The goal of this research was to aid in the fight against the heroin and opioid epidemic by developing new methodology for heroin provenance determination and forensic sample comparison. Over 400 illicit heroin powder samples were analyzed using quadrupole and high-resolution inductively-coupled plasma mass spectrometry (Q-ICP-MS and HR-ICP-MS) in order to measure and identify elemental contaminants useful for associating heroin samples of common origin and differentiating heroin of different geographic origins. Additionally, 198 heroin samples were analyzed by multi-collector ICP-MS (MC-ICP-MS) to measure radiogenic strontium isotope ratios (87Sr/86Sr) with high-precision for heroin provenance determination, for the first time. Supervised discriminant analysis models were constructed to predict heroin origin using elemental composition. The model was able to correctly associate 88% of the samples to their region of origin. When 87Sr/86Sr data were combined with Q-ICP-MS elemental data, the correct association of heroin samples improved to ≥90% for all groups with an average of 93% correct classification. For forensic sample comparisons, quantitative elemental data (11 elements measured) from 120 samples, 30 from each of the four regions, were compared in order to assess the rate of discrimination (5400 total comparisons). Using a match criterion of ±3 standard deviations about the mean, only 14 of the 5400 possible comparison pairs were not discriminated resulting in a discrimination rate of 99.7%. For determining the rate of correct associations, 3 replicates of 24 duplicate samples were prepared and analyzed on separate days. Only 1 of the 24 correct pairs were not associated for a correct association rate of 95.8%. New methods for provenance determination and sample comparison are expected to be incredibly useful to intelligence agencies and law enforcement working to reduce the proliferation of heroin.

In Experiment 1, four subjects were trained to match two visual samples (A) and their respective nonidentical visual comparisons (B); i.e., A-B matching. During nonreinforced test trials, all subjects demonstrated stimulus equivalences within the context of sample-comparison reversibility (B-A matching): When B stimuli were used as samples, appropriate responding to A comparisons occurred. A-B and B-A matching persisted given novel stimuli as alternate comparisons. However, the novel comparisons were consistently selected in the presence of nonmatching stimuli: i.e., during trials comprised of a novel comparison, an A or B sample from one stimulus class, and an "incorrect" comparison from the other, B or A stimuli respectively. In Experiment 2, three groups of subjects were trained under three different mediated transfer paradigms (e.g., A-B, C-B matching). Tests for reversibility (e.g., B0A, B0C matching) and mediated transfer (e.g., A-C, C-A matching)evinced stimulus equivalences for 11 of 12 subjects. The 11 subjects also matched the mediated equivalences given novel comparisons; whereas, they selected the novel comparisons when combined with nonmatching stimuli. Overall, the demonstrated stimulus equivalences favor a concept learning interpretation of non-identity matching-to-sample. Additionally, the trained and mediated matching relations were comprised of complementary sets of S+ and S- rules: Any stimulus of a given class used as a sample designated both the "correct" and "incorrect" comparisons.

<p>Two-sample comparison is a fundamental problem in statistics. Given two samples of data, the interest lies in understanding whether the two samples were generated by the same distribution or not. Traditional two-sample comparison methods are not suitable for modern data where the underlying distributions are multivariate and highly multi-modal, and the differences across the distributions are often locally concentrated. The focus of this thesis is to develop novel statistical methodology for two-sample comparison which is effective in such scenarios. Tools from the nonparametric Bayesian literature are used to flexibly describe the distributions. Additionally, the two-sample comparison problem is decomposed into a collection of local tests on individual parameters describing the distributions. This strategy not only yields high statistical power, but also allows one to identify the nature of the distributional difference. In many real-world applications, detecting the nature of the difference is as important as the existence of the difference itself. Generalizations to multi-sample comparison and more complex statistical problems, such as multi-way analysis of variance, are also discussed.</p> / Dissertation

Developing nonparametric statistical methods and inference procedures for high-dimensional large data have been a challenging frontier problem of statistics. To attack this problem, in recent years, a clear rising trend has been observed with a radically different viewpoint--``Graph-based Nonparametrics," which is the main research focus of this dissertation. The basic idea consists of two steps: (i) representation step: code the given data using graphs, (ii) analysis step: apply statistical methods on the graph-transformed problem to systematically tackle various types of data structures. Under this general framework, this dissertation develops two major research directions. Chapter 2—based on Mukhopadhyay and Wang (2019a)—introduces a new nonparametric method for high-dimensional k-sample comparison problem that is distribution-free, robust, and continues to work even when the dimension of the data is larger than the sample size. The proposed theory is based on modern LP-nonparametrics tools and unexplored connections with spectral graph theory. The key is to construct a specially-designed weighted graph from the data and to reformulate the k-sample problem into a community detection problem. The procedure is shown to possess various desirable properties along with a characteristic exploratory flavor that has practical consequences. The numerical examples show surprisingly well performance of our method under a broad range of realistic situations. Chapter 3—based on Mukhopadhyay and Wang (2019b)—revisits some foundational questions about network modeling that are still unsolved. In particular, we present unified statistical theory of the fundamental spectral graph methods (e.g., Laplacian, Modularity, Diffusion map, regularized Laplacian, Google PageRank model), which are often viewed as spectral heuristic-based empirical mystery facts. Despite half a century of research, this question has been one of the most formidable open issues, if not the core problem in modern network science. Our approach integrates modern nonparametric statistics, mathematical approximation theory (of integral equations), and computational harmonic analysis in a novel way to develop a theory that unifies and generalizes the existing paradigm. From a practical standpoint, it is shown that this perspective can provide adequate guidance for designing next-generation computational tools for large-scale problems. As an example, we have described the high-dimensional change-point detection problem. Chapter 4 discusses some further extensions and application of our methodologies to regularized spectral clustering and spatial graph regression problems. The dissertation concludes with the a discussion of two important areas of future studies. / Statistics