Hypothesis Testing Using Spatially Dependent Heavy-Tailed Multisensor Data

The detection of spatially dependent heavy-tailed signals is considered in this dissertation. While the central limit theorem, and its implication of asymptotic normality of interacting random processes, is generally useful for the theoretical characterization of a wide variety of natural and man-made signals, sensor data from many different applications, in fact, are characterized by non-Gaussian distributions. A common characteristic observed in non-Gaussian data is the presence of heavy-tails or fat tails. For such data, the probability density function (p.d.f.) of extreme values decay at a slower-than-exponential rate, implying that extreme events occur with greater probability. When these events are observed simultaneously by several sensors, their observations are also spatially dependent. In this dissertation, we develop the theory of detection for such data, obtained through heterogeneous sensors. In order to validate our theoretical results and proposed algorithms, we collect and analyze the behavior of indoor footstep data using a linear array of seismic sensors. We characterize the inter-sensor dependence using copula theory. Copulas are parametric functions which bind univariate p.d.f. s, to generate a valid joint p.d.f. We model the heavy-tailed data using the class of alpha-stable distributions. We consider a two-sided test in the Neyman-Pearson framework and present an asymptotic analysis of the generalized likelihood test (GLRT). Both, nested and non-nested models are considered in the analysis. We also use a likelihood maximization-based copula selection scheme as an integral part of the detection process. Since many types of copula functions are available in the literature, selecting the appropriate copula becomes an important component of the detection problem. The performance of the proposed scheme is evaluated numerically on simulated data, as well as using indoor seismic data. With appropriately selected models, our results demonstrate that a high probability of detection can be achieved for false alarm probabilities of the order of 10^-4. These results, using dependent alpha-stable signals, are presented for a two-sensor case. We identify the computational challenges associated with dependent alpha-stable modeling and propose alternative schemes to extend the detector design to a multisensor (multivariate) setting. We use a hierarchical tree based approach, called vines, to model the multivariate copulas, i.e., model the spatial dependence between multiple sensors. The performance of the proposed detectors under the vine-based scheme are evaluated on the indoor footstep data, and significant improvement is observed when compared against the case when only two sensors are deployed. Some open research issues are identified and discussed

Copulae: On the Crossroads of Mathematics and Economics

The central focus of the workshop was on copula theory as well as applications to multivariate stochastic modelling. The programme was intrinsically interdisciplinary and represented areas with much recent progress. The workshop included talks and dynamic discussions on construction, estimation and various applications of copulas to finance, insurance, hydrology, medicine, risk management and related fields

A flexible Clayton-like spatial copula with application to bounded support data

The Gaussian copula is a powerful tool that has been widely used to model spatial and/or temporal correlated data with arbitrary marginal distributions. However, this kind of model can potentially be too restrictive since it expresses a reflection symmetric dependence. In this paper, we propose a new spatial copula model that makes it possible to obtain random fields with arbitrary marginal distributions with a type of dependence that can be reflection symmetric or not. Particularly, we propose a new random field with uniform marginal distributions that can be viewed as a spatial generalization of the classical Clayton copula model. It is obtained through a power transformation of a specific instance of a beta random field which in turn is obtained using a transformation of two independent Gamma random fields. For the proposed random field, we study the second-order properties and we provide analytic expressions for the bivariate distribution and its correlation. Finally, in the reflection symmetric case, we study the associated geometrical properties. As an application of the proposed model we focus on spatial modeling of data with bounded support. Specifically, we focus on spatial regression models with marginal distribution of the beta type. In a simulation study, we investigate the use of the weighted pairwise composite likelihood method for the estimation of this model. Finally, the effectiveness of our methodology is illustrated by analyzing point-referenced vegetation index data using the Gaussian copula as benchmark. Our developments have been implemented in an open-source package for the \textsf{R} statistical environment

THE ROLE OF FIRE AND A FIRE-FREE INTERVAL IN THE RESTORATION OF UPLAND OAK COMMUNITIES ON THE CUMBERLAND PLATEAU, KENTUCKY

The decline of upland oak (Quercus spp.) communities in our eastern forests has been attributed to the loss of periodic disturbance after decades of fire suppression. As land managers have begun to reintroduce fire, effects on oak regeneration and species composition have varied widely, making it apparent that our understanding of how fire can aid in oak forest management needs refinement. Restoring upland oak communities requires decreasing stand density and opening of the canopy to release shade-intolerant oaks in the understory. This necessitates an extended fire-free interval to allow these oaks to be recruited into larger size classes and develop resistance to future fires. The ability of prescribed fire alone to create these structural changes is uncertain due to the low intensity of prescribed burns which for the most part do not kill larger diameter trees. In this work, I examined the utility of a fire-free interval following repeated fire alone as a management tool, as well as the combined effects of fire and mechanical removal in the form of midstory mastication. Where forest structure is significantly reduced by fire or mechanical removal, restoration of oak communities is complicated by both prolific sprouting and ingrowth of competitor species and the introduction of invasive species. The results of this study suggest that, in the absence of mechanical removals, reductions in stem density necessary to restore conditions for oak regeneration might be limited to sites that experience higher fire severity and/or drier landscape positions. Additionally, the rapid response of competing non-oak stems such as maple (Acer spp.), yellow-poplar (Liriodendron tulipifera) and sassafras (Sassafras albidum) during the fire-free interval and the increasingly severe invasion of Japanese stiltgrass (Microstegium vimineum) following disturbance are severe hindrances to successful restoration of upland oak ecosystems. Despite these management concerns, results of the research reported in this thesis indicate that restoring disturbance regimes slows the process of mesophication, improves size and stature of oak regeneration, and increases community diversity across the landscape

Stochastic processes for graphs, extreme values and their causality: inference, asymptotic theory and applications

This thesis provides some theoretical and practical statistical inference tools for multivariate stochastic processes to better understand the behaviours and properties present in the data. In particular, we focus on the modelling of graphs, that is a family of nodes linked together by a collection of edges, and extreme values, that are values above a high threshold to have their own dynamics compared to the typical behaviour of the process. We develop an ensemble of statistical models, statistical inference methods and their asymptotic study to ensure the good behaviour of estimation schemes in a wide variety of settings. We also devote a chapter to the formulation of a methodology based on pre-existing theory to unveil the causal dependency structure behind high-impact events.Open Acces

The Genetic Basis of Adaptation and Speciation in Benthic and Limnetic Threespine Stickleback (Gasterosterus aculeatus)

Sympatrische benthische (am Grund des Sees lebende) und limnische (im offenen Wasser lebende) Stichlinge entwickelten sich unabhängig voneinander in fünf Seen in Britisch-Kolumbien, Kanada. Da sie sich an unterschiedliche Nischen in ihrem Lebensraum anpassten, divergierten der benthische und limnische Stichlingsökotyp in ihrer Morphologie. Diese Evolution des benthischen und limnischen Stichlingsökotyps fand parallel in allen fünf Seen statt. Die Stichlinge dieser Seen bieten somit ein exzellentes Modell zur Untersuchung, welche Rolle die natürlicher Selektion bei der Speziation und der Anpassung spielt. Obwohl die Ökologie der Speziation und der Anpassung der benthischen und limnischen Stichlinge ausführlich untersucht wurde, fehlen bislang die genetischen Grundlagen dieser Mechanismen. Ich verwendete Gesamt-Genom-Sequenzierung, um die Speziation und Anpassung von benthischen und limnischen Stichlingen in vier Seen (Paxton Lake, Priest Lake, Little Quarry Lake, Enos Lake) in Britisch-Kolumbien, Kanada, zu untersuchen. Benthische und limnische Stichlinge aller vier Seen zeigen parallele genetische Divergenz. Benthische und limnische Stichlingsökotypen waren stark divergierender natürlicher Selektion ausgesetzt, bei der abgeleitete und angestammte Allele in jeweils einer der Stichlingsökotypen selektiv favorisiert wurden. Im benthischen Ökotyp wurden erheblich mehr Genomregionen selektiert als im limnischen Ökotyp. Indem ich unterschiedliche statistische Ansätze kombinierte, identifizierte ich mit noch nie dagewesener Auflösung Genomregionen, die zur Anpassung des benthischen und limnischen Ökotyps beitragen. Dies ermöglicht mir die Identifizierung und Charakterisierung von Genen, die für die Anpassung der Ökotypen wichtige phänotypische Merkmale und biologische Prozesse kontrollieren. Durch die Verwendung von high-density genetischen Markern, die durch die Sequenzierung des gesamten Genoms generiert wurden, untersuchte ich die Abstammung der benthischen und limnischen Ökotypen und leitete daraus ein demographisches Modell für die benthischen und limnischen Stichlinge im Paxton Lake ab. Die benthischen und limnischen Stichlinge im Paxton Lake entstanden durch allopratrische Speziation gefolgt xii von sekundärem Kontakt, wobei die Populationsgröße jeweils vor 5.000 und 7.000 Jahren reduziert wurde. Ich verwendete RNA-Sequenzierung, um die Divergenz in der Genexpression zwischen dem benthischen und limnischen Ökotyp im Paxton Lake zu erforschen und deckte auf, dass genetische Veränderungen in cis-regulierenden Elementen eine wichtige Rolle in der Anpassung von benthischen und limnischen Ökotypen spielte. Bisherige Studien zeigten, dass benthische und limnische Stichlingsökotypen im Enos Lake auf Grund von erhöhter Hybridisierung in einen Hybridschwarm „kollabiert“ waren. Die genetischen Grundlagen dieses Prozess sind jedoch größtenteils unbekannt. Durch Untersuchung der Gesamt-Genom-Sequenzierdaten zeigte ich, dass der Zusammenfall des Artenpaars im Enos Lake früher begann als bisher vorhergesagt wurde. Einige Genomregionen wurden bei diesem Prozess homogenisiert, andere nicht. Letzteres ist möglicherweise auf anhaltende divergente Selektion und/oder geringe Rekombinationsraten dieser Regionen zurückzuführen