Mutual Information Covariance

PortfolioOptimisers.MutualInfoCovariance Type

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

$$\begin{array}{rl}{\hat{\mathit{\rho }}}_{ij}& =\mathrm{MI}(X_i,X_j),\\ {\hat{\mathbf{\Sigma }}}_{ij}& ={\hat{\mathit{\rho }}}_{ij}{\hat{\sigma }}_i{\hat{\sigma }}_j.\end{array}$$

Where:

• ${\hat{\mathit{\rho }}}_{ij}$: Mutual information-based correlation between assets $i$ and $j$.

• ${\hat{\mathbf{\Sigma }}}_{ij}$: Covariance between assets $i$ and $j$.

• $\mathrm{MI}(X_i,X_j)$: (Optionally normalised) mutual information between assets $i$ and $j$.

• ${\hat{\sigma }}_i$: 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

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, 0 < bins <= RESOURCE_LIMITS[].max_bins (the joint histogram is bins × bins; see RESOURCE_LIMITS).

Examples

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

Related

• AbstractVarianceEstimator

• AbstractBins

source

Statistics.cor Method

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

• MutualInfoCovariance

• mutual_info

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

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