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Correlation

Correlation[v,w]

gives the correlation between the vectors v and w.

Correlation[a,b]

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) correlation for the multivariate symbolic distribution dist.

Details

Correlation is typically used to measure covariation, i.e. whether one variable tends to vary similarly to another.
For vectors, the correlation estimate Correlation[v,w] is given by with σ_{v w}=Covariance[v,w] and σ_v=StandardDeviation[v].
The correlation is a normalized covariance with .

For matrices and with dimensions and and columns indexed as and , respectively, Correlation[a,b] is a matrix with elements given by :

where Σ_{a b}=Covariance[a,b] and σ_a=StandardDeviation[a] etc.
For a matrix a with columns, Correlation[a] is a matrix given by Correlation[a, a].
Correlation works with any vector that is VectorQ or matrix that is MatrixQ.
Correlation[dist,i,j] gives Covariance[dist,i,j]/(σ_i σ_j), where σ_i=StandardDeviation[dist]〚i〛.

Correlation[dist] gives a correlation matrix with the (i,j) entry given by Correlation[dist,i,j].

Examples

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Basic Examples (3)

Correlation between two vectors:

Real values:

Correlation matrix for a matrix:

Real values:

Correlation matrix for two matrices:

Real values:

Scope (14)

Data (8)

Exact input yields exact output:

Approximate input yields approximate output:

Correlation between vectors of complexes:

Works with large arrays:

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

Find the correlation for data involving quantities:

Correlation between lists of dates:

Correlation between matrices of times:

Distributions and Processes (6)

Correlation for a continuous multivariate distribution:

Correlation for a discrete multivariate distribution:

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

Correlation for derived distributions:

Data distribution:

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

Correlation matrix for TemporalData at times and :

Applications (3)

Compute the correlation of two financial time series:

Correlation can be used to measure linear association:

Correlation can only detect monotonic relationships:

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

Properties & Relations (7)

The correlation matrix is symmetric and positive semidefinite:

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

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

SpearmanRho is Correlation applied to ranks:

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

Correlation and Covariance are the same for standardized vectors:

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

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