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Mathematica > Data Manipulation > Statistical Data Analysis > Descriptive Statistics > Covariance >

Mathematica > Mathematics and Algorithms > Statistical Data Analysis > Descriptive Statistics > Covariance >

BUILT-IN MATHEMATICA SYMBOL

Covariance

gives the covariance between the vectors

and

Covariance[m]
gives the covariance matrix for the matrix m.

Covariance

gives the covariance matrix for the matrices

and

Covariance[dist]
gives the covariance matrix for the multivariate symbolic distribution dist.

Covariance

gives the

covariance for the multivariate symbolic distribution dist.

MORE INFORMATION

Covariance gives the unbiased estimate of the covariance between and .

The lists and must be the same length.

Covariance 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 and an × matrix , Covariance is a × matrix of the covariances between columns of and columns of .

Covariance works with SparseArray objects.

EXAMPLES

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Basic Examples (3)

Covariance between two vectors:

Covariance matrix for a matrix:

Covariance matrix for two matrices:

Covariance between two vectors:

In[1]:=

Out[1]=

Covariance matrix for a matrix:

In[1]:=

Out[1]//MatrixForm=

Covariance matrix for two matrices:

In[1]:=

Out[1]//MatrixForm=

Scope (4)

Covariances for machine-precision reals:

Use arbitrary precision:

Find the covariance between vectors of complexes:

Compute the covariance of a SparseArray:

Applications (1)

Compute the covariance of two financial time series:

Properties & Relations (4)

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

The covariance of a list with itself is the variance:

The diagonal of a covariance matrix is the variance:

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

Neat Examples (1)

Compute the covariance for a GCD array: