Simple variance and standard deviation

The variance is used throughout the library, it can be used as part of the expected return, covariance estimation, performance analysis, and constraint generation. It is trivial to compute the standard deviation from the variance, so we provide those too.

PortfolioOptimisers.SimpleVariance Type

julia

struct SimpleVariance{__T_me, __T_w, __T_corrected} <: AbstractVarianceEstimator

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

Fields

me: Optional expected returns estimator. It is not needed when used on a vector. If nothing and used on a matrix, defaults to SimpleExpectedReturns.
w: Observation weights vector observations × 1.
corrected: Whether to apply Bessel's correction.

Constructors

julia

SimpleVariance(;
    me::Option{<:AbstractExpectedReturnsEstimator} = SimpleExpectedReturns(),
    w::Option{<:ObsWeights} = nothing,
    corrected::Bool = true
) -> SimpleVariance

Keywords correspond to the struct's fields.

Validation

If w is not nothing, !isempty(w).

Examples

julia

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

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

Related

source

PortfolioOptimisers.factory Method

julia

factory(
    ve::SimpleVariance,
    w::ObsWeights
) -> SimpleVariance

Return a new SimpleVariance estimator with the specified observation weights.

Arguments

ve: Variance estimator.
w: Observation weights vector observations × 1.

Returns

ve: New variance estimator of the same type as the argument, with the new weights applied.

Details

The mean estimator is updated using factory(ve.me, w) for consistency.
Sets w to the new weights.
The bias correction flag is preserved from the original estimator.

Examples

julia

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

julia> factory(sv, StatsBase.Weights([0.2, 0.3, 0.5]))
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

Related

source

Statistics.std Method

julia

Statistics.std(
    ve::SimpleVariance,
    X::MatNum;
    dims::Int = 1,
    mean = nothing,
    kwargs...,
) -> ArrNum

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

This method computes the standard deviation of the input matrix X using the configuration specified in ve.

Mathematical definition

Unweighted:

\begin{aligned} {\hat{σ}}_{j} & = \sqrt{{\hat{σ}}_{j}^{2}} . \end{aligned}

Where:

${\hat{σ}}_{j}$ : Standard deviation of asset $j$ .
${\hat{σ}}_{j}^{2}$ : Variance of asset $j$ .

For corrected = true:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{1}{T - 1} \sum_{t = 1}^{T} (r_{t j} - {\hat{μ}}_{j})^{2} . \end{aligned}

For corrected = false:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{1}{T} \sum_{t = 1}^{T} (r_{t j} - {\hat{μ}}_{j})^{2} . \end{aligned}

Weighted:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{\sum_{t = 1}^{T} w_{t} (r_{t j} - {\hat{μ}}_{j})^{2}}{\sum_{t = 1}^{T} w_{t} - c} . \end{aligned}

Where:

${\hat{σ}}_{j}^{2}$ : Estimated variance of asset $j$ .
$r_{t j}$ : Return of asset $j$ at time $t$ .
${\hat{μ}}_{j}$ : Estimated mean of asset $j$ .
$T$ : Number of observations.
$w_{t}$ : Observation weight at time $t$ .
$c$ : Bias correction factor.

Arguments

ve: Variance estimator.
X: Data matrix observations × features if the dims keyword does not exist or dims = 1, features × observations when dims = 2.
dims: Dimension along which to perform the computation.
mean: Optional mean value to use for centering.
kwargs...: Additional keyword arguments passed to the mean estimator.

Returns

sd::ArrNum: Standard deviation vector of X, reshaped to be consistent with the dimension along which the value is computed.

Examples

julia

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

source

Statistics.std Method

julia

std(
    ve::SimpleVariance{Nothing},
    X::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}};
    dims,
    mean,
    kwargs...
) -> Any

SimpleVariance{Nothing} overload of std(ve::SimpleVariance, X::MatNum; dims::Int = 1, mean = nothing, kwargs...). Uses SimpleExpectedReturns to compute the mean when none is provided, ignoring the me field.

source

Statistics.std Method

julia

Statistics.std(
    ve::SimpleVariance,
    X::VecNum;
    mean = nothing
) -> Number

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.

Arguments

ve: Variance estimator.
X: Data vector observations × 1.
mean: Optional mean value to use for centering.

Returns

vr::Number: Standard deviation of X

Examples

julia

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> svw = SimpleVariance(; w = StatsBase.Weights([0.2, 0.3, 0.5]), 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

source

Statistics.var Method

julia

Statistics.var(
    ve::SimpleVariance,
    X::MatNum;
    dims::Int = 1,
    mean = nothing,
    kwargs...
) -> ArrNum

Compute the variance using a SimpleVariance estimator for a matrix.

This method computes the variance of the input matrix X using the configuration specified in ve.

Mathematical definition

Unweighted, corrected = true:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{1}{T - 1} \sum_{t = 1}^{T} (r_{t j} - {\hat{μ}}_{j})^{2} . \end{aligned}

Unweighted, corrected = false:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{1}{T} \sum_{t = 1}^{T} (r_{t j} - {\hat{μ}}_{j})^{2} . \end{aligned}

Weighted:

\begin{aligned} {\hat{σ}}_{j}^{2} & = \frac{\sum_{t = 1}^{T} w_{t} (r_{t j} - {\hat{μ}}_{j})^{2}}{\sum_{t = 1}^{T} w_{t} - c} . \end{aligned}

Where:

${\hat{σ}}_{j}^{2}$ : Estimated variance of asset $j$ .
$r_{t j}$ : Return of asset $j$ at time $t$ .
${\hat{μ}}_{j}$ : Estimated mean of asset $j$ .
$T$ : Number of observations.
$w_{t}$ : Observation weight at time $t$ .
$c$ : Bias correction factor.

Arguments

ve: Variance estimator.
X: Data matrix observations × features if the dims keyword does not exist or dims = 1, features × observations when dims = 2.
dims: Dimension along which to perform the computation.
mean: Optional mean value to use for centering.
kwargs...: Additional keyword arguments passed to the mean estimator.

Returns

vr::ArrNum: Variance vector of X, reshaped to be consistent with the dimension along which the value is computed.

Examples

julia

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

source

Statistics.var Method

julia

var(
    ve::SimpleVariance{Nothing},
    X::AbstractMatrix{<:Union{var"#s20", var"#s19"} where {var"#s20"<:Number, var"#s19"<:AbstractJuMPScalar}};
    dims,
    mean,
    kwargs...
) -> Any

SimpleVariance{Nothing} overload of var(ve::SimpleVariance, X::MatNum; dims::Int = 1, mean = nothing, kwargs...). Uses SimpleExpectedReturns to compute the mean when none is provided, ignoring the me field.

source

Statistics.var Method

julia

Statistics.var(
    ve::SimpleVariance,
    X::VecNum;
    mean = nothing
) -> Number

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.

Arguments

ve: Variance estimator.
X: Data vector observations × 1.
mean: Optional mean value to use for centering.

Returns

vr::Number: Variance of X

Examples

julia

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> svw = SimpleVariance(; w = StatsBase.Weights([0.2, 0.3, 0.5]), 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

source

PortfolioOptimisers.port_opt_view Method

julia

port_opt_view(
    ve::SimpleVariance,
    i,
    args...
) -> SimpleVariance{__T_me, __T_w, Bool} where {__T_me, __T_w}

Gets the view of the simple variance for the i-th element(s).

Arguments

ve: Variance estimator.
i: Index or indices to view.

Returns

ve: New variance estimator of the same type as the argument, for the new view.

Related

SimpleVariance

source

Simple variance and standard deviation ​

Simple variance and standard deviation