The standard approach to analyzing brain electrical activity is to examine the spectral density function (SDF) and identify frequency bands, defined apriori, that have the most substantial relative contributions to the overall variance of the signal. However, a limitation of this approach is that the precise frequency and bandwidth of oscillations are not uniform across cognitive demands. Thus, these bands should not be arbitrarily set in any analysis. To overcome this limitation, we propose three Bayesian Non-parametric models for time series decomposition which are data-driven approaches that identifies (i) the number of prominent spectral peaks, (ii) the frequency peak locations, and (iii) their corresponding bandwidths (or spread of power around the peaks). The standardized SDF is represented as a Dirichlet process mixture based on a kernel derived from second-order auto-regressive processes which completely characterize the location (peak) and scale (bandwidth) parameters. A Metropolis-Hastings within Gibbs algorithm is developed for sampling from the posterior distribution of the mixture parameters for each project. Simulation studies demonstrate the robustness and performance of the proposed methods. The methods developed were applied to analyze local field potential (LFP) activity from the hippocampus of laboratory rats across different conditions in a non-spatial sequence memory experiment to identify the most prominent frequency bands and examine the link between specific patterns of brain oscillatory activity and trial-specific cognitive demands. The second application study 61 EEG channels from two subjects performing a visual recognition task to discover frequency-specific oscillations present across brain zones. The third application extends the model to characterize the data coming from 10 alcoholics and 10 controls across three experimental conditions across 30 trials. The proposed models generate a framework to condense the oscillatory behavior of populations across different tasks isolating the target fundamental components allowing the practitioner different perspectives of analysis.

This thesis focuses on the comprehensive and thorough studies on the geospatial processing of airborne (full) waveform lidar data, including waveform modeling, direct georeferencing, and precise georeferencing with self-calibration.<div><br></div><div>Both parametric and nonparametric approaches of waveform decomposition are studied. The traditional parametric approach assumes that the returned waveforms follow a Gaussian mixture model where each component is a Gaussian. However, many real examples show that the waveform components can be neither Gaussian nor symmetric. To address the problem, this thesis proposes a nonparametric mixture model to represent lidar waveforms without any constraints on the shape of the waveform components. To decompose the waveforms, a fuzzy mean-shift algorithm is then developed. This approach has the following properties: 1) it does not assume that the waveforms follow any parametric or functional distributions; 2) the waveform decomposition is treated as a fuzzy data clustering problem and the number of components is determined during the process of decomposition; 3) neither peak selection nor noise floor filtering prior to the decomposition is needed; and 4) the range measurement is not affected by the process of noise filtering. In addition, the fuzzy mean-shift approach is about three times faster than the conventional expectationmaximization algorithm and tends to lead to fewer artifacts in the resultant digital elevation model. <br></div><div><br></div><div>This thesis also develops a framework and methodology of self-calibration that simultaneously determines the waveform geospatial position and boresight angles. Besides using the flight trajectory and plane attitude recorded by the onboard GPS receiver and inertial measurement unit, the framework makes use of the publically accessible digital elevation models as control over the study area. Compared to the conventional calibration and georeferencing method, the new development has minimum requirements on ground truth: no extra ground control, no planar objects, and no overlap flight strips are needed. Furthermore, it can also solve the problem of clock synchronization and boresight calibration simultaneously. Through a developed two-stage optimization strategy, the self-calibration approach can resolve both the time synchronization bias and boresight misalignment angles to achieve a stable and correct solution. As a result, a consistency of 0.8662 meter is achieved between the waveform derived digital elevation model and the reference one without systematic trend. Such experiments demonstrate the developed method is a necessary and more economic alternative to the conventional, high demanding georeferencing and calibration approach, especially when no or limited ground control is available.<br></div>