Covariance [v,w]
gives the covariance between the vectors v and w.
Covariance [a,b]
gives the cross-covariance matrix for the matrices a and b.
Covariance [a]
gives the auto-covariance matrix for observations in matrix a.
Covariance [dist]
gives the auto-covariance matrix for the multivariate symbolic distribution dist.
Covariance [dist,i,j]
gives the (i,j)^(th) covariance for the multivariate symbolic distribution dist.
Covariance
Covariance [v,w]
gives the covariance between the vectors v and w.
Covariance [a,b]
gives the cross-covariance matrix for the matrices a and b.
Covariance [a]
gives the auto-covariance matrix for observations in matrix a.
Covariance [dist]
gives the auto-covariance matrix for the multivariate symbolic distribution dist.
Covariance [dist,i,j]
gives the (i,j)^(th) covariance for the multivariate symbolic distribution dist.
Details
- Covariance is typically used to measure covariation, i.e. whether one variable tends to vary similarly to another.
- Covariance [v,w] gives the unbiased estimate of the covariance between v and w.
- For vectors and of length , the covariance estimate Covariance [v,w] is given by with mu^^_v=Mean [v].
- For matrices and with dimensions and and columns indexed as and , respectively, Covariance [a,b] is a matrix with elements given by :
- where is an -vector of ones, is Mean [a] and is Mean [b].
- For a matrix a with columns, Covariance [a] is a matrix given by Covariance [a, a].
- Covariance works with any vector that is VectorQ or matrix that is MatrixQ .
- Covariance [dist,i,j] gives Expectation [(xi-μi)(xj-μj),{x1,x2,…}∈dist], where μi is the i^(th) component of the mean of dist. »
- Covariance [dist] gives a covariance matrix with the (i,j)^(th) entry given by Covariance [dist,i,j]. »
Examples
open all close allBasic Examples (3)
Covariance between two vectors:
Covariance matrix for a matrix:
Covariance matrix for two matrices:
Scope (13)
Data (8)
Exact input yields exact output:
Approximate input yields approximate output:
Covariance between vectors of complexes:
Works with large arrays:
A structured array can be used (see the guide):
Find the covariance for data involving quantities:
Covariance between lists of dates:
Covariance between matrices of times:
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:
See Also
Variance Correlation AbsoluteCorrelation CovarianceFunction CentralMoment Expectation PositiveSemidefiniteMatrixQ
Function Repository: TotalVariation WhiteningTransform
History
Introduced in 2007 (6.0) | Updated in 2010 (8.0) ▪ 2023 (13.3) ▪ 2024 (14.1)
Text
Wolfram Research (2007), Covariance, Wolfram Language function, https://reference.wolfram.com/language/ref/Covariance.html (updated 2024).
CMS
Wolfram Language. 2007. "Covariance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. 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
BibTeX
@misc{reference.wolfram_2025_covariance, author="Wolfram Research", title="{Covariance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Covariance.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_covariance, organization={Wolfram Research}, title={Covariance}, year={2024}, url={https://reference.wolfram.com/language/ref/Covariance.html}, note=[Accessed: 05-December-2025]}