Rank Covariances

PortfolioOptimisers.KendallCovariance Type

struct KendallCovariance{__T_ve} <: RankCovarianceEstimator

Robust covariance estimator based on Kendall's tau rank correlation.

KendallCovariance implements a covariance estimator that uses Kendall's tau rank correlation to measure the monotonic association between pairs of asset returns. This estimator is robust to outliers and non-Gaussian data, making it suitable for financial applications where heavy tails or non-linear dependencies are present.

Fields

• ve: Variance estimator.

Constructors

KendallCovariance(;
    ve::AbstractVarianceEstimator = SimpleVariance()
) -> KendallCovariance

Keywords correspond to the struct's fields.

Examples

julia> KendallCovariance()
KendallCovariance
  ve ┼ SimpleVariance
     │          me ┼ SimpleExpectedReturns
     │             │   w ┴ nothing
     │           w ┼ nothing
     │   corrected ┴ Bool: true

Related

• RankCovarianceEstimator

• SpearmanCovariance

• AbstractVarianceEstimator

• SimpleVariance

source

PortfolioOptimisers.factory Method

factory(
    ce::StatsBase.CovarianceEstimator,
    args...;
    kwargs...
) -> StatsBase.CovarianceEstimator

Fallback for covariance estimator factory methods.

Arguments

• ce: Covariance estimator.

• args...: Optional arguments (ignored).

• kwargs...: Optional keyword arguments (ignored).

Returns

• ce::StatsBase.CovarianceEstimator: The original covariance estimator.

Related

• factory

• AbstractCovarianceEstimator

• StatsBase.CovarianceEstimator

source

Statistics.cov Method

Statistics.cov(ce::KendallCovariance, X::MatNum; dims::Int = 1, kwargs...)

Compute the Kendall's tau rank covariance matrix using a KendallCovariance estimator.

This method computes the covariance matrix for the input data matrix X by combining the Kendall's tau rank correlation matrix with the marginal standard deviations estimated by the variance estimator in ce. This approach is robust to outliers and non-Gaussian data.

Arguments

• ce: Kendall's tau-based covariance estimator.

• X: Data matrix of asset returns (observations × assets).

• dims: Dimension along which to perform the computation.

• kwargs...: Additional keyword arguments passed to the variance estimator.

Returns

• sigma::Matrix{<:Number}: Symmetric matrix of Kendall's tau rank covariances.

Validation

• dims is either 1 or 2.

Examples

julia> X = [0.01 0.02; 0.03 0.04; 0.02 0.03];

julia> cov(KendallCovariance(), X)
2×2 Matrix{Float64}:
 0.0001  0.0001
 0.0001  0.0001

Related

• KendallCovariance

• corkendall

source

Statistics.cor Method

Statistics.cor(::KendallCovariance, X::MatNum; dims::Int = 1, kwargs...)

Compute the Kendall's tau rank correlation matrix using a KendallCovariance estimator.

This method computes the pairwise Kendall's tau rank correlation matrix for the input data matrix X. Kendall's tau measures the monotonic association between pairs of asset returns and is robust to outliers and non-Gaussian data.

Mathematical definition

For two asset return series $(x_1,\dots ,x_T)$ and $(y_1,\dots ,y_T)$, Kendall's $\tau$ is:

$$\begin{array}{rl}{\hat{\tau }}_{ij}& =\frac{C-D}{(\genfrac{}{}{0ex}{}{T}{2})}.\end{array}$$

Where:

• ${\hat{\tau }}_{ij}$: Kendall's $\tau$ rank correlation between assets $i$ and $j$.

• $C$: Number of concordant pairs; a pair $(t,s)$ is concordant if $(x_t-x_s)(y_t-y_s)>0$.

• $D$: Number of discordant pairs; a pair $(t,s)$ is discordant if $(x_t-x_s)(y_t-y_s)<0$.

• $T$: Number of observations.

Arguments

• ce: Kendall's tau-based covariance estimator.

• X: Data matrix of asset returns (observations × assets).

• dims: Dimension along which to perform the computation.

• kwargs...: Additional keyword arguments (currently unused).

Returns

• rho::Matrix{<:Number}: Symmetric matrix of Kendall's tau rank correlation coefficients.

Validation

• dims is either 1 or 2.

Examples

julia> X = [0.01 0.02; 0.03 0.04; 0.02 0.03];

julia> cor(KendallCovariance(), X)
2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

Related

• KendallCovariance

• corkendall

source

PortfolioOptimisers.port_opt_view Method

port_opt_view(x, i, args...; kwargs...) -> nothing_scalar_array_view(x, i)

Universal fallback for port_opt_view. Any value that has no more specific port_opt_view method is treated as leaf data: it is delegated to nothing_scalar_array_view, which slices arrays/VecScalars and passes scalars, nothing, estimators, and algorithms through unchanged. Composed structs that need to recurse into children define their own (more specific) method — emitted by the @vprop tag or hand-written.

The threaded tail args... (typically the returns matrix X for the JuMP families) and any kwargs are accepted and dropped here, so a macro-threaded port_opt_view(child, i, X) never MethodErrors on a leaf field.

Related

• nothing_scalar_array_view

• @vprop

source

port_opt_view(
    ce::CovarianceEstimator,
    _,
    args...
) -> CorrelationCovariance{<:CovarianceEstimator}

No-op fallback for getting the view of a covariance estimator.

Arguments

• ce: Covariance estimator.

• args...: Optional arguments (ignored).

Returns

• ce::StatsBase.CovarianceEstimator: The original covariance estimator.

Related

• AbstractCovarianceEstimator

• StatsBase.CovarianceEstimator

source

PortfolioOptimisers.SpearmanCovariance Type

struct SpearmanCovariance{__T_ve} <: RankCovarianceEstimator

Robust covariance estimator based on Spearman's rho rank correlation.

SpearmanCovariance implements a covariance estimator that uses Spearman's rho rank correlation to measure the monotonic association between pairs of asset returns. This estimator is robust to outliers and non-Gaussian data, making it suitable for financial applications where heavy tails or non-linear dependencies are present.

Fields

• ve: Variance estimator.

Constructors

SpearmanCovariance(;
    ve::AbstractVarianceEstimator = SimpleVariance()
) -> SpearmanCovariance

Keywords correspond to the struct's fields.

Examples

julia> SpearmanCovariance()
SpearmanCovariance
  ve ┼ SimpleVariance
     │          me ┼ SimpleExpectedReturns
     │             │   w ┴ nothing
     │           w ┼ nothing
     │   corrected ┴ Bool: true

Related

• RankCovarianceEstimator

• KendallCovariance

• AbstractVarianceEstimator

• SimpleVariance

source

PortfolioOptimisers.factory Method

factory(
    ce::StatsBase.CovarianceEstimator,
    args...;
    kwargs...
) -> StatsBase.CovarianceEstimator

Fallback for covariance estimator factory methods.

Arguments

• ce: Covariance estimator.

• args...: Optional arguments (ignored).

• kwargs...: Optional keyword arguments (ignored).

Returns

• ce::StatsBase.CovarianceEstimator: The original covariance estimator.

Related

• factory

• AbstractCovarianceEstimator

• StatsBase.CovarianceEstimator

source

Statistics.cov Method

Statistics.cov(ce::SpearmanCovariance, X::MatNum; dims::Int = 1, kwargs...)

Compute the Spearman's rho rank covariance matrix using a SpearmanCovariance estimator.

This method computes the covariance matrix for the input data matrix X by combining the Spearman's rho rank correlation matrix with the marginal standard deviations estimated by the variance estimator in ce. This approach is robust to outliers and non-Gaussian data.

Arguments

• ce: Spearman's rho-based covariance estimator.

• X: Data matrix of asset returns (observations × assets).

• dims: Dimension along which to perform the computation.

• kwargs...: Additional keyword arguments passed to the variance estimator.

Returns

• sigma::Matrix{<:Number}: Symmetric matrix of Spearman's rho rank covariances.

Validation

• dims is either 1 or 2.

Examples

julia> X = [0.01 0.02; 0.03 0.04; 0.02 0.03];

julia> cov(SpearmanCovariance(), X)
2×2 Matrix{Float64}:
 0.0001  0.0001
 0.0001  0.0001

Related

• SpearmanCovariance

• corspearman

source

Statistics.cor Method

Statistics.cor(::SpearmanCovariance, X::MatNum; dims::Int = 1, kwargs...)

Compute the Spearman's rho rank correlation matrix using a SpearmanCovariance estimator.

This method computes the pairwise Spearman's rho rank correlation matrix for the input data matrix X. Spearman's rho measures the monotonic association between pairs of asset returns and is robust to outliers and non-Gaussian data.

Mathematical definition

Spearman's $\rho$ is the Pearson correlation of the rank-transformed data. Let $\mathrm{rk}(x_t)$ denote the rank of observation $x_t$ among $x_1,\dots ,x_T$:

$$\begin{array}{rl}{\hat{\rho }}_{ij}^S& =1-\frac{6\sum _{t=1}^T{d}_t^2}{T(T^2-1)},d_t=\mathrm{rk}(x_{ti})-\mathrm{rk}(x_{tj}).\end{array}$$

Where:

• ${\hat{\rho }}_{ij}^S$: Spearman's $\rho$ rank correlation between assets $i$ and $j$.

• $T$: Number of observations.

• $x_{ti}$: Return of asset $i$ at time $t$.

• $d_t$: Difference in ranks between assets $i$ and $j$ at time $t$.

• $\mathrm{rk}(\cdot )$: Rank function.

Arguments

• ce: Spearman's rho-based covariance estimator.

• X: Data matrix of asset returns (observations × assets).

• dims: Dimension along which to perform the computation.

• kwargs...: Additional keyword arguments (currently unused).

Returns

• rho::Matrix{<:Number}: Symmetric matrix of Spearman's rho rank correlation coefficients.

Validation

• dims is either 1 or 2.

Examples

julia> X = [0.01 0.02; 0.03 0.04; 0.02 0.03];

julia> cor(SpearmanCovariance(), X)
2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

Related

• SpearmanCovariance

• corspearman

source

PortfolioOptimisers.port_opt_view Method

port_opt_view(x, i, args...; kwargs...) -> nothing_scalar_array_view(x, i)

Universal fallback for port_opt_view. Any value that has no more specific port_opt_view method is treated as leaf data: it is delegated to nothing_scalar_array_view, which slices arrays/VecScalars and passes scalars, nothing, estimators, and algorithms through unchanged. Composed structs that need to recurse into children define their own (more specific) method — emitted by the @vprop tag or hand-written.

The threaded tail args... (typically the returns matrix X for the JuMP families) and any kwargs are accepted and dropped here, so a macro-threaded port_opt_view(child, i, X) never MethodErrors on a leaf field.

Related

• nothing_scalar_array_view

• @vprop

source

port_opt_view(
    ce::CovarianceEstimator,
    _,
    args...
) -> CorrelationCovariance{<:CovarianceEstimator}

No-op fallback for getting the view of a covariance estimator.

Arguments

• ce: Covariance estimator.

• args...: Optional arguments (ignored).

Returns

• ce::StatsBase.CovarianceEstimator: The original covariance estimator.

Related

• AbstractCovarianceEstimator

• StatsBase.CovarianceEstimator

source

PortfolioOptimisers.RankCovarianceEstimator Type

abstract type RankCovarianceEstimator <: AbstractCovarianceEstimator

Abstract supertype for all rank-based covariance estimators in PortfolioOptimisers.jl.

All concrete and/or abstract types implementing rank-based covariance estimation algorithms should be subtypes of RankCovarianceEstimator.

Related

• KendallCovariance

• SpearmanCovariance

• AbstractCovarianceEstimator

source