Given the recent popularity and clear evidence of wide applicability of wavelets, this thesis is devoted to several statistical applications of Wavelet transforms. Statistical multiscale modeling has, in the most recent decade, become a well-established area in both theoretical and applied statistics, with impact on developments in statistical methodology. Wavelet-based methods are important in statistics in areas such as regression, density and function estimation, factor analysis, modeling and forecasting in time series analysis, assessing self-similarity and fractality in data, and spatial statistics. In this thesis we show applicability of the wavelets by considering three problems: First, we consider a binary wavelet-based linear classifier. Both consistency results and implemental issues are addressed. We show that under mild assumptions wavelet-based classification rule is both weakly and strongly universally consistent. The proposed method is illustrated on synthetic data sets in which the truth is known and on applied classification problems from the industrial and bioengineering fields. Second, we develop wavelet shrinkage methodology based on testing multiple hypotheses in the wavelet domain. The shrinkage/thresholding approach by implicit or explicit simultaneous testing of many hypotheses had been considered by many researchers and goes back to the early 1990's. We propose two new approaches to wavelet shrinkage/thresholding based on local False Discovery Rate (FDR), Bayes factors and ordering of posterior probabilities. Finally, we propose a novel method for the analysis of straight-line alignment of features in the images based on Hough and Wavelet transforms. The new method is designed to work specifically with Transmission Electron Microscope (TEM) images taken at nanoscale to detect linear structure formed by the atomic lattice.

The objective of this dissertation is to develop a suitable statistical methodology for parameter estimation in long memory process. Time series data with complex covariance structure are shown up in various fields such as finance, computer network, and econometrics. Many researchers suggested the various methodologies defined in different domains: frequency domain and time domain. However, many traditional statistical methods are not working well in complicated case, for example, nonstationary process. The development of the robust methodologies against nonstationarity is the main focus of my dissertation. We suggest a wavelet-based Bayesian method which shares good properties coming from both wavelet-based method and Bayesian approach. To check the robustness of the method, we consider ARFIMA(0, d, 0) with linear trend. Also, we compare the result of the method with that of several existing methods, which are defined in different domains, i.e. time domain estimators, frequency domain estimators. Also, we apply the method to functional magnetic resonance imaging (fMRI) data to find some connection between brain activity and long memory parameter. Another objective of this dissertation is to develop a wavelet-based denoising technique when there is heterogeneous variance noise in high throughput data, especially protein mass spectrometry data. Since denoising technique pretty much depends on threshold value, it is very important to get a proper threshold value which involves estimate of standard deviation. To this end, we detect variance change point first and get suitable threshold values in each segment. After that, we apply local wavelet thresholding to each segment, respectively. For comparison, we consider several existing global thresholding methods.

Automated detection of geological boundaries and determination of cyclic events controlling deposition can facilitate stratigraphic analysis and reservoir characterization. This study applies the wavelet transformation, a recent advance in signal analysis techniques, to interpret cyclicity, determine its controlling factors, and detect zone boundaries. We tested the cyclostratigraphic assessments using well log and core data from a well in a fluvio-eolian sequence in the Ormskirk Sandstone, Irish Sea. The boundary detection technique was tested using log data from 10 wells in the Apiay field, Colombia. We processed the wavelet coefficients for each zone of the Ormskirk Formation and determined the wavelengths of the strongest cyclicities. Comparing these periodicities with Milankovitch cycles, we found a strong correspondence of the two. This suggests that climate exercised an important control on depositional cyclicity, as had been concluded in previous studies of the Ormskirk Sandstone. The wavelet coefficients from the log data in the Apiay field were combined to form features. These vectors were used in conjunction with pattern recognition techniques to perform detection in 7 boundaries. For the upper two units, the boundary was detected within 10 feet of their actual depth, in 90% of the wells. The mean detection performance in the Apiay field is 50%. We compared our method with other traditional techniques which do not focus on selecting optimal features for boundary identification. Those methods resulted in detection performances of 40% for the uppermost boundary, which lag behind the 90% performance of our method. Automated determination of geologic boundaries will expedite studies, and knowledge of the controlling deposition factors will enhance stratigraphic and reservoir characterization models. We expect that automated boundary detection and cyclicity analysis will prove to be valuable and time-saving methods for establishing correlations and their uncertainties in many types of oil and gas reservoirs, thus facilitating reservoir exploration and management.

Univ., Habil.-Schr.--Gießen. / Literaturverz. S. [127] - 133.

Thèse doctorat : Images et Systèmes : Villeurbanne, INSA : 2005. / Thèse rédigée en anglais. Titre provenant de l'écran-titre. Bibliogr. p. 87-99.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 112-120). Available also in a digital version from Dissertation Abstracts.

Thesis (M.S. in Math.)--School of Mathematics, Georgia Institute of Technology, 2004. Directed by Yang Wang. / Includes bibliographical references (leaves 37-38).

published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

The ability to efficiently and sparsely represent seismic data is becoming an increasingly important problem in geophysics. Over the last thirty years many transforms such as wavelets, curvelets, contourlets, surfacelets, shearlets, and many other types of ‘x-lets’ have been developed. Such transform were leveraged to resolve this issue of sparse representations. In this work we compare the properties of four of these commonly used transforms, namely the shift-invariant wavelets, complex wavelets, curvelets and surfacelets. We also explore the performance of these transforms for the problem of recovering seismic wavefields from incomplete measurements.

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