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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.
