Arbitrary Scaling

As we mentioned, the mean of vector $ {\bf x}$ is assumed to be equal to the degrees of freedom $ {\bf k}$. If the reference hypothesis $ H_0$ assumes a different mean value, then we can regard the feature transformation as having been applied to a scaled variable. This is easily handled by the change of values theorem. Let $ {\bf x}$ have the prescribed degrees of freedom, but mean $ \mu$. Then, we write that

$\displaystyle {\bf z}= {\bf A}^\prime {\bf x}= {\bf A}^\prime$   $ \Lambda$$\displaystyle \tilde{{\bf x}},$

where $ \tilde{{\bf x}}$ has the standard distribution with mean $ {\bf k}$, and $ \Lambda$ is the diagonal matrix with elements $ \Lambda$$ _i = \mu_i/k_i.$ Then, we simply replace matrix $ {\bf A}$ with $ \Lambda$$ {\bf A}$ and solve the problem assuming $ {\bf x}$ has the standard distribution with mean $ {\bf k}$.

Baggenstoss 2017-05-19