# 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-μx],Sign[y-μy]], where μx and μy are the medians of x and y, respectively.
• The arguments v1 and v2 can be any realvalued 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-μx)(y-μy)>0,{x,y}disti,j]-Probability[(x-μx)(y-μy)<0,{x,y}disti,j] where disti,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:

Wolfram Research (2012), BlomqvistBeta, Wolfram Language function, https://reference.wolfram.com/language/ref/BlomqvistBeta.html.

#### Text

Wolfram Research (2012), BlomqvistBeta, Wolfram Language function, https://reference.wolfram.com/language/ref/BlomqvistBeta.html.

#### CMS

Wolfram Language. 2012. "BlomqvistBeta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BlomqvistBeta.html.

#### APA

Wolfram Language. (2012). BlomqvistBeta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BlomqvistBeta.html

#### BibTeX

@misc{reference.wolfram_2024_blomqvistbeta, author="Wolfram Research", title="{BlomqvistBeta}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/BlomqvistBeta.html}", note=[Accessed: 13-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_blomqvistbeta, organization={Wolfram Research}, title={BlomqvistBeta}, year={2012}, url={https://reference.wolfram.com/language/ref/BlomqvistBeta.html}, note=[Accessed: 13-July-2024 ]}