CS395T/CAM383M Computational Statistics > HW 5 HW5 - Isaac
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#1
03-03-2011, 06:35 PM
 ilevy Member Join Date: Jan 2011 Posts: 43
HW5 - Isaac

I'm confused about multivariate chi-square. Is our chi square model (the one we wish to minimize):

$\Large \sum_i (\mathbf{x}_i - \mu)^T \Sigma^{-1} (\mathbf{x}_i - \mu)$, given data vectors, $\Large \mathbf{x}_1, \mathbf{x}_2 \ldots$ ?

Our data is already Gaussian, but we need to generate an empirical chi-square for multivariates...
#2
03-05-2011, 10:20 AM
 Courtney Member Join Date: Jan 2011 Posts: 14

So in lecture 10 there's a slide that says that the chisquared is the above formula when the entries of data vector x are normal but not necessarily independent. What we need is to get something in the form

chi^2 = sum (yi) where the yi's ARE independent.

Here's the strategy:
1. Use the fact that Sigma is positive definite and use Cholesky decomposition to put the RHS into the form mentioned.

2. Prove that those yi's are independent

Siavash did a good job with the proof.
#3
03-06-2011, 08:58 PM
 ilevy Member Join Date: Jan 2011 Posts: 43

Thanks Courtney.

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