2/15/2013

Covariance matrix example

The definition of covariance is

Cov(x, y) = E[ (X-E(X)) * (Y-E[Y]) ]

where E is abbreviation of expectation. It is same with Mean.
so..
X = [1 2 3 4 5];
E(X) -> 3 or 3.75
3 is the result of "sum(X)/5"
3.75 is the result of "sum(X)/(5-1)"

In the statistics, mean is divided by N-1 to avoid outlier data affection.


*Example of Covariance

X = [ 2 3 4 2 1 4]
Y = [ 2 4 2 1 6 8]

meanX = sum(X) / 6 -> 2.667
(X - meanX) -> [-0.6667    0.3333    1.3333   -0.6667   -1.6667    1.3333]
(X - meanX) * (X -meanX) -> [ 0.4444    0.1111    1.7778    0.4444    2.7778    1.7778]

cov(x, x) -> sum( ( (X - meanX) * (X -meanX) ) ) / (N-1)
              -> 1.4667

cov of X, Y is like that
-> cov(x,x)   cov(x,y)
     cov(x,y)   cov(y,y)

-> 1.4667  0.5333
     0.5333  7.3667


in the matlab...