Smyth-Broby Covariance

julia

struct SmythBroby0 <: SmythBrobyCovarianceAlgorithm

Implements the original Smyth-Broby covariance algorithm.

Related

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PortfolioOptimisers.SmythBroby1 Type

julia

struct SmythBroby1 <: SmythBrobyCovarianceAlgorithm

Implements the first variant of the Smyth-Broby covariance algorithm.

Related

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PortfolioOptimisers.SmythBroby2 Type

julia

struct SmythBroby2 <: SmythBrobyCovarianceAlgorithm

Implements the second variant of the Smyth-Broby covariance algorithm.

Related

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PortfolioOptimisers.SmythBrobyGerber0 Type

julia

struct SmythBrobyGerber0 <: SmythBrobyCovarianceAlgorithm

Implements the original Smyth-Broby covariance algorithm scaled by vote counts.

Related

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PortfolioOptimisers.SmythBrobyGerber1 Type

julia

struct SmythBrobyGerber1 <: SmythBrobyCovarianceAlgorithm

Implements the first variant of the Smyth-Broby covariance algorithm scaled by vote counts.

Related

source

PortfolioOptimisers.SmythBrobyGerber2 Type

julia

struct SmythBrobyGerber2 <: SmythBrobyCovarianceAlgorithm

Implements the second variant of the Smyth-Broby covariance algorithm scaled by vote counts.

Related

source

PortfolioOptimisers.SmythBrobyCount0 Type

julia

struct SmythBrobyCount0 <: SmythBrobyCovarianceAlgorithm

Implements the original Smyth-Broby covariance algorithm using vote counts only.

Related

julia

- [`SmythBrobyCovarianceAlgorithm`](@ref)
- [`SmythBrobyCovariance`](@ref)
- [`SmythBrobyCount1`](@ref)
- [`SmythBrobyCount2`](@ref)

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PortfolioOptimisers.SmythBrobyCount1 Type

julia

struct SmythBrobyCount1 <: SmythBrobyCovarianceAlgorithm

Implements the first variant of the Smyth-Broby covariance algorithm using vote counts only.

Related

julia

- [`SmythBrobyCovarianceAlgorithm`](@ref)
- [`SmythBrobyCovariance`](@ref)
- [`SmythBrobyCount0`](@ref)
- [`SmythBrobyCount2`](@ref)

source

PortfolioOptimisers.SmythBrobyCount2 Type

julia

struct SmythBrobyCount2 <: SmythBrobyCovarianceAlgorithm

Implements the second variant of the Smyth-Broby covariance algorithm using vote counts only.

Related

julia

- [`SmythBrobyCovarianceAlgorithm`](@ref)
- [`SmythBrobyCovariance`](@ref)
- [`SmythBrobyCount0`](@ref)
- [`SmythBrobyCount1`](@ref)

source

PortfolioOptimisers.SmythBrobyCovariance Type

julia

struct SmythBrobyCovariance{__T_ve, __T_me, __T_pdm, __T_c1, __T_c2, __T_c3, __T_n, __T_alg, __T_ex} <: BaseSmythBrobyCovariance

A flexible container type for configuring and applying Smyth-Broby covariance estimators in PortfolioOptimisers.jl.

SmythBrobyCovariance encapsulates all components required for Smyth-Broby-based covariance or correlation estimation, including the expected returns estimator, variance estimator, positive definite matrix estimator, algorithm parameters, and the specific Smyth-Broby algorithm variant.

Fields

ve: Variance estimator.
me: Expected returns estimator. Used for optionally centering the returns.
pdm: Positive definite matrix estimator.
c1: Zone of confusion parameter.
c2: Zone of indecision lower bound.
c3: Zone of indecision upper bound.
n: Exponent parameter for the Smyth-Broby kernel.
alg: Smyth-Broby covariance algorithm.
ex: Parallel execution strategy.

Constructors

julia

SmythBrobyCovariance(;
    ve::StatsBase.CovarianceEstimator = SimpleVariance(),
    me::AbstractExpectedReturnsEstimator = SimpleExpectedReturns(),
    pdm::Option{<:Posdef} = Posdef(),
    c1::Number = 0.5,
    c2::Number = 0.5,
    c3::Number = 4,
    n::Number = 2,
    alg::SmythBrobyCovarianceAlgorithm = SmythBrobyGerber1(),
    ex::FLoops.Transducers.Executor = ThreadedEx()
) -> SmythBrobyCovariance

Keywords correspond to the struct's fields.

Validation

c3 > c2.

Examples

julia

julia> SmythBrobyCovariance()
SmythBrobyCovariance
   ve ┼ SimpleVariance
      │          me ┼ SimpleExpectedReturns
      │             │   w ┴ nothing
      │           w ┼ nothing
      │   corrected ┴ Bool: true
   me ┼ SimpleExpectedReturns
      │   w ┴ nothing
  pdm ┼ Posdef
      │      alg ┼ UnionAll: NearestCorrelationMatrix.Newton
      │   kwargs ┴ @NamedTuple{}: NamedTuple()
   c1 ┼ Float64: 0.5
   c2 ┼ Float64: 0.5
   c3 ┼ Int64: 4
    n ┼ Int64: 2
  alg ┼ SmythBrobyGerber1()
   ex ┴ Transducers.ThreadedEx{@NamedTuple{}}: Transducers.ThreadedEx()

Related

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Statistics.cov Method

julia

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

Compute the Smyth-Broby covariance matrix.

This method computes the Smyth-Broby covariance matrix for the input data matrix X. The mean and standard deviation vectors are computed using the estimator's expected returns and variance estimators. The Smyth-Broby covariance is then computed via smythbroby.

Arguments

ce: Smyth-Broby covariance estimator.
- ce::SmythBrobyCovariance: Compute the unstandardised Smyth-Broby covariance matrix.
X: Data matrix (observations × assets).
dims: Dimension along which to perform the computation.
kwargs...: Additional keyword arguments passed to the mean and standard deviation estimators.

Returns

sigma::Matrix{<:Number}: The Smyth-Broby covariance matrix.

Validation

dims is either 1 or 2.

Related

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Statistics.cor Method

julia

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

Compute the Smyth-Broby correlation matrix.

This method computes the Smyth-Broby correlation matrix for the input data matrix X. The mean and standard deviation vectors are computed using the estimator's expected returns and variance estimators. The Smyth-Broby correlation is then computed via smythbroby.

Arguments

ce: Smyth-Broby covariance estimator.
- ce::SmythBrobyCovariance: Compute the unstandardised Smyth-Broby correlation matrix.
X: Data matrix (observations × assets).
dims: Dimension along which to perform the computation.
kwargs...: Additional keyword arguments passed to the mean and standard deviation estimators.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix.

Validation

dims is either 1 or 2.

Related

source

PortfolioOptimisers.BaseSmythBrobyCovariance Type

julia

abstract type BaseSmythBrobyCovariance <: BaseGerberCovariance

Abstract supertype for all Smyth-Broby covariance estimators in PortfolioOptimisers.jl.

All concrete and/or abstract types implementing Smyth-Broby covariance estimation algorithms should be subtypes of BaseSmythBrobyCovariance.

Related

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PortfolioOptimisers.SmythBrobyCovarianceAlgorithm Type

julia

abstract type SmythBrobyCovarianceAlgorithm <: AbstractMomentAlgorithm

Abstract supertype for all Smyth-Broby covariance algorithm types in PortfolioOptimisers.jl.

All concrete and/or abstract types implementing specific Smyth-Broby covariance algorithms should be subtypes of SmythBrobyCovarianceAlgorithm.

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

Related

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PortfolioOptimisers.sb_delta Function

julia

sb_delta(ri::Number, rj::Number, n::Number) -> Number

Smyth-Broby kernel function for covariance and correlation computation.

This function computes the kernel value for a pair of asset returns, applying the Smyth-Broby logic for zones of confusion and indecision. It is used to aggregate positive and negative co-movements in Smyth-Broby covariance algorithms. It assumes the returns are centered around zero.

Arguments

ri: Absolute standardised return for asset i.
rj: Absolute standardised return for asset j.
n: Exponent parameter for the kernel.

Returns

score::Number: The computed score for the pair (xi, xj).

Details

Returns (sqrt((1 + ri) * (1 + rj)) / (1 + abs(ri - rj)^n), 1).

Related

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PortfolioOptimisers.smythbroby Function

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBroby0}, X::MatNum, mu::ArrNum,
           sd::ArrNum)

Implements the original Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the original SmythBroby0 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive and negative co-movements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBroby0 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions.
The correlation is computed as (pos - neg) / (pos + neg) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBroby1}, X::MatNum, mu::ArrNum,
           sd::ArrNum)

Implements the first variant of the Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the original SmythBroby1 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive, negative, and neutral co-movements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBroby1 algorithm.
X: Data matrix observations × features if the dims keyword does not exist or dims = 1, features × observations when dims = 2.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, use the returns for assets i and j.
Apply the threshold to classify joint positive, negative, and neutral co-movements.
Use the sb_delta kernel to accumulate positive (pos), negative (neg), and neutral (nn) contributions.
The correlation is computed as (pos - neg) / (pos + neg + nn) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBroby2}, X::MatNum, mu::ArrNum,
           sd::ArrNum)

Implements the second variant of the Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the SmythBroby2 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive and negative co-movements. The resulting matrix is then standardised by the geometric mean of its diagonal elements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBroby2 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, standardised and projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions.
The raw correlation is computed as pos - neg.
The resulting matrix is standardised by dividing each element by the geometric mean of the corresponding diagonal elements.
The matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyGerber0}, X::MatNum,
           mu::ArrNum, sd::ArrNum)

Implements the original Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the SmythBrobyGerber0 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive and negative co-movements, with additional weighting by the count of co-movements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber0 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions, and count the number of positive (cpos) and negative (cneg) co-movements.
The correlation is computed as (pos * cpos - neg * cneg) / (pos * cpos + neg * cneg) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyGerber1}, X::MatNum,
           mu::ArrNum, sd::ArrNum)

Implements the first Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the SmythBrobyGerber1 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive, negative, and neutral co-movements, with additional weighting by the count of co-movements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber1 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive, negative, and neutral co-movements.
Use the sb_delta kernel to accumulate positive (pos), negative (neg), and neutral (nn) contributions, and count the number of positive (cpos), negative (cneg), and neutral (cnn) co-movements.
The correlation is computed as (pos * cpos - neg * cneg) / (pos * cpos + neg * cneg + nn * cnn) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyGerber2}, X::MatNum,
           mu::ArrNum, sd::ArrNum)

Implements the second Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

This method computes the Smyth-Broby correlation or covariance matrix for the input data matrix X using the SmythBrobyGerber2 algorithm. The computation is based on thresholding the data, applying the Smyth-Broby kernel, and aggregating positive and negative co-movements, with additional weighting by the count of co-movements. The resulting matrix is then standardised by the geometric mean of its diagonal elements.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber2 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, standardised and projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions, and count the number of positive (cpos) and negative (cneg) co-movements.
The raw correlation is computed as pos * cpos - neg * cneg.
The resulting matrix is standardised by dividing each element by the geometric mean of the corresponding diagonal elements.
The matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyCount0}, X::MatNum,
           mu::ArrNum, sd::ArrNum)

Implements the original Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber0 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions, and count the number of positive (cpos) and negative (cneg) co-movements.
The correlation is computed as (pos * cpos - neg * cneg) / (pos * cpos + neg * cneg) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyCount1}, X::MatNum,
            mu::ArrNum, sd::ArrNum)

Implements the first Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber1 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive, negative, and neutral co-movements.
Use the sb_delta kernel to accumulate positive (pos), negative (neg), and neutral (nn) contributions, and count the number of positive (cpos), negative (cneg), and neutral (cnn) co-movements.
The correlation is computed as (pos * cpos - neg * cneg) / (pos * cpos + neg * cneg + nn * cnn) if the denominator is nonzero, otherwise zero.
The resulting matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

julia

smythbroby(ce::SmythBrobyCovariance{<:Any, <:Any, <:Any, <:Any, <:Any, <:Any,
                                    <:Any, <:SmythBrobyCount2}, X::MatNum,
            mu::ArrNum, sd::ArrNum)

Implements the second Gerber-style variant of the Smyth-Broby covariance/correlation algorithm.

Arguments

ce: Smyth-Broby covariance estimator configured with the SmythBrobyGerber2 algorithm.
X: Data matrix (observations × assets).
sd: Vector of standard deviations for each asset, used for scaling and thresholding.

Returns

rho::Matrix{<:Number}: The Smyth-Broby correlation matrix, standardised and projected to be positive definite using the estimator's pdm field.

Details

The algorithm proceeds as follows:

For each pair of assets (i, j), iterate over all observations.
For each observation, compute the centered and scaled returns for assets i and j.
Apply the threshold to classify joint positive and negative co-movements.
Use the sb_delta kernel to accumulate positive (pos) and negative (neg) contributions, and count the number of positive (cpos) and negative (cneg) co-movements.
The raw correlation is computed as pos * cpos - neg * cneg.
The resulting matrix is standardised by dividing each element by the geometric mean of the corresponding diagonal elements.
The matrix is projected to the nearest positive definite matrix using posdef!.

Related

source

Smyth-Broby Covariance ​

Smyth-Broby Covariance