# AbsoluteCorrelation

AbsoluteCorrelation[v1,v2]

gives the absolute correlation between the vectors v1 and v2.

gives the absolute correlation matrix for the matrix m.

AbsoluteCorrelation[m1,m2]

gives the absolute correlation matrix for the matrices m1 and m2.

AbsoluteCorrelation[dist]

gives the absolute correlation matrix for the multivariate symbolic distribution dist.

AbsoluteCorrelation[dist,i,j]

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

# Details • AbsoluteCorrelation[v1,v2] gives the unbiased estimate of the absolute correlation between v1 and v2.
• The lists v1 and v2 must be the same length.
• AbsoluteCorrelation[v1,v2] is equivalent to v1. Conjugate[v2]/Length[v1].
• For a matrix m with columns, is a × matrix of the absolute correlations between columns of m.
• For an × matrix m1 and an × matrix m2, AbsoluteCorrelation[m1,m2] is a × matrix of the absolute correlations between columns of m1 and columns of m2.
• AbsoluteCorrelation[dist,i,j] gives Expectation[xixj,{x1,x2,}dist].
• AbsoluteCorrelation[dist] gives an absolute correlation matrix with the (i,j) entry given by AbsoluteCorrelation[dist,i,j].

# Examples

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## Basic Examples(3)

Absolute correlation between two vectors:

Absolute correlation matrix for a matrix:

Absolute correlation matrix for two matrices:

## Scope(10)

### Data(6)

Exact input yields exact output:

Approximate input yields approximate output:

Absolute correlation between vectors of complexes:

Works with large arrays:

SparseArray data can be used:

Works with data involving quantities:

### Distributions and Processes(4)

Absolute correlation for a continuous multivariate distribution:

Absolute correlation for a discrete multivariate distribution:

Absolute correlation for derived distributions:

Data distribution:

Absolute correlation matrix for a random process at times s and t:

## Applications(3)

Compute the absolute correlation of two financial time series:

AbsoluteCorrelation can be used to measure linear association:

AbsoluteCorrelation can only detect monotonic relationships:

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

## Properties & Relations(8)

The absolute correlation matrix is symmetric and positive semidefinite:

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

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

AbsoluteCorrelationFunction is the off-diagonal entry in the absolute correlation matrix:

AbsoluteCorrelationFunction for a list can be calculated using absolute correlation:

Calculate absolute correlation function for the data:

Use absolute correlation:

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

The absolute correlation of a list with itself is proportional to the second moment:

The diagonal of an absolute correlation matrix is the second moment:

## Neat Examples(1)

Compute the absolute correlation for a LCM array: