Introduction
My thesis work for the Honors Tutorial College of Ohio University centered on the wavelet transform and its application to frequency estimation in various data series, particularly series from condensed matter physics. I worked closely with advisor David Drabold of the Department of Physics and Astronomy at OU and drew data and advice from collaborators across departments at Ohio University, Texas Tech University, and Cambridge University. For his help with the thesis and his mentorship in general, Dr. Drabold received the Distinguished Mentor Award from the Honors Tutorial College!
The focus of my thesis was the mathematic foundations of and methods for using the continuous wavelet transform for frequency estimation. This was supplemented with analysis of time series examples to demonstrate the validity and power of the technique and included reproduction of known results and original applications. It can easily be used as a primer to the subject or as reference and refresher for those already familiar with the topic.
Thesis Abstract
One of the key problems of data analysis is spectral estimation: the determination of frequencies and amplitudes present in experimental and simulation data. The most popular method is the Fourier transform; however, the Fourier transform has important limitations. Chiefly, it is accurate and reliable only when one frequency is present and when that frequency is stationary (i.e., does not vary with time). Despite these limitations, many continue to use the Fourier transform in suboptimal situations risking nonsensical results. An attempt to overcome these limitations is the continuous wavelet transform, which incorporates temporal information into the analysis, allowing non-stationary (i.e., time-dependent) frequencies to be resolved. We have developed and improved techniques for spectral estimation with the wavelet transform using the Mathematica computing environment and have applied these techniques to a classic time series to demonstrate our method. In addition, we have applied our technique to new time series from molecular dynamics simulations of carbon trimer molecules and of excited hydrogen atoms in a silicon lattice. Energy decay from the local vibrational modes of the hydrogen into the lattice can be inferred from changes in the amplitudes of the spectral components.
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