Throw an error when passing empty input to cov (#93)

74897fe unintentionnally changed the behavior of `cov`,
which now allows passing two empty vectors instead of throwing an error.
This is inconsistent with `cor`. If we wanted to return a value, `NaN`
would be more appropriate and consistent with `var`,
but for now make this an error again.

Also add tests to cover all empty inputs for similar functions.
Unfortunately it turns out we are quite inconsistent already, as
`cor` and `cov` allow empty matrices already, but `cov` returns `-0.0`
while `cor`, `std` and `var` return `NaN`.

src/Statistics.jl

@@ -519,7 +519,11 @@ unscaled_covzm(x::AbstractVector{<:Number})    = sum(abs2, x)
 unscaled_covzm(x::AbstractVector)              = sum(t -> t*t', x)
 unscaled_covzm(x::AbstractMatrix, vardim::Int) = (vardim == 1 ? _conj(x'x) : x * x')
 
-unscaled_covzm(x::AbstractVector, y::AbstractVector) = dot(y, x)
+function unscaled_covzm(x::AbstractVector, y::AbstractVector)
+    (isempty(x) || isempty(y)) &&
+        throw(ArgumentError("covariance only defined for non-empty vectors"))
+    return dot(y, x)
+end
 unscaled_covzm(x::AbstractVector, y::AbstractMatrix, vardim::Int) =
     (vardim == 1 ? *(transpose(x), _conj(y)) : *(transpose(x), transpose(_conj(y))))
 unscaled_covzm(x::AbstractMatrix, y::AbstractVector, vardim::Int) =

test/runtests.jl

@@ -899,3 +899,31 @@ end
         @test isfinite.(cov_sparse) == isfinite.(cov_dense)
     end
 end
+
+@testset "var, std, cov and cor on empty inputs" begin
+    @test isnan(var(Int[]))
+    @test isnan(std(Int[]))
+    @test isequal(cov(Int[]), -0.0)
+    @test isequal(cor(Int[]), 1.0)
+
+    @test_throws ArgumentError cov(Int[], Int[])
+    @test_throws ArgumentError cor(Int[], Int[])
+
+    mx = Matrix{Int}(undef, 0, 2)
+    my = Matrix{Int}(undef, 0, 3)
+
+    @test isequal(var(mx, dims=1), fill(NaN, 1, 2))
+    @test isequal(std(mx, dims=1), fill(NaN, 1, 2))
+    @test isequal(var(mx, dims=2), fill(NaN, 0, 1))
+    @test isequal(std(mx, dims=2), fill(NaN, 0, 1))
+
+    @test isequal(cov(mx, my), fill(-0.0, 2, 3))
+    @test isequal(cor(mx, my), fill(NaN, 2, 3))
+    @test isequal(cov(my, mx, dims=1), fill(-0.0, 3, 2))
+    @test isequal(cor(my, mx, dims=1), fill(NaN, 3, 2))
+
+    @test isequal(cov(mx, Int[]), fill(-0.0, 2, 1))
+    @test isequal(cor(mx, Int[]), fill(NaN, 2, 1))
+    @test isequal(cov(Int[], my), fill(-0.0, 1, 3))
+    @test isequal(cor(Int[], my), fill(NaN, 1, 3))
+end