Relationship between Extremal and Sum Processes Generated by the same Point Process

2000 Mathematics Subject Classification: Primary 60G51, secondary 60G70, 60F17.

Option pricing in market models driven by telegraph processes with jumps

Esta tesis está dividida en dos partes: en la primera parte se presentan y estudian los procesos telegráficos, los procesos de Poisson con compensador telegráfico y los procesos telegráficos con saltos. El estudio presentado en esta primera parte incluye el cálculo de las distribuciones de cada proceso, las medias y varianzas, así como las funciones generadoras de momentos entre otras propiedades. Utilizando estas propiedades en la segunda parte se estudian los modelos de valoración de opciones basados en procesos telegráficos con saltos. En esta parte se da una descripción de cómo calcular las medidas neutrales al riesgo, se encuentra la condición de no arbitraje en este tipo de modelos y por último se calcula el precio de las opciones Europeas de compra y venta.

Gibbs point process approximation: Total variation bounds using Stein's method

We obtain upper bounds for the total variation distance between the distributions of two Gibbs point processes in a very general setting. Applications are provided to various well-known processes and settings from spatial statistics and statistical physics, including the comparison of two Lennard-Jones processes, hard core approximation of an area interaction process and the approximation of lattice processes by a continuous Gibbs process. Our proof of the main results is based on Stein's method. We construct an explicit coupling between two spatial birth-death processes to obtain Stein factors, and employ the Georgii-Nguyen-Zessin equation for the total bound.

Three-dimensional distribution of cortical synapses: a replicated point pattern-based analysis

The biggest problem when analyzing the brain is that its synaptic connections are extremely complex. Generally, the billions of neurons making up the brain exchange information through two types of highly specialized structures: chemical synapses (the vast majority) and so-called gap junctions (a substrate of one class of electrical synapse). Here we are interested in exploring the three-dimensional spatial distribution of chemical synapses in the cerebral cortex. Recent research has showed that the three-dimensional spatial distribution of synapses in layer III of the neocortex can be modeled by a random sequential adsorption (RSA) point process, i.e., synapses are distributed in space almost randomly, with the only constraint that they cannot overlap. In this study we hypothesize that RSA processes can also explain the distribution of synapses in all cortical layers. We also investigate whether there are differences in both the synaptic density and spatial distribution of synapses between layers. Using combined focused ion beam milling and scanning electron microscopy (FIB/SEM), we obtained three-dimensional samples from the six layers of the rat somatosensory cortex and identified and reconstructed the synaptic junctions. A total volume of tissue of approximately 4500μm3 and around 4000 synapses from three different animals were analyzed. Different samples, layers and/or animals were aggregated and compared using RSA replicated spatial point processes. The results showed no significant differences in the synaptic distribution across the different rats used in the study. We found that RSA processes described the spatial distribution of synapses in all samples of each layer. We also found that the synaptic distribution in layers II to VI conforms to a common underlying RSA process with different densities per layer. Interestingly, the results showed that synapses in layer I had a slightly different spatial distribution from the other layers.

Higher-order co-occurrences for exploratory point pattern analysis and decision tree clustering on spatial data

Analyzing geographical patterns by collocating events, objects or their attributes has a long history in surveillance and monitoring, and is particularly applied in environmental contexts, such as ecology or epidemiology. The identification of patterns or structures at some scales can be addressed using spatial statistics, particularly marked point processes methodologies. Classification and regression trees are also related to this goal of finding "patterns" by deducing the hierarchy of influence of variables on a dependent outcome. Such variable selection methods have been applied to spatial data, but, often without explicitly acknowledging the spatial dependence. Many methods routinely used in exploratory point pattern analysis are2nd-order statistics, used in a univariate context, though there is also a wide literature on modelling methods for multivariate point pattern processes. This paper proposes an exploratory approach for multivariate spatial data using higher-order statistics built from co-occurrences of events or marks given by the point processes. A spatial entropy measure, derived from these multinomial distributions of co-occurrences at a given order, constitutes the basis of the proposed exploratory methods. © 2010 Elsevier Ltd.

Higher-order co-occurrences for exploratory point pattern analysis and decision tree clustering on spatial data

Analyzing geographical patterns by collocating events, objects or their attributes has a long history in surveillance and monitoring, and is particularly applied in environmental contexts, such as ecology or epidemiology. The identification of patterns or structures at some scales can be addressed using spatial statistics, particularly marked point processes methodologies. Classification and regression trees are also related to this goal of finding "patterns" by deducing the hierarchy of influence of variables on a dependent outcome. Such variable selection methods have been applied to spatial data, but, often without explicitly acknowledging the spatial dependence. Many methods routinely used in exploratory point pattern analysis are2nd-order statistics, used in a univariate context, though there is also a wide literature on modelling methods for multivariate point pattern processes. This paper proposes an exploratory approach for multivariate spatial data using higher-order statistics built from co-occurrences of events or marks given by the point processes. A spatial entropy measure, derived from these multinomial distributions of co-occurrences at a given order, constitutes the basis of the proposed exploratory methods. © 2010 Elsevier Ltd.

Spatial analysis of urban violence based on emergency room data

OBJECTIVE: To estimate the spatial intensity of urban violence events using wavelet-based methods and emergency room data. METHODS: Information on victims attended at the emergency room of a public hospital in the city of São Paulo, Southeastern Brazil, from January 1, 2002 to January 11, 2003 were obtained from hospital records. The spatial distribution of 3,540 events was recorded and a uniform random procedure was used to allocate records with incomplete addresses. Point processes and wavelet analysis technique were used to estimate the spatial intensity, defined as the expected number of events by unit area. RESULTS: Of all georeferenced points, 59% were accidents and 40% were assaults. There is a non-homogeneous spatial distribution of the events with high concentration in two districts and three large avenues in the southern area of the city of São Paulo. CONCLUSIONS: Hospital records combined with methodological tools to estimate intensity of events are useful to study urban violence. The wavelet analysis is useful in the computation of the expected number of events and their respective confidence bands for any sub-region and, consequently, in the specification of risk estimates that could be used in decision-making processes for public policies.

A two-dimensional fuzzy random model of soil pore structure

A new conceptual model for soil pore-solid structure is formalized. Soil pore-solid structure is proposed to comprise spatially abutting elements each with a value which is its membership to the fuzzy set ''pore,'' termed porosity. These values have a range between zero (all solid) and unity (all pore). Images are used to represent structures in which the elements are pixels and the value of each is a porosity. Two-dimensional random fields are generated by allocating each pixel a porosity by independently sampling a statistical distribution. These random fields are reorganized into other pore-solid structural types by selecting parent points which have a specified local region of influence. Pixels of larger or smaller porosity are aggregated about the parent points and within the region of interest by controlled swapping of pixels in the image. This creates local regions of homogeneity within the random field. This is similar to the process known as simulated annealing. The resulting structures are characterized using one-and two-dimensional variograms and functions describing their connectivity. A variety of examples of structures created by the model is presented and compared. Extension to three dimensions presents no theoretical difficulties and is currently under development.

Spatial analysis of urban violence based on emergency room data

OBJECTIVE: To estimate the spatial intensity of urban violence events using wavelet-based methods and emergency room data. METHODS: Information on victims attended at the emergency room of a public hospital in the city of São Paulo, Southeastern Brazil, from January 1, 2002 to January 11, 2003 were obtained from hospital records. The spatial distribution of 3,540 events was recorded and a uniform random procedure was used to allocate records with incomplete addresses. Point processes and wavelet analysis technique were used to estimate the spatial intensity, defined as the expected number of events by unit area. RESULTS: Of all georeferenced points, 59% were accidents and 40% were assaults. There is a non-homogeneous spatial distribution of the events with high concentration in two districts and three large avenues in the southern area of the city of São Paulo. CONCLUSIONS: Hospital records combined with methodological tools to estimate intensity of events are useful to study urban violence. The wavelet analysis is useful in the computation of the expected number of events and their respective confidence bands for any sub-region and, consequently, in the specification of risk estimates that could be used in decision-making processes for public policies.

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.

Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spaces

Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L2-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.