cov2dist: Conversion of covariance to distance matrix

In MarcooLopez/SFSI_data: Sparse Family and Selection Index

Description

Computes a squared Euclidean distance matrix from a covariance matrix among p variables with mean zero

Usage

cov2dist(COV, void = FALSE)

Arguments

COV	Symmetric variance-covariance matrix among p (centered) variables
void	TRUE or FALSE to whether return or not return the output. When FALSE no result is displayed but the input COV is modified. Default void=FALSE

Details

For any variables X_i and X_j with mean zero and with sample vectors x_i = (x_i1,...,x_in)' and x_j = (x_j1,...,x_jn)' , their (sample) variances are equal (up-to a constant) to their cross-products, this is, var(X_i) = x'_ix_i and var(X_j) = x'_jx_j. Likewise, the covariance is cov(X_i,X_j) = x'_ix_j.

The distance d(X_i,X_j) between the variables expressed in terms of cross-products is

d²(X_i,X_j) = x'_ix_i + x'_jx_j - 2x'_ix_j

Therefore, the output distance matrix will contain as off-diagonal entries

d²(X_i,X_j) = var(X_i) + var(X_j) - 2cov(X_i,X_j)

while in the diagonal, the distance between one variable with itself is d²(X_i,X_i) = 0

Value

A matrix D containing the squared Euclidean distances

Author(s)

Marco Lopez-Cruz ([e-mail]) and Gustavo de los Campos

Examples

  require(SFSI)
  
  # Simulate matrix
  n = 100; p=5
  X = scale(matrix(rnorm(n*p),ncol=p))

  # Distance matrix from a cross-product
  COV = crossprod(X)   # Cross-product X'X
  cov2dist(COV)
  # it must equal (but faster) to:
  as.matrix(dist(t(X)))^2

  # Distance matrix from a variance-covariance matrix
  COV = cov(X)   # Variance matrix of X
  (n-1)*cov2dist(COV)
  # it must equal (but faster) to:
  as.matrix(dist(t(X)))^2

  # Using void=TRUE
  cov2dist(COV,void=TRUE)
  (n-1)*COV   # notice that COV was modified

MarcooLopez/SFSI_data documentation built on April 15, 2021, 10:53 a.m.