Consider a sequence of random variables $\overline{z} = \{z(k-M+1), ..., z(k)\}$, with probability density depending upon a scalar parameter $\theta$. It is intended to decide between two hypotheses

$$ H_0: \theta = \theta_0 \hspace{3em} for \hspace{3em} k-M+1 \leq i \leq k \\ H_1: \theta = \theta_1 \hspace{3em} for \hspace{3em} k-M+1 \leq i \leq k \tag{1} $$

It is assumed that all the parameters are known except for $\theta_1$ which is replaced by its MLE estimate. Then, the GLR test is

$$L(\overline{z}) = \frac{p\left(\overline{z}, \; \hat{\theta_1}; \; H_1\right)} {p \left(\overline{z}, \; {\theta_0}; \; H_0 \right)} \tag{2}$$

where $\overline{z} = \{z(k-M+1), ..., z(k)\}$ is iid.

Then GLR test can be simplified to

$$L(\overline{z}) = \max\limits_{\theta_1} \left( \sum\limits_{i = k-M+1}^{k} ln \frac{p(z(i)|\theta_1)}{p(z(i)|\theta_0)} \right)\tag{3} $$

But what if $\overline{z} = \{z(k-M+1), ..., z(k)\}$ are correlated. I need to simplify $(2)$ for simplified Gaussian process.

Any help, referred references for correlated GLR test is highly appreciated.


A simple approach would be to transform your observations $\bar{z}$ into independent random variables is by using whitening transformation, i.e., if the covariance matrix $\Sigma$ is known, then find $W$ such that $W^TW = \Sigma^{-1}$. Next, define a new random variable $\bar{y}$ as $W \bar{z}$. The above transformation is only possible when $\Sigma$ is invertible.

  • $\begingroup$ Thanks Maxtron for the answer. However, I need kinda to derive it directly. I mean to derive equivalent of equation 3 for simple case of correlated Gaussian process (only change in mean value). $\endgroup$ – Mahdi Ghe Jan 22 '20 at 13:40
  • $\begingroup$ maxtron's approach will still give you the same likelihoods. it just transforms the response so that the covariance matrix is the identity. But the procedure needs the covariance matrix to be known and it wasn't clear from your question whether that's the case. $\endgroup$ – mark leeds Nov 14 '20 at 0:39

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