Covariance matrix

cov

Covariance matrix

Syntax

• C = cov(X)
  C = cov(x,y)
  

Description

C = cov(x) where x is a vector returns the variance of the vector elements. For matrices where each row is an observation and each column a variable, cov(x) is the covariance matrix. diag(cov(x)) is a vector of variances for each column, and sqrt(diag(cov(x))) is a vector of standard deviations.

C = cov(x,y), where x and y are column vectors of equal length, is equivalent to cov([x y]).

Remarks

cov removes the mean from each column before calculating the result.

The covariance function is defined as

• 

where is the mathematical expectation and .

Examples

Consider A = [-1 1 2 ; -2 3 1 ; 4 0 3]. To obtain a vector of variances for each column of A:

• v = diag(cov(A))'
  v =
     10.3333    2.3333    1.0000
  

Compare vector v with covariance matrix C:

• C =
     10.3333   -4.1667    3.0000
     -4.1667    2.3333   -1.5000
      3.0000   -1.5000    1.0000
  

The diagonal elements C(i,i) represent the variances for the columns of A. The off-diagonal elements C(i,j) represent the covariances of columns i and j.