WOLFRAM

Wolfram Language & System Documentation Center
Wolfram Language Home Page »

Correlation
✖
Correlation

Updated in 14.1

✖

gives the correlation between the vectors v and w.

✖

gives the cross-correlation matrix for the matrices a and b.

✖

gives the auto-correlation matrix for observations in matrix a.

✖
Correlation[dist]

gives the correlation matrix for the multivariate symbolic distribution dist.

✖
Correlation[dist,i,j]

gives the (i,j)^(th) correlation for the multivariate symbolic distribution dist.

Details

Examples

open allclose all

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

Correlation between two vectors:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-bgsw6i
Out[1]=1

Real values:

In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-ebsvf6
Out[2]=2

Correlation matrix for a matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-bobjfi
Out[1]=1

Real values:

In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-3zvd
Out[2]=2

Correlation matrix for two matrices:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-de1ufn

Real values:

In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-ib0k4r
Out[2]=2

Scope  (14)Survey of the scope of standard use cases

Data  (8)

Exact input yields exact output:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-eta06h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-mszqlu
Out[2]=2

Approximate input yields approximate output:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-bxeync
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-04fgr
Out[2]=2

Correlation between vectors of complexes:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-dbahpf
Out[1]=1

Works with large arrays:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-kt50m4
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-cce1ir
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-v51wbj
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-eoq11l
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-6l5n0m

Find the correlation for data involving quantities:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-rui6zn
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-e8c21s
Out[2]=2

Correlation between lists of dates:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-vvw1aa
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-4bj6vf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-p2rqb3
Out[3]=3

Correlation between matrices of times:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-vu0sut

Distributions and Processes  (6)

Correlation for a continuous multivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-fnv00k
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-tekwa
Out[2]=2

Correlation for a discrete multivariate distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-epq4i
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-fmicv5
Out[2]=2

Correlation controls the orientation and sharpness of a multivariate probability distribution:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-djrchi
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-qd90y8
Out[2]=2

Correlation for derived distributions:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-c3a9y
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-cy97of
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-etkwz

Data distribution:

In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-hxjash
In[5]:=5
✖
https://wolfram.com/xid/0bimg2hm-gm1a47
In[6]:=6
✖
https://wolfram.com/xid/0bimg2hm-edj4ah

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

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-hv1ygl

Correlation matrix for TemporalData at times and :

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-butvlp
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-k1q1tr

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

Compute the correlation of two financial time series:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-zdbiqi
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-3a8yoh
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-2q6d2
Out[3]=3

Correlation can be used to measure linear association:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-bcl86
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-g1rlnp
Out[2]=2

Correlation can only detect monotonic relationships:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-7nvfr
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-nazb6e
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-j9qi3
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-gtfa0z
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0bimg2hm-jor85u
Out[5]=5

Properties & Relations  (7)Properties of the function, and connections to other functions

The correlation matrix is symmetric and positive semidefinite:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-31bc2
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-mmzf9d
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-c9urx7
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-o3dafo
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-ce9r5a
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-j71cv
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-b8mdab
In[5]:=5
✖
https://wolfram.com/xid/0bimg2hm-bj1ijg
In[6]:=6
✖
https://wolfram.com/xid/0bimg2hm-b703nn
Out[6]=6

Correlation and AbsoluteCorrelation agree for zero mean and unit marginal variances:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-b4nkgn
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-cbk1sj
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-fi0mpw
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-dqzrcn
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bimg2hm-h10yhb
Out[5]=5

SpearmanRho is Correlation applied to ranks:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-cbzhhv
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-pzf7jv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-j0lyo
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-bvkzz
Out[4]=4

CorrelationFunction for a process is the off-diagonal entry in the correlation matrix:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-c8x3e0
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-cs2zy3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-hpmt
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bimg2hm-dmok3u
Out[4]=4

Correlation and Covariance are the same for standardized vectors:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-bwn1ui
In[2]:=2
✖
https://wolfram.com/xid/0bimg2hm-bhgwb3
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bimg2hm-7gmaf
Out[3]=3

The diagonal elements of a correlation matrix are equal to 1:

In[1]:=1
✖
https://wolfram.com/xid/0bimg2hm-l00b5
Out[1]=1
Wolfram Research (2007), Correlation, Wolfram Language function, https://reference.wolfram.com/language/ref/Correlation.html (updated 2024).
✖
Wolfram Research (2007), Correlation, Wolfram Language function, https://reference.wolfram.com/language/ref/Correlation.html (updated 2024).

Text

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

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

CMS

Wolfram Language. 2007. "Correlation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Correlation.html.

✖
Wolfram Language. 2007. "Correlation." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Correlation.html.

APA

Wolfram Language. (2007). Correlation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Correlation.html

✖
Wolfram Language. (2007). Correlation. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Correlation.html

BibTeX

@misc{reference.wolfram_2025_correlation, author="Wolfram Research", title="{Correlation}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Correlation.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_correlation, author="Wolfram Research", title="{Correlation}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Correlation.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_correlation, organization={Wolfram Research}, title={Correlation}, year={2024}, url={https://reference.wolfram.com/language/ref/Correlation.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_correlation, organization={Wolfram Research}, title={Correlation}, year={2024}, url={https://reference.wolfram.com/language/ref/Correlation.html}, note=[Accessed: 29-March-2025 ]}