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.

In this paper, we focus on the goodness of fit test for self-similar property of two well-known processes: the fractional Brownian motion and the fractional autoregressive integrated moving average process. The Hurst parameter of the self-similar process is estimated by the embedding branching process method proposed by Jones and Shen (2004). The goodness of fit test for self-similarity is based on the Pearson chi-square test statistic. We approximate the null distribution of the test statistic by a scaled chi-square distribution to correct the size bias problem of the conventional chi-square distribution. The scale parameter and degrees of freedom of the test statistic are determined via regression method. Simulations are performed to show the finite sample size and power of the proposed test. Empirical applications are conducted for the high frequency financial data and human heart rate data.

Recently there have been reports that certain physiological data seem to have the properties of long-range correlation and self-similarity. These two properties can be characterized by a long-range dependent parameter d, as well as a self-similar parameter H. In Peng et al (1995), the alteration of long-range correlations with life-threatening pathologies are studied by analyzing the heart rate data of different groups of subjects. The self-similarity properties of two well-known processes, namely the Fractional Brownian Motion (FBM) and the Fractional ARIMA (FARIMA), are of interest to see if it is suitable to be used to model the heart rate data in order to examine the health conditions of some patients. The Embedded Branching Process (EBP) method for estimating parameter $H$ and a goodness of fit test for examining the self-similarity of a process based on the EBP method are proposed in Jones and Shen (2004). In this work, the performance of the goodness of fit test are examined using simulated data from the FBM and FARIMA processes. A modification of the distribution of the test statistics under null hypothesis is proposed and has been modified to be more appropriate. Some simulation comparisons of different estimation methods of the parameter $H$ for some FARIMA processes are also presented and applied to heart rate data obtained from Kaohsiung Veterans General Hospital.

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