This chapter introduces two new empirical methods for obtaining optimal smoothing of noise-ridden stationary and nonstationary, linear and nonlinear signals. Both methods utilize an application of the spectral representation theorem (SRT) for signal decomposition that exploits the dynamic properties of optimal control. The methods, named as SRT1 and SRT2, produce a low-resolution and a high-resolution ilter, which may be utilized for optimal long-and short-run tracking as well as forecasting

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... ices. Monte Carlo simulation applied to three broad classes of signals enables comparing the dual SRT methods with a similarly optimized version of the well-known and reputed empirical Hilbert-Huang transform (HHT). The results point to a more satisfactory performance of the SRT methods and especially the second, in terms of low and high resolution as compared to the HHT for any of the three signal classes, in many cases also for nonlinear and stationary/nonstationary signals. Finally, all of the three methods undergo statistical experimenting on eight select real-time data sets, which include climatic, seismological, economic and solar time series. Signal decomposition and statistical taxonomy, linearity and stationarity testing Any observed signal, by means of additive decomposition [3, 13] , may be modeled as the sum of three diferent random variables of unknown distribution such that where Y t is the time sequence of the real-valued observations for the discrete-time notation t ∈ [ 1, T ] , T < ∞ . Moreover, y t is the information set available at t − j , j ∈ [ 1, J ] for J ≤ T a maximum preselect lag [14, 15] . The irst term of Eq. (1) Advances in Statistical Methodologies and Their Application to Real Problems 76