Bigsimr.Kendall — ConstantKendallKendall's $τ$ rank correlation
Bigsimr.Pearson — ConstantPearsonPearson's $r$ product-moment correlation
Bigsimr.Spearman — ConstantSpearmanSpearman's $ρ$ rank correlation
Bigsimr.CorType — TypeBigsimr.cor_bounds — Methodcor_bounds(margins::AbstractVector{<:UnivariateDistribution}, cortype::CorType, samples::Int)Compute the stochastic pairwise lower and upper correlation bounds between a set of marginal distributions.
The possible correlation types are:
Examples
julia> using Distributions
julia> margins = [Normal(78, 10), Binomial(20, 0.2), LogNormal(3, 1)];
julia> lower, upper = cor_bounds(margins, Pearson);
julia> lower
3×3 Matrix{Float64}:
1.0 -0.983111 -0.768184
-0.983111 1.0 -0.702151
-0.768184 -0.702151 1.0
julia> upper
3×3 Matrix{Float64}:
1.0 0.982471 0.766727
0.982471 1.0 0.798379
0.766727 0.798379 1.0Bigsimr.cor_bounds — Methodcor_bounds(d1::UnivariateDistribution, d2::UnivariateDistribution, cortype::CorType, samples::Int)Compute the stochastic lower and upper correlation bounds between two marginal distributions.
This method relies on sampling from each distribution and then estimating the specified correlation between the sorted samples. Because the samples are random, there will be some variation in the answer for each call to cor_bounds. Increasing the number of samples will increase the accuracy of the estimate, but will also take longer to sort. Therefore ≈100,000 samples (the default) are recommended so that it runs fast while still returning a good estimate.
The possible correlation types are:
Examples
julia> using Distributions
julia> A = Normal(78, 10); B = LogNormal(3, 1);
julia> cor_bounds(A, B)
(lower = -0.7646512417819491, upper = 0.7649206069306482)
julia> cor_bounds(A, B, 1e6)
(lower = -0.765850375468031, upper = 0.7655170605697716)
julia> cor_bounds(A, B, Pearson)
(lower = -0.7631871539735006, upper = 0.7624398609255689)
julia> cor_bounds(A, B, Spearman, 250_000)
(lower = -1.0, upper = 1.0)Bigsimr.cor_constrain! — Methodcor_constrain!(X::AbstractMatrix{<:Real})Same as cor_constrain, except that the matrix is updated in place to save memory.
Bigsimr.cor_constrain — Methodcor_constrain(X::AbstractMatrix{<:Real})Constrain a matrix so that its diagonal elements are 1, off-diagonal elements are bounded between -1 and 1, and a symmetric view of the upper triangle is made.
See also: cor_constrain!
Examples
julia> a = [ 0.802271 0.149801 -1.1072 1.13451
0.869788 -0.824395 0.38965 0.965936
-1.45353 -1.29282 0.417233 -0.362526
0.638291 -0.682503 1.12092 -1.27018];
julia> cor_constrain(a)
4×4 Matrix{Float64}:
1.0 0.149801 -1.0 1.0
0.149801 1.0 0.38965 0.965936
-1.0 0.38965 1.0 -0.362526
1.0 0.965936 -0.362526 1.0Bigsimr.cor_convert — Methodcor_convert(X, from, to)Convert from one type of correlation matrix to another.
The role of conversion in this package is typically used from either Spearman or Kendall to Pearson where the Pearson correlation is used in the generation of random multivariate normal samples. After converting, the correlation matrix may not be positive semidefinite, so it is recommended to check using LinearAlgebra.isposdef, and subsequently calling cor_nearPD.
See also: cor_nearPD, cor_fastPD
The possible correlation types are:
Examples
julia> r = [ 1.0 -0.634114 0.551645 0.548993
-0.634114 1.0 -0.332105 -0.772114
0.551645 -0.332105 1.0 0.143949
0.548993 -0.772114 0.143949 1.0];
julia> cor_convert(r, Pearson, Spearman)
4×4 Matrix{Float64}:
1.0 -0.616168 0.533701 0.531067
-0.616168 1.0 -0.318613 -0.756979
0.533701 -0.318613 1.0 0.13758
0.531067 -0.756979 0.13758 1.0
julia> cor_convert(r, Spearman, Kendall)
4×4 Matrix{Float64}:
1.0 -0.452063 0.385867 0.383807
-0.452063 1.0 -0.224941 -0.576435
0.385867 -0.224941 1.0 0.0962413
0.383807 -0.576435 0.0962413 1.0
julia> r == cor_convert(r, Pearson, Pearson)
trueBigsimr.cor_fast — Methodcor_fast(X::AbstractMatrix{<:Real}, C::CorType=Pearson)Calculate the correlation matrix in parallel using available threads.
Bigsimr.cor_fastPD — Functioncor_fastPD(X::AbstractMatrix{<:Real}, tau=1e-6)Return a positive definite correlation matrix that is close to X. tau is a tuning parameter that controls the minimum eigenvalue of the resulting matrix. τ can be set to zero if only a positive semidefinite matrix is needed.
See also: cor_nearPD, cor_nearPSD
Bigsimr.cor_fastPD! — Functioncor_fastPD!(X::AbstractMatrix{<:Real}, tau=1e-6)Same as cor_fastPD, but saves space by overwriting the input X instead of creating a copy.
See also: cor_nearPD!, cor_nearPSD!
Bigsimr.cor_nearPD! — Methodcor_nearPD!(X::AbstractMatrix{<:Real})Same as cor_nearPD, but saves space by overwriting the input X instead of creating a copy.
See also: cor_nearPSD!, cor_fastPD!
Bigsimr.cor_nearPD — Methodcor_nearPD(X::AbstractMatrix{<:Real})Return the nearest positive definite correlation matrix to X.
See also: cor_nearPSD, cor_fastPD
Bigsimr.cor_nearPSD! — Methodcor_nearPSD!(X::AbstractMatrix{<:Real})Same as cor_nearPSD, but saves space by overwriting the input X instead of creating a copy.
See also: cor_nearPD!, cor_fastPD!
Bigsimr.cor_nearPSD — Methodcor_nearPSD(X::AbstractMatrix{<:Real})Return the nearest positive [semi-] definite correlation matrix to X.
See also: cor_nearPD, cor_fastPD
Bigsimr.cor_randPD — Methodcor_randPD(T::Type{<:Real}, d::Int, k::Int=d-1)The same as cor_randPSD, but calls cor_fastPD to ensure that the returned matrix is positive definite.
Examples
julia> cor_randPD(Float64, 4, 2)
4×4 Matrix{Float64}:
1.0 0.458549 -0.33164 0.492572
0.458549 1.0 -0.280873 0.62544
-0.33164 -0.280873 1.0 -0.315011
0.492572 0.62544 -0.315011 1.0
julia> cor_randPD(4, 1)
4×4 Matrix{Float64}:
1.0 -0.0406469 -0.127517 -0.133308
-0.0406469 1.0 0.265604 0.277665
-0.127517 0.265604 1.0 0.871089
-0.133308 0.277665 0.871089 1.0
julia> cor_randPD(4)
4×4 Matrix{Float64}:
1.0 0.356488 0.701521 -0.251671
0.356488 1.0 0.382787 -0.117748
0.701521 0.382787 1.0 -0.424952
-0.251671 -0.117748 -0.424952 1.0Bigsimr.cor_randPSD — Methodcor_randPSD(T::Type{<:Real}, d::Int, k::Int=d-1)Return a random positive semidefinite correlation matrix where d is the dimension ($d ≥ 1$) and k is the number of factor loadings ($1 ≤ k < d$).
See also: cor_randPD
Examples
julia> cor_randPSD(Float64, 4, 2)
4×4 Matrix{Float64}:
1.0 0.276386 0.572837 0.192875
0.276386 1.0 0.493806 -0.352386
0.572837 0.493806 1.0 -0.450259
0.192875 -0.352386 -0.450259 1.0
julia> cor_randPSD(4, 1)
4×4 Matrix{Float64}:
1.0 -0.800513 0.541379 -0.650587
-0.800513 1.0 -0.656411 0.788824
0.541379 -0.656411 1.0 -0.533473
-0.650587 0.788824 -0.533473 1.0
julia> cor_randPSD(4)
4×4 Matrix{Float64}:
1.0 0.81691 -0.27188 0.289011
0.81691 1.0 -0.44968 0.190938
-0.27188 -0.44968 1.0 -0.102597
0.289011 0.190938 -0.102597 1.0Bigsimr.cov2cor! — Methodcov2cor!(X::AbstractMatrix{<:Real})Same as cov2cor, except that the matrix is updated in place to save memory.
Bigsimr.cov2cor — Methodcov2cor(X::AbstractMatrix{<:Real})Transform a covariance matrix into a correlation matrix.
Details
If $X \in \mathbb{R}^{n \times n}$ is a covariance matrix, then
\[\tilde{X} = D^{-1/2} X D^{-1/2}, \quad D = \mathrm{diag(X)}\]
is the associated correlation matrix.
Bigsimr.is_correlation — Methodis_correlation(X)Check if the given matrix passes all the checks required to be a valid correlation matrix.
Criteria
A matrix is a valid correlation matrix if:
- Square
- Symmetric
- Diagonal elements are equal to
1 - Off diagonal elements are between
-1and1 - Is positive definite
Examples
julia> x = rand(3, 3)
3×3 Matrix{Float64}:
0.834446 0.183285 0.837872
0.637295 0.270709 0.458703
0.626566 0.736907 0.61903
julia> is_correlation(x)
false
julia> x = cor_randPD(3)
3×3 Matrix{Float64}:
1.0 0.190911 0.449104
0.190911 1.0 0.636305
0.449104 0.636305 1.0
julia> is_correlation(x)
true
julia> r_negdef = [
1.00 0.82 0.56 0.44
0.82 1.00 0.28 0.85
0.56 0.28 1.00 0.22
0.44 0.85 0.22 1.00
];
julia> is_correlation(r_negdef)
falseBigsimr.rmvn — Methodrmvn(n::Int[, μ::AbstractVector{<:Real}], Σ::AbstractMatrix{<:Real})Utilizes available threads for fast generation of multivariate normal samples.
Examples
julia> μ = [-3, 1, 10];
julia> S = cor_randPD(3)
3×3 Matrix{Float64}:
1.0 -0.663633 -0.0909108
-0.663633 1.0 0.151582
-0.0909108 0.151582 1.0
julia> rmvn(10, μ, S)
10×3 Matrix{Float64}:
-1.32616 -1.02602 11.0202
-3.59396 2.84145 8.84367
-0.441537 -1.53279 8.82931
-4.69202 2.84618 10.5977
-2.63359 2.65779 9.8374
-3.75917 2.07208 8.90139
-3.00716 0.00897664 10.1173
-3.00928 0.851214 9.74029
-3.43021 0.402382 9.51274
-1.77849 0.157933 9.15944Bigsimr.rvec — Methodrvec(n::Int, rho::AbstractMatrix{<:Real}, margins::AbstractVector{<:UnivariateDistribution})Generate samples for a list of marginal distributions and a correaltion structure.
Examples
julia> using Distributions
julia> margins = [Normal(3, 1), LogNormal(3, 1), Exponential(3)]
julia> R = [
1.00 -0.23 0.12
-0.23 1.00 -0.46
0.12 -0.46 1.00
];
julia> rvec(10, R, margins)
10×3 Matrix{Float64}:
3.89423 38.6339 1.30088
5.87344 11.5582 5.25233
3.62383 20.4001 3.25627
3.65075 3.8316 4.48547
1.62223 9.95032 1.48367
3.42208 35.0998 0.644814
1.82689 58.417 0.580125
4.73678 4.75506 11.2741
1.92511 9.44913 0.651013
3.19883 39.3707 0.581781Statistics.cor — Methodcor(x[, y], ::CorType)Compute the correlation matrix of a given type.
The possible correlation types are:
Examples
julia> x = [-1.62169 0.0158613 0.500375 -0.794381
2.50689 3.31666 -1.3049 2.16058
0.495674 0.348621 -0.614451 -0.193579
2.32149 2.18847 -1.83165 2.08399
-0.0573697 0.39908 0.270117 0.658458
0.365239 -0.321493 -1.60223 -0.199998
-0.55521 -0.898513 0.690267 0.857519
-0.356979 -1.03724 0.714859 -0.719657
-3.38438 -1.93058 1.77413 -1.23657
1.57527 0.836351 -1.13275 -0.277048];
julia> cor(x, Pearson)
4×4 Matrix{Float64}:
1.0 0.86985 -0.891312 0.767433
0.86985 1.0 -0.767115 0.817407
-0.891312 -0.767115 1.0 -0.596762
0.767433 0.817407 -0.596762 1.0
julia> cor(x, Spearman)
4×4 Matrix{Float64}:
1.0 0.866667 -0.854545 0.709091
0.866667 1.0 -0.781818 0.684848
-0.854545 -0.781818 1.0 -0.612121
0.709091 0.684848 -0.612121 1.0
julia> cor(x, Kendall)
4×4 Matrix{Float64}:
1.0 0.733333 -0.688889 0.555556
0.733333 1.0 -0.688889 0.555556
-0.688889 -0.688889 1.0 -0.422222
0.555556 0.555556 -0.422222 1.0