<div>Many important scientific, engineering, and societal challenges involve large systems of individual agents or components interacting in complex ways. For example, to understand the emergence of consciousness, we study the dendritic integration in neurons; to prevent disease and rumor outbreaks, we trace the dynamics of social networks; to perform complicated scientific experiments, we separate and control the independent variables. Collectively, the interactions between individual neurons/agents/variables are often non-linear, i.e., a subset of the agents jointly behave in a manner unlike the marginal behaviors of the individuals.</div><div><br></div><div>The goal of this thesis is to construct a theoretical framework for measuring, comparing, and representing complex interactions in stochastic systems. Specifically, tools from information theory, differential geometry, lattice theory, and linear algebra are used to identify and characterize higher-order interactions among random variables.</div><div><br></div><div>We first propose measures of unique, redundant, and synergistic interactions for small stochastic systems using information projections for the exponential family. Their magnitudes are endowed with information theoretical meanings naturally, since they are measured by the Kullback-Leibler divergence. We prove that these quantities satisfy various desired properties.</div><div><br></div><div>We next apply these measures to hypothesis testing and network communication. We interpret the unique information as the two types of error components in a hypothesis testing problem. We analytically show that there is a duality between the synergistic and redundant information in Gaussian Multiple Access Channels (MAC) and Broadcast Channels (BC). We establish a novel duality between the partial information decomposition components for MAC and BC in the general case.</div><div><br></div><div>We lastly propose a new concept of representing the partial information decomposition framework with random variables. We give necessary and sufficient conditions for the representation under the assumption of Gaussianity and develop a construction method.</div><div><br></div><div>This research has the potential to advance the fields of information theory, statistics, and machine learning by contributing novel ideas, implementing these ideas with innovative tools, and constructing new simulation methods.</div>

Indiana University-Purdue University Indianapolis (IUPUI) / Solid phase micro-extraction is a powerful and versatile technique, well-suited to the analysis of numerous samples of forensic interest. The exceptional sensitivity of the SPME platform, combined with its adaptability to traditional GC-MS systems and its ability to extract samples with minimal work-up, make it appropriate to applications in forensic laboratories. In a series of research projects, solid phase micro-extraction was employed for the analysis of explosives, commercial tire treatments, and entomological specimens. In the first project, the volatile organic compounds emanating from two brands of pseudo-explosive training aids for use in detector dog imprinting were determined by SPME-GC-MS, and the efficacy of these training materials was tested in live canine trials. In the second project, the headspace above various plasticizers was analyzed comparative to that of Composition C-4 in order to draw conclusions about the odor compound, 2- ethyl-1-hexnaol, with an eye toward the design of future training aids. In the third, automobile tires which had participated in professional race events were analyzed for the presence of illicit tire treatments, and in the fourth, a novel SPME-GC-MS method was developed for the analysis of blowfly (Diptera) liquid extracts. In the fifth and final project, the new method was put to the task of performing a chemotaxonomic analysis on pupa specimens, seeking to chemically characterize them according to their age, generation, and species.

Animal and human wastes introduce pathogens into rivers and streams, creating human health and economic burdens. While direct monitoring for pathogens is possible, it is impractical due to the sporadic distribution of pathogens, cost to identify, and health risks to laboratory workers. To overcome these issues, fecal indicator organisms are used to estimate the presence of pathogens. Although fecal indicators generally protect public health, they fall short in their utility because of difficulties in public health risk characterization, inconsistent correlations with pathogens, weak source identification, and their potential to persist in environments with no point sources of fecal pollution. This research focuses on characterizing the ecology of fecal indicators using both modeling and metabolic indicators to better understand the processes that drive fecal pollution. Fecal indicator impairment was modeled in Sinking Creek, a 303 (d) listed stream in Northeast Tennessee, using the ecological niche model, Maxent, for two different fecal indicators. While the use of Maxent has been well demonstrated at the macroscale, this study introduces its application to ecological niches at the microscale. Stream impairment seasonality was exhibited in two different indicators over multiple years and different resolutions (quarterly versus monthly sampling programs). This stresses the need for multiple year and month sampling to capture heterogeneity in fecal indicator concentrations. Although discharge is strongly associated with dissolved solutes, fecal indicator impairment was governed by other ecological factors such as populations of heterotrophic bacteria, enzyme activity, nutrient conditions, and other metabolic indicators. This research also incorporated metabolic indicators to characterize spatiotemporal variability in microbial community function, making connections to fecal and other pollution gradients. Communities differed in their ability to use a wide variety of substrates, and metabolic inhibition in sediments captured most of the interaction of aquatic and benthic communities. Sediment substrate activity was also indicative of degrees of pollution, suggesting that sediment is a potential reservoir for Escherichia coli in this stream, and there is possibility for resuspension, extended residence times, and increased duration for exposure. This research highlights the benefit of using models and other microbial indicators to better understand how environment shapes the niche of fecal indicators.

A 2009 report published by the National Research Council addressed the need for improvements in the field of forensic science. In the report emphasis was placed on the need for more rigorous scientific analysis within many forensic science disciplines and for established limitations and determination of error rates from statistical analysis. This research focused on multivariate statistical techniques for the analysis of spectral data obtained for multiple forensic applications which include samples from: automobile float glasses and paints, bones, metal transfers, ignitable liquids and fire debris, and organic compounds including explosives. The statistical techniques were used for two types of data analysis: classification and discrimination. Statistical methods including linear discriminant analysis and a novel soft classification method were used to provide classification of forensic samples based on a compiled library. The novel soft classification method combined three statistical steps: Principal Component Analysis (PCA), Target Factor Analysis (TFA), and Bayesian Decision Theory (BDT) to provide classification based on posterior probabilities of class membership. The posterior probabilities provide a statistical probability of classification which can aid a forensic analyst in reaching a conclusion. The second analytical approach applied nonparametric methods to provide the means for discrimination between samples. Nonparametric methods are performed as hypothesis test and do not assume normal distribution of the analytical figures of merit. The nonparametric iv permutation test was applied to forensic applications to determine the similarity between two samples and provide discrimination rates. Both the classification method and discrimination method were applied to data acquired from multiple instrumental methods. The instrumental methods included: Laser Induced-Breakdown Spectroscopy (LIBS), Fourier Transform Infrared Spectroscopy (FTIR), Raman spectroscopy, and Gas Chromatography-Mass Spectrometry (GC-MS). Some of these instrumental methods are currently applied to forensic applications, such as GC-MS for the analysis of ignitable liquid and fire debris samples; while others provide new instrumental methods to areas within forensic science which currently lack instrumental analysis techniques, such as LIBS for the analysis of metal transfers. The combination of the instrumental techniques and multivariate statistical techniques is investigated in new approaches to forensic applications in this research to assist in improving the field of forensic science.

A series of recent excavations on the 2nd Century AD Antonine frontier forts of the Midland Scottish valley, have produced results which suggest that the army was making its own pottery on an appreciable scale. This was at a time when pottery production was thought to have moved almost exclusively into civilian hands. The possible local ware groups identified by the excavations were largely independent of firm source indicators such as kiln and waster material and the number of available samples was often-small. A program of Neutron Activation and Thin Section petrological analyses was undertaken along with an investigation into Textural Analysis, a facet of the Petrological toolkit. The aim was both to define the site ware groups and a group of specialist vessels thought to be local to Scotland, the Mortaria, and to make statements as to their provenance. Although the Mortaria analysis was limited by problems of sample group size and availability, by improving the objectivity of the statistical handling of the derived data sets and developing methods for the high level study of textural data, the site ware groups were defined successfully at both the "intrall and "inter" site levels. The analyses also furnished interpretations as to the mode and nature of the site production schemes. Through the full analysis of' site Daub samples linked to more traditional provenancing techniques, in all but one case the ware groups could be assigned to the source sites, where contrasting production modes could be identified with military as opposed to civilian production. / SERC

No description available.

Dissolved organic matter (DOM) is a major component of natural waters and provides essential nutrients for aquatic organisms. However, excess DOM in the water results in water quality issues and affects the aquatic life negatively. The present research evaluated the source, composition, reactivity, dynamics, and the spatial distribution of DOM in diverse water bodies using spectrofluorometric methods in tandem with multivariate statistics. The study was conducted in the inland and coastal water bodies of Mississippi, Louisiana, and Alabama over a period of three years (2016 to 2018). Surface water samples were collected from spatially separated waterbodies with diverse land use and land cover classes. In addition, reactivity of DOM was assessed by conducting a series of laboratory experiments at varying magnitudes of sunlight and bacterial activity. Spatial distribution and mobility of DOM, nutrients and trace elements with respect to land cover classes and hydrology was evaluated using watershed delineation and multivariate statistics. Results suggest that microbial humic-like or protein-like DOM compositions derived from microbial/anthropogenic sources were less reactive than the terrestrial humic-like compositions originated from forests and woody wetlands. Furthermore, the sunlight was the major factor causing the degradation of DOM in the water bodies, while temperature had a minor effect. Additionally, the results also suggest that livestock fields in the pastoral and rangelands release a high amount of microbial humic-like DOM along with nutrients such as phosphates and nitrates into the water bodies. Present research identified the presence of four types of DOM in the study areas and were terrestrial humic-like, microbial humic-like, soil-derived humic-like and protein-like compositions. Additionally, trace element availability and mobility of coastal areas is influenced by local hydrology and precipitation. Research also identified forested areas as the major source of DOM to the water bodies of Mississippi. In conclusion, present research found that watershed land use and land cover, hydrology, and climate control the dynamics of DOM, other nutrients, and trace element delivery to the water bodies, while combined effects of light and bacteria are more efficient in reprocessing DOM chemistry within the waterbody.

Diese Dissertation stellt drei sich ergänzende Fortschritte in der statistischen Modellierung multivariater Daten vor und behandelt Herausforderungen im Bereich des modellbasierten Clusterings für hochdimensionale Daten, der Analyse longitudinaler Daten sowie der multivariaten Verteilungsregression. Der erste Forschungszweig konzentriert sich auf tiefe Gaußsche Mischmodelle, eine leistungsfähige Erweiterung herkömmlicher Gaußscher Mischmodelle. Wir erforschen Bayessche Inferenz mit Sparsamkeitsprioris zur Regularisierung der Schätzung tiefer Mischmodelle und stellen ein innovatives tiefes Mischmodell von Faktormodellen vor, das in der Lage ist, hochdimensionale Probleme zu bewältigen. Der zweite Forschungsstrang erweitert tiefe Mischmodelle von Clustering zu Regression. Unter Verwendung des tiefen Mischmodells von Faktormodellen als Prior für Zufallseffekte stellen wir einen innovativen Ansatz vor: tiefe Mischmodelle von linearen gemischten Modellen, der lineare gemischte Modelle so erweitert, dass er den Komplexitäten longitudinaler Daten mit vielen Beobachtungen pro Subjekt und komplexen zeitlichen Trends gerecht wird. Dieser Forschungszweig überwindet Beschränkungen gegenwärtiger Modelle und präsentiert eine anpassungsfähige Lösung für hochdimensionale Szenarien. Der dritte Forschungszweig setzt sich mit der Herausforderung auseinander, wahrhaft multivariate Verteilungen im Kontext von Generalisierten Additiven Modellen für Ort, Skala und Form zu modellieren. Wir präsentieren einen innovativen Ansatz, der Copula-Regression nutzt, um die Abhängigkeitsstruktur mittels einer Gauß-Copula zu modellieren. Dies ermöglicht die gemeinsame Modellierung hochdimensionaler Vektoren mit flexiblen marginalen Verteilungen. Hier erleichtert bayessche Inferenz die effiziente Schätzung des stark parametrisierten Modells und führt zu einem äußerst flexiblen Ansatz im Vergleich zu bestehenden Modellen. / This thesis introduces three complementary advancements in statistical modeling for multivariate data, addressing challenges in model-based clustering for high-dimensional data, longitudinal data analysis, and multivariate distributional regression. The first research strand focuses on deep Gaussian mixture models, a powerful extension of ordinary Gaussian mixture models. We explore the application of Bayesian inference with sparsity priors to regularize the estimation of deep mixtures, presenting a novel Bayesian deep mixtures of factor analyzers model capable of handling high-dimensional problems. The inclusion of sparsity-inducing priors in the model contributes to improved clustering results. A scalable natural gradient variational inference algorithm is developed to enhance computational efficiency. The second research strand extends deep mixture models from clustering towards regression. Leveraging the deep mixtures of factor analyzers model as a prior for random effects, we introduce a novel framework, deep mixtures of linear mixed models that extends mixtures of linear mixed models to accommodate the complexities of longitudinal data with many observations per subject and intricate temporal trends. We describe an efficient variational inference approach. This research addresses the limitations of current models and provides a flexible solution for high-dimensional settings. The third research strand tackles the challenge of modeling truly multivariate distributions in the context of Generalized Additive Models for Location, Scale, and Shape. We propose a novel approach utilizing copula regression to model the dependence structure through a Gaussian copula, allowing for joint modeling of high-dimensional response vectors with flexible marginal distributions. Here, Bayesian inference facilitates efficient estimation of the highly parameterized model, introducing a highly flexible and complementary approach to existing models.

In this study the relationships among the land use and soil types in the entire river basin as well as in the buffer strip on the one hand, and the Lithuanian river water quality physico-chemical parameters on the other hand have been identified. The spatially based Factor analysis has been tested on river hydrochemical data from 108 sites for the period of 1999-2004. The Factor analysis enabled the identification of the main natural and anthropogenic processes (factors) determining water quality during each season of a year. As a result, monitoring stations were grouped into clusters each representing a group of stations mostly affected by a relevant factor. In addition, the suitability of world-wide used diatom-based water status assessment methods to apply under Lithuanian conditions has been assessed. The results of established water quality determining factors and the tested multivariate statistical procedures can be applied in practice when the reasons for water quality impairments are to be investigated or river monitoring network is to be optimized. The results reveal that Wastewater factor is prominent in small rivers downstream larger towns; Agro-geological factor – in northern Lithuania‘s rivers of heavy carbonated soils and intensive agriculture lands as well as in south-eastern Lithuania‘s rivers of more acidic soils; Hardly degradable organics factor – in northern and middle Lithuania‘s rivers of heavy-textured and fertile agricultural soils. The... [to full text] / Pastaruoju metu tiek Lietuvoje, tiek ir visoje Europos Sąjungoje labai aktualu tinkamai įvertinti vandens telkinių būklę (ypač pagal biologinius kokybės elementus), nustatyti vandens telkinių būklės problemas, jas lemiančius veiksnius bei imtis adekvačių būklės gerinimo priemonių, nes to reikalauja 2000 m. įsigaliojusi ES Bendroji vandens politikos direktyva – pagrindinis vandens sritį reguliuojantis teisinis dokumentas. Šio darbo tikslas - nustatyti žemėnaudos, dirvožemių ir upių vandens fizikinių, cheminių ir biologinių parametrų sąveikos dėsningumus vertinant vandens būklę. Įgyvendinant šį tikslą buvo siekiama sistemiškai įvertinti įvairių veiksnių poveikį Lietuvos upių vandens būklei ir nustatyti jų teritorinius dėsningumus bei įvertinti upių vandens būklės pagal titnagdumblius vertinimo metodų tinkamumą Lietuvos sąlygomis. Šiam tikslui pasiekti naudoti Lietuvos upių valstybinio monitoringo, dirvožemio ir žemės dangos duomenys pasitelkiant faktorinę analizę bei atitinkamus vienmatės statistikos metodus. Rezultatai parodė, kad upių būklę lemiančias priežastis galima patikimai pažinti, kartu vertinant hidrocheminius ir baseinų fizinių charakteristikų, dirvožemių ir žemėnaudos duomenis, jų kompleksiškai tarpusavio ryšių analizei pasitelkus tinkamus daugiamatės ir vienmatės statistikos metodus. Tokia analizė leidžia pažinti ne tik atskirus vandens ekosistemos komponentus, bet ir jų visumą kaip vieną sąveikaujančią sistemą ir priimti kokybiškesnius vandens kokybės valdymo... [toliau žr. visą tekstą]

Die Gewichte eines Portfolios liegen meist als als Kombination des Produkts der Präzisionsmatrix und des Erwartungswertvektors vor. In der Praxis müssen diese Parameter geschätzt werden, allerdings ist die Beschreibung der damit verbundenen Schätzunsicherheit über eine Verteilung dieses Produktes eine Herausforderung. In dieser Arbeit wird demonstriert, dass ein geeignetes bayesianisches Modell nicht nur zu einer leicht zugänglichen Posteriori-Verteilung führt, sondern auch zu leicht interpretierbaren Beschreibungen des Portfoliorisikos, wie beispielsweise einer Ausfallwahrscheinlichkeit des gesamten Portfolios zu jedem Zeitpunkt. Dazu werden die Parameter mit ihren konjugierten Prioris ausgestatet. Mit Hilfe bekannter Ergebnisse aus der Theorie multivariater Verteilungen ist es möglich, eine stochastische Darstellung für relevante Ausdrücke wie den Portfoliogewichten oder des effizienten Randes zu geben. Diese Darstellungen ermöglichen nicht nur die Bestimmung von Bayes-Schätzern der Parameter, sondern sind auch noch rechentechnisch hoch effizient, da Zufallszahlen nur aus bekannten und leicht zugänglichen Verteilungen gezogen werden. Insbesondere aber werden Markov-Chain-Monte-Carlo Methoden nicht benötigt. Angewendet wird diese Methodik an einem mehrperiodigen Portfoliomodell für eine exponentielle Nutzenfunktion, am Tangentialportfolio, zur Schätzung des effizienten Randes, des globalen Minimum-Varianz-Portfolios wie auch am gesamten Mittelwert-Varianz Ansatzes. Für alle behandelten Portfoliomodelle werden für wichtige Größen stochastische Darstellungen oder Bayes-Schätzer gefunden. Die Praktikabilität und Flexibilität wie auch bestimmte Eigenschaften werden in Anwendungen mit realen Datensätzen oder Simulationen illustriert. / Usually, the weights of portfolio assets are expressed as a comination of the product of the precision matrix and the mean vector. These parameters have to be estimated in practical applications. But it is a challenge to describe the associated estimation risk of this product. It is demonstrated in this thesis, that a suitable Bayesian approach does not only lead to an easily accessible posteriori distribution, but also leads to easily interpretable risk measures. This also includes for example the default probability of the portfolio at all relevant points in time. To approach this task, the parameters are endowed with their conjugate priors. Using results from the theory of multivariate distributions, stochastic representations for the portfolio parameter are derived, for example for the portfolio weights or the efficient frontier. These representations not only allow to derive Bayes estimates of these parameters, but are computationally highly efficient since all th necessary random variables are drawn from well known and easily accessible distributions. Most importantly, Markov-Chain-Monte-Carlo methods are not necessary. These methods are applied to a multi-period portfolio for an exponential utility function, to the tangent portfolio, to estimate the efficient frontier and also to a general mean-variance approach. Stochastic representations and Bayes estimates are derived for all relevant parameters. The practicability and flexibility as well as specific properties are demonstrated using either real data or simulations.