统计
统计模块包含了基本的统计函数。
Statistics.std
— Functionstd(itr; corrected::Bool=true, mean=nothing[, dims])
Compute the sample standard deviation of collection itr
.
The algorithm returns an estimator of the generative distribution's standard deviation under the assumption that each entry of itr
is a sample drawn from the same unknown distribution, with the samples uncorrelated. For arrays, this computation is equivalent to calculating sqrt(sum((itr .- mean(itr)).^2) / (length(itr) - 1))
. If corrected
is true
, then the sum is scaled with n-1
, whereas the sum is scaled with n
if corrected
is false
with n
the number of elements in itr
.
If itr
is an AbstractArray
, dims
can be provided to compute the standard deviation over dimensions, and means
may contain means for each dimension of itr
.
A pre-computed mean
may be provided. When dims
is specified, mean
must be an array with the same shape as mean(itr, dims=dims)
(additional trailing singleton dimensions are allowed).
If array contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if array contains both). Use the skipmissing
function to omit missing
entries and compute the standard deviation of non-missing values.
Statistics.stdm
— Functionstdm(itr, mean; corrected::Bool=true[, dims])
Compute the sample standard deviation of collection itr
, with known mean(s) mean
.
The algorithm returns an estimator of the generative distribution's standard deviation under the assumption that each entry of itr
is a sample drawn from the same unknown distribution, with the samples uncorrelated. For arrays, this computation is equivalent to calculating sqrt(sum((itr .- mean(itr)).^2) / (length(itr) - 1))
. If corrected
is true
, then the sum is scaled with n-1
, whereas the sum is scaled with n
if corrected
is false
with n
the number of elements in itr
.
If itr
is an AbstractArray
, dims
can be provided to compute the standard deviation over dimensions. In that case, mean
must be an array with the same shape as mean(itr, dims=dims)
(additional trailing singleton dimensions are allowed).
If array contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if array contains both). Use the skipmissing
function to omit missing
entries and compute the standard deviation of non-missing values.
Statistics.var
— Functionvar(itr; corrected::Bool=true, mean=nothing[, dims])
Compute the sample variance of collection itr
.
The algorithm returns an estimator of the generative distribution's variance under the assumption that each entry of itr
is a sample drawn from the same unknown distribution, with the samples uncorrelated. For arrays, this computation is equivalent to calculating sum((itr .- mean(itr)).^2) / (length(itr) - 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
is the number of elements in itr
.
If itr
is an AbstractArray
, dims
can be provided to compute the variance over dimensions.
A pre-computed mean
may be provided. When dims
is specified, mean
must be an array with the same shape as mean(itr, dims=dims)
(additional trailing singleton dimensions are allowed).
If array contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if array contains both). Use the skipmissing
function to omit missing
entries and compute the variance of non-missing values.
Statistics.varm
— Functionvarm(itr, mean; dims, corrected::Bool=true)
Compute the sample variance of collection itr
, with known mean(s) mean
.
The algorithm returns an estimator of the generative distribution's variance under the assumption that each entry of itr
is a sample drawn from the same unknown distribution, with the samples uncorrelated. For arrays, this computation is equivalent to calculating sum((itr .- mean(itr)).^2) / (length(itr) - 1)
. If corrected
is true
, then the sum is scaled with n-1
, whereas the sum is scaled with n
if corrected
is false
with n
the number of elements in itr
.
If itr
is an AbstractArray
, dims
can be provided to compute the variance over dimensions. In that case, mean
must be an array with the same shape as mean(itr, dims=dims)
(additional trailing singleton dimensions are allowed).
If array contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if array contains both). Use the skipmissing
function to omit missing
entries and compute the variance of non-missing values.
Statistics.cor
— Functioncor(x::AbstractVector)
Return the number one.
cor(X::AbstractMatrix; dims::Int=1)
Compute the Pearson correlation matrix of the matrix X
along the dimension dims
.
cor(x::AbstractVector, y::AbstractVector)
Compute the Pearson correlation between the vectors x
and y
.
cor(X::AbstractVecOrMat, Y::AbstractVecOrMat; dims=1)
Compute the Pearson correlation between the vectors or matrices X
and Y
along the dimension dims
.
Statistics.cov
— Functioncov(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::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::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::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)
.
Statistics.mean!
— Functionmean!(r, v)
Compute the mean of v
over the singleton dimensions of r
, and write results to r
.
Examples
julia> using Statistics
julia> v = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> mean!([1., 1.], v)
2-element Vector{Float64}:
1.5
3.5
julia> mean!([1. 1.], v)
1×2 Matrix{Float64}:
2.0 3.0
Statistics.mean
— Functionmean(itr)
Compute the mean of all elements in a collection.
If itr
contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if array contains both). Use the skipmissing
function to omit missing
entries and compute the mean of non-missing values.
Examples
julia> using Statistics
julia> mean(1:20)
10.5
julia> mean([1, missing, 3])
missing
julia> mean(skipmissing([1, missing, 3]))
2.0
mean(f::Function, itr)
Apply the function f
to each element of collection itr
and take the mean.
julia> using Statistics
julia> mean(√, [1, 2, 3])
1.3820881233139908
julia> mean([√1, √2, √3])
1.3820881233139908
mean(f::Function, A::AbstractArray; dims)
Apply the function f
to each element of array A
and take the mean over dimensions dims
.
This method requires at least Julia 1.3.
julia> using Statistics
julia> mean(√, [1, 2, 3])
1.3820881233139908
julia> mean([√1, √2, √3])
1.3820881233139908
julia> mean(√, [1 2 3; 4 5 6], dims=2)
2×1 Matrix{Float64}:
1.3820881233139908
2.2285192400943226
mean(A::AbstractArray; dims)
Compute the mean of an array over the given dimensions.
mean
for empty arrays requires at least Julia 1.1.
Examples
julia> using Statistics
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> mean(A, dims=1)
1×2 Matrix{Float64}:
2.0 3.0
julia> mean(A, dims=2)
2×1 Matrix{Float64}:
1.5
3.5
Statistics.median!
— Functionmedian!(v)
Like median
, but may overwrite the input vector.
Statistics.median
— Functionmedian(itr)
Compute the median of all elements in a collection. For an even number of elements no exact median element exists, so the result is equivalent to calculating mean of two median elements.
If itr
contains NaN
or missing
values, the result is also NaN
or missing
(missing
takes precedence if itr
contains both). Use the skipmissing
function to omit missing
entries and compute the median of non-missing values.
Examples
julia> using Statistics
julia> median([1, 2, 3])
2.0
julia> median([1, 2, 3, 4])
2.5
julia> median([1, 2, missing, 4])
missing
julia> median(skipmissing([1, 2, missing, 4]))
2.0
median(A::AbstractArray; dims)
Compute the median of an array along the given dimensions.
Examples
julia> using Statistics
julia> median([1 2; 3 4], dims=1)
1×2 Matrix{Float64}:
2.0 3.0
Statistics.middle
— Functionmiddle(x)
Compute the middle of a scalar value, which is equivalent to x
itself, but of the type of middle(x, x)
for consistency.
middle(x, y)
Compute the middle of two numbers x
and y
, which is equivalent in both value and type to computing their mean ((x + y) / 2
).
middle(range)
Compute the middle of a range, which consists of computing the mean of its extrema. Since a range is sorted, the mean is performed with the first and last element.
julia> using Statistics
julia> middle(1:10)
5.5
middle(a)
Compute the middle of an array a
, which consists of finding its extrema and then computing their mean.
julia> using Statistics
julia> a = [1,2,3.6,10.9]
4-element Vector{Float64}:
1.0
2.0
3.6
10.9
julia> middle(a)
5.95
Statistics.quantile!
— Functionquantile!([q::AbstractArray, ] v::AbstractVector, p; sorted=false, alpha::Real=1.0, beta::Real=alpha)
Compute the quantile(s) of a vector v
at a specified probability or vector or tuple of probabilities p
on the interval [0,1]. If p
is a vector, an optional output array q
may also be specified. (If not provided, a new output array is created.) The keyword argument sorted
indicates whether v
can be assumed to be sorted; if false
(the default), then the elements of v
will be partially sorted in-place.
By default (alpha = beta = 1
), quantiles are computed via linear interpolation between the points ((k-1)/(n-1), v[k])
, for k = 1:n
where n = length(v)
. This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R and NumPy default.
The keyword arguments alpha
and beta
correspond to the same parameters in Hyndman and Fan, setting them to different values allows to calculate quantiles with any of the methods 4-9 defined in this paper:
- Def. 4:
alpha=0
,beta=1
- Def. 5:
alpha=0.5
,beta=0.5
- Def. 6:
alpha=0
,beta=0
(ExcelPERCENTILE.EXC
, Python default, Stataaltdef
) - Def. 7:
alpha=1
,beta=1
(Julia, R and NumPy default, ExcelPERCENTILE
andPERCENTILE.INC
, Python'inclusive'
) - Def. 8:
alpha=1/3
,beta=1/3
- Def. 9:
alpha=3/8
,beta=3/8
An ArgumentError
is thrown if v
contains NaN
or missing
values.
References
Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365
Quantile on Wikipedia details the different quantile definitions
Examples
julia> using Statistics
julia> x = [3, 2, 1];
julia> quantile!(x, 0.5)
2.0
julia> x
3-element Vector{Int64}:
1
2
3
julia> y = zeros(3);
julia> quantile!(y, x, [0.1, 0.5, 0.9]) === y
true
julia> y
3-element Vector{Float64}:
1.2000000000000002
2.0
2.8000000000000003
Statistics.quantile
— Functionquantile(itr, p; sorted=false, alpha::Real=1.0, beta::Real=alpha)
Compute the quantile(s) of a collection itr
at a specified probability or vector or tuple of probabilities p
on the interval [0,1]. The keyword argument sorted
indicates whether itr
can be assumed to be sorted.
Samples quantile are defined by Q(p) = (1-γ)*x[j] + γ*x[j+1]
, where $x[j]$ is the j-th order statistic, and γ
is a function of j = floor(n*p + m)
, m = alpha + p*(1 - alpha - beta)
and g = n*p + m - j
.
By default (alpha = beta = 1
), quantiles are computed via linear interpolation between the points ((k-1)/(n-1), v[k])
, for k = 1:n
where n = length(itr)
. This corresponds to Definition 7 of Hyndman and Fan (1996), and is the same as the R and NumPy default.
The keyword arguments alpha
and beta
correspond to the same parameters in Hyndman and Fan, setting them to different values allows to calculate quantiles with any of the methods 4-9 defined in this paper:
- Def. 4:
alpha=0
,beta=1
- Def. 5:
alpha=0.5
,beta=0.5
- Def. 6:
alpha=0
,beta=0
(ExcelPERCENTILE.EXC
, Python default, Stataaltdef
) - Def. 7:
alpha=1
,beta=1
(Julia, R and NumPy default, ExcelPERCENTILE
andPERCENTILE.INC
, Python'inclusive'
) - Def. 8:
alpha=1/3
,beta=1/3
- Def. 9:
alpha=3/8
,beta=3/8
An ArgumentError
is thrown if v
contains NaN
or missing
values. Use the skipmissing
function to omit missing
entries and compute the quantiles of non-missing values.
References
Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365
Quantile on Wikipedia details the different quantile definitions
Examples
julia> using Statistics
julia> quantile(0:20, 0.5)
10.0
julia> quantile(0:20, [0.1, 0.5, 0.9])
3-element Vector{Float64}:
2.0
10.0
18.000000000000004
julia> quantile(skipmissing([1, 10, missing]), 0.5)
5.5