The paper is dedicated to the modeling and the simulation of random processes and fields. Using the concept and the theory of weakly correlated functions a consistent representation of sufficiently smooth random processes will be derived. Special applications will be given with respect to the simulation of road surfaces in vehicle dynamics and to the confirmation of theoretical results with respect to the zeros of random polynomials.

Aggregated data appears in many areas such as econimics, sociology, geography, etc. This motivates an importance of studying the (dis)aggregation problem. One of the most important reasons why the contemporaneous aggregation become an object of research is the possibility of obtaining the long memory phenomena in processes. The aggregation provides an explanation of the long-memory effect in time series and a simulation method of such series as well. Accumulation of short-memory non-ergodic random processes can lead to the long memory ergodic process, that can be used for the forecasts of the macro and micro variables. We explore the aggregation scheme of AR(1) processes and nearest-neighbour random fields with infinite variance. We provide results on the existence of limit aggregated processes, and find conditions under which it has long memory properties in certain sense. For the random fields on Z^2, we introduce the notion of (an)isotropic long memory based on the behavior of partial sums. In L_2 case, the known aggregation of independent AR(1) processes leads to the Gaussian limit. While we describe a new model of aggregation based on independent triangular arrays. This scheme gives the limit aggregated process with finite variance which is not necessary Gaussian. We study a discrete time risk insurance model with stationary claims, modeled by the aggregated heavy-tailed process. We establish the asymptotic properties of the ruin probability and the dependence structure... [to full text] / Agreguoti duomenys naudojami daugelyje mokslo sričių tokių kaip ekonomika, sociologija, geografija ir kt. Tai motyvuoja tirti (de)agregavimo uždavinį. Viena iš pagrindinių priežasčių kodėl vienalaikis agregavimas tapo tyrimų objektu yra galimybė gauti ilgos atminties procesus. Agregavimas paaiškina ilgos atminties atsiradima procesuose ir yra vienas iš būdų tokius procesus generuoti. Agreguodami trumpos atminties neergodiškus atsitiktinius procesus, galime gauti ilgos atminties ergodišką procesą, kuris gali būti naudojamas mikro ir makro kintamųjų prognozavimui. Disertacijoje nagrinėjama AR(1) procesų bei artimiausio kaimyno atsitiktinių laukų, turinčių begalinę dispersiją, agregavimo schema, randamos sąlygos, kurioms esant ribinis agreguotas procesas egzistuoja, ir turi ilgąją atmintį tam tikra prasme. Atsitiktinių laukų atveju, įvedamas anizotropinės/izotropinės ilgos atminties apibrėžimas, kuris yra paremtas dalinių sumų elgesiu. Baigtinės dispersijos atveju yra gerai žinoma nepriklausomų AR(1) procesų schema, kuri rezultate duoda Gauso ribinį agreguotą procesą. Disertacijoje aprašoma trikampio masyvo agregavimo modelis, kuris baigtinės dispersijos atveju duoda nebūtinai Gauso ribinį agreguotą procesą. Taip pat disertacijoje nagrinėjama bankroto tikimybės asimptotika, kai žalos yra aprašomos sunkiauodegiu agreguotu procesu, nusakoma priklausomybė tarp žalų, apibūdinama žalų ilga atmintis.

Agreguoti duomenys naudojami daugelyje mokslo sričių tokių kaip ekonomika, sociologija, geografija ir kt. Tai motyvuoja tirti (de)agregavimo uždavinį. Viena iš pagrindinių priežasčių kodėl vienalaikis agregavimas tapo tyrimų objektu yra galimybė gauti ilgos atminties procesus. Agregavimas paaiškina ilgos atminties atsiradima procesuose ir yra vienas iš būdų tokius procesus generuoti. Agreguodami trumpos atminties neergodiškus atsitiktinius procesus, galime gauti ilgos atminties ergodišką procesą, kuris gali būti naudojamas mikro ir makro kintamųjų prognozavimui. Disertacijoje nagrinėjama AR(1) procesų bei artimiausio kaimyno atsitiktinių laukų, turinčių begalinę dispersiją, agregavimo schema, randamos sąlygos, kurioms esant ribinis agreguotas procesas egzistuoja, ir turi ilgąją atmintį tam tikra prasme. Atsitiktinių laukų atveju, įvedamas anizotropinės/izotropinės ilgos atminties apibrėžimas, kuris yra paremtas dalinių sumų elgesiu. Baigtinės dispersijos atveju yra gerai žinoma nepriklausomų AR(1) procesų schema, kuri rezultate duoda Gauso ribinį agreguotą procesą. Disertacijoje aprašoma trikampio masyvo agregavimo modelis, kuris baigtinės dispersijos atveju duoda nebūtinai Gauso ribinį agreguotą procesą. Taip pat disertacijoje nagrinėjama bankroto tikimybės asimptotika, kai žalos yra aprašomos sunkiauodegiu agreguotu procesu, nusakoma priklausomybė tarp žalų, apibūdinama žalų ilga atmintis. / Aggregated data appears in many areas such as econimics, sociology, geography, etc. This motivates an importance of studying the (dis)aggregation problem. One of the most important reasons why the contemporaneous aggregation become an object of research is the possibility of obtaining the long memory phenomena in processes. The aggregation provides an explanation of the long-memory effect in time series and a simulation method of such series as well. Accumulation of short-memory non-ergodic random processes can lead to the long memory ergodic process, that can be used for the forecasts of the macro and micro variables. We explore the aggregation scheme of AR(1) processes and nearest-neighbour random fields with infinite variance. We provide results on the existence of limit aggregated processes, and find conditions under which it has long memory properties in certain sense. For the random fields on Z^2, we introduce the notion of (an)isotropic long memory based on the behavior of partial sums. In L_2 case, the known aggregation of independent AR(1) processes leads to the Gaussian limit. While we describe a new model of aggregation based on independent triangular arrays. This scheme gives the limit aggregated process with finite variance which is not necessary Gaussian. We study a discrete time risk insurance model with stationary claims, modeled by the aggregated heavy-tailed process. We establish the asymptotic properties of the ruin probability and the dependence structure... [to full text]

Thesis (Ph.D.)--University of Melbourne, Dept. of Computer Science and Software Engineering, Faculty of Engineering, 2007. / Typescript. Includes bibliographical references (leaves 171-179).

Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: This thesis provides an introduction to the statistical modeling technique known as graphical models. Since graph theory and probability theory are the two legs of graphical models, these two topics are presented, and then combined to produce two examples of graphical models: Bayesian Networks and Markov Random Fields. Furthermore, the max-sum, sum-product and junction tree algorithms are discussed. The graphical modeling technique is then applied to the specific problem of finding coplanar points in stereo images, taken with an uncalibrated camera. Although it is discovered that graphical models might not be the best method, in terms of speed, to use for this appliation, it does illustrate how to apply this technique in a real-life problem. / AFRIKAANSE OPSOMMING: Hierdie tesis stel die leser voor aan die statistiese modelerings-tegniek genoemd grafiese modelle. Aangesien grafiek teorie en waarskynlikheidsleer die twee bene van grafiese modelle is, word hierdie areas aangespreek en dan gekombineer om twee voorbeelde van grafiese modelle te vind: Bayesian Netwerke en Markov Lukrake Liggaam. Die maks-som, som-produk en aansluitboom algoritmes word ook bestudeer. Nadat die teorie van grafiese modelle en hierdie drie algoritmes afgehandel is, word grafiese modelle dan toegepas op ’n spesifieke probleem— om punte op ’n gemeenskaplike vlak in stereo beelde te vind, wat met ’n ongekalibreerde kamera geneem is. Alhoewel gevind is dat grafiese modelle nie die optimale metode is om punte op ’n gemeenskaplike vlak te vind, in terme van spoed, word die gebruik van grafiese modelle wel ten toongestel met hierdie praktiese voorbeeld. / National Research Foundation (South Africa)

Casamento de padrões de pontos em Espaços Euclidianos é um dos problemas fundamentais em reconhecimento de padrões, tendo aplicações que vão desde Visão Computacional até Química Computacional. Sempre que dois padrões complexos estão codi- ficados em termos de dois conjuntos de pontos que identificam suas características fundamentais, sua comparação pode ser vista como um problema de casamento de padrões de pontos. Este trabalho propõe uma abordagem unificada para os problemas de casamento exato e inexato de padrões de pontos em Espaços Euclidianos de dimensão arbitrária. No caso de casamento exato, é garantida a obtenção de uma solução ótima. Para casamento inexato (quando ruído está presente), resultados experimentais confirmam a validade da abordagem. Inicialmente, considera-se o problema de casamento de padrões de pontos como um problema de casamento de grafos ponderados. O problema de casamento de grafos ponderados é então formulado como um problema de inferência Bayesiana em um modelo gráfico probabilístico. Ao explorar certos vínculos fundamentais existentes em padrões de pontos imersos em Espaços Euclidianos, provamos que, para o casamento exato de padrões de pontos, um modelo gráfico simples é equivalente ao modelo completo. É possível mostrar que inferência probabilística exata neste modelo simples tem complexidade polinomial para qualquer dimensionalidade do Espaço Euclidiano em consideração. Experimentos computacionais comparando esta técnica com a bem conhecida baseada em relaxamento probabilístico evidenciam uma melhora significativa de desempenho para casamento inexato de padrões de pontos. A abordagem proposta é signi- ficativamente mais robusta diante do aumento do tamanho dos padrões envolvidos. Na ausência de ruído, os resultados são sempre perfeitos. / Point pattern matching in Euclidean Spaces is one of the fundamental problems in Pattern Recognition, having applications ranging from Computer Vision to Computational Chemistry. Whenever two complex patterns are encoded by two sets of points identifying their key features, their comparison can be seen as a point pattern matching problem. This work proposes a single approach to both exact and inexact point set matching in Euclidean Spaces of arbitrary dimension. In the case of exact matching, it is assured to find an optimal solution. For inexact matching (when noise is involved), experimental results confirm the validity of the approach. We start by regarding point pattern matching as a weighted graph matching problem. We then formulate the weighted graph matching problem as one of Bayesian inference in a probabilistic graphical model. By exploiting the existence of fundamental constraints in patterns embedded in Euclidean Spaces, we prove that for exact point set matching a simple graphical model is equivalent to the full model. It is possible to show that exact probabilistic inference in this simple model has polynomial time complexity with respect to the number of elements in the patterns to be matched. This gives rise to a technique that for exact matching provably finds a global optimum in polynomial time for any dimensionality of the underlying Euclidean Space. Computational experiments comparing this technique with well-known probabilistic relaxation labeling show significant performance improvement for inexact matching. The proposed approach is significantly more robust under augmentation of the sizes of the involved patterns. In the absence of noise, the results are always perfect.

L'imagerie hyperspectrale, grâce à un nombre élevé de bandes spectrales très fines et contigües, est capable d'associer àchaque pixel d'une image une signature spectrale caractéristique du comportement réflectif du matériau ou du mélange dematériaux présents dans ce pixel. La plupart des algorithmes de classification tirent profit de cette grande profusiond'information spectrale mais exploitent très peu l'information contextuelle existant entre les pixels appartenant à un mêmevoisinage. L'objectif de cette thèse est de réaliser de nouveaux algorithmes utilisant simultanément les informations spectraleet spatiale à des fins de classification et d'étudier la complémentarité de ces deux types d'information dans divers contextes.Dans cette optique nous avons développé trois scénarios de classification sensiblement différents, chacun étant adapté à untype d'application particulier.Nous avons tout d'abord développé un procédé d'extraction puis de classification vectorielle d'un ensemble de caractéristiquesspectrales et spatiales. Les caractéristiques spectrales sont extraites au moyen de méthodes visant à réduire la dimension desimages hyperspectrales tout en conservant une majorité de l'information utile. Les caractéristiques spatiales sont quant àelles produites par l'intermédiaire d'outils de caractérisation de la texture (matrices de co-occurrence et spectres de texture)ou de la forme (profils morphologiques). Nous nous sommes ensuite intéressés à la modélisation markovienne et avonsentrepris d'adapter un algorithme de classification de type Conditional Random Field à un contexte hyperspectral. Notretroisième et dernière approche s'appuie sur une segmentation préalable de l'image afin de réaliser une classification parzones et non plus par pixels.L'information spectrale pure permet de regrouper efficacement des pixels présentant des signatures spectrales similaires etsuffit généralement dans le cadre de problèmes de classification ne faisant intervenir que des classes sémantiquement trèsprécises, liées à un unique type de matériau. Les classes plus générales (utilisées par exemple pour des applicationsd'aménagement des sols) se composent en revanche de plusieurs matériaux parfois communs à plusieurs classes et agencésselon des motifs qui se répètent. Caractérisables à la fois spatialement et spectralement, ces classes sont susceptibles d'êtreplus complètement décrites par une utilisation simultanée de ces deux types d'information. Pour conclure cette étude, nousavons effectué une comparaison des trois méthodes d'intégration de l'information spatiale au processus de classification selonles trois critères sont la précision de classification, la complexité algorithmique et la robustesse / Thanks to a high number of thin and contiguous spectral bands, the hyperpectral imagery can associate to each pixel of animage a spectral signature representing the reflective behaviour of the materials composing the pixel. Most of theclassification algorithms use this great amount of spectral information without noticing the contextual information betweenthe pixels that belong to the same neighborhood. This study aims to realize new algorithms using simultaneously the spectraland spatial informations in order to classify hyperspectral images, and to study their complementarity in several contexts. Forthis purpose, we have developped three different classification scenarios, each one adapted to a particular type of application.The first scenario consists in a vectorial classification processus. Several spectral and spatial characteristics are extracted andmerged in order to form a unique data set, which is classified using a Support Vector Machine method or a Gaussian MixingModel algorithm. The spectral characteristics are extracted using dimension reduction method, such as PCA or MNF, while thespatial characteristics are extracted using textural characterization tools (co-occurrence matrices and texture spectra) ormorphological tools (morphological profiles). For the second scenario, we adapted a Conditional Random Field algorithm tothe hyperspectral context. Finally, the last scenario is an area-wise classification algorithm relying on a textural segmentationmethod as a pre-processing step.The spectral information is generally sufficient to deal with semantically simple classes, linked to a unique type of material.Complex classes (such as ground amenagment classes) are composed of several materials which potentially belong to morethan one class. Those classes can be characterized both spectrally and spatially, which means that they can be morecompletly described using both spectral and spatial informations. To conclude this study, we compared the threespectral/spatial classification scenarios using three criterions : classification accuracy, algorithmic complexity and strength.

Casamento de padrões de pontos em Espaços Euclidianos é um dos problemas fundamentais em reconhecimento de padrões, tendo aplicações que vão desde Visão Computacional até Química Computacional. Sempre que dois padrões complexos estão codi- ficados em termos de dois conjuntos de pontos que identificam suas características fundamentais, sua comparação pode ser vista como um problema de casamento de padrões de pontos. Este trabalho propõe uma abordagem unificada para os problemas de casamento exato e inexato de padrões de pontos em Espaços Euclidianos de dimensão arbitrária. No caso de casamento exato, é garantida a obtenção de uma solução ótima. Para casamento inexato (quando ruído está presente), resultados experimentais confirmam a validade da abordagem. Inicialmente, considera-se o problema de casamento de padrões de pontos como um problema de casamento de grafos ponderados. O problema de casamento de grafos ponderados é então formulado como um problema de inferência Bayesiana em um modelo gráfico probabilístico. Ao explorar certos vínculos fundamentais existentes em padrões de pontos imersos em Espaços Euclidianos, provamos que, para o casamento exato de padrões de pontos, um modelo gráfico simples é equivalente ao modelo completo. É possível mostrar que inferência probabilística exata neste modelo simples tem complexidade polinomial para qualquer dimensionalidade do Espaço Euclidiano em consideração. Experimentos computacionais comparando esta técnica com a bem conhecida baseada em relaxamento probabilístico evidenciam uma melhora significativa de desempenho para casamento inexato de padrões de pontos. A abordagem proposta é signi- ficativamente mais robusta diante do aumento do tamanho dos padrões envolvidos. Na ausência de ruído, os resultados são sempre perfeitos. / Point pattern matching in Euclidean Spaces is one of the fundamental problems in Pattern Recognition, having applications ranging from Computer Vision to Computational Chemistry. Whenever two complex patterns are encoded by two sets of points identifying their key features, their comparison can be seen as a point pattern matching problem. This work proposes a single approach to both exact and inexact point set matching in Euclidean Spaces of arbitrary dimension. In the case of exact matching, it is assured to find an optimal solution. For inexact matching (when noise is involved), experimental results confirm the validity of the approach. We start by regarding point pattern matching as a weighted graph matching problem. We then formulate the weighted graph matching problem as one of Bayesian inference in a probabilistic graphical model. By exploiting the existence of fundamental constraints in patterns embedded in Euclidean Spaces, we prove that for exact point set matching a simple graphical model is equivalent to the full model. It is possible to show that exact probabilistic inference in this simple model has polynomial time complexity with respect to the number of elements in the patterns to be matched. This gives rise to a technique that for exact matching provably finds a global optimum in polynomial time for any dimensionality of the underlying Euclidean Space. Computational experiments comparing this technique with well-known probabilistic relaxation labeling show significant performance improvement for inexact matching. The proposed approach is significantly more robust under augmentation of the sizes of the involved patterns. In the absence of noise, the results are always perfect.

La corrélation de bruit sismique est aujourd'hui une technique largement utilise en sismologie. L'aspect attrayante de cette technique est lie a la possibilité d'obtenir une estimation de la fonction Green partout sur la surface de la Terre, aussi en absence de source sismique comme le tremblement de terre. Les ondes des surface ils sont facilement obtenu par corrélation de bruit, et elles sont très utiliser pour l'imagerie de la croute et du manteau superieur. Dans la possible observation des ondes de volume par corrélation de bruit, il peut avoir le potentielle pour résoudre aussi les structures de la Terre profonde.Dans la première partie de la thèse, on utilise les ondes de surface estime par corrélation de bruit sismique au fin de produire en modèle tomographique de la croute en Finlande.Dans la deuxième partie, on se focalise sur la nouvelle possibilité d'obtenir des ondes de volume par corrélation de bruit. Suite a la démonstration aue des ondes des volume son bien observe par corrélation de bruit a toutes les échelles, on utilise ces ondes pour l'imagerie de la Terre profonde. Ces nouveaux et prometteur résultats ils permettent d’incrémenter la connaissance de la structure profonde de la Terre, avec un incroyable augmentation de la résolution de la méthode d'imagerie. / Ambient seismic noise correlation technique is today widely applied in seismology. The attractive aspect of this method relies in the possibility of obtaining an estimated Green's function everywhere in the world, also in absence of explosions or earthquakes. As surface waves dominate the estimated Green's fucntion, they can be used for high-resolution imaging of the shallow Earth. Similar observation of body waves would provide the required resolution to solve the deeper Earth structures.In the first part of our work we focus on resolving the crustal structure of northern Finland. Using surface waves reconstructed from noise correlation, we reconstructed a threedimensional S waves model that gives acces to the ancient structures of the crust.In the second part we analyse how body waves can emerge from seismic noise correlation. We show that body waves travelling from regional to teleseismic distances are well reconstructed from noise correlation. As these waves contain information about the Earth structure they represent a new and original dataset to improve the knowledge of our planet.

Casamento de padrões de pontos em Espaços Euclidianos é um dos problemas fundamentais em reconhecimento de padrões, tendo aplicações que vão desde Visão Computacional até Química Computacional. Sempre que dois padrões complexos estão codi- ficados em termos de dois conjuntos de pontos que identificam suas características fundamentais, sua comparação pode ser vista como um problema de casamento de padrões de pontos. Este trabalho propõe uma abordagem unificada para os problemas de casamento exato e inexato de padrões de pontos em Espaços Euclidianos de dimensão arbitrária. No caso de casamento exato, é garantida a obtenção de uma solução ótima. Para casamento inexato (quando ruído está presente), resultados experimentais confirmam a validade da abordagem. Inicialmente, considera-se o problema de casamento de padrões de pontos como um problema de casamento de grafos ponderados. O problema de casamento de grafos ponderados é então formulado como um problema de inferência Bayesiana em um modelo gráfico probabilístico. Ao explorar certos vínculos fundamentais existentes em padrões de pontos imersos em Espaços Euclidianos, provamos que, para o casamento exato de padrões de pontos, um modelo gráfico simples é equivalente ao modelo completo. É possível mostrar que inferência probabilística exata neste modelo simples tem complexidade polinomial para qualquer dimensionalidade do Espaço Euclidiano em consideração. Experimentos computacionais comparando esta técnica com a bem conhecida baseada em relaxamento probabilístico evidenciam uma melhora significativa de desempenho para casamento inexato de padrões de pontos. A abordagem proposta é signi- ficativamente mais robusta diante do aumento do tamanho dos padrões envolvidos. Na ausência de ruído, os resultados são sempre perfeitos. / Point pattern matching in Euclidean Spaces is one of the fundamental problems in Pattern Recognition, having applications ranging from Computer Vision to Computational Chemistry. Whenever two complex patterns are encoded by two sets of points identifying their key features, their comparison can be seen as a point pattern matching problem. This work proposes a single approach to both exact and inexact point set matching in Euclidean Spaces of arbitrary dimension. In the case of exact matching, it is assured to find an optimal solution. For inexact matching (when noise is involved), experimental results confirm the validity of the approach. We start by regarding point pattern matching as a weighted graph matching problem. We then formulate the weighted graph matching problem as one of Bayesian inference in a probabilistic graphical model. By exploiting the existence of fundamental constraints in patterns embedded in Euclidean Spaces, we prove that for exact point set matching a simple graphical model is equivalent to the full model. It is possible to show that exact probabilistic inference in this simple model has polynomial time complexity with respect to the number of elements in the patterns to be matched. This gives rise to a technique that for exact matching provably finds a global optimum in polynomial time for any dimensionality of the underlying Euclidean Space. Computational experiments comparing this technique with well-known probabilistic relaxation labeling show significant performance improvement for inexact matching. The proposed approach is significantly more robust under augmentation of the sizes of the involved patterns. In the absence of noise, the results are always perfect.