README.md

IndependentRandomSequences

This package implements distribution types (in the vein of the Distributions package) for independent random sequences. These sequences can have elements which are identically distributed (IID) or with elements with non necessarily identically distributed (INID) elements.

Types

The distributions types (IIDRandomSequence and INIDRandomSequence)are vector valued random variables (MultivariateDistribution in the parlance of Distributions).

Basic Usage

To create an IID random sequence and compute some quantities of interest:

julia> using Distributions
julia> using IndependentRandomSequences

julia> srand(163)
julia> W,N = Uniform(-1,1),3
julia> Y = IIDRandomSequence(W,N)

julia> show(rand(Y))  
[-0.404987, 0.633975, 0.308448]

julia> entropy(Y)
2.0794415416798357

The approach is similar for INID random sequences:

julia> using Distributions
julia> using IndependentRandomSequences
julia> srand(163)
julia> W,X = Bernoulli(.3),Bernoulli(.8)
julia> Y = INIDRandomSequence([W,X])

julia> rand(Y,10)  
 2×10 Array{Int64,2}:
  1  0  0  0  0  0  0  0  0  0
  0  1  1  0  1  1  1  1  1  1

julia> cov(Y)
 2×2 Diagonal{Float64}:
  0.21   ⋅  
   ⋅    0.16

However, it should be noted that INID random sequence can be composed of heterogenous univariate distribution types

julia> using Distributions
julia> using IndependentRandomSequences

julia> srand(163)
julia> W,X = Binomial(3,.5),Bernoulli(.5)
julia> Y = INIDRandomSequence([W,X])

julia> rand(Y,10)
 2×10 Array{Int64,2}:
  3  1  1  2  1  2  0  3  2  1
  1  0  0  0  0  1  0  0  0  1

Why would anyone need this package?

By itself, this package allows one to save just a bit of typing when sampling from and computing quantities of interest for independent random sequences. More importantly, however, it provides a specification of IID/INID types to be used in other packages, which may implement non-trivial functionality. Actual and possible examples include:

• order statistics of independent random variables (OrderStatistics.jl)
• Basic arithmetic for independent random variables (coming soon)
• As containers for affine transformed INID sequence in the sense of independent component analysis