The RandStream
class allows you to create a
random number stream. This is useful for several reasons:
You can generate random values without affecting the state of the global stream.
You can separate sources of randomness in a simulation.
You can use a generator that is configured differently than the one MATLAB^{®} software uses at startup.
With a RandStream
object,
you can create your own stream, set the writable properties, and use the stream to
generate random numbers. You can control the stream you create the same way you control
the global stream. You can even replace the global stream with the stream you
create.
To create a stream, use the RandStream
function.
myStream = RandStream('mlfg6331_64');
rand(myStream,1,5)
ans = 0.6986 0.7413 0.4239 0.6914 0.7255
The random stream myStream
acts separately from the global stream.
If you call the rand
, randn
, randi
, and randperm
functions with myStream
as the first
argument, they draw from the stream you created. If you call rand
, randn
, randi
, and randperm
without
myStream
, they draw from the global stream.
You can make myStream
the global stream using the RandStream.setGlobalStream
method.
RandStream.setGlobalStream(myStream) RandStream.getGlobalStream
ans = mlfg6331_64 random stream (current global stream) Seed: 0 NormalTransform: Ziggurat
RandStream.getGlobalStream == myStream
ans = 1
You can use substreams to get different results that are statistically independent from a stream. Unlike seeds, where the locations along the sequence of random numbers are not exactly known, the spacing between substreams is known, so any chance of overlap can be eliminated. In short, substreams are a more-controlled way to do many of the same things that seeds have traditionally been used for. Substreams are also a more lightweight solution than parallel streams.
Substreams provide a quick and easy way to ensure that you get different results
from the same code at different times. To use the Substream
property of a RandStream
object, create a stream using a
generator that supports substreams. For a list of generator algorithms that support
substreams and their properties, see the table in the next section. For example,
generate several random numbers in a loop.
myStream = RandStream('mlfg6331_64'); RandStream.setGlobalStream(myStream) for i = 1:5 myStream.Substream = i; z = rand(1,i) end
z = 0.6986 z = 0.9230 0.2489 z = 0.0261 0.2530 0.0737 z = 0.3220 0.7405 0.1983 0.1052 z = 0.2067 0.2417 0.9777 0.5970 0.4187
In another loop, you can generate random values that are independent from the first set of 5 iterations.
for i = 6:10 myStream.Substream = i; z = rand(1,11-i) end
z = 0.2650 0.8229 0.2479 0.0247 0.4581 z = 0.3963 0.7445 0.7734 0.9113 z = 0.2758 0.3662 0.7979 z = 0.6814 0.5150 z = 0.5247
Substreams are useful in serial computation. Substreams can recreate all or part of a simulation by returning to a particular checkpoint in stream. For example, you can return to the 6th substream in the loop. The result contains the same values as the 6th output above.
myStream.Substream = 6; z = rand(1,5)
z = 0.2650 0.8229 0.2479 0.0247 0.4581
MATLAB offers several generator algorithm options. The table summarizes the
key properties of the available generator algorithms and the keywords used to create
them. To return a list of all the available generator algorithms, use the RandStream.list
method.
Keyword | Generator | Multiple Stream and Substream Support | Approximate Period In Full Precision |
---|---|---|---|
mt19937ar | Mersenne twister (used by default stream at MATLAB startup) | No | 2^{19937}-1 |
dsfmt19937 | SIMD-oriented fast Mersenne twister | No | 2^{19937}-1 |
mcg16807 | Multiplicative congruential generator | No | 2^{31}-2 |
mlfg6331_64 | Multiplicative lagged Fibonacci generator | Yes | 2^{124} (2^{51} streams of length 2^{72}) |
mrg32k3a | Combined multiple recursive generator | Yes | 2^{191} (2^{63} streams of length 2^{127}) |
philox4x32_10 | Philox 4x32 generator with 10 rounds | Yes | 2^{193} (2^{64} streams of length 2^{129}) |
threefry4x64_20 | Threefry 4x64 generator with 20 rounds | Yes | 2^{514} (2^{256} streams of length 2^{258}) |
shr3cong | Shift-register generator summed with linear congruential generator | No | 2^{64} |
swb2712 | Modified subtract with borrow generator | No | 2^{1492} |
The generators mcg16807
, shr3cong
, and
swb2712
provide for backwards compatibility with earlier
versions of MATLAB. mt19937ar
and dsfmt19937
are
designed primarily for sequential applications. The remaining generators provide
explicit support for parallel random number generation.
Depending on the application, some generators might be faster or return values with more precision. All pseudorandom number generators are based on deterministic algorithms, and all generators pass a sufficiently specific statistical test for randomness. One way to check the results of a Monte Carlo simulation is to rerun the simulation with two or more different generator algorithms, and the choice of generators in MATLAB provides you with the means to do that. Although it is unlikely that your results will differ by more than the Monte Carlo sampling error when using different generators, there are examples in the literature where this kind of validation has turned up flaws in a particular generator algorithm. (See [13] for an example.)
mt19937ar
The Mersenne Twister, as described in [11], has period $${2}^{19937}-1$$ and each U(0,1) value is created using two 32-bit
integers. The possible values are multiples of $${2}^{-53}$$ in the interval (0, 1). This generator does not
support multiple streams or substreams. The
randn
algorithm used by default for
mt19937ar
streams is the ziggurat algorithm
[7], but with the mt19937ar
generator
underneath.
Note
This generator is identical to the one used by the
rand
function beginning in MATLAB Version 7, activated by
rand('twister',s)
.
dsfmt19937
The double precision SIMD-oriented Fast Mersenne Twister, as described in [12], is a faster implementation of the Mersenne Twister algorithm. The period is $${2}^{19937}-1$$ and the possible values are multiples of $${2}^{-52}$$ in the interval (0, 1). The generator produces double precision values in [1, 2) natively, which are transformed to create U(0,1) values. This generator does not support multiple streams or substreams.
mcg16807
A 32-bit multiplicative congruential generator, as described in
[14], with multiplier $$a={7}^{5}$$, modulo $$m={2}^{31}-1$$. This generator has a period of $${2}^{31}-2$$ and does not support multiple streams or
substreams. Each U(0,1) value is created using a single 32-bit
integer from the generator; the possible values are all multiples of $${({2}^{31}-1)}^{-1}$$ strictly within the interval (0, 1). For
mcg16807
streams, the default algorithm used
by randn
is the polar algorithm (described in
[1]).
Note
This generator is identical to the one used beginning in
MATLAB Version 4 by both the rand
and randn
functions, activated using
rand('seed',s)
or
randn('seed',s)
.
mlfg6331_64
A 64-bit multiplicative lagged Fibonacci generator, as described
in [10], with lags $$l=63$$, $$k=31$$. This generator is similar to the MLFG implemented
in the SPRNG package. It has a period of approximately $${2}^{124}$$. It supports up to $${2}^{61}$$ parallel streams, via parameterization, and $${2}^{51}$$ substreams each of length $${2}^{72}$$. Each U(0,1) value is created using one 64-bit
integer from the generator; the possible values are all multiples of $${2}^{-64}$$ strictly within the interval (0, 1). The
randn
algorithm used by default for
mlfg6331_64
streams is the ziggurat algorithm
[7], but with the mlfg6331_64
generator
underneath.
mrg32k3a
A 32-bit combined multiple recursive generator, as described in
[2]. This generator is similar to the CMRG implemented in the
RngStreams package in C. It has a period of $${2}^{191}$$ and supports up to $${2}^{63}$$ parallel streams through sequence splitting, each
of length $${2}^{127}$$. It also supports $${2}^{51}$$ substreams, each of length $${2}^{76}$$. Each U(0,1) value is created using two 32-bit
integers from the generator; the possible values are multiples of $${2}^{-53}$$ strictly within the interval (0, 1). The
randn
algorithm used by default for
mrg32k3a
streams is the ziggurat algorithm
[7], but with the mrg32k3a
generator underneath.
philox4x32_10
A 4x32 generator with 10 rounds as described in [15]. This generator uses a Feistel network and integer multiplication. The generator is specifically designed for high performance in highly parallel systems such as GPUs. It has a period of 2^{193} (2^{64} streams of length 2^{129}).
threefry4x64_20
A 4x64 generator with 20 rounds as described in [15]. This generator is a non-cryptographic adaptation of the Threefish block cipher from the Skein Hash Function. It has a period of 2^{514} (2^{256} streams of length 2^{258}).
shr3cong
Marsaglia's SHR3 shift-register generator summed with a linear
congruential generator with multiplier $$a=69069$$, addend $$b=1234567$$, and modulus $${2}^{-32}$$. SHR3 is a 3-shift-register generator defined as $$u=u(I+{L}^{13})(I+{R}^{17})(I+{L}^{5})$$, where $$I$$ is the identity operator, $$L$$ is the left shift operator, and R
is the right shift operator. The combined generator (the SHR3 part
is described in [7]) has a period of approximately $${2}^{64}$$. This generator does not support multiple streams
or substreams. Each U(0,1) value is created using one 32-bit integer
from the generator; the possible values are all multiples of $${2}^{-32}$$ strictly within the interval (0, 1). The
randn
algorithm used by default for
shr3cong
streams is the earlier form of the
ziggurat algorithm [9], but with the shr3cong
generator underneath.
This generator is identical to the one used by the
randn
function beginning in MATLAB Version 5, activated using
randn('state',s)
.
swb2712
A modified Subtract-with-Borrow generator, as described in [8]. This generator is similar to an additive lagged Fibonacci
generator with lags 27 and 12, but it is modified to have a much
longer period of approximately $${2}^{1492}$$. The generator works natively in double precision
to create U(0,1) values, and all values in the open interval (0, 1)
are possible. The randn
algorithm used by
default for swb2712
streams is the ziggurat
algorithm [7], but with the swb2712
generator
underneath.
Note
This generator is identical to the one used by the
rand
function beginning in MATLAB Version 5, activated using
rand('state',s)
.
Inversion
Computes a normal random variate by applying the standard normal inverse cumulative distribution function to a uniform random variate. Exactly one uniform value is consumed per normal value.
Polar
The polar rejection algorithm, as described in [1]. Approximately 1.27 uniform values are consumed per normal value, on average.
Ziggurat
The ziggurat algorithm, as described in [7]. Approximately 2.02 uniform values are consumed per normal value, on average.
A random number stream s
has properties that control its
behavior. To access or change a property, use the syntax p =
s.Property
and s.Property = p
.
For example, you can configure the transformation algorithm to generate normally
distributed pseudorandom values when using randn
. Generate normally distributed pseudorandom values using the
default Ziggurat
transformation algorithm.
s1 = RandStream('mt19937ar');
s1.NormalTransform
ans = 'Ziggurat'
r1 = randn(s1,1,10);
Configure the stream to use the Polar
transformation algorithm
to generate normally distributed pseudorandom values.
s1.NormalTransform = 'Polar'
s1 = mt19937ar random stream Seed: 0 NormalTransform: Polar
r2 = randn(s1,1,10);
When generating random numbers with uniform distribution using rand
, you can also configure the
stream to generate antithetic pseudorandom values, that is, the usual values
subtracted from 1 for uniform values.
Create 6 random numbers with uniform distribution from the stream s.
s2 = RandStream('mt19937ar');
r1 = rand(s2,1,6)
r1 = 0.8147 0.9058 0.1270 0.9134 0.6324 0.0975
Restore the initial state of the stream. Create another 6 random numbers with the
Antithetic
property set to true. Check that these 6 random
numbers are equal to the previously generated random numbers subtracted from
1.
reset(s2) s2.Antithetic = true; r2 = rand(s2,1,6)
r2 = 0.1853 0.0942 0.8730 0.0866 0.3676 0.9025
isequal(r1,1 - r2)
ans = 1
Instead of setting the properties of a stream one-by-one, you can save and restore
all properties of a stream s
by using A =
get(s)
and set(s,A)
, respectively. For example,
configure all properties of the stream s2
to be the same as the
stream s1
.
A = get(s1)
A = Type: 'mt19937ar' NumStreams: 1 StreamIndex: 1 Substream: 1 Seed: 0 State: [625x1 uint32] NormalTransform: 'Polar' Antithetic: 0 FullPrecision: 1
set(s2,A)
Type: 'mt19937ar' NumStreams: 1 StreamIndex: 1 Substream: 1 Seed: 0 State: [625x1 uint32] NormalTransform: 'Polar' Antithetic: 0 FullPrecision: 1
The get
and set
functions enable you to save
and restore a stream's entire configuration so that results are exactly reproducible
later on.
The State
property is the internal state of the random number
generator. You can save the state of the global stream at a certain point in your
simulation when generating random numbers to reproduce the results later on.
Use RandStream.getGlobalStream
to return a handle to the global
stream, that is, the current global stream that rand
generates
random numbers from. Save the state of the global stream.
globalStream = RandStream.getGlobalStream; myState = globalStream.State;
Using myState
, you can restore the state of
globalStream
and reproduce previous
results.
A = rand(1,100); globalStream.State = myState; B = rand(1,100); isequal(A,B)
ans = logical 1
rand
, randi
, randn
, and
randperm
access the global stream. Since all of these
functions access the same underlying stream, a call to one affects the values
produced by the others at subsequent calls.
globalStream.State = myState; A = rand(1,100); globalStream.State = myState; C = randi(100); B = rand(1,100); isequal(A,B)
ans = logical 0
You can also reset a stream to its initial settings using the reset
function.
reset(globalStream) A = rand(1,100); reset(globalStream) B = rand(1,100); isequal(A,B)
ans = logical 1
[1] Devroye, L. Non-Uniform Random Variate Generation, Springer-Verlag, 1986.
[2] L’Ecuyer, P. “Good Parameter Sets for Combined Multiple Recursive Random Number Generators”, Operations Research, 47(1): 159–164. 1999.
[3] L'Ecuyer, P. and S. Côté. “Implementing A Random Number Package with Splitting Facilities”, ACM Transactions on Mathematical Software, 17: 98–111. 1991.
[4] L'Ecuyer, P. and R. Simard. “TestU01: A C Library for Empirical Testing of Random Number Generators,” ACM Transactions on Mathematical Software, 33(4): Article 22. 2007.
[5] L'Ecuyer, P., R. Simard, E. J. Chen, and W. D. Kelton. “An Objected-Oriented Random-Number Package with Many Long Streams and Substreams.” Operations Research, 50(6):1073–1075. 2002.
[6] Marsaglia, G. “Random numbers for C: The
END?” Usenet posting to sci.stat.math. 1999. Available online at https://groups.google.com/group/sci.crypt/browse_thread/
.
thread/ca8682a4658a124d/
[7] Marsaglia G., and W. W. Tsang. “The ziggurat method
for generating random variables.” Journal of Statistical
Software, 5:1–7. 2000. Available online at https://www.jstatsoft.org/v05/i08
.
[8] Marsaglia, G., and A. Zaman. “A new class of random number generators.” Annals of Applied Probability 1(3):462–480. 1991.
[9] Marsaglia, G., and W. W. Tsang. “A fast, easily implemented method for sampling from decreasing or symmetric unimodal density functions.” SIAM J. Sci. Stat. Comput. 5(2):349–359. 1984.
[10] Mascagni, M., and A. Srinivasan. “Parameterizing Parallel Multiplicative Lagged-Fibonacci Generators.” Parallel Computing, 30: 899–916. 2004.
[11] Matsumoto, M., and T. Nishimura.“Mersenne Twister: A 623-Dimensionally Equidistributed Uniform Pseudorandom Number Generator.” ACM Transactions on Modeling and Computer Simulation, 8(1):3–30. 1998.
[12] Matsumoto, M., and M. Saito.“A PRNG Specialized in Double Precision Floating Point Numbers Using an Affine Transition.” Monte Carlo and Quasi-Monte Carlo Methods 2008, 10.1007/978-3-642-04107-5_38. 2009.
[13] Moler, C.B. Numerical Computing with
MATLAB. SIAM, 2004. Available online at https://www.mathworks.com/moler
[14] Park, S.K., and K.W. Miller. “Random Number Generators: Good Ones Are Hard to Find.” Communications of the ACM, 31(10):1192–1201. 1998.
[15] Salmon, J. K., M. A. Moraes, R. O. Dror, and D. E. Shaw. "Parallel Random Numbers: As Easy As 1, 2, 3." In Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis (SC11). New York, NY: ACM, 2011.