Gerber Covariance

PortfolioOptimisers.BaseGerberCovariance — Type
abstract type BaseGerberCovariance <: AbstractCovarianceEstimator end

Abstract supertype for all Gerber covariance estimators in PortfolioOptimisers.jl.

All concrete types implementing Gerber covariance estimation algorithms should subtype BaseGerberCovariance. This enables a consistent interface for Gerber-based covariance estimators throughout the package.

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PortfolioOptimisers.GerberCovarianceAlgorithm — Type
abstract type GerberCovarianceAlgorithm <: AbstractMomentAlgorithm end

Abstract supertype for all Gerber covariance algorithm types in PortfolioOptimisers.jl.

All concrete types implementing specific Gerber covariance algorithms should subtype GerberCovarianceAlgorithm. This enables flexible extension and dispatch of Gerber covariance routines.

These types are used to specify the algorithm when constructing a GerberCovariance estimator.

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PortfolioOptimisers.NormalisedGerberCovarianceAlgorithm — Type
abstract type NormalisedGerberCovarianceAlgorithm <: GerberCovarianceAlgorithm end

Abstract supertype for all normalised Gerber covariance algorithm types. These Z-transform the data before applying the Gerber covariance algorithm.

Concrete types implementing normalised Gerber covariance algorithms should subtype NormalisedGerberCovarianceAlgorithm.

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PortfolioOptimisers.NormalisedGerber0 — Type
struct NormalisedGerber0{T1} <: NormalisedGerberCovarianceAlgorithm
    me::T1
end

Implements the original Gerber covariance algorithm on Z-transformed data.

Fields

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering prior to normalisation.

Constructor

  • NormalisedGerber0(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns()): Construct the original normalised Gerber covariance algorithm.

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PortfolioOptimisers.NormalisedGerber0 — Method
NormalisedGerber0(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns())

Creates a NormalisedGerber0 object using the specified expected returns estimator for mean-centering prior to normalisation.

Arguments

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering before normalisation.

ReturnsResult

  • NormalisedGerber0: An instance of the original normalised Gerber covariance algorithm.

Examples

julia> ng0 = NormalisedGerber0()
NormalisedGerber0
  me | SimpleExpectedReturns
     |   w | nothing

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PortfolioOptimisers.NormalisedGerber1 — Type
struct NormalisedGerber1{T1} <: NormalisedGerberCovarianceAlgorithm
    me::T1
end

Implements the first variant of the Gerber covariance algorithm on Z-transformed data.

Fields

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering prior to normalisation.

Constructor

  • NormalisedGerber1(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns()): Construct the first variant of the normalised Gerber covariance algorithm.

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PortfolioOptimisers.NormalisedGerber1 — Method
NormalisedGerber1(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns())

Creates a NormalisedGerber1 object using the specified expected returns estimator for mean-centering prior to normalisation.

Arguments

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering before normalisation.

ReturnsResult

  • NormalisedGerber1: An instance of the original normalised Gerber covariance algorithm.

Examples

julia> ng0 = NormalisedGerber1()
NormalisedGerber1
  me | SimpleExpectedReturns
     |   w | nothing

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PortfolioOptimisers.NormalisedGerber2 — Type
struct NormalisedGerber2{T1} <: NormalisedGerberCovarianceAlgorithm
    me::T1
end

Implements the second variant of the Gerber covariance algorithm on Z-transformed data.

Fields

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering prior to normalisation.

Constructors

  • NormalisedGerber2(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns()): Construct the second variant of the normalised Gerber covariance algorithm.

These types are used to specify the algorithm when constructing a GerberCovariance estimator with normalisation.

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PortfolioOptimisers.NormalisedGerber2 — Method
NormalisedGerber2(; me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns())

Creates a NormalisedGerber2 object using the specified expected returns estimator for mean-centering prior to normalisation.

Arguments

  • me::AbstractExpectedReturnsEstimator: Expected returns estimator used for mean-centering before normalisation.

ReturnsResult

  • NormalisedGerber2: An instance of the original normalised Gerber covariance algorithm.

Examples

julia> ng0 = NormalisedGerber2()
NormalisedGerber2
  me | SimpleExpectedReturns
     |   w | nothing

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PortfolioOptimisers.GerberCovariance — Type
struct GerberCovariance{T1, T2, T3, T4} <: BaseGerberCovariance
    ve::T1
    pdm::T2
    threshold::T3
    alg::T4
end

A flexible container type for configuring and applying Gerber covariance estimators in PortfolioOptimisers.jl.

GerberCovariance encapsulates all components required for Gerber-based covariance or correlation estimation, including the variance estimator, positive definite matrix estimator, threshold parameter, and the specific Gerber algorithm variant. This enables modular and extensible workflows for robust covariance estimation using Gerber statistics.

Fields

  • ve::StatsBase.CovarianceEstimator: Variance estimator.
  • pdm::Posdef: Positive definite matrix estimator (see Posdef).
  • threshold::Real: Threshold parameter for Gerber covariance computation (typically in (0, 1)).
  • alg::GerberCovarianceAlgorithm: Gerber covariance algorithm variant.

Constructor

GerberCovariance(; alg::GerberCovarianceAlgorithm = Gerber1(),
                  ve::StatsBase.CovarianceEstimator = SimpleVariance(),
                  pdm::Union{Nothing, <:Posdef} = Posdef(),
                  threshold::Real = 0.5)

Construct a GerberCovariance estimator with the specified algorithm, variance estimator, positive definite estimator, and threshold.

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PortfolioOptimisers.GerberCovariance — Method
GerberCovariance(; ve::StatsBase.CovarianceEstimator = SimpleVariance(),
                 pdm::Union{Nothing, <:Posdef} = Posdef(),
                 threshold::Real = 0.5,
                 alg::GerberCovarianceAlgorithm = Gerber1())

Construct a GerberCovariance estimator for robust Gerber-based covariance or correlation estimation.

This constructor creates a GerberCovariance object using the specified Gerber algorithm, variance estimator, positive definite matrix estimator, and threshold parameter. The estimator is highly modular, allowing users to select from different Gerber algorithm variants, and positive definite projections.

Arguments

  • ve::StatsBase.CovarianceEstimator: Variance estimator.
  • pdm::Union{Nothing, <:Posdef}: Positive definite matrix estimator.
  • threshold::Real: Threshold parameter for Gerber covariance computation.
  • alg::GerberCovarianceAlgorithm: Gerber covariance algorithm variant.

ReturnsResult

  • GerberCovariance: A configured Gerber covariance estimator.

Validation

  • Asserts that threshold is strictly in (0, 1).

Examples

julia> gc = GerberCovariance()
GerberCovariance
         ve | SimpleVariance
            |          me | SimpleExpectedReturns
            |             |   w | nothing
            |           w | nothing
            |   corrected | Bool: true
        pdm | Posdef
            |   alg | UnionAll: NearestCorrelationMatrix.Newton
  threshold | Float64: 0.5
        alg | Gerber1()

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Statistics.cov — Method
cov(ce::GerberCovariance{<:Any, <:Any, <:Any, <:UnNormalisedGerberCovarianceAlgorithm},
    X::AbstractMatrix; dims::Int = 1, kwargs...)

Compute the Gerber covariance matrix using an unnormalised Gerber covariance estimator.

This method computes the Gerber covariance matrix for the input data matrix X using the specified unnormalised Gerber covariance estimator. The standard deviation vector is computed using the estimator's variance estimator. The Gerber correlation is computed via gerber, and the result is rescaled to a covariance matrix using the standard deviation vector.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:UnNormalisedGerberCovarianceAlgorithm}: Gerber covariance estimator.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • dims::Int: Dimension along which to compute the covariance.
  • kwargs...: Additional keyword arguments passed to the standard deviation estimator.

ReturnsResult

  • sigma::Matrix{Float64}: The Gerber covariance matrix.

Validation

  • Asserts that dims is either 1 or 2.

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Statistics.cov — Method
cov(ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerberCovarianceAlgorithm},
    X::AbstractMatrix; dims::Int = 1, kwargs...)

Compute the Gerber covariance matrix using a normalised Gerber covariance estimator.

This method computes the Gerber covariance matrix for the input data matrix X using the specified normalised Gerber covariance estimator. The standard deviation vector is computed using the estimator's variance estimator. The Gerber correlation is computed via gerber, and the result is rescaled to a covariance matrix using the standard deviation vector.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerberCovarianceAlgorithm}: Gerber covariance estimator.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • dims::Int: Dimension along which to compute the covariance.
  • kwargs...: Additional keyword arguments passed to the standard deviation estimator.

ReturnsResult

  • sigma::Matrix{Float64}: The Gerber covariance matrix.

Validation

  • Asserts that dims is either 1 or 2.

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Statistics.cor — Method
cor(ce::GerberCovariance{<:Any, <:Any, <:Any, <:UnNormalisedGerberCovarianceAlgorithm},
    X::AbstractMatrix; dims::Int = 1, kwargs...)

Compute the Gerber correlation matrix using an unnormalised Gerber covariance estimator.

This method computes the Gerber correlation matrix for the input data matrix X using the specified unnormalised Gerber covariance estimator. The standard deviation vector is computed using the estimator's variance estimator. The Gerber correlation is then computed via gerber.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:UnNormalisedGerberCovarianceAlgorithm}: Gerber covariance estimator.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • dims::Int: Dimension along which to compute the correlation.
  • kwargs...: Additional keyword arguments passed to the standard deviation estimator.

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix.

Validation

  • Asserts that dims is either 1 or 2.

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Statistics.cor — Method
cor(ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerberCovarianceAlgorithm},
    X::AbstractMatrix; dims::Int = 1, kwargs...)

Compute the Gerber correlation matrix using a normalised Gerber covariance estimator.

This method computes the Gerber correlation matrix for the input data matrix X using the specified normalised Gerber covariance estimator. The standard deviation vector is computed using the estimator's variance estimator. The Gerber correlation is then computed via gerber.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerberCovarianceAlgorithm}: Gerber covariance estimator.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • dims::Int: Dimension along which to compute the correlation.
  • kwargs...: Additional keyword arguments passed to the standard deviation estimator.

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix.

Validation

  • Asserts that dims is either 1 or 2.

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber0}, X::AbstractMatrix, std_vec::AbstractArray)

Implements the original Gerber correlation algorithm.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the original Gerber0 algorithm. The computation is based on thresholding the standardized data and counting co-occurrences of threshold exceedances.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber0}: Gerber correlation estimator configured with the Gerber0 algorithm.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • std_vec::AbstractArray: Vector of standard deviations for each asset, used to scale the threshold.

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute two Boolean matrices:

    • U: Entries where X exceeds threshold * std_vec.
    • D: Entries where X is less than -threshold * std_vec.

  2. Compute UmD = U - D and UpD = U + D.

  3. The Gerber correlation is given by (UmD' * UmD) ⊘ (UpD' * UpD).

  4. The result is projected to the nearest positive definite matrix using posdef!.

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber0}, X::AbstractMatrix)

Implements the original Gerber correlation algorithm on Z-transformed data.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the original NormalisedGerber0 algorithm. The computation is performed on data that has already been Z-transformed (mean-centered and standardised), and is based on thresholding and counting co-occurrences of threshold exceedances.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber0}: Gerber correlation estimator configured with the NormalisedGerber0 algorithm.
  • X::AbstractMatrix: Z-transformed data matrix (observations × assets).

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute two Boolean matrices:

    • U: Entries where X exceeds ce.threshold.
    • D: Entries where X is less than -ce.threshold.

  2. Compute UmD = U - D and UpD = U + D.

  3. The Gerber correlation is given by (UmD' * UmD) ⊘ (UpD' * UpD).

  4. The result is projected to the nearest positive definite matrix using posdef!.

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber1}, X::AbstractMatrix, std_vec::AbstractArray)

Implements the first variant of the Gerber correlation algorithm.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the Gerber1 algorithm. The computation is based on thresholding the standardized data, counting co-occurrences of threshold exceedances, and adjusting for non-exceedance events.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber1}: Gerber correlation estimator configured with the Gerber1 algorithm.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • std_vec::AbstractArray: Vector of standard deviations for each asset, used to scale the threshold.

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute three Boolean matrices:

    • U: Entries where X exceeds threshold * std_vec.
    • D: Entries where X is less than -threshold * std_vec.
    • N: Entries where X is within [-threshold * std_vec, threshold * std_vec] (i.e., neither up nor down).

  2. Compute UmD = U - D.

  3. The Gerber1 correlation is given by (UmD' * UmD) ⊘ (T .- (N' * N)), where T is the number of observations.

  4. The result is projected to the nearest positive definite matrix using posdef!.

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber1}, X::AbstractMatrix)

Implements the first variant of the Gerber correlation algorithm on Z-transformed data.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the NormalisedGerber1 algorithm. The computation is performed on data that has already been Z-transformed (mean-centered and standardised), and is based on thresholding, counting co-occurrences of threshold exceedances, and adjusting for non-exceedance events.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber1}: Gerber correlation estimator configured with the NormalisedGerber1 algorithm.
  • X::AbstractMatrix: Z-transformed data matrix (observations × assets).

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute three Boolean matrices:

    • U: Entries where X exceeds ce.threshold.
    • D: Entries where X is less than -ce.threshold.
    • N: Entries where X is within [-ce.threshold, ce.threshold] (i.e., neither up nor down).

  2. Compute UmD = U - D.

  3. The Gerber1 correlation is given by (UmD' * UmD) ⊘ (T .- (N' * N)), where T is the number of observations.

  4. The result is projected to the nearest positive definite matrix using posdef!.

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber2}, X::AbstractMatrix, std_vec::AbstractArray)

Implements the second variant of the Gerber correlation algorithm.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the Gerber2 algorithm. The computation is based on thresholding the standardized data, constructing a signed indicator matrix, and normalizing by the geometric mean of diagonal elements.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:Gerber2}: Gerber correlation estimator configured with the Gerber2 algorithm.
  • X::AbstractMatrix: Data matrix (observations × assets).
  • std_vec::AbstractArray: Vector of standard deviations for each asset, used to scale the threshold.

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation or correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute two Boolean matrices:

    • U: Entries where X exceeds threshold * std_vec.
    • D: Entries where X is less than -threshold * std_vec.

  2. Compute the signed indicator matrix UmD = U - D.

  3. Compute the raw Gerber2 matrix H = UmD' * UmD.

  4. Normalize: rho = H ⊘ (h * h'), where h = sqrt.(diag(H)).

  5. The result is projected to the nearest positive definite matrix using posdef!.

Related

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PortfolioOptimisers.gerber — Method
gerber(ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber2}, X::AbstractMatrix)

Implements the second variant of the Gerber correlation algorithm on Z-transformed data.

This method computes the Gerber correlation or correlation matrix for the input data matrix X using the NormalisedGerber2 algorithm. The computation is performed on data that has already been Z-transformed (mean-centered and standardised), and is based on thresholding, constructing a signed indicator matrix, and normalizing by the geometric mean of diagonal elements.

Arguments

  • ce::GerberCovariance{<:Any, <:Any, <:Any, <:NormalisedGerber2}: Gerber correlation estimator configured with the NormalisedGerber2 algorithm.
  • X::AbstractMatrix: Z-transformed data matrix (observations × assets).

ReturnsResult

  • rho::Matrix{Float64}: The Gerber correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

  1. For each entry in X, compute two Boolean matrices:

    • U: Entries where X exceeds ce.threshold.
    • D: Entries where X is less than -ce.threshold.

  2. Compute the signed indicator matrix UmD = U - D.

  3. Compute the raw Gerber2 matrix H = UmD' * UmD.

  4. Normalize: rho = H ⊘ (h * h'), where h = sqrt.(diag(H)).

  5. The result is projected to the nearest positive definite matrix using posdef!.

Related

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