statistics.jl
# This file is a part of Julia. License is MIT: https://julialang.org/license
using 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 @inferred(var(1.0:8.0)) == 6.
@test varm(1.0:8.0,1.0) == varm(collect(1.0:8.0),1)
@test isnan(varm(1.0:1.0,1.0))
@test isnan(var(1.0:1.0))
@test isnan(var(1.0:-1.0))
@test @inferred(var(1.0f0:8.0f0)) === 6.f0
@test varm(1.0f0:8.0f0,1.0f0) == varm(collect(1.0f0:8.0f0),1)
@test isnan(varm(1.0f0:1.0f0,1.0f0))
@test isnan(var(1.0f0:1.0f0))
@test isnan(var(1.0f0:-1.0f0))
@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]')
let 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)
end
# 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, corrected=cr) :
cov(x1, corrected=cr)
@test isa(c, Float64)
@test c ≈ Cxx[1,1]
@inferred cov(x1, corrected=cr)
@test cov(X) == Base.covm(X, mean(X, 1))
C = zm ? Base.covm(X, 0, vd, corrected=cr) :
cov(X, vd, corrected=cr)
@test size(C) == (k, k)
@test C ≈ Cxx
@inferred cov(X, vd, corrected=cr)
@test cov(x1, y1) == Base.covm(x1, mean(x1), y1, mean(y1))
c = zm ? Base.covm(x1, 0, y1, 0, corrected=cr) :
cov(x1, y1, corrected=cr)
@test isa(c, Float64)
@test c ≈ Cxy[1,1]
@inferred cov(x1, y1, corrected=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, corrected=cr) :
cov(x1, Y, vd, corrected=cr)
@test size(C) == (1, k)
@test vec(C) ≈ Cxy[1,:]
@inferred cov(x1, Y, vd, corrected=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, corrected=cr) :
cov(X, y1, vd, corrected=cr)
@test size(C) == (k, 1)
@test vec(C) ≈ Cxy[:,1]
@inferred cov(X, y1, vd, corrected=cr)
@test cov(X, Y) == Base.covm(X, mean(X, 1), Y, mean(Y, 1))
C = zm ? Base.covm(X, 0, Y, 0, vd, corrected=cr) :
cov(X, Y, vd, corrected=cr)
@test size(C) == (k, k)
@test C ≈ Cxy
@inferred cov(X, Y, vd, corrected=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
@test quantile([1, 2, 3, 4],[]) == []
@test quantile([1, 2, 3, 4], (0.5,)) == (2.5,)
@test quantile([4, 9, 1, 5, 7, 8, 2, 3, 5, 17, 11], (0.1, 0.2, 0.4, 0.9)) == (2.0, 3.0, 5.0, 11.0)
@test quantile([1, 2, 3, 4], ()) == ()
# StatsBase issue 164
let 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)))
end
# 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
# cov and cor of complex arrays (issue #21093)
let x = [2.7 - 3.3im, 0.9 + 5.4im, 0.1 + 0.2im, -1.7 - 5.8im, 1.1 + 1.9im],
y = [-1.7 - 1.6im, -0.2 + 6.5im, 0.8 - 10.0im, 9.1 - 3.4im, 2.7 - 5.5im]
@test cov(x, y) ≈ 4.8365 - 12.119im
@test cov(y, x) ≈ 4.8365 + 12.119im
@test cov(x, reshape(y, :, 1)) ≈ reshape([4.8365 - 12.119im], 1, 1)
@test cov(reshape(x, :, 1), y) ≈ reshape([4.8365 - 12.119im], 1, 1)
@test cov(reshape(x, :, 1), reshape(y, :, 1)) ≈ reshape([4.8365 - 12.119im], 1, 1)
@test cov([x y]) ≈ [21.779 4.8365-12.119im;
4.8365+12.119im 54.548]
@test cor(x, y) ≈ 0.14032104449218274 - 0.35160772008699703im
@test cor(y, x) ≈ 0.14032104449218274 + 0.35160772008699703im
@test cor(x, reshape(y, :, 1)) ≈ reshape([0.14032104449218274 - 0.35160772008699703im], 1, 1)
@test cor(reshape(x, :, 1), y) ≈ reshape([0.14032104449218274 - 0.35160772008699703im], 1, 1)
@test cor(reshape(x, :, 1), reshape(y, :, 1)) ≈ reshape([0.14032104449218274 - 0.35160772008699703im], 1, 1)
@test cor([x y]) ≈ [1.0 0.14032104449218274-0.35160772008699703im
0.14032104449218274+0.35160772008699703im 1.0]
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
# dimensional correctness
isdefined(Main, :TestHelpers) || @eval Main include("TestHelpers.jl")
using Main.TestHelpers.Furlong
@testset "Unitful elements" begin
r = Furlong(1):Furlong(1):Furlong(2)
a = collect(r)
@test sum(r) == sum(a) == Furlong(3)
@test cumsum(r) == Furlong.([1,3])
@test mean(r) == mean(a) == median(a) == median(r) == Furlong(1.5)
@test var(r) == var(a) == Furlong{2}(0.5)
@test std(r) == std(a) == Furlong{1}(sqrt(0.5))
# Issue #21786
A = [Furlong{1}(rand(-5:5)) for i in 1:2, j in 1:2]
@test mean(mean(A, 1), 2)[1] === mean(A)
@test var(A, 1)[1] === var(A[:, 1])
@test_broken std(A, 1)[1] === std(A[:, 1])
end
# Issue #22901
@testset "var and quantile of Any arrays" begin
x = Any[1, 2, 4, 10]
y = Any[1, 2, 4, 10//1]
@test var(x) === 16.25
@test var(y) === 65//4
@test std(x) === sqrt(16.25)
@test quantile(x, 0.5) === 3.0
@test quantile(x, 1//2) === 3//1
end
@testset "Promotion in covzm. Issue #8080" begin
A = [1 -1 -1; -1 1 1; -1 1 -1; 1 -1 -1; 1 -1 1]
@test Base.covzm(A) - mean(A, 1)'*mean(A, 1)*size(A, 1)/(size(A, 1) - 1) ≈ cov(A)
A = [1//1 -1 -1; -1 1 1; -1 1 -1; 1 -1 -1; 1 -1 1]
@test (A'A - size(A, 1)*Base.mean(A, 1)'*Base.mean(A, 1))/4 == cov(A)
end
@testset "Mean along dimension of empty array" begin
a0 = zeros(0)
a00 = zeros(0, 0)
a01 = zeros(0, 1)
a10 = zeros(1, 0)
@test isequal(mean(a0, 1) , fill(NaN, 1))
@test isequal(mean(a00, (1, 2)), fill(NaN, 1, 1))
@test isequal(mean(a01, 1) , fill(NaN, 1, 1))
@test isequal(mean(a10, 2) , fill(NaN, 1, 1))
end
@testset "cov/var/std of Vector{Vector}" begin
x = [[2,4,6],[4,6,8]]
@test var(x) ≈ vec(var([x[1] x[2]], 2))
@test std(x) ≈ vec(std([x[1] x[2]], 2))
@test cov(x) ≈ cov([x[1] x[2]], 2)
end
