统计

std(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.

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).

stdm(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).

var(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).

varm(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).

cor(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.

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::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).

mean!(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

mean(itr)

Compute the mean of all elements in a collection.

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, 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, A::AbstractArray; dims)

Apply the function f to each element of array A and take the mean over dimensions dims.

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.

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

median!(v)

Like median, but may overwrite the input vector.

median(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.

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

middle(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(a::AbstractArray)

Compute the middle of an array a, which consists of finding its extrema and then computing their mean.

julia> using Statistics

julia> middle(1:10)
5.5

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

quantile!([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.

Samples quantile are defined by Q(p) = (1-γ)*x[j] + γ*x[j+1], where x[j] is the j-th order statistic of v, j = floor(n*p + m), m = alpha + p*(1 - alpha - beta) and γ = n*p + m - j.

By default (alpha = beta = 1), quantiles are computed via linear interpolation between the points ((k-1)/(n-1), x[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 (Excel PERCENTILE.EXC, Python default, Stata altdef)
• Def. 7: alpha=1, beta=1 (Julia, R and NumPy default, Excel PERCENTILE and PERCENTILE.INC, Python 'inclusive')
• Def. 8: alpha=1/3, beta=1/3
• Def. 9: alpha=3/8, beta=3/8

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

quantile(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 of itr, j = floor(n*p + m), m = alpha + p*(1 - alpha - beta) and γ = n*p + m - j.

By default (alpha = beta = 1), quantiles are computed via linear interpolation between the points ((k-1)/(n-1), x[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 (Excel PERCENTILE.EXC, Python default, Stata altdef)
• Def. 7: alpha=1, beta=1 (Julia, R and NumPy default, Excel PERCENTILE and PERCENTILE.INC, Python 'inclusive')
• Def. 8: alpha=1/3, beta=1/3
• Def. 9: alpha=3/8, beta=3/8

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