m.multivariance {multivariance}

m distance multivariance

Description

Computes m distance multivariance.

Usage

m.multivariance(
  x,
  vec = NA,
  m = 2,
  Nscale = TRUE,
  Escale = TRUE,
  squared = TRUE,
  ...
)

Arguments

x

either a data matrix or a list of doubly centered distance matrices

vec

if x is a matrix, then this indicates which columns are treated together as one sample; if x is a list, these are the indexes for which the multivariance is calculated. The default is all columns and all indexes, respectively.

m

=2 or 3 the m-multivariance will be computed.

Nscale

if TRUE the multivariance is scaled up by the sample size (and thus it is exactly as required for the test of independence)

Escale

if TRUE then it is scaled by the number of multivariances which are theoretically summed up (in the case of independence this yields for normalized distance matrices an estimator with expectation 1)

squared

if FALSE it returns the actual multivariance, otherwise the squared multivariance (less computation)

...

these are passed to cdms (which is only invoked if x is a matrix)

Details

m-distance multivariance is per definition the scaled sum of certain distance multivariances, and it characterize m-dependence.

As a rough guide to interpret the value of total distance multivariance note:

• Large values indicate dependence.

• If the random variables are (m-1)-independent and Nscale = TRUE, values close to 1 and smaller indicate m-independence, larger values indicate dependence. In fact, in the case of independence the test statistic is a Gaussian quadratic form with expectation 1 and samples of it can be generated by resample.multivariance.

• If the random variables are (m-1)-independent and Nscale = FALSE, small values (close to 0) indicate m-independence, larger values indicate dependence.

Since random variables are always 1-independent, the case m=2 characterizes pairwise independence.

Finally note, that due to numerical (in)precision the value of m-multivariance might become negative. In these cases it is set to 0. A warning is issued, if the value is negative and further than the usual (used by all.equal) tolerance away from 0.

References

For the theoretic background see the reference [3] given on the main help page of this package: multivariance-package.

Examples

x = matrix(rnorm(3*30),ncol = 3)

# the following values are identical
m.multivariance(x,m =2)
1/choose(3,2)*(multivariance(x[,c(1,2)]) +
               multivariance(x[,c(1,3)]) +
               multivariance(x[,c(2,3)]))

# the following values are identical
m.multivariance(x,m=3)
multivariance(x)

# the following values are identical
1/4*(3*(m.multivariance(x,m=2)) + m.multivariance(x,m=3))
total.multivariance(x, Nscale = TRUE)
1/4*(multivariance(x[,c(1,2)], Nscale = TRUE) +
     multivariance(x[,c(1,3)], Nscale = TRUE) +
     multivariance(x[,c(2,3)], Nscale = TRUE) + multivariance(x, Nscale = TRUE))