Advisor: Professor Brani Vidakovic

Committee Members: Professor David Goldsman, Professor Kamran Paynabar, Professor Ben Haaland, Professor Eberhard O. Voit (The Wallace H. Coulter Department of Biomedical Engineering)

Date and Time: Friday, May 6, 2016, 1:00PM

Location: ISyE Groseclose – 226A

Abstract:

The advancement of sensor technology enables us to collect a massive amount of data and at the same time, poses a challenge of summarizing such data in useful features. In this thesis, the focus is on the summary of complex real-life signals that possess self-similarity, which indicates that the signal behaves similarly in a range of scales, or resolutions. Such signals can be well characterized with a scaling index, or self-similarity index, that represents essential scaling characteristics.

This thesis proposes four novel methods that facilitate and improve the assessment of scaling in signals based on non-decimated wavelet transform (NDWT). NDWT's are preferred to the standard orthogonal wavelet transforms in a number of data analytic tasks because of their time-invariance and redundancy. To facilitate NDWT, Chapter 3 of this thesis devises an NDWT matrix that efficiently maps an input signal from an acquisition domain to the wavelet domain with a simple matrix multiplication. Applying the proposed NDWT matrix provides four advantages: It yields a transform compressive in summarizing information, it is faster in computation, and it processes inputs of any size (non-dyadic). Such advantages of an NDWT matrix are illustrated with various example applications. An NDWT matrix is used for all NDWT's in Chapters 4 to 6. Chapter 4 introduces a method for scaling estimation based on NDWT and its wavelet spectrum. The method utilizes a distinctive character of NDWT that does not decimate wavelet coefficients, which enables us to obtain local spectra and more accurate scaling estimators. The method applied to simulated signals for which scaling is known in advance yields estimators with lower mean squared errors. An example application with mammographic images for breast cancer detection yields the best diagnostic accuracy in excess of 80%. In Chapter 5, we shift the focus to some real-life signals for which a theoretical scaling index is known and fixed. Based on Bayesian statistics, the method proposed in the chapter incorporates prior information about Hurst exponent H of signals into a likelihood model and estimates H with maximum a posteriori (MAP) estimation. The method yields estimators with lower mean squared errors even when the mean value of the prior distribution is slightly different from the true theoretical value.

In the assessment of scaling of one-dimensional data based on NDWT, the regression method, which is standardly used, yields biased estimators because of autocorrelations present within wavelet coefficients. This autocorrelation is a result of redundancy of NDWT. Chapter 6 illustrates two robust methods for estimation of scaling that decrease the autocorrelation with a logarithmic transform and a resampling approach. The proposed methods yield lower mean squared errors with decreased bias, and the resulting estimators are asymptotically normal and unbiased with variance that is independent of the multiresolution levels.