1

Is it possible to get a simple formula linking a central moment to the same moment in a rotated frame, such as the relation between the central moment and the moment about the origin?

The formula I am using is http://en.wikipedia.org/wiki/Central_moment#Multivariate_moments

Flavian
  • 267

1 Answers1

3

Assuming you are speaking of second order moments (covariance), and assuming the rotation is about the mean (or equivalently, that the variable has zero mean - this does not loose generality), the rotation can be expressed as a matrix multiplication, so ${\bf y} = Q {\bf x}$ and then

$\Sigma_y = E[{\bf y}{\bf y^T}]=Q E[{\bf x}{\bf x^T}] Q^t= Q \, \Sigma_x Q^t $

leonbloy
  • 1,458
  • 1
    This is correct, however I would like something as simple as your example for higher moments. – Flavian Jun 21 '12 at 20:06
  • 1
    The same principles apply, Flav, but to fully write out the calculations requires some form of higher-rank tensor notation. Do you really want that? – whuber Jun 22 '12 at 04:03