number Generator</a>, 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 2<sup>512</sup> - 1 |
| to 2<sup>44497</sup> - 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 <a |
| href="http://www.iro.umontreal.ca/~lecuyer/myftp/papers/wellrng.pdf">Improved Long-Period |
| Generators Based on Linear Recurrences Modulo 2</a> ACM Transactions on Mathematical Software, |
| 32, 1 (2006). The errata for the paper are in <a |
| href="http://www.iro.umontreal.ca/~lecuyer/myftp/papers/wellrng-errata.txt">wellrng-errata.txt</a>. |
| </p> |
| |
| <p> |
| 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 2<sup>n</sup>-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. |
| </p> |
| |
| <p> |
| 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 <i>escape zeroland</i> 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). |
| </p> |
| |
| <p> |
| 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. |
| </p> |
| |
| <p> |
| 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: |
| </p> |
| |
| <p> |
| <table border="1" align="center"> |
| <tr BGCOLOR="#CCCCFF"><td colspan="2"><font size="+2">Example of performances</font></td></tr> |
| <tr BGCOLOR="#EEEEFF"><font size="+1"><td>Name</td><td>generation rate (relative to MersenneTwister)</td></font></tr> |
| <tr><td>{@link org.apache.commons.math.random.MersenneTwister MersenneTwister}</td><td>1</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.JDKRandomGenerator JDKRandomGenerator}</td><td>between 0.96 and 1.16</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well512a Well512a}</td><td>between 0.85 and 0.88</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well1024a Well1024a}</td><td>between 0.63 and 0.73</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well19937a Well19937a}</td><td>between 0.70 and 0.71</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well19937c Well19937c}</td><td>between 0.57 and 0.71</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well44497a Well44497a}</td><td>between 0.69 and 0.71</td></tr> |
| <tr><td>{@link org.apache.commons.math.random.Well44497b Well44497b}</td><td>between 0.65 and 0.71</td></tr> |
| </table> |
| </p> |
| |
| <p> |
| So for most simulation problems, the better generators like {@link |
| org.apache.commons.math.random.Well19937c Well19937c} and {@link |
| org.apache.commons.math.random.Well44497b Well44497b} are probably very good choices. |
| </p> |
| |
| <p> |
| Note that <em>none</em> 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. |
| </p> |
| |
| </body> |
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