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SSA: choice of parameters

The parameters of the SSA decomposition are the lag parameter $L$ and the set of indices $I$. Their choice must depend on the properties of the original series and the purpose of the analysis. For a properly made SSA decomposition, the component $z_t$ in ([*]) can be identified as a trend of the original series, as an oscillatory series (for example, seasonality) or a sum of those; the component $\varepsilon_t$ can often be associated with noise. An oscillatory series is a periodic or quasi-periodic series which can be either pure or amplitude-modulated. Noise is any aperiodic series. The trend of the series is, roughly speaking, a slowly varying additive component of the series with all oscillations removed. Note that no parametric model for the components in ([*]) is fixed and these components are produced by the series itself. Thus, when analysing real-life series with the help of SSA one can hardly hope to obtain $z_t$ as an exact periodicity or linear trend, for example, even if this periodicity or linear trend are indeed present in the series. This is an influence of noise and a consequence of the non-parametric nature of the method. In many cases, however, we can get a good approximation to these series. In many cases some of the eigenvectors $U_i$ and the principal components $V_i$ could be identified if considered as time series. The structure of those selected for the group $I$ on the third step of the SSA algorithm is inherited by the series $z_t$. In the ideal situation, the components in ([*]) must be `independent'. Achieving `independence' (or `separability') of the components $z_t$ and $\varepsilon_t$ in the SSA decomposition ([*]) is of prime importance in SSA. One of the characteristics of separabilty is the so-called$\bf w$-correlation which is defined as
\begin{displaymath}\frac{\sum_{t=1}^N w_tz_t \varepsilon_t}{\left(\sum_{t=1}^N......z_t^2 \sum_{t=1}^N w_t\varepsilon_t^2\right)^{\frac12}}\, ,\end{displaymath}


where, see Golyandina, Nekrutkin and Zhigljavsky (2001),

\begin{displaymath}w_t=\left\{ \begin{array}{ll}t &\;\; {\rm for } \; 1 \l......eq N=K+M-1\\0 & \;\;{\rm otherwise }\end{array}\right.\end{displaymath}


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