Mutual Information Covariance

PortfolioOptimisers.MutualInfoCovariance Type

julia

struct MutualInfoCovariance{__T_ve, __T_bins, __T_normalise} <: AbstractCovarianceEstimator

Covariance estimator based on mutual information.

MutualInfoCovariance implements a robust covariance estimator that uses mutual information (MI) to capture both linear and nonlinear dependencies between asset returns. This estimator is particularly useful for identifying complex relationships that are not detected by traditional correlation-based methods. The MI matrix is optionally normalised and then rescaled by marginal standard deviations to produce a covariance matrix.

Mathematical definition

\begin{aligned} {\hat{ρ}}_{i j} & = MI (X_{i}, X_{j}), \\ {\hat{Σ}}_{i j} & = {\hat{ρ}}_{i j} {\hat{σ}}_{i} {\hat{σ}}_{j} . \end{aligned}

Where:

${\hat{ρ}}_{i j}$ : Mutual information-based correlation between assets $i$ and $j$ .
${\hat{Σ}}_{i j}$ : Covariance between assets $i$ and $j$ .
$MI (X_{i}, X_{j})$ : (Optionally normalised) mutual information between assets $i$ and $j$ .
${\hat{σ}}_{i}$ : Marginal standard deviation of asset $i$ from the variance estimator ve.

Fields

ve: Variance estimator.
bins: Binning algorithm or fixed number of bins.
normalise: Whether to normalise the mutual and/or variation of information calculation.

Constructors

julia

MutualInfoCovariance(;
    ve::AbstractVarianceEstimator = SimpleVariance(),
    bins::Int_Bin = HacineGharbiRavier(),
    normalise::Bool = true
) -> MutualInfoCovariance

Keywords correspond to the struct's fields.

Validation

If bins is an integer, bins > 0.

Examples

julia

julia> MutualInfoCovariance()
MutualInfoCovariance
         ve ┼ SimpleVariance
            │          me ┼ SimpleExpectedReturns
            │             │   w ┴ nothing
            │           w ┼ nothing
            │   corrected ┴ Bool: true
       bins ┼ HacineGharbiRavier()
  normalise ┴ Bool: true

Related

source

PortfolioOptimisers.factory Method

julia

factory(ce::MutualInfoCovariance, w::ObsWeights) -> MutualInfoCovariance

Return a new MutualInfoCovariance estimator with observation weights w applied to the underlying variance estimator.

Arguments

ce: Covariance estimator.
w: Observation weights vector observations × 1.

Returns

ce: New covariance estimator of the same type as the argument, with the new weights applied.

Examples

julia

julia> ce = MutualInfoCovariance();

julia> factory(ce, StatsBase.Weights([0.2, 0.3, 0.5]))
MutualInfoCovariance
         ve ┼ SimpleVariance
            │          me ┼ SimpleExpectedReturns
            │             │   w ┴ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.3, 0.5]
            │           w ┼ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.3, 0.5]
            │   corrected ┴ Bool: true
       bins ┼ HacineGharbiRavier()
  normalise ┴ Bool: true

Related

source

Statistics.cov Method

julia

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

Compute the mutual information (MI) covariance matrix using a MutualInfoCovariance estimator.

This method computes the pairwise mutual information covariance matrix for the input data matrix X, using the binning strategy and normalisation specified in ce. The MI covariance matrix is obtained by rescaling the MI correlation matrix by the marginal standard deviations, as estimated by the variance estimator in ce.

Arguments

ce: Mutual information-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 mutual information-based covariances.

Validation

dims is either 1 or 2.

Related

source

Statistics.cor Method

julia

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

Compute the mutual information (MI) correlation matrix using a MutualInfoCovariance estimator.

This method computes the pairwise mutual information correlation matrix for the input data matrix X, using the binning strategy and normalisation specified in ce. The MI correlation captures both linear and nonlinear dependencies between asset returns, making it robust to complex relationships that may not be detected by traditional correlation measures.

Arguments

ce: Mutual information-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 mutual information-based correlation coefficients.

Validation

dims is either 1 or 2.

Related

source

PortfolioOptimisers.port_opt_view Method

julia

port_opt_view(
    ce::MutualInfoCovariance,
    i,
    args...
) -> Union{MutualInfoCovariance{<:AbstractVarianceEstimator, <:Integer, Bool}, MutualInfoCovariance{<:AbstractVarianceEstimator, <:AbstractBins, Bool}}

Gets the view of the covariance estimator for the i-th element(s).

Arguments

ce: Covariance estimator.
i: Index or indices to view.

Returns

ce: New covariance estimator of the same type as the argument, for the new view.

Related

MutualInfoCovariance

source

Mutual Information Covariance ​

Mutual Information Covariance