cdm: computes a doubly centered distance matrix
In multivariance: Measuring Multivariate Dependence Using Distance Multivariance

Description Usage Arguments Details References Examples

View source: R/multivariance-functions.R

computes the doubly centered distance matrix

cdm(
  x,
  normalize = TRUE,
  psi = NULL,
  p = NULL,
  isotropic = FALSE,
  external.dm.fun = NULL
)

`x`	matrix, each row of the matrix is treated as one sample
`normalize`	logical, indicates if the matrix should be normalized
`psi`	if it is `NULL`, the euclidean distance will be used. In the case of `isotropic = TRUE`: a real valued negative definite function of one variable (accepting vectors as arguments; returning a vector of the same length). In the case of `isotropic = FALSE`: a real valued function of two variables (or vectors) to compute the distance of two samples based on a continuous negative definite function.
`p`	numeric, if it is a value between 1 and 2 then the Minkowski distance with parameter p is used.
`isotropic`	logical, indicates if psi of the Euclidean distance matrix should be computed, i.e., if an isotropic distance should be used.
`external.dm.fun`	here one can supply an external function, which computes the distance matrix given `x`.

The doubly centered distance matrices are required for the computation of (total / m-) multivariance.

If normalize = TRUE then the value of multivariance is comparable and meaningful. It can be compared to the rejection.level or its p-value multivariance.pvalue can be computed.

More details: If normalize = TRUE the matrix is scaled such that the multivariance based on it, times the sample size, has in the limit - in the case of independence - the distribution of an L^2 norm of a Gaussian process with known expectation.

As default the Euclidean distance is used. The parameters psi, p, isotropic and external.dm.fun can be used to select a different distance. In particular, external.dm.fun can be used to provide any function which calculates a distance matrix for the rows of a given matrix.

For the theoretic background see the references given on the main help page of this package: multivariance-package.

x = coins(100)
cdm(x) # fast euclidean distances
cdm(x,psi = function(x,y) sqrt(sum((x-y)^2))) # this is identical to the previous (but slower)

# the function cdm does the following three lines in a faster way
N = nrow(x)
C = diag(N) - matrix(1/N,nrow = N,ncol = N)
A = - C %*% as.matrix(stats::dist(x,method="euclidean")) %*% C #'
all(abs(A- cdm(x,normalize = FALSE)) < 10^(-12))