Variance and Standard Deviation

PortfolioOptimisers.SimpleVariance Type

struct SimpleVariance{T1, T2, T3} <: AbstractVarianceEstimator
    me::T1
    w::T2
    corrected::T3
end

A flexible variance estimator for PortfolioOptimisers.jl supporting optional expected returns estimators, observation weights, and bias correction.

SimpleVariance enables users to specify an expected returns estimator (for mean-centering), optional observation weights, and whether to apply bias correction (Bessel's correction). This type is suitable for both unweighted and weighted variance estimation workflows.

Fields

• me: Optional expected returns estimator. If nothing, the mean is not estimated.

• w: Optional observation weights. If nothing, the estimator is unweighted.

• corrected: Whether to apply Bessel's correction (unbiased variance).

Constructor

SimpleVariance(;
               me::Union{Nothing, <:AbstractExpectedReturnsEstimator} = SimpleExpectedReturns(),
               w::Union{Nothing, <:AbstractWeights} = nothing, corrected::Bool = true)

Keyword arguments correspond to the fields above.

Validation

• If w is provided, !isempty(w).

Examples

julia> using StatsBase

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

julia> w = Weights([0.2, 0.3, 0.5]);

julia> SimpleVariance(; w = w, corrected = false)
SimpleVariance
         me ┼ SimpleExpectedReturns
            │   w ┴ nothing
          w ┼ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.3, 0.5]
  corrected ┴ Bool: false

Related

• AbstractVarianceEstimator

• AbstractExpectedReturnsEstimator

• SimpleExpectedReturns

• StatsBase.AbstractWeights

• std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• std(ve::SimpleVariance, X::AbstractVector; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractVector; mean = nothing)

source

Statistics.std Method

std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

Compute the standard deviation using a SimpleVariance estimator for an array.

This method computes the standard deviation of the input array X using the configuration specified in ve, including optional mean-centering (via ve.me), observation weights (ve.w), and bias correction (ve.corrected). If a mean is not provided, it is estimated using the expected returns estimator in ve.me.

Arguments

• ve: Variance estimator specifying the mean estimator, weights, and bias correction.

• X: Data array (vector or matrix) for which to compute the standard deviation.

• dims: Dimension along which to compute the standard deviation (for matrices).

• mean: Optional mean value or vector for centering. If not provided, estimated using ve.me.

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

Returns

• sd::Vector{<:Real}: Standard deviation vector of X.

Examples

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

julia> Xmat = [1.0 2.0; 3.0 4.0];

julia> std(sv, Xmat; dims = 1)
1×2 Matrix{Float64}:
 1.41421  1.41421

Related

• SimpleVariance

• Statistics.std

• std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• std(ve::SimpleVariance, X::AbstractVector; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractVector; mean = nothing)

source

Statistics.std Method

std(ve::SimpleVariance, X::AbstractVector; mean = nothing)

Compute the standard deviation using a SimpleVariance estimator for a vector.

This method computes the standard deviation of the input vector X using the configuration specified in ve, including optional observation weights (ve.w) and bias correction (ve.corrected). If a mean is provided, it is used for centering; otherwise, the default mean is used.

Arguments

• ve: Variance estimator specifying weights and bias correction.

• X: Data vector for which to compute the standard deviation.

• mean: Optional Mean value for centering. If not provided, the default mean is used.

Returns

• sd::Real: Standard deviation of X.

Examples

julia> using StatsBase

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

julia> X = [1.0, 2.0, 3.0];

julia> std(sv, X)
1.0

julia> w = Weights([0.2, 0.3, 0.5]);

julia> svw = SimpleVariance(; w = w, corrected = false)
SimpleVariance
         me ┼ SimpleExpectedReturns
            │   w ┴ nothing
          w ┼ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.3, 0.5]
  corrected ┴ Bool: false

julia> std(svw, X)
0.7810249675906654

Related

• SimpleVariance

• Statistics.std

• std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractVector; mean = nothing)

source

Statistics.var Method

var(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

Compute the variance using a SimpleVariance estimator for an array.

This method computes the variance of the input array X using the configuration specified in ve, including optional mean-centering (via ve.me), observation weights (ve.w), and bias correction (ve.corrected). If a mean is not provided, it is estimated using the expected returns estimator in ve.me.

Arguments

• ve: Variance estimator specifying the mean estimator, weights, and bias correction.

• X: Data array (vector or matrix) for which to compute the variance.

• dims: Dimension along which to compute the variance (for matrices).

• mean: Optional mean value or vector for centering. If not provided, estimated using ve.me.

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

Returns

• v::Vector{<:Real}: Variance vector of X.

Examples

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

julia> Xmat = [1.0 2.0; 3.0 4.0];

julia> var(sv, Xmat; dims = 1)
1×2 Matrix{Float64}:
 2.0  2.0

Related

• SimpleVariance

• Statistics.var

• std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• std(ve::SimpleVariance, X::AbstractVector; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractVector; mean = nothing)

source

Statistics.var Method

var(ve::SimpleVariance, X::AbstractVector; mean = nothing)

Compute the variance using a SimpleVariance estimator for a vector.

This method computes the variance of the input vector X using the configuration specified in ve, including optional observation weights (ve.w) and bias correction (ve.corrected). If a mean is provided, it is used for centering; otherwise, the default mean is used.

Arguments

• ve: Variance estimator specifying weights and bias correction.

• X: Data vector for which to compute the variance.

• mean: Optional mean value for centering. If not provided, the default mean is used.

Returns

• v::Real: Variance of X.

Examples

julia> using StatsBase

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

julia> X = [1.0, 2.0, 3.0];

julia> var(sv, X)
1.0

julia> w = Weights([0.2, 0.3, 0.5]);

julia> svw = SimpleVariance(; w = w, corrected = false)
SimpleVariance
         me ┼ SimpleExpectedReturns
            │   w ┴ nothing
          w ┼ StatsBase.Weights{Float64, Float64, Vector{Float64}}: [0.2, 0.3, 0.5]
  corrected ┴ Bool: false

julia> var(svw, X)
0.61

Related

• SimpleVariance

• Statistics.var

• std(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

• std(ve::SimpleVariance, X::AbstractVector; dims::Int = 1, mean = nothing, kwargs...)

• var(ve::SimpleVariance, X::AbstractMatrix; dims::Int = 1, mean = nothing, kwargs...)

source