statistics.jl

# This file is a part of Julia. License is MIT: http://julialang.org/license

using Base.Test

# middle

@test middle(3) === 3.0
@test middle(2, 3) === 2.5
let x = ((realmax(1.0)/4)*3)
    @test middle(x, x) === x
end
@test middle(1:8) === 4.5
@test middle([1:8;]) === 4.5

# ensure type-correctness
for T in [Bool,Int8,Int16,Int32,Int64,Int128,UInt8,UInt16,UInt32,UInt64,UInt128,Float16,Float32,Float64]
    @test middle(one(T)) === middle(one(T), one(T))
end


# median
@test median([1.]) === 1.
@test median([1.,3]) === 2.
@test median([1.,3,2]) === 2.

@test median([1,3,2]) === 2.0
@test median([1,3,2,4]) === 2.5

@test median([0.0,Inf]) == Inf
@test median([0.0,-Inf]) == -Inf
@test median([0.,Inf,-Inf]) == 0.0
@test median([1.,-1.,Inf,-Inf]) == 0.0
@test isnan(median([-Inf,Inf]))

X = [2 3 1 -1; 7 4 5 -4]
@test all(median(X, 2) .== [1.5, 4.5])
@test all(median(X, 1) .== [4.5 3.5 3.0 -2.5])
@test X == [2 3 1 -1; 7 4 5 -4] # issue #17153

@test_throws ArgumentError median([])
@test isnan(median([NaN]))
@test isnan(median([0.0,NaN]))
@test isnan(median([NaN,0.0]))
@test isequal(median([NaN 0.0; 1.2 4.5], 2), reshape([NaN; 2.85], 2, 1))

@test median!([1 2 3 4]) == 2.5
@test median!([1 2; 3 4]) == 2.5


@test invoke(median, Tuple{AbstractVector}, 1:10) == median(1:10) == 5.5

# mean
@test_throws ArgumentError mean(())
@test mean((1,2,3)) === 2.
@test mean([0]) === 0.
@test mean([1.]) === 1.
@test mean([1.,3]) == 2.
@test mean([1,2,3]) == 2.
@test mean([0 1 2; 4 5 6], 1) == [2.  3.  4.]
@test mean([1 2 3; 4 5 6], 1) == [2.5 3.5 4.5]
@test mean(i->i+1, 0:2) === 2.
@test mean(isodd, [3]) === 1.
@test mean(x->3x, (1,1)) === 3.

@test isnan(mean([NaN]))
@test isnan(mean([0.0,NaN]))
@test isnan(mean([NaN,0.0]))

@test isnan(mean([0.,Inf,-Inf]))
@test isnan(mean([1.,-1.,Inf,-Inf]))
@test isnan(mean([-Inf,Inf]))
@test isequal(mean([NaN 0.0; 1.2 4.5], 2), reshape([NaN; 2.85], 2, 1))

# test var & std

# edge case: empty vector
# iterable; this has to throw for type stability
@test_throws ArgumentError var(())
@test_throws ArgumentError var((); corrected=false)
@test_throws ArgumentError var((); mean=2)
@test_throws ArgumentError var((); mean=2, corrected=false)
# reduction
@test isnan(var(Int[]))
@test isnan(var(Int[]; corrected=false))
@test isnan(var(Int[]; mean=2))
@test isnan(var(Int[]; mean=2, corrected=false))
# reduction across dimensions
@test isequal(var(Int[], 1), [NaN])
@test isequal(var(Int[], 1; corrected=false), [NaN])
@test isequal(var(Int[], 1; mean=[2]), [NaN])
@test isequal(var(Int[], 1; mean=[2], corrected=false), [NaN])

# edge case: one-element vector
# iterable
@test isnan(@inferred(var((1,))))
@test var((1,); corrected=false) === 0.0
@test var((1,); mean=2) === Inf
@test var((1,); mean=2, corrected=false) === 1.0
# reduction
@test isnan(@inferred(var([1])))
@test var([1]; corrected=false) === 0.0
@test var([1]; mean=2) === Inf
@test var([1]; mean=2, corrected=false) === 1.0
# reduction across dimensions
@test isequal(@inferred(var([1], 1)), [NaN])
@test var([1], 1; corrected=false) ≈ [0.0]
@test var([1], 1; mean=[2]) ≈ [Inf]
@test var([1], 1; mean=[2], corrected=false) ≈ [1.0]

@test var(1:8) == 6.
@test varm(1:8,1) == varm(collect(1:8),1)
@test isnan(varm(1:1,1))
@test isnan(var(1:1))
@test isnan(var(1:-1))

@test varm([1,2,3], 2) ≈ 1.
@test var([1,2,3]) ≈ 1.
@test var([1,2,3]; corrected=false) ≈ 2.0/3
@test var([1,2,3]; mean=0) ≈ 7.
@test var([1,2,3]; mean=0, corrected=false) ≈ 14.0/3

@test varm((1,2,3), 2) ≈ 1.
@test var((1,2,3)) ≈ 1.
@test var((1,2,3); corrected=false) ≈ 2.0/3
@test var((1,2,3); mean=0) ≈ 7.
@test var((1,2,3); mean=0, corrected=false) ≈ 14.0/3
@test_throws ArgumentError var((1,2,3); mean=())

@test var([1 2 3 4 5; 6 7 8 9 10], 2) ≈ [2.5 2.5]'
@test var([1 2 3 4 5; 6 7 8 9 10], 2; corrected=false) ≈ [2.0 2.0]'

@test stdm([1,2,3], 2) ≈ 1.
@test std([1,2,3]) ≈ 1.
@test std([1,2,3]; corrected=false) ≈ sqrt(2.0/3)
@test std([1,2,3]; mean=0) ≈ sqrt(7.0)
@test std([1,2,3]; mean=0, corrected=false) ≈ sqrt(14.0/3)

@test stdm((1,2,3), 2) ≈ 1.
@test std((1,2,3)) ≈ 1.
@test std((1,2,3); corrected=false) ≈ sqrt(2.0/3)
@test std((1,2,3); mean=0) ≈ sqrt(7.0)
@test std((1,2,3); mean=0, corrected=false) ≈ sqrt(14.0/3)

@test std([1 2 3 4 5; 6 7 8 9 10], 2) ≈ sqrt.([2.5 2.5]')
@test std([1 2 3 4 5; 6 7 8 9 10], 2; corrected=false) ≈ sqrt.([2.0 2.0]')

A = Complex128[exp(i*im) for i in 1:10^4]
@test varm(A,0.) ≈ sum(map(abs2,A))/(length(A)-1)
@test varm(A,mean(A)) ≈ var(A)

# test covariance

function safe_cov(x, y, zm::Bool, cr::Bool)
    n = length(x)
    if !zm
        x = x .- mean(x)
        y = y .- mean(y)
    end
    dot(vec(x), vec(y)) / (n - Int(cr))
end

X = [1. 2. 3. 4. 5.; 5. 4. 6. 2. 1.]'
Y = [6. 1. 5. 3. 2.; 2. 7. 8. 4. 3.]'

for vd in [1, 2], zm in [true, false], cr in [true, false]
    # println("vd = $vd: zm = $zm, cr = $cr")
    if vd == 1
        k = size(X, 2)
        Cxx = zeros(k, k)
        Cxy = zeros(k, k)
        for i = 1:k, j = 1:k
            Cxx[i,j] = safe_cov(X[:,i], X[:,j], zm, cr)
            Cxy[i,j] = safe_cov(X[:,i], Y[:,j], zm, cr)
        end
        x1 = vec(X[:,1])
        y1 = vec(Y[:,1])
    else
        k = size(X, 1)
        Cxx = zeros(k, k)
        Cxy = zeros(k, k)
        for i = 1:k, j = 1:k
            Cxx[i,j] = safe_cov(X[i,:], X[j,:], zm, cr)
            Cxy[i,j] = safe_cov(X[i,:], Y[j,:], zm, cr)
        end
        x1 = vec(X[1,:])
        y1 = vec(Y[1,:])
    end

    c = zm ? Base.covm(x1, 0, cr) :
             cov(x1, cr)
    @test isa(c, Float64)
    @test c ≈ Cxx[1,1]
    @inferred cov(x1, cr)

    @test cov(X) == Base.covm(X, mean(X, 1))
    C = zm ? Base.covm(X, 0, vd, cr) :
             cov(X, vd, cr)
    @test size(C) == (k, k)
    @test C ≈ Cxx
    @inferred cov(X, vd, cr)

    @test cov(x1, y1) == Base.covm(x1, mean(x1), y1, mean(y1))
    c = zm ? Base.covm(x1, 0, y1, 0, cr) :
             cov(x1, y1, cr)
    @test isa(c, Float64)
    @test c ≈ Cxy[1,1]
    @inferred cov(x1, y1, cr)

    if vd == 1
        @test cov(x1, Y) == Base.covm(x1, mean(x1), Y, mean(Y, 1))
    end
    C = zm ? Base.covm(x1, 0, Y, 0, vd, cr) :
             cov(x1, Y, vd, cr)
    @test size(C) == (1, k)
    @test vec(C) ≈ Cxy[1,:]
    @inferred cov(x1, Y, vd, cr)

    if vd == 1
        @test cov(X, y1) == Base.covm(X, mean(X, 1), y1, mean(y1))
    end
    C = zm ? Base.covm(X, 0, y1, 0, vd, cr) :
             cov(X, y1, vd, cr)
    @test size(C) == (k, 1)
    @test vec(C) ≈ Cxy[:,1]
    @inferred cov(X, y1, vd, cr)

    @test cov(X, Y) == Base.covm(X, mean(X, 1), Y, mean(Y, 1))
    C = zm ? Base.covm(X, 0, Y, 0, vd, cr) :
             cov(X, Y, vd, cr)
    @test size(C) == (k, k)
    @test C ≈ Cxy
    @inferred cov(X, Y, vd, cr)
end

# test correlation

function safe_cor(x, y, zm::Bool)
    if !zm
        x = x .- mean(x)
        y = y .- mean(y)
    end
    x = vec(x)
    y = vec(y)
    dot(x, y) / (sqrt(dot(x, x)) * sqrt(dot(y, y)))
end

for vd in [1, 2], zm in [true, false]
    # println("vd = $vd: zm = $zm")
    if vd == 1
        k = size(X, 2)
        Cxx = zeros(k, k)
        Cxy = zeros(k, k)
        for i = 1:k, j = 1:k
            Cxx[i,j] = safe_cor(X[:,i], X[:,j], zm)
            Cxy[i,j] = safe_cor(X[:,i], Y[:,j], zm)
        end
        x1 = vec(X[:,1])
        y1 = vec(Y[:,1])
    else
        k = size(X, 1)
        Cxx = zeros(k, k)
        Cxy = zeros(k, k)
        for i = 1:k, j = 1:k
            Cxx[i,j] = safe_cor(X[i,:], X[j,:], zm)
            Cxy[i,j] = safe_cor(X[i,:], Y[j,:], zm)
        end
        x1 = vec(X[1,:])
        y1 = vec(Y[1,:])
    end

    c = zm ? Base.corm(x1, 0) : cor(x1)
    @test isa(c, Float64)
    @test c ≈ Cxx[1,1]
    @inferred cor(x1)

    @test cor(X) == Base.corm(X, mean(X, 1))
    C = zm ? Base.corm(X, 0, vd) : cor(X, vd)
    @test size(C) == (k, k)
    @test C ≈ Cxx
    @inferred cor(X, vd)

    @test cor(x1, y1) == Base.corm(x1, mean(x1), y1, mean(y1))
    c = zm ? Base.corm(x1, 0, y1, 0) : cor(x1, y1)
    @test isa(c, Float64)
    @test c ≈ Cxy[1,1]
    @inferred cor(x1, y1)

    if vd == 1
        @test cor(x1, Y) == Base.corm(x1, mean(x1), Y, mean(Y, 1))
    end
    C = zm ? Base.corm(x1, 0, Y, 0, vd) : cor(x1, Y, vd)
    @test size(C) == (1, k)
    @test vec(C) ≈ Cxy[1,:]
    @inferred cor(x1, Y, vd)

    if vd == 1
        @test cor(X, y1) == Base.corm(X, mean(X, 1), y1, mean(y1))
    end
    C = zm ? Base.corm(X, 0, y1, 0, vd) : cor(X, y1, vd)
    @test size(C) == (k, 1)
    @test vec(C) ≈ Cxy[:,1]
    @inferred cor(X, y1, vd)

    @test cor(X, Y) == Base.corm(X, mean(X, 1), Y, mean(Y, 1))
    C = zm ? Base.corm(X, 0, Y, 0, vd) : cor(X, Y, vd)
    @test size(C) == (k, k)
    @test C ≈ Cxy
    @inferred cor(X, Y, vd)
end

@test cor(repmat(1:17, 1, 17))[2] <= 1.0
@test cor(1:17, 1:17) <= 1.0
@test cor(1:17, 18:34) <= 1.0
let tmp = linspace(1, 85, 100)
    tmp2 = collect(tmp)
    @test cor(tmp, tmp) <= 1.0
    @test cor(tmp, tmp2) <= 1.0
end

@test quantile([1,2,3,4],0.5) == 2.5
@test quantile([1,2,3,4],[0.5]) == [2.5]
@test quantile([1., 3],[.25,.5,.75])[2] == median([1., 3])
@test quantile(100.0:-1.0:0.0, 0.0:0.1:1.0) == collect(0.0:10.0:100.0)
@test quantile(0.0:100.0, 0.0:0.1:1.0, sorted=true) == collect(0.0:10.0:100.0)
@test quantile(100f0:-1f0:0.0, 0.0:0.1:1.0) == collect(0f0:10f0:100f0)
@test quantile([Inf,Inf],0.5) == Inf
@test quantile([-Inf,1],0.5) == -Inf
@test quantile([0,1],1e-18) == 1e-18

# StatsBase issue 164
y = [0.40003674665581906,0.4085630862624367,0.41662034698690303,0.41662034698690303,0.42189053966652057,0.42189053966652057,0.42553514344518345,0.43985732442991354]
@test issorted(quantile(y, linspace(0.01, 0.99, 17)))

# variance of complex arrays (#13309)
let z = rand(Complex128, 10)
    @test var(z) ≈ invoke(var, Tuple{Any}, z) ≈ cov(z) ≈ var(z,1)[1] ≈ sum(abs2, z - mean(z))/9
    @test isa(var(z), Float64)
    @test isa(invoke(var, Tuple{Any}, z), Float64)
    @test isa(cov(z), Float64)
    @test isa(var(z,1), Vector{Float64})
    @test varm(z, 0.0) ≈ invoke(varm, Tuple{Any,Float64}, z, 0.0) ≈ sum(abs2, z)/9
    @test isa(varm(z, 0.0), Float64)
    @test isa(invoke(varm, Tuple{Any,Float64}, z, 0.0), Float64)
    @test cor(z) === 1.0
end
let v = varm([1.0+2.0im], 0; corrected = false)
    @test v ≈ 5
    @test isa(v, Float64)
end

# Issue #17153 and PR #17154
let a = rand(10,10)
    b = deepcopy(a)
    x = median(a, 1)
    @test b == a
    x = median(a, 2)
    @test b == a
    x = mean(a, 1)
    @test b == a
    x = mean(a, 2)
    @test b == a
    x = var(a, 1)
    @test b == a
    x = var(a, 2)
    @test b == a
    x = std(a, 1)
    @test b == a
    x = std(a, 2)
    @test b == a
end