Hurst exponent and variance are two quantities that often characterize real-life, highfrequency
observations. We develop the method for simultaneous estimation of a timechanging
Hurst exponent H(t) and constant scale (variance) parameter C in a multifractional
Brownian motion model in the presence of white noise based on the asymptotic behavior of
the local variation of its sample paths. We also discuss the accuracy of the stable and simultaneous
estimator compared with a few selected methods and the stability of computations
that use adapted wavelet filters.
Multifractals have become popular as flexible models in modeling real-life data of high
frequency. We developed a method of testing whether the data of high frequency is consistent
with monofractality using meaningful descriptors coming from a wavelet-generated multifractal
spectrum. We discuss theoretical properties of the descriptors, their computational
implementation, the use in data mining, and the effectiveness in the context of simulations,
an application in turbulence, and analysis of coding/noncoding regions in DNA sequences.
The wavelet thresholding is a simple and effective operation in wavelet domains that selects
the subset of wavelet coefficients from a noised signal. We propose the selection of this
subset in a semi-supervised fashion, in which a neighbor structure and classification function
appropriate for wavelet domains are utilized. The decision to include an unlabeled coefficient
in the model depends not only on its magnitude but also on the labeled and unlabeled
coefficients from its neighborhood. The theoretical properties of the method are discussed
and its performance is demonstrated on simulated examples.Ph.D.Committee Chair: Brani Vidakovic; Committee Member: Justin Romberg; Committee Member: Ming Yuan; Committee Member: Paul Kvam; Committee Member: Xiaoming Hu
