## 2 Materials and Methods
### Singular Spectrum Analysis (SSA)  
There are many different methods for studying time series. Many of these methods rely on parametric assumptions; for example, they require linearity or nonlinearity with functional forms. An alternative method that uses nonparametric assumptions and which is neutral to usual requirements such as stationarity, linearity, and normality in data is Singular Spectrum Analysis (SSA). SSA is a relatively new nonparametric method that has demonstrated its useful applications in many different fields, from economics to physics, including meteorology. SSA has two main applications, which are filtering and smoothing and forecasting [59]. In this article, we employed both applications. For filtering and smoothing, the technique decomposes the original series in the sum of interpretable components. These components are usually slowly varying trends, oscillatory components for seasonality, and noise. The primary SSA method consists of two complementary phases: decomposition and reconstruction. Each phase has two separate steps. In the first phase, the series is decomposed. This phase comprises two steps: embedding and singular value decomposition. The embedding step can be considered a mapping process that transforms a one dimensional \(Y_{N}=\left\{y_{t};t=1,2,\ldots N\right\}\) time series into a multidimensional series, \(X_{i}=\left(y_{i},\ldots,y_{i+L-1}\right)^{T}\in\mathbb{R}^{L}\), where \(L\) is the window length and \(2\leq L\leq N/2\). As established by [PERSON] [59] and [PERSON] et al. [60], one of the crucial steps in the technique of SSA is the definition of the window period, which is useful in the embedding step. The product of this step is the Hankel matrix, or trajectory matrix, which is the primary resource for the SVD, and this is the basic difference when applying a traditional PCA procedure to a multivariate time series. The widow period was set at \(L=N/2\), as established by [PERSON] [59], obtaining the trajectory matrix: \(\tau_{SSA}(PTPM)=X\). In the second step, the SVD of \(X\) is performed, which is denoted by \(\lambda_{1},\ldots\lambda_{L}\). The eigenvalues of \(XX^{T}\) are arranged in decreasing order \((\lambda_{1}\geq\cdots\geq\lambda_{L}\geq 0)\) and by the corresponding left eigenvectors \((U_{1},\ldots U_{L})\). The SVD of the Hankel matrix can be defined as the collection of \(X=X_{1}+\cdots X_{L}\), where \(X_{i}=\sqrt{\lambda_{i}}U_{i}V_{i}^{T}\), and \(V_{i}=X^{T}U_{i}/\sqrt{\lambda_{i}}\) (if \(\lambda_{i}=0\), then \(X_{i}=0\)). The \(\sqrt{\lambda_{i}}\) are the singular values of \(X\), and \(\{\sqrt{\lambda_{1}},\sqrt{\lambda_{2}},\ldots,\sqrt{\lambda_{L}}\}\) is the spectrum. The collection \((\sqrt{\lambda_{i}},U_{i},V_{i})\) is called the \(i\)-th eigentriple, formed by the eigenvalues and the left and right eigenvectors. Therefore, the SVD of the Hankel matrix produces \(X=X_{1},+\cdots+X_{d}\). The SSA can be seen as a way of obtaining and analyzing this spectrum of singular values to identify and then distinguish between the signal and the noise, usually by visual techniques in each time series [60]. In the second phase, the series, which is filtered and free from noise, is reconstructed. This phase is formed by two steps as well: grouping and diagonal averaging. The goal of the grouping step is to enable the signal and noise to be distinguished. This is done by splitting the elementary matrices into several disjointed subsets, \(I_{1},\ldots I_{m}\), with the representation, \(X\equiv X_{I1}+\cdots+X_{lm}\), which is later summed within each group (tendency or seasonality). The process of choosing the sets \(I_{1},\ldots I_{m}\) is called grouping. Then, the series, which can be interpreted as the proxy of the original series, is used for forecasting purposes [59; 60]. Finally, the purpose of diagonal averaging is to transform a matrix from the Hankel format to a time series. The forecasting stage can be implemented once the decomposition is performed.