Correlation

Correlation[v₁,v₂]

gives the correlation between the vectors v₁ and v₂.

gives the sample correlation matrix for observations in matrix m.

Correlation[m₁,m₂]

gives the correlation matrix for the matrices m₁ and m₂.

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[v₁,v₂] gives Pearson's correlation coefficient between v₁ and v₂.
The lists v₁ and v₂ must be the same length.
Correlation[v₁,v₂] is equivalent to Covariance[v₁,v₂]/(StandardDeviation[v₁]StandardDeviation[v₂]).
For a matrix m with columns, Correlation[m] is a × matrix of the correlations between columns of m.
For an × matrix m₁ and an × matrix m₂, Correlation[m₁,m₂] is a × matrix of the correlations between columns of m₁ and columns of m₂.
Correlation works with SparseArray objects.
Correlation[dist,i,j] gives Covariance[dist,i,j]/(σ_i σ_j), where σ_i is the i component of the standard deviation of dist.
Correlation[dist] gives a correlation matrix with the (i,j) entry given by Correlation[dist,i,j].

Examples

open allclose all

Basic Examples (3)

Correlation between two vectors:

Real values:

Correlation matrix for a matrix:

Real values:

Correlation matrix for two matrices:

Real values:

Scope (12)

Data (7)

Exact input yields exact output:

Approximate input yields approximate output:

Correlation between vectors of complexes:

Works with large arrays:

SparseArray data can be used:

Find the correlation of WeightedData:

Find the correlation for data involving quantities:

Distributions and Processes (5)

Correlation for a continuous multivariate distribution:

Correlation for a discrete multivariate 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:

Top

Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how