Statistics

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).

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).

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.

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).

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> v = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4

julia> mean!([1., 1.], v)
2-element Array{Float64,1}:
 1.5
 3.5

julia> mean!([1. 1.], v)
1×2 Array{Float64,2}:
 2.0  3.0

mean(itr)

Compute the mean of all elements in a collection.

Examples

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> mean(√, [1, 2, 3])
1.3820881233139908

julia> mean([√1, √2, √3])
1.3820881233139908

mean(A::AbstractArray; dims)

Compute the mean of an array over the given dimensions.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4

julia> mean(A, dims=1)
1×2 Array{Float64,2}:
 2.0  3.0

julia> mean(A, dims=2)
2×1 Array{Float64,2}:
 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> 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> median([1 2; 3 4], dims=1)
1×2 Array{Float64,2}:
 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 reals 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> 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> a = [1,2,3.6,10.9]
4-element Array{Float64,1}:
  1.0
  2.0
  3.6
 10.9

julia> middle(a)
5.95

quantile!([q::AbstractArray, ] v::AbstractVector, p; sorted=false)

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.

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 default.

• Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365

Examples

julia> x = [3, 2, 1];

julia> quantile!(x, 0.5)
2.0

julia> x
3-element Array{Int64,1}:
 1
 2
 3

julia> y = zeros(3);

julia> quantile!(y, x, [0.1, 0.5, 0.9]) === y
true

julia> y
3-element Array{Float64,1}:
 1.2
 2.0
 2.8

quantile(itr, p; sorted=false)

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.

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 default.

• Hyndman, R.J and Fan, Y. (1996) "Sample Quantiles in Statistical Packages", The American Statistician, Vol. 50, No. 4, pp. 361-365

Examples

```jldoctest julia> quantile(0:20, 0.5) 10.0

julia> quantile(0:20, [0.1, 0.5, 0.9]) 3-element Array{Float64,1}: 2.0 10.0 18.0

julia> quantile(skipmissing([1, 10, missing]), 0.5) 5.5 ```