WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

Covariance
✖
Covariance

Updated in 14.1

✖
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.

✖

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 allclose all

Basic Examples  (3)Summary of the most common use cases

Covariance between two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bgsw6i
Out[1]=1

Covariance matrix for a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bobjfi

Covariance matrix for two matrices:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-de1ufn

Scope  (13)Survey of the scope of standard use cases

Data  (8)

Exact input yields exact output:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-eta06h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-mszqlu
Out[2]=2

Approximate input yields approximate output:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bxeync
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-04fgr
Out[2]=2

Covariance between vectors of complexes:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-dbahpf
Out[1]=1

Works with large arrays:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-kt50m4
Out[1]=1

A structured array can be used (see the guide):

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-h2htud
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-v51wbj
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-eoq11l
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-6l5n0m

Find the covariance for data involving quantities:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-rui6zn
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-e8c21s
Out[2]=2

Covariance between lists of dates:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-vvw1aa
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-4bj6vf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-p2rqb3
Out[3]=3

Covariance between matrices of times:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-vu0sut

Distributions and Processes  (5)

Covariance for a continuous multivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-fnv00k
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-tekwa
Out[2]=2

Covariance for a discrete multivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-epq4i
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-fmicv5
Out[2]=2

Covariance for derived distributions:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-c3a9y
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-cy97of
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-etkwz
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-tmnd9
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0b0keq7yl5a-kla8wx
Out[5]=5

Data distribution:

In[6]:=6
✖
https://wolfram.com/xid/0b0keq7yl5a-hxjash
In[7]:=7
✖
https://wolfram.com/xid/0b0keq7yl5a-gm1a47
In[8]:=8
✖
https://wolfram.com/xid/0b0keq7yl5a-edj4ah

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

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-hv1ygl

Covariance matrix for TemporalData at times and :

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-butvlp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-k1q1tr

Compare to the covariance of the process slice:

In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-m9xt34
Out[3]=3

Applications  (3)Sample problems that can be solved with this function

Compute the covariance of two financial time series:

In[18]:=18
✖
https://wolfram.com/xid/0b0keq7yl5a-zdbiqi
In[17]:=17
✖
https://wolfram.com/xid/0b0keq7yl5a-3a8yoh
In[19]:=19
✖
https://wolfram.com/xid/0b0keq7yl5a-2q6d2
Out[19]=19

Covariance can be used to measure linear association:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bcl86
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-g1rlnp
Out[2]=2

Covariance can only detect monotonic relationships:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-7nvfr
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-nazb6e
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-j9qi3
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-gtfa0z
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0b0keq7yl5a-jor85u
Out[5]=5

Properties & Relations  (9)Properties of the function, and connections to other functions

The covariance matrix is symmetric and positive semidefinite:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-31bc2
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-mmzf9d
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-c9urx7
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-o3dafo
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-d6zye
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-dz7nph
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-b4nkgn
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-cbk1sj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-dqzrcn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-h10yhb
Out[4]=4

SpearmanRho is related to Covariance applied to ranks:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-cbzhhv
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-pzf7jv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-j0lyo
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-bvkzz
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-c8x3e0
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-cs2zy3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-hpmt
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0b0keq7yl5a-dmok3u
Out[4]=4

Covariance and Correlation are the same for standardized vectors:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bwn1ui
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-bhgwb3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-7gmaf
Out[3]=3

The covariance of a list with itself is the variance:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-cnm6n0
Out[1]=1

The diagonal of a covariance matrix is the variance:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-evvxao
In[2]:=2
✖
https://wolfram.com/xid/0b0keq7yl5a-qppaj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0b0keq7yl5a-ydmht
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-bo1p6w
Out[1]=1

Neat Examples  (1)Surprising or curious use cases

Compute the covariance for a GCD array:

In[1]:=1
✖
https://wolfram.com/xid/0b0keq7yl5a-ish8bj
Out[1]=1
Wolfram Research (2007), Covariance, Wolfram Language function, https://reference.wolfram.com/language/ref/Covariance.html (updated 2024).
✖
Wolfram Research (2007), Covariance, Wolfram Language function, https://reference.wolfram.com/language/ref/Covariance.html (updated 2024).

Text

Wolfram Research (2007), Covariance, Wolfram Language function, https://reference.wolfram.com/language/ref/Covariance.html (updated 2024).

✖
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.

✖
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

✖
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: 01-April-2025 ]}

✖
@misc{reference.wolfram_2025_covariance, author="Wolfram Research", title="{Covariance}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Covariance.html}", note=[Accessed: 01-April-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: 01-April-2025 ]}

✖
@online{reference.wolfram_2025_covariance, organization={Wolfram Research}, title={Covariance}, year={2024}, url={https://reference.wolfram.com/language/ref/Covariance.html}, note=[Accessed: 01-April-2025 ]}