Discrimination of Nonstationary Time Series using the SLEX Model

Statistical discrimination for nonstationary random processes have developed into a widely practiced field with various applications. In some applications, such as signal processing and geophysical data analysis, the generated processes are usually long series. In such cases, a discriminant scheme with computational efficiency and optimal property is of great interest.In this dissertation, a discriminant scheme for nonstationary time series based on the SLEX model (Ombao, Raz, von Sachs and Guo, 2002) is presented. The SLEX model is based on the Smooth Localized complex EXponential (SLEX)[Wickerhauser, 1994] basis functions. SLEX basis functions generalize directly to a library of SLEX basis vectors that are complex-valued, orthonormal, and simultaneously localized in time and frequency domains (Wickerhauser, 1994). Thus, it provides an explicit segmentation of the time-frequency plane and hence is able to represent discrete random processes whose spectral properties change with time. Since the SLEX basis functions can also be considered a generalization of the tapered Fourier vectors, the calculation from SLEX basis functions to a library of SLEX basis vectors (called the SLEX transform) can use the Fast Fourier Transform. That is, the SLEX transform has computational efficiency. Moreover, the SLEX model, with a structure for asymptotic theory, allows the derivation of the optimal properties of the discriminant statistic in this dissertation. A statistical time series classification scheme can be considered a formulation with two steps: extracting features from the data and developing a decision function. For feature extraction, a fast algorithm associated with the SLEX model is formed to extract the features. For developing a decision function, an optimal discriminant statistic based on the Kullback-Leibler divergence (Kullback and Leibler, 1951) of the SLEX model is proposed. The entire scheme will be organized as an algorithm. That is, a computationally efficient and statistically optimal discriminant scheme for nonstationary time series is proposed in this dissertation