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Dans le premier chapitre, la notion d'évaluation des risques sismiques est définie et les caractéristiques sismotectoniques de la région d'étude sont brièvement présentés. Un examen rigoureux des modèles stochastiques, appliqués au domaine de la sismologie est fourni. Dans le chapitre 2, différents modèles semi-Markoviens sont développés pour étudier la sismicité des îles Ioniennes centrales ainsi que le Nord de la mer Egée (Grèce). Les quantités telles que le noyau semi-Markovien et les probabilités de destination sont évaluées, en considérant que les temps de séjour suivent les distributions géométrique, discrète Weibull et Pareto. Des résultats utiles sont obtenus pour l'estimation de la sismicité. Dans le troisième chapitre un nouvel algorithme de Viterbi pour les modèles semi-Markoviens cachés est construit, dont la complexité est une fonction linéaire du nombre d'observations et une fonction quadratique du nombre d'états cachés, la plus basse existante dans la littérature. Une extension de ce nouvel algorithme est développée pour le cas où une observation dépend de l'état caché correspondant, mais aussi de l'observation précédente (cas SM1-M1). Dans le chapitre 4 les modèles semi-Markoviens cachés sont appliquées pour étudier la sismicité du Nord et du Sud de la mer Égée. La séquence d'observation est constituée des magnitudes et des positions d’un tremblement de terre et le nouvel algorithme de Viterbi est mis en œuvre afin de décoder les niveaux des tensions cachés qui sont responsables pour la sismogenèse. Les phases précurseurs (variations des tensions cachées) ont été détectées en avertissant qu’un tremblement de terre pourrait se produire. Ce résultat est vérifié pour 70 sur 88 cas (le score optimal). Les temps de séjour du processus caché étaient supposés suivre les distributions Poisson, logarithmique ou binomiale négative, tandis que les niveaux de tensions cachés ont été classés en 2, 3 ou 4 états. Les modèles de Markov caché ont également été adaptés sans présenter des résultats intéressants concernant les phases précurseurs. Dans le chapitre 5 un algorithme de Viterbi généralisé pour les modèles semi-Markoviens cachés, est construit dans le sens que les transitions au même état caché sont autorisées et peuvent également être décodées. De plus, une extension de cet algorithme généralisé dans le contexte SM1-M1 est présentée. Dans le chapitre 6 nous modifions de manière convenable le modèle Cramér-Lundberg y compris des sinistres négatifs et positifs, afin de décrire l'évolution avec le temps des changements de contraintes de Coulomb (valeurs ΔCFF) calculées pour sept épicentres (M ≥ 6) du Nord de la mer Egée. Formules pour les probabilités de ruine sont définies sous une forme générale. Corollaires sont également formulés pour la distribution exponentielle et Pareto. L'objectif est de mettre en lumière la question suivante qui pose la problématique dans la Sismologie: Au cours d'une année pourquoi un tremblement de terre s’est produit dans une position précise et pas dans une autre position, aux régions sismotectoniquement homogènes ayant valeurs ΔCFF positives. Les résultats montrent que les nouvelles formules de probabilité peuvent contribuer à répondre au problème susmentionné. / In the first chapter the definition of the seismic hazard assessment is provided, the seismotectonic features of the study areas are briefly presented and the already existing mathematical models applied in the field of Seismology are thoroughly reviewed. In chapter 2, different semi-Markov models are developed for studying the seismicity of the areas of the central Ionian Islands and the North Aegean Sea (Greece). Quantities such as the kernel and the destination probabilities are evaluated, considering geometric, discrete-Weibull and Pareto distributed sojourn times. Useful results are obtained for forecasting purposes. In the third chapter a new Viterbi algorithm for hidden semi-Markov models is developed, whose complexity is a linear function of the number of observations and a quadratic function of the number of hidden states, the lowest existing in the literature. Furthermore, an extension of this new algorithm is introduced for the case that an observation depends on the corresponding hidden state but also on the previous observation (SM1-M1 case). In chapter 4, different hidden semi-Markov models (HSMMs) are applied for the study of the North and South Aegean Sea. The earthquake magnitudes and locations comprise the observation sequence and the new Viterbi algorithm is implemented in order to decode the hidden stress field associated with seismogenesis. Precursory phases (variations of the hidden stress field) were detected warning for an anticipated earthquake occurrence for 70 out of 88 cases (the optimal model’s score). The sojourn times of the hidden process were assumed to follow Poisson, logarithmic or negative binomial distributions, whereas the hidden stress levels were classified into 2, 3 or 4 states. HMMs were also adapted without presenting significant results as for the precursory phases. In chapter 5 a generalized Viterbi algorithm for HSMMs is constructed in the sense that now transitions to the same hidden state are allowed and can also be decoded. Furthermore, an extension of this generalized algorithm in the SM1-M1 context is given. In chapter 6 we modify adequately the Cramér-Lundberg model considering negative and positive claims, in order to describe the evolution in time of the Coulomb failure function changes (ΔCFF values) computed at the locations of seven strong (M ≥ 6) earthquakes of the North Aegean Sea. Ruin probability formulas are derived and proved in a general form. Corollaries are also formulated for the exponential and the Pareto distribution. The aim is to shed light to the following problem posed by the seismologists: During a specific year why did an earthquake occur at a specific location and not at another location in seismotectonically homogeneous areas with positive ΔCFF values (stress enhanced areas). The results demonstrate that the new probability formulas can contribute in answering the aforementioned question.

Safety analysis is critical when developing new engineering systems. Many systems have to function under randomly occurring events, making stochastic processes useful in a safety modelling context. However, a general stochastic process is very challenging to analyse mathematically. Therefore, model restrictions are necessary to simplify the mathematical analysis. A popular simplified stochastic model is the Semi-Markov process (SMP), which is a generalization of the "memoryless" continuous-time Markov chain. However, only a subclass of Semi-Markov models can be analysed with non-simulation based methods. In these models, the cumulative density function (cdf) of the random variables describing the system is in the form of expolynomials. This thesis investigates the possibility to extend the number of Semi-Markov models that can be analysed with non-simulation based methods by approximating the non-expolynomial random variables with expolynomials. This thesis focus on approximation of models partially described by LogNormal and Weibull distributed random variables. The result shows that it is possible to approximate some Semi-Markov models with non-expolynomial random variables. However, there is an increasing difficulty in approximating a non-expolynomial random variable when the variability in the distribution increases. / Säkerhetsanalys är avgörande när man utvecklar nya tekniska system. Många system måste fungera under slumpmässigt inträffande händelser, vilket gör stokastiska processer användbara i ett säkerhetsmodellerande sammanhang. En allmän stokastisk process är dock mycket utmanande att analysera matematiskt. Därför är begränsningar på modellen nödvändiga för att förenkla den matematiska analysen. En populär förenklad stokastisk modell är Semi-Markov-processen (SMP), vilket är en generalisering av den "minneslösa" tids-kontinuerliga Markov-kedjan. Dock är det endast en underklass av Semi-Markov-modeller som kan analyseras med icke-simuleringsbaserade metoder. I dessa modeller är den kumulativa densitetsfunktionen (cdf) för de slumpmässiga variablerna som beskriver systemet i form av expolynomials. Denna rapport undersöker möjligheten att utöka antalet Semi-Markov-modeller som kan analyseras med icke-simuleringsbaserade metoder genom att approximera de icke-expolynomial slumpvariablerna med expolynomials. Vi fokuserar på approximering av modeller som delvis beskrivs av LogNormal distribuerade och Weibull distribuerade slumpmässiga variabler. Resultatet visar att det är möjligt att approximera vissa stokastiska variabler som är icke-expolynomial i Semi-Markov-modeller. Resultatet visar dock att det är en ökande svårighet att approximera en icke-expolynomial slumpmässiga variabeln när variabiliteten i fördelningen ökar.

The typical research of Alzheimer's disease includes a series of cognitive states. Multi-state models are often used to describe the history of disease evolvement. Competing risks models are a sub-category of multi-state models with one starting state and several absorbing states. Analyses for competing risks data in medical papers frequently assume independent risks and evaluate covariate effects on these events by modeling distinct proportional hazards regression models for each event. Jeong and Fine (2007) proposed a parametric proportional sub-distribution hazard (SH) model for cumulative incidence functions (CIF) without assumptions about the dependence among the risks. We modified their model to assure that the sum of the underlying CIFs never exceeds one, by assuming a proportional SH model for dementia only in the Nun study. To accommodate left censored data, we computed non-parametric MLE of CIF based on Expectation-Maximization algorithm. Our proposed parametric model was applied to the Nun Study to investigate the effect of genetics and education on the occurrence of dementia. After including left censored dementia subjects, the incidence rate of dementia becomes larger than that of death for age < 90, education becomes significant factor for incidence of dementia and standard errors for estimates are smaller. Multi-state Markov model is often used to analyze the evolution of cognitive states by assuming time independent transition intensities. We consider both constant and duration time dependent transition intensities in BRAiNS data, leading to a mixture of Markov and semi-Markov processes. The joint probability of observing a sequence of same state until transition in a semi-Markov process was expressed as a product of the overall transition probability and survival probability, which were simultaneously modeled. Such modeling leads to different interpretations in BRAiNS study, i.e., family history, APOE4, and sex by head injury interaction are significant factors for transition intensities in traditional Markov model. While in our semi-Markov model, these factors are significant in predicting the overall transition probabilities, but none of these factors are significant for duration time distribution.

L'étude de l'évolution du pronostic de santé d'un patient constitue un domaine important en recherche clinique. Récemment, le développement des modèles multi-états a permis d'étudier cette dynamique en prenant en compte plusieurs états de santé. Dans ce manuscrit, nous utilisons plus particulièrement les modèles semi-markoviens. Ce type de processus distingue les temps de séjour dans les états et les trajectoires des transitions, contrairement à l'approche markovienne classique. Nous avons proposé plusieurs adaptations pour pouvoir appliquer ce type de modèle : la censure par intervalle, le choix des distributions des temps d'attente et l'introduction des covariables. Un test d'adéquation est aussi proposé pour vérifier l'hypothèse de stationnarité. Enfin, une méthode originale, incluant la théorie des courbes ROC, est présentée pour définir des états de santé pertinents au regard du pronostic. Ces développements sont principalement appliqués à une cohorte de patients greffés rénaux (base de données DIVAT).

A transition model (MOSAIC) is used to describe forest dynamics at the landscape scale. The model uses a semi-Markov framework by considering transition probabilities and Erlang distributed holding times in each transition. Parameters for the transition model are derived from a gap model (ZELIG). This procedure ensures conceptual consistency of the landscape model with the fine scale ecological detail represented by the forest gap model. Spatial heterogeneity in the transition model is driven by maps of terrain with characteristics contained in a Geographic Information System (GIS) database. The results of the transition model simulations, percent cover forest type maps, are exported to grid-maps in the GIS. These cover type maps can be classified and used to describe forest dynamics using landscape statistics metrics. The linkage model-GIS enhances the transition model spatial analytical capabilities. A parameterization algorithm was developed that takes as input gap model tracer files which contain the percent occupation of each cover type through time. As output, the algorithm produces a file that contains the parameter values needed for MOSAIC for each one of the possible transitions. Parameters for the holding time distribution were found by calculating an empirical estimate of the cumulative probability function and using a non-linear least squares method to fit this estimate to an Erlang distribution. The algorithm provided good initial estimates of the transitions parameters that can be refined with few additional simulations. A method for deriving classification criteria to designate cover types is presented. The method uses cluster analysis to detect the number and type of forest classes and Classification and Regression Tree (CART) analysis to explain the forest classes in term of stand attributes. This method provided a precise and objective approach for forest cover type definition and classification. The H. J. Andrews forest in Oregon was used to demonstrate the methods and procedures developed in this study.

Orchards demand large areas of land, thus they are often situated far from major population centres. As a result it is often difficult to obtain the necessary personnel, limiting both growth and productivity. However, if autonomous robots could be integrated into the operation of the orchard, the manpower demand could be reduced. A key problem for any autonomous robot is localisation; how does the robot know where it is? In agriculture robots, the most common approach is to use GPS positioning. However, in an orchard environment, the dense and tall vegetation restricts the usage to large robots that reach above the surroundings. In order to enable the use of smaller robots, it is instead necessary to use a GPS independent system. However, due to the similarity of the environment and the lack of strong recognisable features, it appears unlikely that typical non-GPS solutions will prove successful. Therefore we present a GPS independent localisation system, specifically aimed for orchards, that utilises the inherent structure of the surroundings. Furthermore, we examine and individually evaluate three related sub-problems. The proposed system utilises a 3D point cloud created from a 2D LIDAR and the robot’s movement. First, we show how the data can be segmented into individual trees using a Hidden Semi-Markov Model. Second, we introduce a set of descriptors for describing the geometric characteristics of the individual trees. Third, we present a robust localisation method based on Hidden Markov Models. Finally, we propose a method for detecting segmentation errors when associating new tree measurements with previously measured trees. Evaluation shows that the proposed segmentation method is accurate and yields very few segmentation errors. Furthermore, the introduced descriptors are determined to be consistent and informative enough to allow localisation. Third, we show that the presented localisation method is robust both to noise and segmentation errors. Finally it is shown that a significant majority of all segmentation errors can be detected without falsely labeling correct segmentations as incorrect. / Eftersom fruktodlingar kräver stora markområden är de ofta belägna långt från större befolkningscentra. Detta gör det svårt att finna tillräckligt med arbetskraft och begränsar expansionsmöjligheterna. Genom att integrera autonoma robotar i drivandet av odlingarna skulle arbetet kunna effektiviseras och behovet av arbetskraft minska. Ett nyckelproblem för alla autonoma robotar är lokalisering; hur vet roboten var den är? I jordbruksrobotar är standardlösningen att använda GPS-positionering. Detta är dock problematiskt i fruktodlingar, då den höga och täta vegetationen begränsar användandet till större robotar som når ovanför omgivningen. För att möjliggöra användandet av mindre robotar är det istället nödvändigt att använda ett GPS-oberoende lokaliseringssystem. Detta problematiseras dock av den likartade omgivningen och bristen på distinkta riktpunkter, varför det framstår som osannolikt att existerande standardlösningar kommer fungera i denna omgivning. Därför presenterar vi ett GPS-oberoende lokaliseringssystem, speciellt riktat mot fruktodlingar, som utnyttjar den naturliga strukturen hos omgivningen.Därutöver undersöker vi och utvärderar tre relaterade delproblem. Det föreslagna systemet använder ett 3D-punktmoln skapat av en 2D-LIDAR och robotens rörelse. Först visas hur en dold semi-markovmodell kan användas för att segmentera datasetet i enskilda träd. Därefter introducerar vi ett antal deskriptorer för att beskriva trädens geometriska form. Vi visar därefter hur detta kan kombineras med en dold markovmodell för att skapa ett robust lokaliseringssystem.Slutligen föreslår vi en metod för att detektera segmenteringsfel när nya mätningar av träd associeras med tidigare uppmätta träd. De föreslagna metoderna utvärderas individuellt och visar på goda resultat. Den föreslagna segmenteringsmetoden visas vara noggrann och ge upphov till få segmenteringsfel. Därutöver visas att de introducerade deskriptorerna är tillräckligt konsistenta och informativa för att möjliggöra lokalisering. Ytterligare visas att den presenterade lokaliseringsmetoden är robust både mot brus och segmenteringsfel. Slutligen visas att en signifikant majoritet av alla segmenteringsfel kan detekteras utan att felaktigt beteckna korrekta segmenteringar som inkorrekta.

Multi-state models are often used to evaluate the effect of death as a competing event to the development of dementia in a longitudinal study of the cognitive status of elderly subjects. In this dissertation, both multi-state Markov model and semi-Markov model are used to characterize the flow of subjects from intact cognition to dementia with mild cognitive impairment and global impairment as intervening transient, cognitive states and death as a competing risk. Firstly, a multi-state Markov model with three transient states: intact cognition, mild cognitive impairment (M.C.I.) and global impairment (G.I.) and one absorbing state: dementia is used to model the cognitive panel data. A Weibull model and a Cox proportional hazards (Cox PH) model are used to fit the time to death based on age at entry and the APOE4 status. A shared random effect correlates this survival time with the transition model. Secondly, we further apply a Semi-Markov process in which we assume that the wait- ing times are Weibull distributed except for transitions from the baseline state, which are exponentially distributed and we assume no additional changes in cognition occur between two assessments. We implement a quasi-Monte Carlo (QMC) method to calculate the higher order integration needed for the likelihood based estimation. At the end of this dissertation we extend a non-parametric “local EM algorithm” to obtain a smooth estimator of the cause-specific hazard function (CSH) in the presence of competing risk. All the proposed methods are justified by simulation studies and applications to the Nun Study data, a longitudinal study of late life cognition in a cohort of 461 subjects.

Autonomous vehicles is soon a reality in the every-day life. Though before it is used commercially the vehicles need to be proven safe. The current standard for functional safety on roads, ISO 26262, does not include autonomous vehicles at the moment, which is why in this project an approach using semi-Markov models is used to assess safety. A semi-Markov process is a stochastic process modelled by a state space model where the transitions between the states of the model can be arbitrarily distributed. The approach is realized as a MATLAB tool where the user can use a steady-state based analysis called a Loss and Risk based measure of safety to assess safety. The tool works and can assess safety of semi-Markov systems as long as they are irreducible and positive recurrent. For systems that fulfill these properties, it is possible to draw conclusions about the safety of the system through a risk analysis and also about which autonomous driving level the system is in through a sensitivity analysis. The developed tool, or the approach with the semi-Markov model, might be a good complement to ISO 26262.

京都大学 / 新制・課程博士 / 博士(情報学) / 甲第23311号 / 情博第747号 / 新制||情||128(附属図書館) / 京都大学大学院情報学研究科知能情報学専攻 / (主査)准教授 吉井 和佳, 教授 河原 達也, 教授 西野 恒, 教授 鹿島 久嗣 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM