2. Covariance matrix transformations {SFSI}

Conversion of a covariance matrix to a distance/correlation matrix

Description

Transformation into correlation matrix or distance matrix from a covariance matrix

Usage

cov2dist(A, a = 1, inplace = FALSE)

cov2cor2(A, a = 1, inplace = FALSE)

Arguments

A

(numeric matrix) Variance-covariance matrix

inplace

TRUE or FALSE to whether operate directly on the input matrix. When TRUE no result is produced but the input A is modified. Default inplace=FALSE

a

(numeric) A number to multiply the whole resulting matrix by. Default a = 1

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.

Distance.

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

d(X_i,X_j) = (x'_ix_i + x'_jx_j - 2x'_ix_j)^1/2

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

d(X_i,X_j) = (var(X_i) + var(X_j) - 2cov(X_i,X_j))^1/2

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

Correlation.

The correlation between the variables is obtained from variances and covariances as

cor(X_i,X_j) = cov(X_i,X_j)/(sd(X_i)sd(X_j))

where sd(X_i)=sqrt(var(X_i)); while in the diagonal, the correlation between one variable with itself is cor(X_i,X_i) = 1

Variances are obtained from the diagonal values while covariances are obtained from the out-diagonal.

Value

Function 'cov2dist' returns a matrix containing the Euclidean distances. Function 'cov2cor2' returns a correlation matrix

Examples

  require(SFSI)
  data(wheatHTP)
  
  X = scale(M)[1:100,]/sqrt(ncol(M))
  A = tcrossprod(X)        # A 100x100 covariance matrix 
  
  # Covariance matrix to distance matrix
  D = cov2dist(A)
  # (it must equal (but faster) to:)
  D0 = as.matrix(dist(X))
  max(D-D0)
  
  # Covariance to a correlation matrix
  R = cov2cor2(A)
  # (it must equal (but faster) to:)
  R0 = cov2cor(A)
  max(R-R0)
  
  # Inplace calculation
  A[1:5,1:5]
  cov2dist(A, inplace=TRUE)
  A[1:5,1:5]  # notice that A was modified