On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable configurations are generated with positive probability Lundh calls this percolation diffusion. An integral condition for percolation diffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

On Lundh's percolation diffusion

A collection of spherical obstacles in the unit ball in Euclidean space is said to be avoidable for Brownian motion if there is a positive probability that Brownian motion diffusing from some point in the ball will avoid all the obstacles and reach the boundary of the ball. The centres of the spherical obstacles are generated according to a Poisson point process while the radius of an obstacle is a deterministic function. If avoidable con gurations are generated with positive probability Lundh calls this percolation di usion. An integral condition for percolation di ffusion is derived in terms of the intensity of the point process and the function that determines the radii of the obstacles.

Insurance with frequent trading

This paper looks at the dynamic management of risk in an economy with discrete time consumption and endowments and continuous trading. I study how agents in such an economy deal with all the risk in the economy and attain their Pareto optimal allocations by trading in a few natural securities: private insurance contracts and a common set of derivatives on the aggregate endowment. The parsimonious nature ofthe implied securities needed for Pareto optimality suggests that insuch contexts complete markets is a very reasonable assumption.

Unraveling dynamics of human physical activity patterns in chronic pain conditions.

Chronic pain is a complex disabling experience that negatively affects the cognitive, affective and physical functions as well as behavior. Although the interaction between chronic pain and physical functioning is a well-accepted paradigm in clinical research, the understanding of how pain affects individuals' daily life behavior remains a challenging task. Here we develop a methodological framework allowing to objectively document disruptive pain related interferences on real-life physical activity. The results reveal that meaningful information is contained in the temporal dynamics of activity patterns and an analytical model based on the theory of bivariate point processes can be used to describe physical activity behavior. The model parameters capture the dynamic interdependence between periods and events and determine a 'signature' of activity pattern. The study is likely to contribute to the clinical understanding of complex pain/disease-related behaviors and establish a unified mathematical framework to quantify the complex dynamics of various human activities.

Detection of clusters using space-time scan statistics

This paper aims at detecting spatio-temporal clustering in fire sequences using space?time scan statistics, a powerful statistical framework for the analysis of point processes. The methodology is applied to active fire detection in the state of Florida (US) identified by MODIS (Moderate Resolution Imaging Spectroradiometer) during the period 2003?06. Results of the present study show that statistically significant clusters can be detected and localized in specific areas and periods of the year. Three out of the five most likely clusters detected for the entire frame period are localized in the north of the state, and they cover forest areas; the other two clusters cover a large zone in the south, corresponding to agricultural land and the prairies in the Everglades. In order to analyze if the wildfires recur each year during the same period, the analyses have been performed separately for the 4 years: it emerges that clusters of forest fires are more frequent in hot seasons (spring and summer), while in the southern areas, they are widely present during the whole year. The recognition of overdensities of events and the ability to locate them in space and in time can help in supporting fire management and focussing on prevention measures.

Spatial pattern of landslides in Swiss Rhone valley

The present study analyses the spatial pattern of quaternary gravitational slope deformations (GSD) and historical/present-day instabilities (HPI) inventoried in the Swiss Rhone Valley. The main objective is to test if these events are clustered (spatial attraction) or randomly distributed (spatial independency). Moreover, analogies with the cluster behaviour of earthquakes inventoried in the same area were examined. The Ripley's K-function was applied to measure and test for randomness. This indicator allows describing the spatial pattern of a point process at increasing distance values. To account for the non-constant intensity of the geological phenomena, a modification of the K-function for inhomogeneous point processes was adopted. The specific goal is to explore the spatial attraction (i.e. cluster behaviour) among landslide events and between gravitational slope deformations and earthquakes. To discover if the two classes of instabilities (GSD and HPI) are spatially independently distributed, the cross K-function was computed. The results show that all the geological events under study are spatially clustered at a well-defined distance range. GSD and HPI show a similar pattern distribution with clusters in the range 0.75?9 km. The cross K-function reveals an attraction between the two classes of instabilities in the range 0?4 km confirming that HPI are more prone to occur within large-scale slope deformations. The K-function computed for GSD and earthquakes indicates that both present a cluster tendency in the range 0?10 km, suggesting that earthquakes could represent a potential predisposing factor which could influence the GSD distribution.

Analysis of space-time environmental data in complex regions

This thesis develops a comprehensive and a flexible statistical framework for the analysis and detection of space, time and space-time clusters of environmental point data. The developed clustering methods were applied in both simulated datasets and real-world environmental phenomena; however, only the cases of forest fires in Canton of Ticino (Switzerland) and in Portugal are expounded in this document. Normally, environmental phenomena can be modelled as stochastic point processes where each event, e.g. the forest fire ignition point, is characterised by its spatial location and occurrence in time. Additionally, information such as burned area, ignition causes, landuse, topographic, climatic and meteorological features, etc., can also be used to characterise the studied phenomenon. Thereby, the space-time pattern characterisa- tion represents a powerful tool to understand the distribution and behaviour of the events and their correlation with underlying processes, for instance, socio-economic, environmental and meteorological factors. Consequently, we propose a methodology based on the adaptation and application of statistical and fractal point process measures for both global (e.g. the Morisita Index, the Box-counting fractal method, the multifractal formalism and the Ripley's K-function) and local (e.g. Scan Statistics) analysis. Many measures describing the space-time distribution of environmental phenomena have been proposed in a wide variety of disciplines; nevertheless, most of these measures are of global character and do not consider complex spatial constraints, high variability and multivariate nature of the events. Therefore, we proposed an statistical framework that takes into account the complexities of the geographical space, where phenomena take place, by introducing the Validity Domain concept and carrying out clustering analyses in data with different constrained geographical spaces, hence, assessing the relative degree of clustering of the real distribution. Moreover, exclusively to the forest fire case, this research proposes two new methodologies to defining and mapping both the Wildland-Urban Interface (WUI) described as the interaction zone between burnable vegetation and anthropogenic infrastructures, and the prediction of fire ignition susceptibility. In this regard, the main objective of this Thesis was to carry out a basic statistical/- geospatial research with a strong application part to analyse and to describe complex phenomena as well as to overcome unsolved methodological problems in the characterisation of space-time patterns, in particular, the forest fire occurrences. Thus, this Thesis provides a response to the increasing demand for both environmental monitoring and management tools for the assessment of natural and anthropogenic hazards and risks, sustainable development, retrospective success analysis, etc. The major contributions of this work were presented at national and international conferences and published in 5 scientific journals. National and international collaborations were also established and successfully accomplished. -- Cette thèse développe une méthodologie statistique complète et flexible pour l'analyse et la détection des structures spatiales, temporelles et spatio-temporelles de données environnementales représentées comme de semis de points. Les méthodes ici développées ont été appliquées aux jeux de données simulées autant qu'A des phénomènes environnementaux réels; nonobstant, seulement le cas des feux forestiers dans le Canton du Tessin (la Suisse) et celui de Portugal sont expliqués dans ce document. Normalement, les phénomènes environnementaux peuvent être modélisés comme des processus ponctuels stochastiques ou chaque événement, par ex. les point d'ignition des feux forestiers, est déterminé par son emplacement spatial et son occurrence dans le temps. De plus, des informations tels que la surface bru^lée, les causes d'ignition, l'utilisation du sol, les caractéristiques topographiques, climatiques et météorologiques, etc., peuvent aussi être utilisées pour caractériser le phénomène étudié. Par conséquent, la définition de la structure spatio-temporelle représente un outil puissant pour compren- dre la distribution du phénomène et sa corrélation avec des processus sous-jacents tels que les facteurs socio-économiques, environnementaux et météorologiques. De ce fait, nous proposons une méthodologie basée sur l'adaptation et l'application de mesures statistiques et fractales des processus ponctuels d'analyse global (par ex. l'indice de Morisita, la dimension fractale par comptage de boîtes, le formalisme multifractal et la fonction K de Ripley) et local (par ex. la statistique de scan). Des nombreuses mesures décrivant les structures spatio-temporelles de phénomènes environnementaux peuvent être trouvées dans la littérature. Néanmoins, la plupart de ces mesures sont de caractère global et ne considèrent pas de contraintes spatiales com- plexes, ainsi que la haute variabilité et la nature multivariée des événements. A cet effet, la méthodologie ici proposée prend en compte les complexités de l'espace géographique ou le phénomène a lieu, à travers de l'introduction du concept de Domaine de Validité et l'application des mesures d'analyse spatiale dans des données en présentant différentes contraintes géographiques. Cela permet l'évaluation du degré relatif d'agrégation spatiale/temporelle des structures du phénomène observé. En plus, exclusif au cas de feux forestiers, cette recherche propose aussi deux nouvelles méthodologies pour la définition et la cartographie des zones périurbaines, décrites comme des espaces anthropogéniques à proximité de la végétation sauvage ou de la forêt, et de la prédiction de la susceptibilité à l'ignition de feu. A cet égard, l'objectif principal de cette Thèse a été d'effectuer une recherche statistique/géospatiale avec une forte application dans des cas réels, pour analyser et décrire des phénomènes environnementaux complexes aussi bien que surmonter des problèmes méthodologiques non résolus relatifs à la caractérisation des structures spatio-temporelles, particulièrement, celles des occurrences de feux forestières. Ainsi, cette Thèse fournit une réponse à la demande croissante de la gestion et du monitoring environnemental pour le déploiement d'outils d'évaluation des risques et des dangers naturels et anthro- pogéniques. Les majeures contributions de ce travail ont été présentées aux conférences nationales et internationales, et ont été aussi publiées dans 5 revues internationales avec comité de lecture. Des collaborations nationales et internationales ont été aussi établies et accomplies avec succès.

Sistemas de informação geográfica e análise espacial de dados como ferramentas para determinação de agregação espacial de doenças: a hanseníase como modelo

This work demonstrates the importance of using tools used in geographic information systems (GIS) and spatial data analysis (SDA) for the study of infectious diseases. Analysis methods were used to describe more fully the spatial distribution of a particular disease by incorporating the geographical element in the analysis. In Chapter 1, we report the historical evolution of these techniques in the field of human health and use Hansen s disease (leprosy) in Rio Grande do Norte as an example. In Chapter 2, we introduced a few basic theoretical concepts on the methodology and classified the types of spatial data commonly treated. Chapters 3 and 4 defined and demonstrated the use of the two most important techniques for analysis of health data, which are data point processes and data area. We modelled the case distribution of Hansen s disease in the city of Mossoró - RN. In the analysis, we used R scripts and made available routines and analitical procedures developed by the author. This approach can be easily used by researchers in several areas. As practical results, major risk areas in Mossoró leprosy were detected, and its association with the socioeconomic profile of the population at risk was found. Moreover, it is clearly shown that his approach could be of great help to be used continuously in data analysis and processing, allowing the development of new strategies to work might increase the use of such techniques in data analysis in health care

Análise espacial do risco de leptospirose canina na Vila Pantanal, Curitiba, Paraná

A leptospirose é uma grave zoonose associada às áreas de baixa renda dos centros urbanos. Embora roedores urbanos sejam considerados como principal reservatório para a leptospirose, o cão também pode desenvolver a doença e se tornar carreador assintomático. O objetivo do presente trabalho foi utilizar a metodologia estatística baseada na teoria de processos pontuais espaciais, buscando identificar a forma como se distribuem os cães sororreagentes para a leptospirose e seus determinantes de risco em uma vila na cidade de Curitiba. A análise do modelo possibilitou identificar as regiões de sobre-risco, onde o risco de soropositividade canina à leptospirose é significativamente maior. A relação significativa do efeito espacial no desenvolvimento da doença, além das variáveis estudadas, revela que não apenas um, mas a ação conjunta dos fatores relacionados ao animal, ao proprietário e ao ambiente influencia o risco maior da doença nos locais de maior efeito espacial. O resultado da análise indica claramente os territórios em maior risco na região da Vila Pantanal, possibilitando o planejamento de ações mais específicas e dirigidas a essas áreas em um contexto de vigilância da saúde.

Harmonic deformation of Delaunay triangulations

We construct harmonic functions on random graphs given by Delaunay triangulations of ergodic point processes as the limit of the zero-temperature harness process. (C) 2012 Elsevier B.V All rights reserved.

Ausbreitung einer Infektion durch ein System Brownscher Bewegungen

In dieser Arbeit werden wir ein Modell untersuchen, welches die Ausbreitung einer Infektion beschreibt. Bei diesem Modell werden zunächst Partikel gemäß eines Poissonschen Punktprozesses auf der reellen Achse verteilt. Bis zu einem gewissen Punkt auf der reellen Achse sind alle Partikel von einer Infektion befallen. Während sich nicht infizierte Partikel nicht bewegen, folgen die infizierten Partikel den Pfaden von voneinander unabhängigen Brownschen Bewegungen und verbreitet die Infektion dabei an den Orten, welche sie betreten. Wenn sie dabei auf ein nicht infiziertes Partikel treffen, ist dieses von diesem Moment an auch infiziert und beginnt ebenfalls, dem Pfad einer Brownschen Bewegung zu folgen und die Infektion auszubreiten. Auf diese Art verschiebt sich nun der am weitesten rechts liegende Ort R_t, an dem die Infektion bereits verbreitet wurde. Wir werden mit Hilfe des subadditiven Ergodensatzes zeigen, dass sich dieser Ort mit linearer Geschwindigkeit fortbewegt. Ferner werden wir eine obere und eine untere Schranke für die Ausbreitungsgeschwindkeit angeben. Danach werden wir zeigen, dass der Prozess Regenerationszeiten hat, nämlich solche zufällige Zeiten, zu denen er eine Art Neustart unter speziellen Startbedingungen durchführt. Wir werden diese für eine weitere Charakterisierung der Ausbreitungsgeschwingkeit nutzen. Ferner erhalten wir durch die Regenerationszeiten auch einen Zentralen Grenzwertsatz für R_t und können zeigen, dass die Verteilung der infizierten Partikel aus Sicht des am weitesten rechts liegenden infizierten Ortes gegen eine invariante Verteilung konvergiert.

GEOSTATISTICAL INFERENCE UNDER PREFERENTIAL SAMPLING

Geostatistics involves the fitting of spatially continuous models to spatially discrete data (Chil`es and Delfiner, 1999). Preferential sampling arises when the process that determines the data-locations and the process being modelled are stochastically dependent. Conventional geostatistical methods assume, if only implicitly, that sampling is non-preferential. However, these methods are often used in situations where sampling is likely to be preferential. For example, in mineral exploration samples may be concentrated in areas thought likely to yield high-grade ore. We give a general expression for the likelihood function of preferentially sampled geostatistical data and describe how this can be evaluated approximately using Monte Carlo methods. We present a model for preferential sampling, and demonstrate through simulated examples that ignoring preferential sampling can lead to seriously misleading inferences. We describe an application of the model to a set of bio-monitoring data from Galicia, northern Spain, in which making allowance for preferential sampling materially changes the inferences.

Detecting determinism with improved sensitivity in time series: Rank-based nonlinear predictability score

The rank-based nonlinear predictability score was recently introduced as a test for determinism in point processes. We here adapt this measure to time series sampled from time-continuous flows. We use noisy Lorenz signals to compare this approach against a classical amplitude-based nonlinear prediction error. Both measures show an almost identical robustness against Gaussian white noise. In contrast, when the amplitude distribution of the noise has a narrower central peak and heavier tails than the normal distribution, the rank-based nonlinear predictability score outperforms the amplitude-based nonlinear prediction error. For this type of noise, the nonlinear predictability score has a higher sensitivity for deterministic structure in noisy signals. It also yields a higher statistical power in a surrogate test of the null hypothesis of linear stochastic correlated signals. We show the high relevance of this improved performance in an application to electroencephalographic (EEG) recordings from epilepsy patients. Here the nonlinear predictability score again appears of higher sensitivity to nonrandomness. Importantly, it yields an improved contrast between signals recorded from brain areas where the first ictal EEG signal changes were detected (focal EEG signals) versus signals recorded from brain areas that were not involved at seizure onset (nonfocal EEG signals).

Mitochondrial DNA reveals hidden diversity for tardigrades from across the Antarctic realm

Antarctica contains some of the most challenging environmental conditions on the planet due to freezing temperatures, prolonged winters and lack of liquid water. Whereas 99.7% of Antarctica is permanently covered by ice and snow, some coastal areas and mountain ridges have remained ice-free and are able to sustain populations of microinvertebrates. Tardigrades are one of the more dominant groups of microfauna in soil and limno-terrestrial habitats, but little is known of their diversity and distribution across Antarctica. Here, we examine tardigrades sampled from across an extensive region of continental Antarctica, and analyse and compare their partial mitochondrial cytochrome c oxidase subunit 1 (COI) gene sequences with those from the Antarctic Peninsula, maritime and sub-Antarctica, Tierra del Fuego and other worldwide locations in order to recognise operational taxonomic units (OTUs). From 439 new tardigrade COI sequences, we identified 98 unique haplotypes (85 from Antarctica) belonging to Acutuncus, Diphascon, Echiniscus, Macrobiotus, Milnesium and unidentified Parachela. Operational taxonomic units were delimited by Poisson tree processes and general mixed Yule coalescent methods, resulting in 58 and 55 putative species, respectively. Most tardigrades appear to be locally endemic (i.e. restricted to a single geographic region), but some (e.g. Acutuncus antarcticus (Richters, 1904)) are widespread across continental Antarctica. Our molecular results reveal: (i) greater diversity than has previously been appreciated with distinct OTUs that potentially represent undescribed species, and (ii) a lack of connectivity between most OTUs from continental Antarctica and those from other Antarctic geographical zones.

A distribution of tumor size at detection and its limiting form.

A distribution of tumor size at detection is derived within the framework of a mechanistic model of carcinogenesis with the object of estimating biologically meaningful parameters of tumor latency. Its limiting form appears to be a generalization of the distribution that arises in the length-biased sampling from stationary point processes. The model renders the associated estimation problems tractable. The usefulness of the proposed approach is illustrated with an application to clinical data on premenopausal breast cancer.