Parameter Estimation, Forecasting, Gap Filling

Abstract

Similarly to Chap. 2, this chapter is devoted to applications of SSA for one-dimensional series; that is, to 1D-SSA. The SSA analysis of time series, which is considered in Chap. 2, can be classified as model-free. In this chapter, on the contrary, we consider the methodologies within the 1D-SSA approach, which require a model. These methodologies include the common problems of forecasting, interpolation, low-rank approximation, and parameter estimation. The model used is based on properties of the approximating subspace constructed in the process of 1D-SSA analysis of Chap. 2 and so the methodologies of this chapter belong to the class of subspace-based methods of time series analysis and signal processing. The main parametric model of 1D-SSA is a linear recurrence relation (LRR) which a time series should approximately satisfy. In Sect. 3.1, we describe how to estimate the LRR coefficients and parameters of a series component satisfying such LRR. Section 3.2 is devoted to forecasting, the most practically important application of time series analysis. In 1D-SSA, the problem of forecasting coincides with the problem of continuation of the signal S extracted from the observed series S + R, where R is the residual (or noise). To do that, we estimate the trajectory space of S and make the continuation based on the estimated subspace. A straightforward manner to make a forecast is to directly use the parametric form of the signal estimated using the methods of Sect. 3.1. However, the class of series suitable for forecasting is much wider than the class of series where the parametric model is adequate and some of the forecasting methods of Sect. 3.2 only use certain features of the estimated subspaces and not the estimators of the signal parameters; hence, a medium-term forecast could be quite accurate even if the given series cannot be approximated by a signal which globally satisfies an LRR. Section 3.2 also thoroughly discusses the problem of assessing stability of forecasts, which is the key issue in understanding of how much the forecasts can be trusted.