# Information Criteria for Signal Extraction Using Singular Spectrum Analysis: White and Red Noise
## 3 Modifications of Information Criteria for the Case of Hankel Noise
### White Noise  
Let \(\mathsf{N}\) be a Gaussian white noise with zero expectation and variance \(\sigma^{2}\). The noise series \(\mathsf{N}\) can be estimated as \(\mathcal{T}^{-1}\circ\Pi_{\mathcal{H}}(\mathbf{D}_{r})\). In the proposed approach, the estimation of \(\sigma^{2}\) will be conducted without proceeding from the matrix \(\mathbf{D}_{r}\) to a time series.  
Let \(\mathbf{D}_{r}=(d_{i,j}^{(r)})=(d_{i,j})\). Then the estimate of \(\sigma^{2}\) can be given as follows (let us call this version 'SVD'):  
\[\tilde{\sigma}^{2}=\frac{\sum_{i=1}^{L}\sum_{j=1}^{K}d_{i,j}^{2}}{LK}. \tag{8}\]  
By employing the singular value decomposition, the same noise variance estimate can be obtained as \(\tilde{\sigma}^{2}=\sum_{i=r+1}^{L}\lambda_{i}/(LK)\), where \(\lambda_{i}\) are the squares of the singular values of the trajectory matrix \(\mathbf{X}\).  
It can be seen from (1) that the operator \(\mathcal{T}_{L}\) repeats each time series element in the trajectory matrix as many times as there are elements on the corresponding anti-diagonal. Consequently, given the Hankel structure of the input matrices, we put forward a more accurate weighted version of the \(\tilde{\sigma}^{2}\) estimator (which we shall henceforth refer to as 'TRMAT'):  
\[\tilde{\sigma}^{2}=\frac{1}{N}\sum_{i=1}^{L}\sum_{j=1}^{K}\frac{1}{w_{i+j-1}} d_{i,j}^{2}, \tag{9}\]  
where \(w_{i}\), \(i=1\),..., \(N\), are the numbers of elements on the \(i\)-th anti-diagonals of an \(L\times K\) matrix. The division by \(N\) is a consequence of the fact that the number of such diagonals is \(N\). Note, that if the window length \(L\) is a relatively small value compared to \(N\), there is a negligible difference with the SVD criterion, since the weights \(w_{L},w_{L+1},\ldots,w_{N-L}\) are the same for both methods.  
In both cases, the alternative estimate of \(k\) (instead of \(k=2r\)) to substitute into the Formula (4) is  
\[k=\frac{N(r(K+L)-r^{2})}{LK}, \tag{10}\]that is, it is the dimension \(r(K+L)-r^{2}\) of the smooth manifold \(\mathcal{M}_{r}\), reweighted taking into account the \"replacement\" of the dimension \(LK\) of the matrix space by the dimension \(N\) of the time series space. Let us explain this normalization. One can look at the contribution of number \(k\) of parameters to the penalty in the AIC and BIC values in the form (4) by dividing the expression by \(N\). This shows that the parameter penalty depends on \(k/N\).  
We have numerically checked that the non-normalized number of parameters \(k=2r\) that corresponds to the non-Hankel case leads to a severe underestimation of the penalty term \(\phi(k)\) in (4) when (8) or (9) is taken as an estimate of \(\sigma^{2}\); therefore, we will not consider this case further.  
We will consider (8) (SVD) and (9) (TRMAT) with the normalized number of parameters given by (10) using the information criteria (4). Recall that the best rank corresponds to the maximum value of a criterion. Preliminary experiments have shown that only the BIC-penalty criteria turned out to be working, and we will consider it further. A graph of \(\mathrm{AIC}(k)\) flattens out after growth to the correct rank, so the maximum point is determined unstably and the rank is usually overestimated.