| In this dissertation, we address the statistical analysis under the multiscale framework for the self-similar process. Motivated by the problems arising from geophysics and health informatics, we develop a set of statistical measures as discriminative summaries of the self-similar process. These measures include Multiscale Schur Monotone (MSM) measures, Geometric Attributes of Multifractal Spectrum (GAMFS), Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model. These measures are used as methods to quantify the difference (or similarities) or as input (feature) vectors in the classification model. As the cornstone of GAMFS, we study the estimation of multifractal spectrum and adopt a Weighted Least Squares (WLS) schemes in the wavelet domain to minimize the heteroskedastic effects, which is inherent because the sample variances of the wavelet coefficients depend on the scale. We also propose a Combined K-Nearest-Neighbor classifier (Comb-K-NN) to address the inhomogeneity of the class attributes, which is indicated by the large variations between subsets of input vectors. The Comb-K-NN classifier stabilizes the variations in the sense of reducing the misclassification rates. Bayesian justifications of Comb-K-NN classifier are provided.; GAMFS, Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model are used in the study of assessing the effects of atmospheric stability on the turbulence measurements in the inertial subrange. We also formulate the criteria for success in evaluating how atmospheric stability alters the MFS of a single flow variable time series as a statistical classification model. We use the multifractal discriminate model as the solution of this problem. Also, high frequency pupil-diameter dynamic measurements, which are well documented as measures of mental workload, are summarized using both GAMFS and MSM. These summaries are further used as the feature vector in the Comb-K-NN classifier. The serious inhomogeneity among subjects in the same user group makes classification difficult. These difficulties are overcome by using Comb-K-NN classifier. |
