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AbsoluteCorrelation

AbsoluteCorrelation[v,w]

gives the absolute correlation between the vectors v and w.

AbsoluteCorrelation[a,b]

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

AbsoluteCorrelation[a]

gives the absolute correlation matrix for the matrix a.

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[v,w] gives the unbiased estimate of the absolute correlation.
For centralized (zero mean) vectors, AbsoluteCorrelation computes Covariance.
For standardized (zero mean and unit variance) vectors, AbsoluteCorrelation computes Correlation.
For vectors and of length , the absolute correlation estimate AbsoluteCorrelation[v,w] is given by .
For matrices and with dimensions and and columns indexed as and , respectively, AbsoluteCorrelation[a,b] is a matrix with elements given by .
For a matrix with columns, AbsoluteCorrelation[a] is a matrix given by AbsoluteCorrelation[a, a].
AbsoluteCorrelation works with any vector that is VectorQ or matrix that is MatrixQ.
AbsoluteCorrelation[dist,i,j] gives Expectation[x_ix_j,{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:

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

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 the second moment:

For random data:

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

Neat Examples (1)

Compute the absolute correlation for a LCM array:

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