Complex Uniform Distribution between 1+0im and 2+0im

Specific Domains Statistics
1

I am trying to extend Uniform so rand can generate random numbers for Complex types (to test IntelVectorMath library for Complex types)

# zeros like syntax
function Distributions.Uniform(::Type{T}, a, b) where {T <: Real}
    return Uniform(T(a), T(b))
end

# Complex support
function Distributions.Uniform(T::Type{Complex{P}}, a, b) where {P}
    aR, aI =reim(T(a))
    bR, bI =reim(T(b))
    if aR == bR
        outR = P(0) #?? Ask about this
    else
        outR = Uniform(P, min(aR,bR), max(aR,bR))
    end
    if aI == bI
        outI = P(0)  #?? Ask about this
    else
        outI = Uniform(P, min(aI,bI), max(aI,bI))
    end
    return complex(outR, outI)
end

# doesn't work:
Uniform(Complex{Float64},1,2)

The problem arises when a and b are equal. In literature, the probability is 0. However, in Julia’s implementation, it is not apparent that when we define Unifrom(a,b), is it a [a, b] or (a,b).

If it is [a, b] then Uniform(a, a) should always give a
if is is (a, b) then Uniform(a, a) should give a constant 0 distribution that rand always gives some null situation.

My implementation is faulty, but I can’t find anything in Distributions doc anything about Complex types. Probably it should be a multi-dimensional distribution.

2

Here is my opinion on the issue.

Uniform(a, b) as per the docs returns [a, b] but demands that a < b. I feel like this should stay.

We should introduce a new distribution d = Singleton{X} that rand(d) returns the constant X. Then the product measure defined by OP will work as intended. This notation is used in https://github.com/MikeInnes/Poirot.jl .

Last, we should resolve https://github.com/JuliaStats/Distributions.jl/issues/142 and Ill like to hijack the issue and say that generic convolution should also work.

1 Like
3

I updated OP. We also need extend complex() so it can get Distributions types.

Another issue that is related is my problem with the Uniform definition and its API. I talked about it here:

https://github.com/JuliaStats/Distributions.jl/issues/1041

I created a PR to solve it here:

https://github.com/JuliaStats/Distributions.jl/pull/1045/files
From the mathematical point of view, a and b bounds may not be part of the distribution and so their type shouldn’t be part of the Distribution definition. Actually their probability is equal to 0!

4

Having probability 0 does not mean it does not occur. Either definition is self-consistent.

To see this more clearly, the probability for any value on an interval [a, b] gets returned by a uniform random distribution on it is 0.

3 Likes
5

I imagine that the problem arises before this: since complex numbers have no canonical ordering, it’s hard to make sense of a < x < b. Commonly, complex uniform distributions are defined over unit circles.

As for the problem of singular/meaningless parameters: I think it is preferable if constructors simply error, instead of doing something clever. Currently

julia> Uniform(0,0)
ERROR: ArgumentError: Uniform: the condition a < b is not satisfied.

which is fine, but

julia> Normal(1, 0)
Normal{Float64}(μ=1.0, σ=0.0)
3 Likes
6

I think the reason why Uniform does not permit a = b is due to the docs being written as

  Uniform(a,b)

  The continuous uniform distribution over an interval [a, b] has probability
  density function

f(x; a, b) = \frac{1}{b - a}, \quad a \le x \le b

which would imply that a \neq b due to the division by 0. But returning the Singleton as described in my first reply is perfectly sensible to, with some rewriting of the docs.

7

No, it would be type unstable. While the compiler can handle this to a limited extent, it is not good design without a compelling reason.

Also, it is a convention for T constructors to return T objects — this is not enforced, but it is not good style to violate this.

1 Like
8

You’re right I didn’t think this through in that direction.

1 Like