total.multivariance {multivariance}

total distance multivariance

Description

computes the total distance multivariance

Usage

total.multivariance(
  x,
  vec = NA,
  lambda = 1,
  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.

lambda

a scaling parameter >0. Each k-tuple multivariance gets weight lambda^(n-k).

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

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

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

• Large values indicate dependence.

• For Nscale = TRUE values close to 1 and smaller indicate 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.

• For Nscale = FALSE small values (close to 0) indicate independence, larger values indicate dependence.

Finally note, that due to numerical (in)precision the value of total 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 references given on the main help page of this package: multivariance-package.

Examples

x = matrix(rnorm(100*3),ncol = 3)
total.multivariance(x) #for an independent sample
# the value coincides with
(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))/4

total.multivariance(coins(100)) #value for a dependent sample which is 2-independent