# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
## 3 Modifications of Information Criteria for the Case of Hankel Noise
### Red Noise  
Let the noise be stationary and Gaussian. The procedure that makes the noise white is called whitening and consists of multiplying the time series by the square root of the inverse of the noise autocovariance matrix. The whitening operation affects both the signal and the noise. Since the matrix-form model is stable concerning a linear transformation (multiplication by a full-rank matrix), we can apply the methods of the previous section to the result of the whitening. To apply the criterion, it is sufficient to know what the variance of the white noise after whitening will be; more precisely, it is sufficient to know how the variance of the noise after whitening is expressed through the variance of the original noise. To conduct this, we need to know the covariance matrix of the noise.  
Let us denote the variance estimate using the Formula (8) or (9) as \(\vartheta_{0}^{2}\). Then, we need to substitute the variance \(\tilde{\sigma}^{2}\)of the whitened noise into the Formula (4). For example, in the case of an AR(1) model with coefficient \(\phi\),  
\[\vartheta^{2}=\vartheta_{0}^{2}(1-\phi^{2}). \tag{11}\]  
Recall that red noise is an AR(1) process with a positive coefficient.  
Let  
\[c=\sum_{i=1}^{L}\sum_{j=1}^{K-1}d_{i,j}d_{i,j+1},\qquad v_{1}=\sum_{i=1}^{L} \sum_{j=1}^{K-1}d_{i,j}^{2},\qquad v_{2}=\sum_{i=1}^{L}\sum_{j=1}^{K-1}d_{i,j +1}^{2}.\]  
Then, as an estimate of \(\phi\) we take  
\[c/\sqrt{v_{1}v_{2}}. \tag{12}\]  
**Remark 2**: _The idea behind TRMAT can also be applied to the evaluation of \(\phi\) by considering the same weights \(\frac{1}{v_{i+j-1}}\) in each sum as in (9). We will apply such an estimate with weights to the algorithm TRMAT._  
An alternative is to estimate \(\phi\) as the correlation between successive observations in the series \(\mathsf{D}_{r}=\mathcal{T}^{-1}\circ\Pi_{\mathcal{H}}(\mathbf{D}_{r})\), but we will not consider it since it did not lead to an improvement in preliminary numerical experiments.  
Unfortunately, especially if the signal is not stationary and even more so if it is not a low-rank series, the calculated estimate of \(\phi\) based on nonstationary residuals in the case of wrong rank can be accidental and lead to an incorrect maximum of an information criterion. Therefore, we consider the following variant in which the information criterion is assigned \(-\infty\) when the residual is clearly nonstationary (recall that the best model corresponds to the maximum of criteria):  
1. Check the series \(\mathsf{D}_{r}\) for nonstationarity (this is optional, not necessary), e.g., using the KPSS test [36]. If the hypothesis is rejected (e.g., \(p\)-value \(<0.05\)), then \(\phi=1\) and STOP.
2. Find the better model for \(\mathsf{D}_{r}\) from white and red noise models. If it is detected as white noise, then \(\phi=0\), and if it is closer to red noise, then estimate parameter \(\phi\) in the red-noise model, for example, by the MLE method without requiring the model to be stationary.
3. If \(\phi\geq 1\), then the criterion returns \(-\infty\); otherwise, the value is calculated using the formula of the corresponding information criterion.  
The option of criteria with the adjustment (11) according to the estimated \(\phi\) will be referred to by adding _AR.