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### normal likelihood function &covariance matrix &parameter estimation

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```Hi,
I have a question which has made me confused for a long time.

Here is the question:
I  want to estimate the parameters in the normal likelihood function, the parameters are only in the covariance matrix. I try to use the conjugate gradient method to get the values of the parameters which make the negative loglikelihood  function minimum.

The problem is that a lot  of  initial  values of the parameters would make this covariance matrix  "effectively singular ",  I mean the condition numbers  are  large.
Then how to get the determinant and inverse of the covariance matrix is a problem.

I&#12288;use  "p==0 & rcond(A)>1e-10"  to  choose  the parameters  which would make the covariance matrix non-singular. In this way, a lot of parameters cannot be chosen,sometimes none of them satisfies  this  condition.

In such a case, what should I do?

The covariance matrix  in  the normal likelihood function  has to be positive definite,right?

Thanks for the help!
Brooke
```
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```> I  want to estimate the parameters in the normal likelihood function, the
> parameters are only in the covariance matrix. I try to use the conjugate
> gradient method to get the values of the parameters which make the
> negative loglikelihood  function minimum.
>
> The problem is that a lot  of  initial  values of the parameters would
> make this covariance matrix  "effectively singular ",  I mean the
> condition numbers  are  large.
> Then how to get the determinant and inverse of the covariance matrix is a
> problem.

Brooke, one thing I have seen is to use the Cholesky factorization C=T'*T.
Optimize for the nonzero values of T. Another thing might be to find T to
optimize the likelihood with covariance having diagonal elements bounded
above zero, such as

0.1*eye(3) + T'*T

There are more details, like converting T between vector and upper
triangular matrix, and so forth, but this is the idea.

-- Tom

```
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