# Scatter Matrix and Covariance

This package implements functions for computing scatter matrix, as well as weighted covariance matrix.

StatsBase.scattermatFunction
scattermat(X, [wv::AbstractWeights]; mean=nothing, dims=1)

Compute the scatter matrix, which is an unnormalized covariance matrix. A weighting vector wv can be specified to weight the estimate.

Arguments

• mean=nothing: a known mean value. nothing indicates that the mean is unknown, and the function will compute the mean. Specifying mean=0 indicates that the data are centered and hence there's no need to subtract the mean.
• dims=1: the dimension along which the variables are organized. When dims = 1, the variables are considered columns with observations in rows; when dims = 2, variables are in rows with observations in columns.
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Statistics.covFunction
cov(X, w::AbstractWeights, vardim=1; mean=nothing, corrected=false)

Compute the weighted covariance matrix. Similar to var and std the biased covariance matrix (corrected=false) is computed by multiplying scattermat(X, w) by $\frac{1}{\sum{w}}$ to normalize. However, the unbiased covariance matrix (corrected=true) is dependent on the type of weights used:

• AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$
• FrequencyWeights: $\frac{1}{\sum{w} - 1}$
• ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)
• Weights: ArgumentError (bias correction not supported)
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Statistics.covMethod
cov(ce::CovarianceEstimator, x::AbstractVector; mean=nothing)

Compute a variance estimate from the observation vector x using the estimator ce.

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Statistics.covMethod
cov(ce::CovarianceEstimator, x::AbstractVector, y::AbstractVector)

Compute the covariance of the vectors x and y using estimator ce.

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Statistics.covMethod
cov(ce::CovarianceEstimator, X::AbstractMatrix, [w::AbstractWeights]; mean=nothing, dims::Int=1)

Compute the covariance matrix of the matrix X along dimension dims using estimator ce. A weighting vector w can be specified. The keyword argument mean can be:

• nothing (default) in which case the mean is estimated and subtracted from the data X,
• a precalculated mean in which case it is subtracted from the data X. Assuming size(X) is (N,M), mean can either be:
• when dims=1, an AbstractMatrix of size (1,M),
• when dims=2, an AbstractVector of length N or an AbstractMatrix of size (N,1).
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Statistics.varMethod
var(ce::CovarianceEstimator, x::AbstractVector; mean=nothing)

Compute the variance of the vector x using the estimator ce.

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Statistics.stdMethod
std(ce::CovarianceEstimator, x::AbstractVector; mean=nothing)

Compute the standard deviation of the vector x using the estimator ce.

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Statistics.corFunction
cor(X, w::AbstractWeights, dims=1)

Compute the Pearson correlation matrix of X along the dimension dims with a weighting w .

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cor(ce::CovarianceEstimator, x::AbstractVector, y::AbstractVector)

Compute the correlation of the vectors x and y using estimator ce.

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cor(
ce::CovarianceEstimator, X::AbstractMatrix, [w::AbstractWeights];
mean=nothing, dims::Int=1
)

Compute the correlation matrix of the matrix X along dimension dims using estimator ce. A weighting vector w can be specified. The keyword argument mean can be:

• nothing (default) in which case the mean is estimated and subtracted from the data X,
• a precalculated mean in which case it is subtracted from the data X. Assuming size(X) is (N,M), mean can either be:
• when dims=1, an AbstractMatrix of size (1,M),
• when dims=2, an AbstractVector of length N or an AbstractMatrix of size (N,1).
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StatsBase.mean_and_covFunction
mean_and_cov(x, [wv::AbstractWeights,] vardim=1; corrected=false) -> (mean, cov)

Return the mean and covariance matrix as a tuple. A weighting vector wv can be specified. vardim that designates whether the variables are columns in the matrix (1) or rows (2). Finally, bias correction is applied to the covariance calculation if corrected=true. See cov documentation for more details.

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StatsBase.cov2corFunction
cov2cor(C::AbstractMatrix, [s::AbstractArray])

Compute the correlation matrix from the covariance matrix C and, optionally, a vector of standard deviations s. Use StatsBase.cov2cor! for an in-place version.

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StatsBase.cor2covFunction
cor2cov(C, s)

Compute the covariance matrix from the correlation matrix C and a vector of standard deviations s. Use StatsBase.cor2cov! for an in-place version.

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StatsBase.SimpleCovarianceType
SimpleCovariance(;corrected::Bool=false)

Simple covariance estimator. Estimation calls cov(x; corrected=corrected), cov(x, y; corrected=corrected) or cov(X, w, dims; corrected=corrected) where x, y are vectors, X is a matrix and w is a weighting vector.

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