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<properties>
<title>The Commons Math User Guide - Data Generation
</properties>
<body>
<section name="2 Data Generation">
<subsection name="2.1 Overview" href="overview">
<p>
The Commons Math random package includes utilities for
<ul>
<li>generating random numbers
<li>generating random vectors
<li>generating random strings
<li>generating cryptographically secure sequences of random numbers or
strings</li>
<li>generating random samples and permutations
<li>analyzing distributions of values in an input file and generating
values "like" the values in the file</li>
<li>generating data for grouped frequency distributions or
histograms</li>
</ul>
<p>
The source of random data used by the data generation utilities is
pluggable. By default, the JDK-supplied PseudoRandom Number Generator
(PRNG) is used, but alternative generators can be "plugged in" using an
adaptor framework, which provides a generic facility for replacing
<code>java.util.Random with an alternative PRNG. Other very
good PRNG suitable for Monte-Carlo analysis (but <strong>not
for cryptography) provided by the library are the Mersenne twister from
Makoto Matsumoto and Takuji Nishimura and the more recent WELL generators
(Well Equidistributed Long-period Linear) from François Panneton, Pierre
L'Ecuyer and Makoto Matsumoto.
</p>
<p>
Sections 2.2-2.6 below show how to use the commons math API to generate
different kinds of random data. The examples all use the default
JDK-supplied PRNG. PRNG pluggability is covered in 2.7. The only
modification required to the examples to use alternative PRNGs is to
replace the argumentless constructor calls with invocations including
a <code>RandomGenerator instance as a parameter.
</p>
</subsection>
<subsection name="2.2 Random numbers" href="deviates">
<p>
The <a href="../apidocs/org/apache/commons/math3/random/RandomDataGenerator.html">
RandomDataGenerator</a> class implements methods for generating random sequences
of numbers. The API contracts of these methods use the following concepts:
<dl>
<dt>Random sequence of numbers from a probability distribution
<dd>There is no such thing as a single "random number." What can be
generated are <i>sequences of numbers that appear to be random. When
using the built-in JDK function <code>Math.random(), sequences of
values generated follow the
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm">
Uniform Distribution</a>, which means that the values are evenly spread
over the interval between 0 and 1, with no sub-interval having a greater
probability of containing generated values than any other interval of the
same length. The mathematical concept of a
<a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda36.htm">
probability distribution</a> basically amounts to asserting that different
ranges in the set of possible values of a random variable have
different probabilities of containing the value. Commons Math supports
generating random sequences from each of the distributions in the
<a href="../apidocs/org/apache/commons/math3/distribution/package-summary.html">
distributions</a> package.
The javadoc for the <code>nextXxx methods in
<a href="../apidocs/org/apache/commons/math3/random/RandomDataGenerator.html">
RandomDataGenerator</a> describes the algorithms used to generate
random deviates.
</dd>
<dt>Cryptographically secure random sequences
<dd>It is possible for a sequence of numbers to appear random, but
nonetheless to be predictable based on the algorithm used to generate the
sequence. If in addition to randomness, strong unpredictability is
required, it is best to use a
<a href="http://www.wikipedia.org/wiki/Cryptographically_secure_pseudo-random_number_generator">
secure random number generator</a> to generate values (or strings). The
nextSecureXxx methods implemented by <code>RandomDataGenerator
use the JDK <code>SecureRandom PRNG to generate cryptographically
secure sequences. The <code>setSecureAlgorithm method allows you to
change the underlying PRNG. These methods are <strong>much slower than
the corresponding "non-secure" versions, so they should only be used when cryptographic
security is required. The <code>ISAACRandom class implements a fast
cryptographically secure pseudorandom number generator.</dd>
<dt>Seeding pseudo-random number generators
<dd>By default, the implementation provided in RandomDataGenerator
uses the JDK-provided PRNG. Like most other PRNGs, the JDK generator
generates sequences of random numbers based on an initial "seed value."
For the non-secure methods, starting with the same seed always produces the
same sequence of values. Secure sequences started with the same seeds will
diverge. When a new <code>RandomDataGenerator is created, the underlying
random number generators are <strong>not initialized. The first
call to a data generation method, or to a <code>reSeed() method
initializes the appropriate generator. If you do not explicitly seed the
generator, it is by default seeded with the current time in milliseconds.
Therefore, to generate sequences of random data values, you should always
instantiate <strong>one RandomDataGenerator and use it
repeatedly instead of creating new instances for subsequent values in the
sequence. For example, the following will generate a random sequence of 50
long integers between 1 and 1,000,000, using the current time in
milliseconds as the seed for the JDK PRNG:
<source>
RandomDataGenerator randomData = new RandomDataGenerator();
for (int i = 0; i < 1000; i++) {
value = randomData.nextLong(1, 1000000);
}
</source>
The following will not in general produce a good random sequence, since the
PRNG is reseeded each time through the loop with the current time in
milliseconds:
<source>
for (int i = 0; i < 1000; i++) {
RandomDataGenerator randomData = new RandomDataGenerator();
value = randomData.nextLong(1, 1000000);
}
</source>
The following will produce the same random sequence each time it is
executed:
<source>
RandomDataGenerator randomData = new RandomDataGenerator();
randomData.reSeed(1000);
for (int i = 0; i = 1000; i++) {
value = randomData.nextLong(1, 1000000);
}
</source>
The following will produce a different random sequence each time it is
executed.
<source>
RandomDataGenerator randomData = new RandomDataGenerator();
randomData.reSeedSecure(1000);
for (int i = 0; i < 1000; i++) {
value = randomData.nextSecureLong(1, 1000000);
}
</source>
</dd>
</p>
</subsection>
<subsection name="2.3 Random Vectors" href="vectors">
<p>
Some algorithms require random vectors instead of random scalars. When the
components of these vectors are uncorrelated, they may be generated simply
one at a time and packed together in the vector. The <a
href="../apidocs/org/apache/commons/math3/random/UncorrelatedRandomVectorGenerator.html">
UncorrelatedRandomVectorGenerator</a> class simplifies this
process by setting the mean and deviation of each component once and
generating complete vectors. When the components are correlated however,
generating them is much more difficult. The <a href="../apidocs/org/apache/commons/math3/random/CorrelatedRandomVectorGenerator.html">
CorrelatedRandomVectorGenerator</a> class provides this service. In this
case, the user must set up a complete covariance matrix instead of a simple
standard deviations vector. This matrix gathers both the variance and the
correlation information of the probability law.
</p>
<p>
The main use for correlated random vector generation is for Monte-Carlo
simulation of physical problems with several variables, for example to
generate error vectors to be added to a nominal vector. A particularly
common case is when the generated vector should be drawn from a <a
href="http://en.wikipedia.org/wiki/Multivariate_normal_distribution">
Multivariate Normal Distribution</a>.
</p>
<p>
<dt>Generating random vectors from a bivariate normal distribution-
<source>
// Create and seed a RandomGenerator (could use any of the generators in the random package here)
RandomGenerator rg = new JDKRandomGenerator();
rg.setSeed(17399225432l); // Fixed seed means same results every time
// Create a GassianRandomGenerator using rg as its source of randomness
GaussianRandomGenerator rawGenerator = new GaussianRandomGenerator(rg);
// Create a CorrelatedRandomVectorGenerator using rawGenerator for the components
CorrelatedRandomVectorGenerator generator =
new CorrelatedRandomVectorGenerator(mean, covariance, 1.0e-12 * covariance.getNorm(), rawGenerator);
// Use the generator to generate correlated vectors
double[] randomVector = generator.nextVector();
... </source>
The <code>mean argument is a double[] array holding the means of the random vector
components. In the bivariate case, it must have length 2. The <code>covariance argument
is a RealMatrix, which needs to be 2 x 2. The main diagonal elements are the
variances of the vector components and the off-diagonal elements are the covariances.
For example, if the means are 1 and 2 respectively, and the desired standard deviations
are 3 and 4, respectively, then we need to use
<source>
double[] mean = {1, 2};
double[][] cov = {{9, c}, {c, 16}};
RealMatrix covariance = MatrixUtils.createRealMatrix(cov); </source>
where c is the desired covariance. If you are starting with a desired correlation,
you need to translate this to a covariance by multiplying it by the product of the
standard deviations. For example, if you want to generate data that will give Pearson's
R of 0.5, you would use c = 3 * 4 * .5 = 6.
</dd>
<p>
In addition to multivariate normal distributions, correlated vectors from multivariate uniform
distributions can be generated by creating a
<a href="../apidocs/org/apache/commons/math3/random/UniformRandomGenerator.html">UniformRandomGenerator
in place of the
<code>GaussianRandomGenerator above. More generally, any
<a href="../apidocs/org/apache/commons/math3/random/NormalizedRandomGenerator.html">NormalizedRandomGenerator
may be used.
</p>
<p>
<dt>Low discrepancy sequences
<dd>
There exist several quasi-random sequences with the property that for all values of N, the subsequence
x<sub>1, ..., xN has low discrepancy, which results in equi-distributed samples.
While their quasi-randomness makes them unsuitable for most applications (i.e. the sequence of values
is completely deterministic), their unique properties give them an important advantage for quasi-Monte Carlo simulations.<br/>
Currently, the following low-discrepancy sequences are supported:
<ul>
<li>Sobol sequence (pre-configured up to dimension 1000)
<li>Halton sequence (pre-configured up to dimension 40)
</ul>
<source>
// Create a Sobol sequence generator for 2-dimensional vectors
RandomVectorGenerator generator = new SobolSequence(2);
// Use the generator to generate vectors
double[] randomVector = generator.nextVector();
... </source>
The figure below illustrates the unique properties of low-discrepancy sequences when generating N samples
in the interval [0, 1]. Roughly speaking, such sequences "fill" the respective space more evenly which leads to faster convergence in
quasi-Monte Carlo simulations.<br/>
<img src="../images/userguide/low_discrepancy_sequences.png" alt="Comparison of low-discrepancy sequences"/>
</dd>
</p>
<p>
</subsection>
<subsection name="2.4 Random Strings" href="strings">
<p>
The methods <code>nextHexString and
nextSecureHexString
can be used to generate random strings of hexadecimal characters. Both
of these methods produce sequences of strings with good dispersion
properties. The difference between the two methods is that the second is
cryptographically secure. Specifically, the implementation of
<code>nextHexString(n) in
RandomDataGenerator uses the
following simple algorithm to generate a string of <code>n hex digits:
<ol>
<li>n/2+1 binary bytes are generated using the underlying RandomGenerator
<li>Each binary byte is translated into 2 hex digits
The <code>RandomDataGenerator implementation of the "secure" version,
<code>nextSecureHexString generates hex characters in 40-byte
"chunks" using a 3-step process:
<ol>
<li>20 random bytes are generated using the underlying
<code>SecureRandom.
<li>SHA-1 hash is applied to yield a 20-byte binary digest.
<li>Each byte of the binary digest is converted to 2 hex digits
Similarly to the secure random number generation methods,
<code>nextSecureHexString is
much slower than
the non-secure version. It should be used only for applications such as
generating unique session or transaction ids where predictability of
subsequent ids based on observation of previous values is a security
concern. If all that is needed is an even distribution of hex characters
in the generated strings, the non-secure method should be used.
</p>
</subsection>
<subsection name="2.5 Random permutations, combinations, sampling"
href="combinatorics">
<p>
To select a random sample of objects in a collection, you can use the
<code>nextSample method implemented by
RandomDataGenerator.
Specifically, if <code>c is a collection containing at least
<code>k objects, and
randomData is a
<code>RandomDataGenerator instance
randomData.nextSample(c, k)
will return an <code>object[] array of length
k
consisting of elements randomly selected from the collection. If
<code>c contains duplicate references, there may be duplicate
references in the returned array; otherwise returned elements will be
unique -- i.e., the sampling is without replacement among the object
references in the collection. </p>
<p>
If <code>randomData is a
RandomDataGenerator instance, and
<code>n and
k are integers with
<code> k <= n, then
<code>randomData.nextPermutation(n, k) returns an
int[]
array of length <code>k whose whose entries are selected randomly,
without repetition, from the integers <code>0 through
<code>n-1 (inclusive), i.e.,
<code>randomData.nextPermutation(n, k) returns a random
permutation of <code>n taken
k at a time.
</p>
</subsection>
<subsection name="2.6 Generating data 'like' an input file" href="empirical">
<p>
Using the <code>ValueServer class, you can generate data based on
the values in an input file in one of two ways:
<dl>
<dt>Replay Mode
<dd> The following code will read data from
url
(a <code>java.net.URL instance), cycling through the values in the
file in sequence, reopening and starting at the beginning again when all
values have been read.
<source>
ValueServer vs = new ValueServer();
vs.setValuesFileURL(url);
vs.setMode(ValueServer.REPLAY_MODE);
vs.resetReplayFile();
double value = vs.getNext();
// ...Generate and use more values...
vs.closeReplayFile();
</source>
The values in the file are not stored in memory, so it does not matter
how large the file is, but you do need to explicitly close the file
as above. The expected file format is \n -delimited (i.e. one per line)
strings representing valid floating point numbers.
</dd>
<dt>Digest Mode
<dd>When used in Digest Mode, the ValueServer reads the entire input file
and estimates a probability density function based on data from the file.
The estimation method is essentially the
<a href="http://nedwww.ipac.caltech.edu/level5/March02/Silverman/Silver2_6.html">
Variable Kernel Method</a> with Gaussian smoothing. Once the density
has been estimated, <code>getNext() returns random values whose
probability distribution matches the empirical distribution -- i.e., if
you generate a large number of such values, their distribution should
"look like" the distribution of the values in the input file. The values
are not stored in memory in this case either, so there is no limit to the
size of the input file. Here is an example:
<source>
ValueServer vs = new ValueServer();
vs.setValuesFileURL(url);
vs.setMode(ValueServer.DIGEST_MODE);
vs.computeDistribution(500); //Read file and estimate distribution using 500 bins
double value = vs.getNext();
// ...Generate and use more values...
</source>
See the javadoc for <code>ValueServer and
<code>EmpiricalDistribution for more details. Note that
<code>computeDistribution() opens and closes the input file
by itself.
</dd>
</dl>
</p>
</subsection>
<subsection name="2.7 PRNG Pluggability" href="pluggability">
<p>
To enable alternative PRNGs to be "plugged in" to the commons-math data
generation utilities and to provide a generic means to replace
<code>java.util.Random in applications, a random generator
adaptor framework has been added to commons-math. The
<a href="../apidocs/org/apache/commons/math3/random/RandomGenerator.html">
RandomGenerator</a> interface abstracts the public interface of
<code>java.util.Random and any implementation of this
interface can be used as the source of random data for the commons-math
data generation classes. An abstract base class,
<a href="../apidocs/org/apache/commons/math3/random/AbstractRandomGenerator.html">
AbstractRandomGenerator</a> is provided to make implementation easier.
This class provides default implementations of "derived" data generation
methods based on the primitive, <code>nextDouble().
To support generic replacement of <code>java.util.Random, the
<a href="../apidocs/org/apache/commons/math3/random/RandomAdaptor.html">
RandomAdaptor</a> class is provided, which extends
<code>java.util.Random and wraps and delegates calls to
a <code>RandomGenerator instance.
</p>
<p>Commons-math provides by itself several implementations of the
JDKRandomGenerator
that extends the JDK provided generator</li>
<li>
AbstractRandomGenerator</a> as a helper for users generators
<li>
BitStreamGenerator</a> which is an abstract class for several generators and
which in turn is extended by:
<ul>
<li>MersenneTwister
<li>
Well512a
<li>
Well1024a
<li>
Well19937a
<li>
Well19937c
<li>
Well44497a
<li>
Well44497b
</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 - 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 - 1
to 2<sup>44497 - 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-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 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 between 25 and 27 millions doubles per second (remember we generate two 32 bits integers for
each double). Generation rates for other PRNG, relative to MersenneTwister:
</p>
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