# Intelligent Identification of Trend Components in Singular Spectrum Analysis
## 3 Separability Improvement
### Methods of Separability Improvement  
The following methods are used to improve separability: FOSSA [9], IOSSA [9]. and EOSSA. Algorithms and implementations of the FOSSA and IOSSA methods can be found in [13; 14]. A separate Section 5 is devoted to EOSSA. The common part of the methods' names is OSSA, which stands for Oblique SSA. All methods are nested (that is, they are add-ons to the SVD step of Algorithm 1) and satisfy the following scheme. **Scheme of applying nested methods**:  
* Perform the SVD step of Algorithm 1. * Select \(t\) of the SVD components, presumably related to the signal; denote their sum \(\mathbf{Y}\). * Perform re-decomposition of \(\mathbf{Y}\) by one of nested methods to improve the separability of the signal components (_nested decomposition_). In all algorithms for improving separability, we first select the components of the expansion to be improved. Consider the case where the leading \(t\) components are selected. The algorithms' input is a matrix \(\mathbf{Y}\) of rank \(t\), recovered from the leading \(t\) components, not necessarily Hankel. Denote \(\mathbf{Y}=\mathcal{T}^{-1}\circ\Pi_{\mathcal{H}}(\mathbf{Y})\). Then the result of the algorithms is the decomposition of the matrix \(\mathbf{Y}\) and of the time series \(\mathbf{Y}\) generated by \(\mathbf{Y}\). Figure 2 illustrates the relationship between the various methods of improving separability and the different types of separability, which are indicated by color. The characteristicsof these methods are briefly listed. In the following paragraphs, we provide a more detailed description of each method. The Filter-adjusted Oblique SSA (**FOSSA**) [9] method is easy to use; however, it can improve only contribution separability, that is, strong separability if the weak separability holds. The idea behind a variation of the method, called DerivSSA (this is the one we will use in this work), is to use the time series derivative (going to consecutive differences in the discrete case) that changes the component contributions. The method calculates \(\Phi(\mathbf{Y})=[Y_{2}-Y_{1}:\cdots:Y_{K}-Y_{K-1}]\), whose column space belongs to the column space of \(\mathbf{Y}\), and then constructs the SVD of the matrix \(\mathbf{Z}=[\mathbf{Y}:\gamma\Phi(\mathbf{Y})]\), \(\gamma>0\). The [14] (Algorithm 2.11) provides an extended version of the algorithm that makes different contributions of the components to the time series, regardless of whether those contributions were the same or different. It is implemented in [13] and that is what we will use. We will take the default value of \(\gamma\), so we will not specify it in the algorithm parameters. Thus, the only parameter of the algorithm is \(t\). We will write \(\text{FOSSA}(t)\). The Iterative Oblique SSA (**IOSSA**) [9] method can improve weak separability. It can also improve the strong separability, but the use of IOSSA is quite time-consuming, so when there is weak separability, it does not make sense to use it. The idea behind the method is to change the standard Euclidean inner product \((X,Y)=X^{\text{T}}\mathbf{Y}\) to the oblique inner product \((X,Y)_{\mathbf{A}}=X^{\text{T}}\mathbf{A}Y\) for a symmetric positive semi-definite matrix \(\mathbf{A}\). This approach allows any two non-collinear vectors to become orthogonal. It is shown in [9] that any two sufficiently long time series governed by LRRs can be made exactly separable by choosing suitable inner products in the column and row signal spaces. The algorithm uses iterations, starting from the standard inner product, to find the appropriate matrices that specify the inner products. The detailed algorithm is given in [14]. We will use the implementation from [13]. The parameter of the algorithm is the initial grouping \(\{1,\ldots,t\}=\bigsqcup_{j=1}^{k}I_{j}\). We will consider two types of grouping. Since our goal is trend extraction, in the first case the initial grouping consists of two groups and is set only by the first group \(I=I_{1}\).