SFSI (version 1.2.0)

covariance_matrix: Conversion of a covariance matrix to a distance or correlation matrix

Description

Computes a correlation matrix or a Euclidean distance matrix from a covariance matrix

Usage

cov2dist(V, void = FALSE)
cov2cor2(V, a = 1, void = FALSE)

Value

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

Arguments

V

(numeric matrix) Symmetric variance-covariance matrix among \(p\) variables. It can be of the "float32" type as per the 'float' R-package

void

TRUE or FALSE to whether return or not return the output. When FALSE no result is displayed but the input V is modified. Default void=FALSE

a

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

Author

Marco Lopez-Cruz () and Gustavo de los Campos

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 square of 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 (square) 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

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.

Examples

  require(SFSI)
  data(wheatHTP)
  
  X = scale(Y[,4:7])
  (V = crossprod(X))             # Covariance matrix 
  
  # Covariance matrix to distance matrix
  (D1 = cov2dist(V))
  # it must equal (but faster) to:
  D0 = as.matrix(dist(t(X)))^2
  max(abs(D0-D1))
  
  # Covariance to a correlation matrix
  (R1 = cov2cor2(V))
  # it must equal (but faster) to:
  R0 = cov2cor(V)
  max(abs(R0-R1))
  
  if(requireNamespace("float")){
   # Using a 'float' type variable
   V2 = float::fl(V)
   D2 = cov2dist(V2)
   max(abs(D1-D2))   # discrepancy with previous matrix
   R2 = cov2cor2(V2)
   max(abs(R1-R2))   # discrepancy with previous matrix
  }
  
  # Using void=TRUE
  cov2dist(V,void=TRUE)
  V       # notice that V was modified
  cov2dist(V2,void=TRUE)
  V2       # notice that V2 was modified