Covariance
Covariance[v1,v2]
gives the covariance between the vectors v1 and v2.
Covariance[m]
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[(xi-μi)(xj-μj),{x1,x2,…}∈dist], where μi 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
open allclose allBasic Examples (3)
Scope (12)
Data (7)
Exact input yields exact output:
Approximate input yields approximate output:
Covariance between vectors of complexes:
SparseArray data can be used:
Find the covariance of WeightedData:
Distributions and Processes (5)
Covariance for a continuous multivariate distribution:
Covariance for a discrete multivariate distribution:
Covariance for derived distributions:
Covariance matrix for a random process at times s and t:
Covariance matrix for TemporalData at times and
:
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:
Text
Wolfram Research (2007), Covariance, Wolfram Language function, https://reference.wolfram.com/language/ref/Covariance.html (updated 2010).
CMS
Wolfram Language. 2007. "Covariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2010. https://reference.wolfram.com/language/ref/Covariance.html.
APA
Wolfram Language. (2007). Covariance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Covariance.html