Random Numbers

Random number generation in Julia uses the Mersenne Twister library via MersenneTwister objects. Julia has a global RNG, which is used by default. Other RNG types can be plugged in by inheriting the AbstractRNG type; they can then be used to have multiple streams of random numbers. Besides MersenneTwister, Julia also provides the RandomDevice RNG type, which is a wrapper over the OS provided entropy.

Most functions related to random generation accept an optional AbstractRNG object as first argument, which defaults to the global one if not provided. Moreover, some of them accept optionally dimension specifications dims... (which can be given as a tuple) to generate arrays of random values.

A MersenneTwister or RandomDevice RNG can generate uniformly random numbers of the following types: Float16, Float32, Float64, BigFloat, Bool, Int8, UInt8, Int16, UInt16, Int32, UInt32, Int64, UInt64, Int128, UInt128, BigInt (or complex numbers of those types). Random floating point numbers are generated uniformly in $[0, 1)$. As BigInt represents unbounded integers, the interval must be specified (e.g. rand(big.(1:6))).

Additionally, normal and exponential distributions are implemented for some AbstractFloat and Complex types, see randn and randexp for details.

Random numbers module

Random

Random generation functions

rand([rng=GLOBAL_RNG], [S], [dims...])

julia> rand(Int, 2)
2-element Array{Int64,1}:
 1339893410598768192
 1575814717733606317

julia> using Random

julia> rand(MersenneTwister(0), Dict(1=>2, 3=>4))
1=>2

julia> rand((2, 3))
3

julia> rand(Float64, (2, 3))
2×3 Array{Float64,2}:
 0.999717  0.0143835  0.540787
 0.696556  0.783855   0.938235

rand!([rng=GLOBAL_RNG], A, [S=eltype(A)])

julia> rng = MersenneTwister(1234);

julia> rand!(rng, zeros(5))
5-element Array{Float64,1}:
 0.5908446386657102
 0.7667970365022592
 0.5662374165061859
 0.4600853424625171
 0.7940257103317943

bitrand([rng=GLOBAL_RNG], [dims...])

julia> rng = MersenneTwister(1234);

julia> bitrand(rng, 10)
10-element BitArray{1}:
 0
 1
 1
 1
 1
 0
 1
 0
 0
 1

randn([rng=GLOBAL_RNG], [T=Float64], [dims...])

julia> using Random

julia> rng = MersenneTwister(1234);

julia> randn(rng, ComplexF64)
0.6133070881429037 - 0.6376291670853887im

julia> randn(rng, ComplexF32, (2, 3))
2×3 Array{Complex{Float32},2}:
 -0.349649-0.638457im  0.376756-0.192146im  -0.396334-0.0136413im
  0.611224+1.56403im   0.355204-0.365563im  0.0905552+1.31012im

randn!([rng=GLOBAL_RNG], A::AbstractArray) -> A

julia> rng = MersenneTwister(1234);

julia> randn!(rng, zeros(5))
5-element Array{Float64,1}:
  0.8673472019512456
 -0.9017438158568171
 -0.4944787535042339
 -0.9029142938652416
  0.8644013132535154

randexp([rng=GLOBAL_RNG], [T=Float64], [dims...])

julia> rng = MersenneTwister(1234);

julia> randexp(rng, Float32)
2.4835055f0

julia> randexp(rng, 3, 3)
3×3 Array{Float64,2}:
 1.5167    1.30652   0.344435
 0.604436  2.78029   0.418516
 0.695867  0.693292  0.643644

randexp!([rng=GLOBAL_RNG], A::AbstractArray) -> A

julia> rng = MersenneTwister(1234);

julia> randexp!(rng, zeros(5))
5-element Array{Float64,1}:
 2.4835053723904896
 1.516703605376473
 0.6044364871025417
 0.6958665886385867
 1.3065196315496677

randstring([rng=GLOBAL_RNG], [chars], [len=8])

julia> Random.seed!(3); randstring()
"4zSHdXlw"

julia> randstring(MersenneTwister(3), 'a':'z', 6)
"bzlhqn"

julia> randstring("ACGT")
"AGGACATT"

Subsequences, permutations and shuffling

randsubseq([rng=GLOBAL_RNG,] A, p) -> Vector

julia> rng = MersenneTwister(1234);

julia> randsubseq(rng, 1:8, 0.3)
2-element Array{Int64,1}:
 7
 8

randsubseq!([rng=GLOBAL_RNG,] S, A, p)

julia> rng = MersenneTwister(1234);

julia> S = Int64[];

julia> randsubseq!(rng, S, 1:8, 0.3)
2-element Array{Int64,1}:
 7
 8

julia> S
2-element Array{Int64,1}:
 7
 8

randperm([rng=GLOBAL_RNG,] n::Integer)

julia> randperm(MersenneTwister(1234), 4)
4-element Array{Int64,1}:
 2
 1
 4
 3

randperm!([rng=GLOBAL_RNG,] A::Array{<:Integer})

julia> randperm!(MersenneTwister(1234), Vector{Int}(undef, 4))
4-element Array{Int64,1}:
 2
 1
 4
 3

randcycle([rng=GLOBAL_RNG,] n::Integer)

julia> randcycle(MersenneTwister(1234), 6)
6-element Array{Int64,1}:
 3
 5
 4
 6
 1
 2

randcycle!([rng=GLOBAL_RNG,] A::Array{<:Integer})

julia> randcycle!(MersenneTwister(1234), Vector{Int}(undef, 6))
6-element Array{Int64,1}:
 3
 5
 4
 6
 1
 2

shuffle([rng=GLOBAL_RNG,] v::AbstractArray)

julia> rng = MersenneTwister(1234);

julia> shuffle(rng, Vector(1:10))
10-element Array{Int64,1}:
  6
  1
 10
  2
  3
  9
  5
  7
  4
  8

shuffle!([rng=GLOBAL_RNG,] v::AbstractArray)

julia> rng = MersenneTwister(1234);

julia> shuffle!(rng, Vector(1:16))
16-element Array{Int64,1}:
  2
 15
  5
 14
  1
  9
 10
  6
 11
  3
 16
  7
  4
 12
  8
 13

Generators (creation and seeding)

seed!([rng=GLOBAL_RNG], seed) -> rng
seed!([rng=GLOBAL_RNG]) -> rng

julia> Random.seed!(1234);

julia> x1 = rand(2)
2-element Array{Float64,1}:
 0.590845
 0.766797

julia> Random.seed!(1234);

julia> x2 = rand(2)
2-element Array{Float64,1}:
 0.590845
 0.766797

julia> x1 == x2
true

julia> rng = MersenneTwister(1234); rand(rng, 2) == x1
true

julia> MersenneTwister(1) == Random.seed!(rng, 1)
true

julia> rand(Random.seed!(rng), Bool) # not reproducible
true

julia> rand(Random.seed!(rng), Bool)
false

julia> rand(MersenneTwister(), Bool) # not reproducible either
true

AbstractRNG

MersenneTwister(seed)
MersenneTwister()

julia> rng = MersenneTwister(1234);

julia> x1 = rand(rng, 2)
2-element Array{Float64,1}:
 0.5908446386657102
 0.7667970365022592

julia> rng = MersenneTwister(1234);

julia> x2 = rand(rng, 2)
2-element Array{Float64,1}:
 0.5908446386657102
 0.7667970365022592

julia> x1 == x2
true

RandomDevice()

Hooking into the Random API

There are two mostly orthogonal ways to extend Random functionalities:

1. generating random values of custom types
2. creating new generators

The API for 1) is quite functional, but is relatively recent so it may still have to evolve in subsequent releases of the Random module. For example, it's typically sufficient to implement one rand method in order to have all other usual methods work automatically.

The API for 2) is still rudimentary, and may require more work than strictly necessary from the implementor, in order to support usual types of generated values.

Generating random values of custom types

Generating random values for some distributions may involve various trade-offs. Pre-computed values, such as an alias table for discrete distributions, or “squeezing” functions for univariate distributions, can speed up sampling considerably. How much information should be pre-computed can depend on the number of values we plan to draw from a distribution. Also, some random number generators can have certain properties that various algorithms may want to exploit.

The Random module defines a customizable framework for obtaining random values that can address these issues. Each invocation of rand generates a sampler which can be customized with the above trade-offs in mind, by adding methods to Sampler, which in turn can dispatch on the random number generator, the object that characterizes the distribution, and a suggestion for the number of repetitions. Currently, for the latter, Val{1} (for a single sample) and Val{Inf} (for an arbitrary number) are used, with Random.Repetition an alias for both.

The object returned by Sampler is then used to generate the random values. When implementing the random generation interface for a value X that can be sampled from, the implementor should define the method

rand(rng, sampler)

for the particular sampler returned by Sampler(rng, X, repetition).

Samplers can be arbitrary values that implement rand(rng, sampler), but for most applications the following predefined samplers may be sufficient:

1. SamplerType{T}() can be used for implementing samplers that draw from type T (e.g. rand(Int)). This is the default returned by Sampler for types.

2. SamplerTrivial(self) is a simple wrapper for self, which can be accessed with []. This is the recommended sampler when no pre-computed information is needed (e.g. rand(1:3)), and is the default returned by Sampler for values.

3. SamplerSimple(self, data) also contains the additional data field, which can be used to store arbitrary pre-computed values, which should be computed in a custom method of Sampler.

We provide examples for each of these. We assume here that the choice of algorithm is independent of the RNG, so we use AbstractRNG in our signatures.

Sampler(rng, x, repetition = Val(Inf))

SamplerType{T}()

SamplerTrivial(x)

SamplerSimple(x, data)

Decoupling pre-computation from actually generating the values is part of the API, and is also available to the user. As an example, assume that rand(rng, 1:20) has to be called repeatedly in a loop: the way to take advantage of this decoupling is as follows:

rng = MersenneTwister()
sp = Random.Sampler(rng, 1:20) # or Random.Sampler(MersenneTwister, 1:20)
for x in X
    n = rand(rng, sp) # similar to n = rand(rng, 1:20)
    # use n
end

This is the mechanism that is also used in the standard library, e.g. by the default implementation of random array generation (like in rand(1:20, 10)).

Generating values from a type

Given a type T, it's currently assumed that if rand(T) is defined, an object of type T will be produced. SamplerType is the default sampler for types. In order to define random generation of values of type T, the rand(rng::AbstractRNG, ::Random.SamplerType{T}) method should be defined, and should return values what rand(rng, T) is expected to return.

Let's take the following example: we implement a Die type, with a variable number n of sides, numbered from 1 to n. We want rand(Die) to produce a Die with a random number of up to 20 sides (and at least 4):

struct Die
    nsides::Int # number of sides
end

Random.rand(rng::AbstractRNG, ::Random.SamplerType{Die}) = Die(rand(rng, 4:20))

# output

Scalar and array methods for Die now work as expected:

julia> rand(Die)
Die(10)

julia> rand(MersenneTwister(0), Die)
Die(16)

julia> rand(Die, 3)
3-element Array{Die,1}:
 Die(5)
 Die(20)
 Die(9)

julia> a = Vector{Die}(undef, 3); rand!(a)
3-element Array{Die,1}:
 Die(11)
 Die(20)
 Die(10)

A simple sampler without pre-computed data

Here we define a sampler for a collection. If no pre-computed data is required, it can be implemented with a SamplerTrivial sampler, which is in fact the default fallback for values.

In order to define random generation out of objects of type S, the following method should be defined: rand(rng::AbstractRNG, sp::Random.SamplerTrivial{S}). Here, sp simply wraps an object of type S, which can be accessed via sp[]. Continuing the Die example, we want now to define rand(d::Die) to produce an Int corresponding to one of d's sides:

julia> Random.rand(rng::AbstractRNG, d::Random.SamplerTrivial{Die}) = rand(rng, 1:d[].nsides);

julia> rand(Die(4))
2

julia> rand(Die(4), 3)
3-element Array{Any,1}:
 1
 4
 2

Given a collection type S, it's currently assumed that if rand(::S) is defined, an object of type eltype(S) will be produced. In the last example, a Vector{Any} is produced; the reason is that eltype(Die) == Any. The remedy is to define Base.eltype(::Type{Die}) = Int.

Generating values for an AbstractFloat type

AbstractFloat types are special-cased, because by default random values are not produced in the whole type domain, but rather in [0,1). The following method should be implemented for T <: AbstractFloat: Random.rand(::AbstractRNG, ::Random.SamplerTrivial{Random.CloseOpen01{T}})

An optimized sampler with pre-computed data

Consider a discrete distribution, where numbers 1:n are drawn with given probabilities that sum to one. When many values are needed from this distribution, the fastest method is using an alias table. We don't provide the algorithm for building such a table here, but suppose it is available in make_alias_table(probabilities) instead, and draw_number(rng, alias_table) can be used to draw a random number from it.

Suppose that the distribution is described by

struct DiscreteDistribution{V <: AbstractVector}
    probabilities::V
end

and that we always want to build an alias table, regardless of the number of values needed (we learn how to customize this below). The methods

Random.eltype(::Type{<:DiscreteDistribution}) = Int

function Random.Sampler(::Type{<:AbstractRNG}, distribution::DiscreteDistribution, ::Repetition)
    SamplerSimple(disribution, make_alias_table(distribution.probabilities))
end

should be defined to return a sampler with pre-computed data, then

function rand(rng::AbstractRNG, sp::SamplerSimple{<:DiscreteDistribution})
    draw_number(rng, sp.data)
end

will be used to draw the values.

Custom sampler types

The SamplerSimple type is sufficient for most use cases with precomputed data. However, in order to demonstrate how to use custom sampler types, here we implement something similar to SamplerSimple.

Going back to our Die example: rand(::Die) uses random generation from a range, so there is an opportunity for this optimization. We call our custom sampler SamplerDie.

import Random: Sampler, rand

struct SamplerDie <: Sampler{Int} # generates values of type Int
    die::Die
    sp::Sampler{Int} # this is an abstract type, so this could be improved
end

Sampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =
    SamplerDie(die, Sampler(RNG, 1:die.nsides, r))
# the `r` parameter will be explained later on

rand(rng::AbstractRNG, sp::SamplerDie) = rand(rng, sp.sp)

It's now possible to get a sampler with sp = Sampler(rng, die), and use sp instead of die in any rand call involving rng. In the simplistic example above, die doesn't need to be stored in SamplerDie but this is often the case in practice.

Of course, this pattern is so frequent that the helper type used above, namely Random.SamplerSimple, is available, saving us the definition of SamplerDie: we could have implemented our decoupling with:

Sampler(RNG::Type{<:AbstractRNG}, die::Die, r::Random.Repetition) =
    SamplerSimple(die, Sampler(RNG, 1:die.nsides, r))

rand(rng::AbstractRNG, sp::SamplerSimple{Die}) = rand(rng, sp.data)

Here, sp.data refers to the second parameter in the call to the SamplerSimple constructor (in this case equal to Sampler(rng, 1:die.nsides, r)), while the Die object can be accessed via sp[].

Like SamplerDie, any custom sampler must be a subtype of Sampler{T} where T is the type of the generated values. Note that SamplerSimple(x, data) isa Sampler{eltype(x)}, so this constrains what the first argument to SamplerSimple can be (it's recommended to use SamplerSimple like in the Die example, where x is simply forwarded while defining a Sampler method). Similarly, SamplerTrivial(x) isa Sampler{eltype(x)}.

Another helper type is currently available for other cases, Random.SamplerTag, but is considered as internal API, and can break at any time without proper deprecations.

Using distinct algorithms for scalar or array generation

In some cases, whether one wants to generate only a handful of values or a large number of values will have an impact on the choice of algorithm. This is handled with the third parameter of the Sampler constructor. Let's assume we defined two helper types for Die, say SamplerDie1 which should be used to generate only few random values, and SamplerDieMany for many values. We can use those types as follows:

Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{1}) = SamplerDie1(...)
Sampler(RNG::Type{<:AbstractRNG}, die::Die, ::Val{Inf}) = SamplerDieMany(...)

Of course, rand must also be defined on those types (i.e. rand(::AbstractRNG, ::SamplerDie1) and rand(::AbstractRNG, ::SamplerDieMany)). Note that, as usual, SamplerTrivial and SamplerSimple can be used if custom types are not necessary.

Note: Sampler(rng, x) is simply a shorthand for Sampler(rng, x, Val(Inf)), and Random.Repetition is an alias for Union{Val{1}, Val{Inf}}.

Creating new generators

The API is not clearly defined yet, but as a rule of thumb:

1. any rand method producing "basic" types (isbitstype integer and floating types in Base) should be defined for this specific RNG, if they are needed;
2. other documented rand methods accepting an AbstractRNG should work out of the box, (provided the methods from 1) what are relied on are implemented), but can of course be specialized for this RNG if there is room for optimization;
3. copy for pseudo-RNGs should return an independent copy that generates the exact same random sequence as the original from that point when called in the same way. When this is not feasible (e.g. hardware-based RNGs), copy must not be implemented.

Concerning 1), a rand method may happen to work automatically, but it's not officially supported and may break without warnings in a subsequent release.

To define a new rand method for an hypothetical MyRNG generator, and a value specification s (e.g. s == Int, or s == 1:10) of type S==typeof(s) or S==Type{s} if s is a type, the same two methods as we saw before must be defined:

1. Sampler(::Type{MyRNG}, ::S, ::Repetition), which returns an object of type say SamplerS
2. rand(rng::MyRNG, sp::SamplerS)

It can happen that Sampler(rng::AbstractRNG, ::S, ::Repetition) is already defined in the Random module. It would then be possible to skip step 1) in practice (if one wants to specialize generation for this particular RNG type), but the corresponding SamplerS type is considered as internal detail, and may be changed without warning.

Specializing array generation

In some cases, for a given RNG type, generating an array of random values can be more efficient with a specialized method than by merely using the decoupling technique explained before. This is for example the case for MersenneTwister, which natively writes random values in an array.

To implement this specialization for MyRNG and for a specification s, producing elements of type S, the following method can be defined: rand!(rng::MyRNG, a::AbstractArray{S}, ::SamplerS), where SamplerS is the type of the sampler returned by Sampler(MyRNG, s, Val(Inf)). Instead of AbstractArray, it's possible to implement the functionality only for a subtype, e.g. Array{S}. The non-mutating array method of rand will automatically call this specialization internally.

Reproducibility

By using an RNG parameter initialized with a given seed, you can reproduce the same pseudorandom number sequence when running your program multiple times. However, a minor release of Julia (e.g. 1.3 to 1.4) may change the sequence of pseudorandom numbers generated from a specific seed. (Even if the sequence produced by a low-level function like rand does not change, the output of higher-level functions like randsubseq may change due to algorithm updates.) Rationale: guaranteeing that pseudorandom streams never change prohibits many algorithmic improvements.

If you need to guarantee exact reproducibility of random data, it is advisable to simply save the data (e.g. as a supplementary attachment in a scientific publication). (You can also, of course, specify a particular Julia version and package manifest, especially if you require bit reproducibility.)

Software tests that rely on specific "random" data should also generally save the data or embed it into the test code. On the other hand, tests that should pass for most random data (e.g. testing A \ (A*x) ≈ x for a random matrix A = randn(n,n)) can use an RNG with a fixed seed to ensure that simply running the test many times does not encounter a failure due to very improbable data (e.g. an extremely ill-conditioned matrix).

The statistical distribution from which random samples are drawn is guaranteed to be the same across any minor Julia releases.