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statistics.jl
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statistics.jl
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##### mean #####
function mean(iterable)
state = start(iterable)
if done(iterable, state)
error("mean of empty collection undefined: $(repr(iterable))")
end
count = 1
total, state = next(iterable, state)
while !done(iterable, state)
value, state = next(iterable, state)
total += value
count += 1
end
return total/count
end
mean(A::AbstractArray) = sum(A) / length(A)
function mean!{T}(R::AbstractArray{T}, A::AbstractArray)
sum!(R, A; init=true)
lenR = length(R)
rs = convert(T, length(A) / lenR)
if rs != 1
for i = 1:lenR
@inbounds R[i] /= rs
end
end
return R
end
momenttype{T}(::Type{T}) = typeof((zero(T) + zero(T)) / 2)
momenttype(::Type{Float32}) = Float32
momenttype{T<:Union(Float64,Int32,Int64,UInt32,UInt64)}(::Type{T}) = Float64
mean{T}(A::AbstractArray{T}, region) =
mean!(Array(momenttype(T), reduced_dims(size(A), region)), A)
##### variances #####
function var(iterable; corrected::Bool=true, mean=nothing)
state = start(iterable)
if done(iterable, state)
error("variance of empty collection undefined: $(repr(iterable))")
end
count = 1
value, state = next(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 = zero(M)
while !done(iterable, state)
value, state = next(iterable, state)
count += 1
new_M = M + (value - M) / count
S = S + (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 = (value - mean::Number)^2
while !done(iterable, state)
value, state = next(iterable, state)
count += 1
sum2 += (value - mean)^2
end
return sum2 / (count - int(corrected))
else
error("invalid value of mean")
end
end
function varzm{T}(A::AbstractArray{T}; corrected::Bool=true)
n = length(A)
n == 0 && return convert(momenttype(T), NaN)
return sumabs2(A) / (n - int(corrected))
end
function varzm!{S}(R::AbstractArray{S}, A::AbstractArray; corrected::Bool=true)
if isempty(A)
fill!(R, convert(S, NaN))
else
rn = div(length(A), length(r)) - int(corrected)
scale!(sumabs2!(R, A; init=true), convert(S, 1/rn))
end
return R
end
varzm{T}(A::AbstractArray{T}, region; corrected::Bool=true) =
varzm!(Array(momenttype(T), reduced_dims(A, region)), A; corrected=corrected)
immutable CentralizedAbs2Fun{T<:Number} <: Func{1}
m::T
end
call(f::CentralizedAbs2Fun, x) = abs2(x - f.m)
centralize_sumabs2(A::AbstractArray, m::Number, ifirst::Int, ilast::Int) =
mapreduce_impl(CentralizedAbs2Fun(m), AddFun(), A, ifirst, ilast)
@ngenerate N typeof(R) function centralize_sumabs2!{S,T,N}(R::AbstractArray{S}, A::AbstractArray{T,N}, means::AbstractArray)
# following the implementation of _mapreducedim! at base/reducedim.jl
lsiz = check_reducdims(R, A)
isempty(R) || fill!(R, zero(S))
isempty(A) && return R
@nextract N sizeR d->size(R,d)
sizA1 = size(A, 1)
if has_fast_linear_indexing(A) && lsiz > 16
# use centralize_sumabs2, which is probably better tuned to achieve higher performance
nslices = div(length(A), lsiz)
ibase = 0
for i = 1:nslices
@inbounds R[i] = centralize_sumabs2(A, means[i], ibase+1, ibase+lsiz)
ibase += lsiz
end
elseif size(R, 1) == 1 && sizA1 > 1
# keep the accumulator as a local variable when reducing along the first dimension
@nloops N i d->(d>1? (1:size(A,d)) : (1:1)) d->(j_d = sizeR_d==1 ? 1 : i_d) begin
@inbounds r = (@nref N R j)
@inbounds m = (@nref N means j)
for i_1 = 1:sizA1
@inbounds r += abs2((@nref N A i) - m)
end
@inbounds (@nref N R j) = r
end
else
# general implementation
@nloops N i A d->(j_d = sizeR_d==1 ? 1 : i_d) begin
@inbounds (@nref N R j) += abs2((@nref N A i) - (@nref N means j))
end
end
return R
end
function varm{T}(A::AbstractArray{T}, m::Number; corrected::Bool=true)
n = length(A)
n == 0 && return convert(momenttype(T), NaN)
n == 1 && return convert(momenttype(T), abs2(A[1] - m)/(1 - int(corrected)))
return centralize_sumabs2(A, m, 1, n) / (n - int(corrected))
end
function varm!{S}(R::AbstractArray{S}, A::AbstractArray, m::AbstractArray; corrected::Bool=true)
if isempty(A)
fill!(R, convert(S, NaN))
else
rn = div(length(A), length(R)) - int(corrected)
scale!(centralize_sumabs2!(R, A, m), convert(S, 1/rn))
end
return R
end
varm{T}(A::AbstractArray{T}, m::AbstractArray, region; corrected::Bool=true) =
varm!(Array(momenttype(T), reduced_dims(size(A), region)), A, m; corrected=corrected)
function var{T}(A::AbstractArray{T}; corrected::Bool=true, mean=nothing)
convert(momenttype(T), mean == 0 ? varzm(A; corrected=corrected) :
mean == nothing ? varm(A, Base.mean(A); corrected=corrected) :
isa(mean, Number) ? varm(A, mean::Number; corrected=corrected) :
error("invalid value of mean"))::momenttype(T)
end
function var(A::AbstractArray, region; corrected::Bool=true, mean=nothing)
mean == 0 ? varzm(A, region; corrected=corrected) :
mean == nothing ? varm(A, Base.mean(A, region), region; corrected=corrected) :
isa(mean, AbstractArray) ? varm(A, mean::AbstractArray, region; corrected=corrected) :
error("invalid value of mean")
end
varm(iterable, m::Number; corrected::Bool=true) =
var(iterable, corrected=corrected, mean=m)
## variances over ranges
varm(v::Range, m::Number) = var(v)
function var(v::Range)
s = step(v)
l = length(v)
if l == 0 || l == 1
return NaN
end
return abs2(s) * (l + 1) * l / 12
end
##### standard deviation #####
function sqrt!(A::AbstractArray)
for i = 1:length(A)
@inbounds A[i] = sqrt(A[i])
end
A
end
stdm(A::AbstractArray, m::Number; corrected::Bool=true) =
sqrt(varm(A, m; corrected=corrected))
std(A::AbstractArray; corrected::Bool=true, mean=nothing) =
sqrt(var(A; corrected=corrected, mean=mean))
std(A::AbstractArray, region; corrected::Bool=true, mean=nothing) =
sqrt!(var(A, region; corrected=corrected, mean=mean))
std(iterable; corrected::Bool=true, mean=nothing) =
sqrt(var(iterable, corrected=corrected, mean=mean))
stdm(iterable, m::Number; corrected::Bool=true) =
std(iterable, corrected=corrected, mean=m)
###### covariance ######
# auxiliary functions
_conj{T<:Real}(x::AbstractArray{T}) = 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, vardim)
# core functions
unscaled_covzm(x::AbstractVector) = dot(x, x)
unscaled_covzm(x::AbstractMatrix, vardim::Int) = (vardim == 1 ? _conj(x'x) : x * x')
unscaled_covzm(x::AbstractVector, y::AbstractVector) = dot(x, y)
unscaled_covzm(x::AbstractVector, y::AbstractMatrix, vardim::Int) =
(vardim == 1 ? At_mul_B(x, _conj(y)) : At_mul_Bt(x, _conj(y)))
unscaled_covzm(x::AbstractMatrix, y::AbstractVector, vardim::Int) =
(c = vardim == 1 ? At_mul_B(x, _conj(y)) : x * _conj(y); reshape(c, length(c), 1))
unscaled_covzm(x::AbstractMatrix, y::AbstractMatrix, vardim::Int) =
(vardim == 1 ? At_mul_B(x, _conj(y)) : A_mul_Bc(x, y))
# covzm (with centered data)
covzm(x::AbstractVector; corrected::Bool=true) = unscaled_covzm(x, x) / (length(x) - int(corrected))
covzm(x::AbstractMatrix; vardim::Int=1, corrected::Bool=true) =
scale!(unscaled_covzm(x, vardim), inv(size(x,vardim) - int(corrected)))
covzm(x::AbstractVector, y::AbstractVector; corrected::Bool=true) =
unscaled_covzm(x, y) / (length(x) - int(corrected))
covzm(x::AbstractVecOrMat, y::AbstractVecOrMat; vardim::Int=1, corrected::Bool=true) =
scale!(unscaled_covzm(x, y, vardim), inv(_getnobs(x, y, vardim) - int(corrected)))
# covm (with provided mean)
covm(x::AbstractVector, xmean; corrected::Bool=true) =
covzm(x .- xmean; corrected=corrected)
covm(x::AbstractMatrix, xmean; vardim::Int=1, corrected::Bool=true) =
covzm(x .- xmean; vardim=vardim, corrected=corrected)
covm(x::AbstractVector, xmean, y::AbstractVector, ymean; corrected::Bool=true) =
covzm(x .- xmean, y .- ymean; corrected=corrected)
covm(x::AbstractVecOrMat, xmean, y::AbstractVecOrMat, ymean; vardim::Int=1, corrected::Bool=true) =
covzm(x .- xmean, y .- ymean; vardim=vardim, corrected=corrected)
# cov (API)
function cov(x::AbstractVector; corrected::Bool=true, mean=nothing)
mean == 0 ? covzm(x; corrected=corrected) :
mean == nothing ? covm(x, Base.mean(x); corrected=corrected) :
isa(mean, Number) ? covm(x, mean; corrected=corrected) :
error("Invalid value of mean.")
end
function cov(x::AbstractMatrix; vardim::Int=1, corrected::Bool=true, mean=nothing)
mean == 0 ? covzm(x; vardim=vardim, corrected=corrected) :
mean == nothing ? covm(x, _vmean(x, vardim); vardim=vardim, corrected=corrected) :
isa(mean, AbstractArray) ? covm(x, mean; vardim=vardim, corrected=corrected) :
error("Invalid value of mean.")
end
function cov(x::AbstractVector, y::AbstractVector; corrected::Bool=true, mean=nothing)
mean == 0 ? covzm(x, y; corrected=corrected) :
mean == nothing ? covm(x, Base.mean(x), y, Base.mean(y); corrected=corrected) :
isa(mean, (Number,Number)) ? covm(x, mean[1], y, mean[2]; corrected=corrected) :
error("Invalid value of mean.")
end
function cov(x::AbstractVecOrMat, y::AbstractVecOrMat; vardim::Int=1, corrected::Bool=true, mean=nothing)
if mean == 0
covzm(x, y; vardim=vardim, corrected=corrected)
elseif mean == nothing
covm(x, _vmean(x, vardim), y, _vmean(y, vardim); vardim=vardim, corrected=corrected)
elseif isa(mean, (Any,Any))
covm(x, mean[1], y, mean[2]; vardim=vardim, corrected=corrected)
else
error("Invalid value of mean.")
end
end
##### correlation #####
# cov2cor!
function cov2cor!{T}(C::AbstractMatrix{T}, xsd::AbstractArray)
nx = length(xsd)
size(C) == (nx, nx) || throw(DimensionMismatch("Inconsistent dimensions."))
for j = 1:nx
for i = 1:j-1
C[i,j] = C[j,i]
end
C[j,j] = one(T)
for i = j+1:nx
C[i,j] /= (xsd[i] * xsd[j])
end
end
return C
end
function cov2cor!(C::AbstractMatrix, xsd::Number, ysd::AbstractArray)
nx, ny = size(C)
length(ysd) == ny || throw(DimensionMismatch("Inconsistent dimensions."))
for j = 1:ny
for i = 1:nx
C[i,j] /= (xsd * ysd[j])
end
end
return C
end
function cov2cor!(C::AbstractMatrix, xsd::AbstractArray, ysd::Number)
nx, ny = size(C)
length(xsd) == nx || throw(DimensionMismatch("Inconsistent dimensions."))
for j = 1:ny
for i = 1:nx
C[i,j] /= (xsd[i] * 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 j = 1:ny
for i = 1:nx
C[i,j] /= (xsd[i] * ysd[j])
end
end
return C
end
# corzm (non-exported, with centered data)
corzm{T}(x::AbstractVector{T}) = float(one(T) * one(T))
corzm(x::AbstractMatrix; vardim::Int=1) =
(c = unscaled_covzm(x, vardim); cov2cor!(c, sqrt!(diag(c))))
function corzm(x::AbstractVector, y::AbstractVector)
n = length(x)
length(y) == n || throw(DimensionMismatch("Inconsistent lengths."))
x1 = x[1]
y1 = y[1]
xx = abs2(x1)
yy = abs2(y1)
xy = x1 * conj(y1)
i = 1
while i < n
i += 1
@inbounds xi = x[i]
@inbounds yi = y[i]
xx += abs2(xi)
yy += abs2(yi)
xy += xi * conj(yi)
end
return xy / (sqrt(xx) * sqrt(yy))
end
corzm(x::AbstractVector, y::AbstractMatrix; vardim::Int=1) =
cov2cor!(unscaled_covzm(x, y, vardim), sqrt(sumabs2(x)), sqrt!(sumabs2(y, vardim)))
corzm(x::AbstractMatrix, y::AbstractVector; vardim::Int=1) =
cov2cor!(unscaled_covzm(x, y, vardim), sqrt!(sumabs2(x, vardim)), sqrt(sumabs2(y)))
corzm(x::AbstractMatrix, y::AbstractMatrix; vardim::Int=1) =
cov2cor!(unscaled_covzm(x, y, vardim), sqrt!(sumabs2(x, vardim)), sqrt!(sumabs2(y, vardim)))
# corm
corm(x::AbstractVector, xmean) = corzm(x .- xmean)
corm(x::AbstractMatrix, xmean; vardim::Int=1) = corzm(x .- xmean; vardim=vardim)
corm(x::AbstractVector, xmean, y::AbstractVector, ymean) = corzm(x .- xmean, y .- ymean)
corm(x::AbstractVecOrMat, xmean, y::AbstractVecOrMat, ymean; vardim::Int=1) =
corzm(x .- xmean, y .- ymean; vardim=vardim)
# cor
function cor(x::AbstractVector; mean=nothing)
mean == 0 ? corzm(x) :
mean == nothing ? corm(x, Base.mean(x)) :
isa(mean, Number) ? corm(x, mean) :
error("Invalid value of mean.")
end
function cor(x::AbstractMatrix; vardim::Int=1, mean=nothing)
mean == 0 ? corzm(x; vardim=vardim) :
mean == nothing ? corm(x, _vmean(x, vardim); vardim=vardim) :
isa(mean, AbstractArray) ? corm(x, mean; vardim=vardim) :
error("Invalid value of mean.")
end
function cor(x::AbstractVector, y::AbstractVector; mean=nothing)
mean == 0 ? corzm(x, y) :
mean == nothing ? corm(x, Base.mean(x), y, Base.mean(y)) :
isa(mean, (Number,Number)) ? corm(x, mean[1], y, mean[2]) :
error("Invalid value of mean.")
end
function cor(x::AbstractVecOrMat, y::AbstractVecOrMat; vardim::Int=1, mean=nothing)
if mean == 0
corzm(x, y; vardim=vardim)
elseif mean == nothing
corm(x, _vmean(x, vardim), y, _vmean(y, vardim); vardim=vardim)
elseif isa(mean, (Any,Any))
corm(x, mean[1], y, mean[2]; vardim=vardim)
else
error("Invalid value of mean.")
end
end
##### median & quantiles #####
# Specialized functions for real types allow for improved performance
middle(x::Union(Bool,Int8,Int16,Int32,Int64,Int128,UInt8,UInt16,UInt32,UInt64,UInt128)) = float64(x)
middle(x::FloatingPoint) = x
middle(x::Float16) = float32(x)
middle(x::Real) = (x + zero(x)) / 1
middle(x::Real, y::Real) = (x + y) / 2
middle(a::Range) = middle(a[1], a[end])
middle(a::AbstractArray) = ((v1, v2) = extrema(a); middle(v1, v2))
function median!{T}(v::AbstractVector{T})
isempty(v) && error("median of an empty array is undefined")
if T<:FloatingPoint
@inbounds for x in v
isnan(x) && return x
end
end
n = length(v)
if isodd(n)
return middle(select!(v,div(n+1,2)))
else
m = select!(v, div(n,2):div(n,2)+1)
return middle(m[1], m[2])
end
end
median{T}(v::AbstractArray{T}) = median!(vec(copy(v)))
median{T}(v::AbstractArray{T}, region) = mapslices(median, v, region)
# for now, use the R/S definition of quantile; may want variants later
# see ?quantile in R -- this is type 7
# TODO: need faster implementation (use select!?)
#
function quantile!(v::AbstractVector, q::AbstractVector)
isempty(v) && error("empty data array")
isempty(q) && error("empty quantile array")
# make sure the quantiles are in [0,1]
q = bound_quantiles(q)
lv = length(v)
lq = length(q)
index = 1 .+ (lv-1)*q
lo = floor(Int,index)
hi = ceil(Int,index)
sort!(v)
isnan(v[end]) && error("quantiles are undefined in presence of NaNs")
i = find(index .> lo)
r = float(v[lo])
h = (index.-lo)[i]
r[i] = (1.-h).*r[i] + h.*v[hi[i]]
return r
end
quantile(v::AbstractVector, q::AbstractVector) = quantile!(copy(v),q)
quantile(v::AbstractVector, q::Number) = quantile(v,[q])[1]
function bound_quantiles(qs::AbstractVector)
epsilon = 100*eps()
if (any(qs .< -epsilon) || any(qs .> 1+epsilon))
error("quantiles out of [0,1] range")
end
[min(1,max(0,q)) for q = qs]
end
##### histogram #####
## nice-valued ranges for histograms
function histrange{T<:FloatingPoint,N}(v::AbstractArray{T,N}, n::Integer)
if length(v) == 0
return 0.0:1.0:0.0
end
lo, hi = extrema(v)
if hi == lo
step = 1.0
else
bw = (hi - lo) / n
e = 10.0^floor(log10(bw))
r = bw / e
if r <= 2
step = 2*e
elseif r <= 5
step = 5*e
else
step = 10*e
end
end
start = step*(ceil(lo/step)-1)
nm1 = ceil(Int,(hi - start)/step)
start:step:(start + nm1*step)
end
function histrange{T<:Integer,N}(v::AbstractArray{T,N}, n::Integer)
if length(v) == 0
return 0:1:0
end
lo, hi = extrema(v)
if hi == lo
step = 1
else
bw = (hi - lo) / n
e = 10^max(0,floor(Int,log10(bw)))
r = bw / e
if r <= 1
step = e
elseif r <= 2
step = 2*e
elseif r <= 5
step = 5*e
else
step = 10*e
end
end
start = step*(ceil(lo/step)-1)
nm1 = ceil(Int,(hi - start)/step)
start:step:(start + nm1*step)
end
## midpoints of intervals
midpoints(r::Range) = r[1:length(r)-1] + 0.5*step(r)
midpoints(v::AbstractVector) = [0.5*(v[i] + v[i+1]) for i in 1:length(v)-1]
## hist ##
function sturges(n) # Sturges' formula
n==0 && return one(n)
ceil(Int,log2(n))+1
end
function hist!{HT}(h::AbstractArray{HT}, v::AbstractVector, edg::AbstractVector; init::Bool=true)
n = length(edg) - 1
length(h) == n || error("length(h) must equal length(edg) - 1.")
if init
fill!(h, zero(HT))
end
for x in v
i = searchsortedfirst(edg, x)-1
if 1 <= i <= n
h[i] += 1
end
end
edg, h
end
hist(v::AbstractVector, edg::AbstractVector) = hist!(Array(Int, length(edg)-1), v, edg)
hist(v::AbstractVector, n::Integer) = hist(v,histrange(v,n))
hist(v::AbstractVector) = hist(v,sturges(length(v)))
function hist!{HT}(H::AbstractArray{HT,2}, A::AbstractMatrix, edg::AbstractVector; init::Bool=true)
m, n = size(A)
size(H) == (length(edg)-1, n) || error("Incorrect size of H.")
if init
fill!(H, zero(HT))
end
for j = 1:n
hist!(sub(H, :, j), sub(A, :, j), edg)
end
edg, H
end
hist(A::AbstractMatrix, edg::AbstractVector) = hist!(Array(Int, length(edg)-1, size(A,2)), A, edg)
hist(A::AbstractMatrix, n::Integer) = hist(A,histrange(A,n))
hist(A::AbstractMatrix) = hist(A,sturges(size(A,1)))
## hist2d
function hist2d!{HT}(H::AbstractArray{HT,2}, v::AbstractMatrix,
edg1::AbstractVector, edg2::AbstractVector; init::Bool=true)
size(v,2) == 2 || error("hist2d requires an Nx2 matrix.")
n = length(edg1) - 1
m = length(edg2) - 1
size(H) == (n, m) || error("Incorrect size of H.")
if init
fill!(H, zero(HT))
end
for i = 1:size(v,1)
x = searchsortedfirst(edg1, v[i,1]) - 1
y = searchsortedfirst(edg2, v[i,2]) - 1
if 1 <= x <= n && 1 <= y <= m
@inbounds H[x,y] += 1
end
end
edg1, edg2, H
end
hist2d(v::AbstractMatrix, edg1::AbstractVector, edg2::AbstractVector) =
hist2d!(Array(Int, length(edg1)-1, length(edg2)-1), v, edg1, edg2)
hist2d(v::AbstractMatrix, edg::AbstractVector) = hist2d(v, edg, edg)
hist2d(v::AbstractMatrix, n1::Integer, n2::Integer) =
hist2d(v, histrange(sub(v,:,1),n1), histrange(sub(v,:,2),n2))
hist2d(v::AbstractMatrix, n::Integer) = hist2d(v, n, n)
hist2d(v::AbstractMatrix) = hist2d(v, sturges(size(v,1)))