Covariance

Covariance[v₁,v₂]

gives the covariance between the vectors v₁ and v₂.

gives the sample covariance matrix for observations in matrix m.

Covariance[m₁,m₂]

gives the covariance matrix for the matrices m₁ and m₂.

Covariance[dist]

gives the covariance matrix for the multivariate symbolic distribution dist.

Covariance[dist,i,j]

gives the (i,j) covariance for the multivariate symbolic distribution dist.

Details

Covariance[v₁,v₂] gives the unbiased estimate of the covariance between v₁ and v₂.
The lists v₁ and v₂ must be the same length.
Covariance[v₁,v₂] is equivalent to (v₁-Mean[v₁]). Conjugate[v₂-Mean[v₂]]/(Length[v₁]-1).
For a matrix m with columns, Covariance[m] is a × matrix of the covariances between columns of m.
For an × matrix m₁ and an × matrix m₂, Covariance[m₁,m₂] is a × matrix of the covariances between columns of m₁ and columns of m₂.
Covariance works with SparseArray objects.
Covariance[dist,i,j] gives Expectation[(x_i-μ_i)(x_j-μ_j),{x₁,x₂,…}∈dist], where μ_i is the i component of the mean of dist.
Covariance[dist] gives a covariance matrix with the (i,j) entry given by Covariance[dist,i,j].

Examples

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Basic Examples (3)

Covariance between two vectors:

Covariance matrix for a matrix:

Covariance matrix for two matrices:

Scope (12)

Data (7)

Exact input yields exact output:

Approximate input yields approximate output:

Covariance between vectors of complexes:

Works with large arrays:

SparseArray data can be used:

Find the covariance of WeightedData:

Find the covariance for data involving quantities:

Distributions and Processes (5)

Covariance for a continuous multivariate distribution:

Covariance for a discrete multivariate distribution:

Covariance for derived distributions:

Data distribution:

Covariance matrix for a random process at times s and t:

Covariance matrix for TemporalData at times and :

Compare to the covariance of the process slice:

Applications (3)

Compute the covariance of two financial time series:

Covariance can be used to measure linear association:

Covariance can only detect monotonic relationships:

HoeffdingD can be used to detect a variety of dependence structures:

Properties & Relations (9)

The covariance matrix is symmetric and positive semidefinite:

A covariance matrix scaled by standard deviations is a correlation matrix:

Covariance and AbsoluteCorrelation are the same for a distribution with zero mean:

SpearmanRho is related to Covariance applied to ranks:

CovarianceFunction for a process is the off-diagonal entry in the covariance matrix:

Covariance and Correlation are the same for standardized vectors:

The covariance of a list with itself is the variance:

The diagonal of a covariance matrix is the variance:

The covariance tends to be large only on the diagonal of a random matrix:

Neat Examples (1)

Compute the covariance for a GCD array:

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