diff --git a/base/statistics.jl b/base/statistics.jl index a5310c3f..7737ee58 100644 --- a/base/statistics.jl +++ b/base/statistics.jl @@ -79,487 +79,6 @@ mean(A::AbstractArray; dims=:) = _mean(A, dims) _mean(A::AbstractArray{T}, region) where {T} = mean!(reducedim_init(t -> t/2, +, A, region), A) _mean(A::AbstractArray, ::Colon) = sum(A) / _length(A) -##### variances ##### - -# faster computation of real(conj(x)*y) -realXcY(x::Real, y::Real) = x*y -realXcY(x::Complex, y::Complex) = real(x)*real(y) + imag(x)*imag(y) - -var(iterable; corrected::Bool=true, mean=nothing) = _var(iterable, corrected, mean) - -function _var(iterable, corrected::Bool, mean) - y = iterate(iterable) - if y === nothing - throw(ArgumentError("variance of empty collection undefined: $(repr(iterable))")) - end - count = 1 - value, state = y - y = iterate(iterable, state) - if mean === nothing - # Use Welford algorithm as seen in (among other places) - # Knuth's TAOCP, Vol 2, page 232, 3rd edition. - M = value / 1 - S = real(zero(M)) - while y !== nothing - value, state = y - y = iterate(iterable, state) - count += 1 - new_M = M + (value - M) / count - S = S + realXcY(value - M, value - new_M) - M = new_M - end - return S / (count - Int(corrected)) - elseif isa(mean, Number) # mean provided - # Cannot use a compensated version, e.g. the one from - # "Updating Formulae and a Pairwise Algorithm for Computing Sample Variances." - # by Chan, Golub, and LeVeque, Technical Report STAN-CS-79-773, - # Department of Computer Science, Stanford University, - # because user can provide mean value that is different to mean(iterable) - sum2 = abs2(value - mean::Number) - while y !== nothing - value, state = y - y = iterate(iterable, state) - count += 1 - sum2 += abs2(value - mean) - end - return sum2 / (count - Int(corrected)) - else - throw(ArgumentError("invalid value of mean, $(mean)::$(typeof(mean))")) - end -end - -centralizedabs2fun(m) = x -> abs2.(x - m) -centralize_sumabs2(A::AbstractArray, m) = - mapreduce(centralizedabs2fun(m), +, A) -centralize_sumabs2(A::AbstractArray, m, ifirst::Int, ilast::Int) = - mapreduce_impl(centralizedabs2fun(m), +, A, ifirst, ilast) - -function centralize_sumabs2!(R::AbstractArray{S}, A::AbstractArray, means::AbstractArray) where S - # following the implementation of _mapreducedim! at base/reducedim.jl - lsiz = check_reducedims(R,A) - isempty(R) || fill!(R, zero(S)) - isempty(A) && return R - - if has_fast_linear_indexing(A) && lsiz > 16 - nslices = div(_length(A), lsiz) - ibase = first(LinearIndices(A))-1 - for i = 1:nslices - @inbounds R[i] = centralize_sumabs2(A, means[i], ibase+1, ibase+lsiz) - ibase += lsiz - end - return R - end - indsAt, indsRt = safe_tail(axes(A)), safe_tail(axes(R)) # handle d=1 manually - keep, Idefault = Broadcast.shapeindexer(indsRt) - if reducedim1(R, A) - i1 = first(indices1(R)) - @inbounds for IA in CartesianIndices(indsAt) - IR = Broadcast.newindex(IA, keep, Idefault) - r = R[i1,IR] - m = means[i1,IR] - @simd for i in axes(A, 1) - r += abs2(A[i,IA] - m) - end - R[i1,IR] = r - end - else - @inbounds for IA in CartesianIndices(indsAt) - IR = Broadcast.newindex(IA, keep, Idefault) - @simd for i in axes(A, 1) - R[i,IR] += abs2(A[i,IA] - means[i,IR]) - end - end - end - return R -end - -function varm!(R::AbstractArray{S}, A::AbstractArray, m::AbstractArray; corrected::Bool=true) where S - if isempty(A) - fill!(R, convert(S, NaN)) - else - rn = div(_length(A), _length(R)) - Int(corrected) - centralize_sumabs2!(R, A, m) - R .= R .* (1 // rn) - end - return R -end - -""" - varm(v, m; dims, corrected::Bool=true) - -Compute the sample variance of a collection `v` with known mean(s) `m`, -optionally over the given dimensions. `m` may contain means for each dimension of -`v`. If `corrected` is `true`, then the sum is scaled with `n-1`, -whereas the sum is scaled with `n` if `corrected` is `false` where `n = length(x)`. - -!!! note - Julia does not ignore `NaN` values in the computation. Use the [`missing`](@ref) type - to represent missing values, and the [`skipmissing`](@ref) function to omit them. -""" -varm(A::AbstractArray, m::AbstractArray; corrected::Bool=true, dims=:) = _varm(A, m, corrected, dims) - -_varm(A::AbstractArray{T}, m, corrected::Bool, region) where {T} = - varm!(reducedim_init(t -> abs2(t)/2, +, A, region), A, m; corrected=corrected) - -varm(A::AbstractArray, m; corrected::Bool=true) = _varm(A, m, corrected, :) - -function _varm(A::AbstractArray{T}, m, corrected::Bool, ::Colon) where T - n = _length(A) - n == 0 && return typeof((abs2(zero(T)) + abs2(zero(T)))/2)(NaN) - return centralize_sumabs2(A, m) / (n - Int(corrected)) -end - - -""" - var(v; dims, corrected::Bool=true, mean=nothing) - -Compute the sample variance of a vector or array `v`, optionally along the given dimensions. -The algorithm will return an estimator of the generative distribution's variance -under the assumption that each entry of `v` is an IID drawn from that generative -distribution. This computation is equivalent to calculating `sum(abs2, v - mean(v)) / -(length(v) - 1)`. If `corrected` is `true`, then the sum is scaled with `n-1`, -whereas the sum is scaled with `n` if `corrected` is `false` where `n = length(x)`. -The mean `mean` over the region may be provided. - -!!! note - Julia does not ignore `NaN` values in the computation. Use the [`missing`](@ref) type - to represent missing values, and the [`skipmissing`](@ref) function to omit them. -""" -var(A::AbstractArray; corrected::Bool=true, mean=nothing, dims=:) = _var(A, corrected, mean, dims) - -_var(A::AbstractArray, corrected::Bool, mean, dims) = - varm(A, coalesce(mean, Base.mean(A, dims=dims)); corrected=corrected, dims=dims) - -_var(A::AbstractArray, corrected::Bool, mean, ::Colon) = - real(varm(A, coalesce(mean, Base.mean(A)); corrected=corrected)) - -varm(iterable, m; corrected::Bool=true) = _var(iterable, corrected, m) - -## variances over ranges - -varm(v::AbstractRange, m::AbstractArray) = range_varm(v, m) -varm(v::AbstractRange, m) = range_varm(v, m) - -function range_varm(v::AbstractRange, m) - f = first(v) - m - s = step(v) - l = length(v) - vv = f^2 * l / (l - 1) + f * s * l + s^2 * l * (2 * l - 1) / 6 - if l == 0 || l == 1 - return typeof(vv)(NaN) - end - return vv -end - -function var(v::AbstractRange) - s = step(v) - l = length(v) - vv = abs2(s) * (l + 1) * l / 12 - if l == 0 || l == 1 - return typeof(vv)(NaN) - end - return vv -end - - -##### standard deviation ##### - -function sqrt!(A::AbstractArray) - for i in eachindex(A) - @inbounds A[i] = sqrt(A[i]) - end - A -end - -stdm(A::AbstractArray, m; corrected::Bool=true) = - sqrt.(varm(A, m; corrected=corrected)) - -""" - std(v; corrected::Bool=true, mean=nothing, dims) - -Compute the sample standard deviation of a vector or array `v`, optionally along the given -dimensions. The algorithm returns an estimator of the generative distribution's standard -deviation under the assumption that each entry of `v` is an IID drawn from that generative -distribution. This computation is equivalent to calculating `sqrt(sum((v - mean(v)).^2) / -(length(v) - 1))`. A pre-computed `mean` may be provided. If `corrected` is `true`, -then the sum is scaled with `n-1`, whereas the sum is scaled with `n` if `corrected` is -`false` where `n = length(x)`. - -!!! note - Julia does not ignore `NaN` values in the computation. Use the [`missing`](@ref) type - to represent missing values, and the [`skipmissing`](@ref) function to omit them. -""" -std(A::AbstractArray; corrected::Bool=true, mean=nothing, dims=:) = _std(A, corrected, mean, dims) - -_std(A::AbstractArray, corrected::Bool, mean, dims) = - sqrt.(var(A; corrected=corrected, mean=mean, dims=dims)) - -_std(A::AbstractArray, corrected::Bool, mean, ::Colon) = - sqrt.(var(A; corrected=corrected, mean=mean)) - -_std(A::AbstractArray{<:AbstractFloat}, corrected::Bool, mean, dims) = - sqrt!(var(A; corrected=corrected, mean=mean, dims=dims)) - -_std(A::AbstractArray{<:AbstractFloat}, corrected::Bool, mean, ::Colon) = - sqrt.(var(A; corrected=corrected, mean=mean)) - -std(iterable; corrected::Bool=true, mean=nothing) = - sqrt(var(iterable, corrected=corrected, mean=mean)) - -""" - stdm(v, m; corrected::Bool=true) - -Compute the sample standard deviation of a vector `v` -with known mean `m`. If `corrected` is `true`, -then the sum is scaled with `n-1`, whereas the sum is -scaled with `n` if `corrected` is `false` where `n = length(x)`. - -!!! note - Julia does not ignore `NaN` values in the computation. Use the [`missing`](@ref) type - to represent missing values, and the [`skipmissing`](@ref) function to omit them. -""" -stdm(iterable, m; corrected::Bool=true) = - std(iterable, corrected=corrected, mean=m) - - -###### covariance ###### - -# auxiliary functions - -_conj(x::AbstractArray{<:Real}) = x -_conj(x::AbstractArray) = conj(x) - -_getnobs(x::AbstractVector, vardim::Int) = _length(x) -_getnobs(x::AbstractMatrix, vardim::Int) = size(x, vardim) - -function _getnobs(x::AbstractVecOrMat, y::AbstractVecOrMat, vardim::Int) - n = _getnobs(x, vardim) - _getnobs(y, vardim) == n || throw(DimensionMismatch("dimensions of x and y mismatch")) - return n -end - -_vmean(x::AbstractVector, vardim::Int) = mean(x) -_vmean(x::AbstractMatrix, vardim::Int) = mean(x, dims=vardim) - -# core functions - -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) = sum(conj(y[i])*x[i] for i in eachindex(y, x)) -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) = - (c = vardim == 1 ? *(transpose(x), _conj(y)) : x * _conj(y); reshape(c, length(c), 1)) -unscaled_covzm(x::AbstractMatrix, y::AbstractMatrix, vardim::Int) = - (vardim == 1 ? *(transpose(x), _conj(y)) : *(x, adjoint(y))) - -# covzm (with centered data) - -covzm(x::AbstractVector; corrected::Bool=true) = unscaled_covzm(x) / (_length(x) - Int(corrected)) -function covzm(x::AbstractMatrix, vardim::Int=1; corrected::Bool=true) - C = unscaled_covzm(x, vardim) - T = promote_type(typeof(first(C) / 1), eltype(C)) - A = convert(AbstractMatrix{T}, C) - b = 1//(size(x, vardim) - corrected) - A .= A .* b - return A -end -covzm(x::AbstractVector, y::AbstractVector; corrected::Bool=true) = - unscaled_covzm(x, y) / (_length(x) - Int(corrected)) -function covzm(x::AbstractVecOrMat, y::AbstractVecOrMat, vardim::Int=1; corrected::Bool=true) - C = unscaled_covzm(x, y, vardim) - T = promote_type(typeof(first(C) / 1), eltype(C)) - A = convert(AbstractArray{T}, C) - b = 1//(_getnobs(x, y, vardim) - corrected) - A .= A .* b - return A -end - -# covm (with provided mean) -## Use map(t -> t - xmean, x) instead of x .- xmean to allow for Vector{Vector} -## which can't be handled by broadcast -covm(x::AbstractVector, xmean; corrected::Bool=true) = - covzm(map(t -> t - xmean, x); corrected=corrected) -covm(x::AbstractMatrix, xmean, vardim::Int=1; corrected::Bool=true) = - covzm(x .- xmean, vardim; corrected=corrected) -covm(x::AbstractVector, xmean, y::AbstractVector, ymean; corrected::Bool=true) = - covzm(map(t -> t - xmean, x), map(t -> t - ymean, y); corrected=corrected) -covm(x::AbstractVecOrMat, xmean, y::AbstractVecOrMat, ymean, vardim::Int=1; corrected::Bool=true) = - covzm(x .- xmean, y .- ymean, vardim; corrected=corrected) - -# cov (API) -""" - cov(x::AbstractVector; corrected::Bool=true) - -Compute the variance of the vector `x`. If `corrected` is `true` (the default) then the sum -is scaled with `n-1`, whereas the sum is scaled with `n` if `corrected` is `false` where `n = length(x)`. -""" -cov(x::AbstractVector; corrected::Bool=true) = covm(x, Base.mean(x); corrected=corrected) - -""" - cov(X::AbstractMatrix; dims::Int=1, corrected::Bool=true) - -Compute the covariance matrix of the matrix `X` along the dimension `dims`. If `corrected` -is `true` (the default) then the sum is scaled with `n-1`, whereas the sum is scaled with `n` -if `corrected` is `false` where `n = size(X, dims)`. -""" -cov(X::AbstractMatrix; dims::Int=1, corrected::Bool=true) = - covm(X, _vmean(X, dims), dims; corrected=corrected) - -""" - cov(x::AbstractVector, y::AbstractVector; corrected::Bool=true) - -Compute the covariance between the vectors `x` and `y`. If `corrected` is `true` (the -default), computes ``\\frac{1}{n-1}\\sum_{i=1}^n (x_i-\\bar x) (y_i-\\bar y)^*`` where -``*`` denotes the complex conjugate and `n = length(x) = length(y)`. If `corrected` is -`false`, computes ``\\frac{1}{n}\\sum_{i=1}^n (x_i-\\bar x) (y_i-\\bar y)^*``. -""" -cov(x::AbstractVector, y::AbstractVector; corrected::Bool=true) = - covm(x, Base.mean(x), y, Base.mean(y); corrected=corrected) - -""" - cov(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims::Int=1, corrected::Bool=true) - -Compute the covariance between the vectors or matrices `X` and `Y` along the dimension -`dims`. If `corrected` is `true` (the default) then the sum is scaled with `n-1`, whereas -the sum is scaled with `n` if `corrected` is `false` where `n = size(X, dims) = size(Y, dims)`. -""" -cov(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims::Int=1, corrected::Bool=true) = - covm(X, _vmean(X, dims), Y, _vmean(Y, dims), dims; corrected=corrected) - -##### correlation ##### - -""" - clampcor(x) - -Clamp a real correlation to between -1 and 1, leaving complex correlations unchanged -""" -clampcor(x::Real) = clamp(x, -1, 1) -clampcor(x) = x - -# cov2cor! - -function cov2cor!(C::AbstractMatrix{T}, xsd::AbstractArray) where T - nx = length(xsd) - size(C) == (nx, nx) || throw(DimensionMismatch("inconsistent dimensions")) - for j = 1:nx - for i = 1:j-1 - C[i,j] = adjoint(C[j,i]) - end - C[j,j] = oneunit(T) - for i = j+1:nx - C[i,j] = clampcor(C[i,j] / (xsd[i] * xsd[j])) - end - end - return C -end -function cov2cor!(C::AbstractMatrix, xsd, ysd::AbstractArray) - nx, ny = size(C) - length(ysd) == ny || throw(DimensionMismatch("inconsistent dimensions")) - for (j, y) in enumerate(ysd) # fixme (iter): here and in all `cov2cor!` we assume that `C` is efficiently indexed by integers - for i in 1:nx - C[i,j] = clampcor(C[i, j] / (xsd * y)) - end - end - return C -end -function cov2cor!(C::AbstractMatrix, xsd::AbstractArray, ysd) - nx, ny = size(C) - length(xsd) == nx || throw(DimensionMismatch("inconsistent dimensions")) - for j in 1:ny - for (i, x) in enumerate(xsd) - C[i,j] = clampcor(C[i,j] / (x * ysd)) - end - end - return C -end -function cov2cor!(C::AbstractMatrix, xsd::AbstractArray, ysd::AbstractArray) - nx, ny = size(C) - (length(xsd) == nx && length(ysd) == ny) || - throw(DimensionMismatch("inconsistent dimensions")) - for (i, x) in enumerate(xsd) - for (j, y) in enumerate(ysd) - C[i,j] = clampcor(C[i,j] / (x * y)) - end - end - return C -end - -# corzm (non-exported, with centered data) - -corzm(x::AbstractVector{T}) where {T} = one(real(T)) -function corzm(x::AbstractMatrix, vardim::Int=1) - c = unscaled_covzm(x, vardim) - return cov2cor!(c, collect(sqrt(c[i,i]) for i in 1:min(size(c)...))) -end -corzm(x::AbstractVector, y::AbstractMatrix, vardim::Int=1) = - cov2cor!(unscaled_covzm(x, y, vardim), sqrt(sum(abs2, x)), sqrt!(sum(abs2, y, dims=vardim))) -corzm(x::AbstractMatrix, y::AbstractVector, vardim::Int=1) = - cov2cor!(unscaled_covzm(x, y, vardim), sqrt!(sum(abs2, x, dims=vardim)), sqrt(sum(abs2, y))) -corzm(x::AbstractMatrix, y::AbstractMatrix, vardim::Int=1) = - cov2cor!(unscaled_covzm(x, y, vardim), sqrt!(sum(abs2, x, dims=vardim)), sqrt!(sum(abs2, y, dims=vardim))) - -# corm - -corm(x::AbstractVector{T}, xmean) where {T} = one(real(T)) -corm(x::AbstractMatrix, xmean, vardim::Int=1) = corzm(x .- xmean, vardim) -function corm(x::AbstractVector, mx, y::AbstractVector, my) - n = length(x) - length(y) == n || throw(DimensionMismatch("inconsistent lengths")) - n > 0 || throw(ArgumentError("correlation only defined for non-empty vectors")) - - @inbounds begin - # Initialize the accumulators - xx = zero(sqrt(abs2(x[1]))) - yy = zero(sqrt(abs2(y[1]))) - xy = zero(x[1] * y[1]') - - @simd for i in eachindex(x, y) - xi = x[i] - mx - yi = y[i] - my - xx += abs2(xi) - yy += abs2(yi) - xy += xi * yi' - end - end - return clampcor(xy / max(xx, yy) / sqrt(min(xx, yy) / max(xx, yy))) -end - -corm(x::AbstractVecOrMat, xmean, y::AbstractVecOrMat, ymean, vardim::Int=1) = - corzm(x .- xmean, y .- ymean, vardim) - -# cor -""" - cor(x::AbstractVector) - -Return the number one. -""" -cor(x::AbstractVector) = one(real(eltype(x))) - -""" - cor(X::AbstractMatrix; dims::Int=1) - -Compute the Pearson correlation matrix of the matrix `X` along the dimension `dims`. -""" -cor(X::AbstractMatrix; dims::Int=1) = corm(X, _vmean(X, dims), dims) - -""" - cor(x::AbstractVector, y::AbstractVector) - -Compute the Pearson correlation between the vectors `x` and `y`. -""" -cor(x::AbstractVector, y::AbstractVector) = corm(x, Base.mean(x), y, Base.mean(y)) - -""" - cor(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims=1) - -Compute the Pearson correlation between the vectors or matrices `X` and `Y` along the dimension `dims`. -""" -cor(x::AbstractVecOrMat, y::AbstractVecOrMat; dims::Int=1) = - corm(x, _vmean(x, dims), y, _vmean(y, dims), dims) - ##### median & quantiles ##### """