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BlomqvistBeta

BlomqvistBeta[v₁,v₂]

gives Blomqvist's medial correlation coefficient β for the vectors v₁ and v₂.

BlomqvistBeta[m]

gives Blomqvist's medial correlation coefficient β for the matrix m.

BlomqvistBeta[m₁,m₂]

gives Blomqvist's medial correlation coefficient β for the matrices m₁ and m₂.

BlomqvistBeta[dist]

gives the medial correlation coefficient matrix for the multivariate symbolic distribution dist.

BlomqvistBeta[dist,i,j]

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

Details

BlomqvistBeta[v₁,v₂] gives Blomqvist's medial correlation coefficient β between v₁ and v₂.
Blomqvist's β between vectors x and y is given by Correlation[Sign[x-μ_x],Sign[y-μ_y]], where μ_x and μ_y are the medians of x and y, respectively.
The arguments v₁ and v₂ can be any real‐valued vectors of equal length.
For a matrix m with columns BlomqvistBeta[m] is a × matrix of the β's between columns of m.
For an × matrix m₁ and an × matrix m₂ BlomqvistBeta[m₁,m₂] is a × matrix of the β's between columns of m₁ and columns of m₂.
BlomqvistBeta[dist,i,j] is Probability[(x-μ_x)(y-μ_y)>0,{x,y}dist_i,j]-Probability[(x-μ_x)(y-μ_y)<0,{x,y}dist_i,j] where dist_i,j is the marginal of dist.
BlomqvistBeta is not well defined for discrete distributions or in the presence of ties.
BlomqvistBeta[dist] gives a matrix β where the entry is given by BlomqvistBeta[dist,i,j].

Examples

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Basic Examples (4)

Blomqvist's β for two vectors:

Blomqvist's β for a matrix:

Blomqvist's β for two matrices:

Compute Blomqvist's β matrix for a bivariate distribution:

Compare to a simulated value:

Scope (7)

Data (4)

Exact input yields exact output:

Approximate input yields approximate output:

Works with large arrays:

SparseArray data can be used:

Distributions and Processes (3)

Blomqvist's β matrix for a continuous multivariate distribution:

Blomqvist's β matrix for derived distributions:

Data distribution:

Blomqvist's β matrix for a random process at times and :

Applications (3)

Blomqvist's β is typically used to detect linear dependence between two vectors:

The absolute magnitude of β tends to 1 given strong linear dependence:

The value tends to 0 for linearly independent vectors:

Blomqvist's β can be used to measure linear association:

Blomqvist's β only detects monotonic dependence structures:

HoeffdingD can be used for a variety of other dependence structures:

Properties & Relations (7)

Blomqvist's β ranges from to for high negative and high positive association, respectively:

Blomqvist's β matrix is symmetric:

The diagonal elements of Blomqvist's β matrix are 1:

Blomqvist's for even sample sizes:

Median-centered data:

Count the number of points in each quadrant:

Blomqvist's β:

Blomqvist's β will yield or if there is perfect monotonic association:

This is in contrast to Correlation, which measures the degree of linear association:

BlomqvistBetaTest can be used to test the value of β:

IndependenceTest can be used to automatically select an appropriate test:

Blomqvist's β for a bivariate distribution:

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