Wir studieren das Verhalten des Maximums, des Minimums und des Endwerts zeithomogener eindimensionaler Diffusionen auf endlichen Zeitintervallen. Zuerst beweisen wir mit Hilfe des Malliavin-Kalküls ein Existenzresultat für die gemeinsamen Dichten. Außerdem leiten wir Entwicklungen der gemeinsamen Momente des Tripels (H,L,X) zur Zeit Delta bzgl. Delta her. Dabei steht X für die zugrundeliegende Diffusion, und H und L bezeichnen ihr fortlaufendes Maximum bzw. Minimum. Ein erster Ansatz, der vollständig auf den elementaren Abschätzungen der Doob’schen und der Cauchy-Schwarz’schen Ungleichung beruht, liefert eine Entwicklung bis zur Ordnung 2 bzgl. der Wurzel der Zeitvariablen Delta. Ein komplexerer Ansatz benutzt Partielle-Differentialgleichungstechniken, um eine Entwicklung der einseitigen Austrittswahrscheinlichkeit für gepinnte Diffusionen zu bestimmen. Da eine Entwicklung der Übergangsdichten von Diffusionen bekannt ist, erhält man eine vollständige Entwicklung der gemeinsamen Wahrscheinlichkeit von (H,X) bzgl. Delta. Die entwickelten Verteilungseigenschaften erlauben es uns, eine Theorie für Martingalschätzfunktionen, die aus wertebereich-basierten Daten konstruiert werden, in einem parameterisierten Diffusionsmodell, herzuleiten. Ein Small-Delta-Optimalitätsansatz, der die approximierten Momente benutzt, liefert eine Vereinfachung der vergleichsweise komplizierten Schätzprozedur und wir erhalten asymptotische Optimalitätsresultate für gegen 0 gehende Sampling-Frequenz. Beim Schätzen des Drift-Koeffizienten ist der wertebereich-basierte Ansatz der Methode, die auf equidistanten Beobachtungen der Diffusion beruht, nicht überlegen. Der Effizienzgewinn im Fall des Schätzens des Diffusionskoeffizienten ist hingegen enorm. Die Maxima und Minima in die Analyse miteinzubeziehen senkt die Varianz des Schätzers für den Parameter in diesem Szenario erheblich. / We study the behavior of the maximum, the minimum and the terminal value of time-homogeneous one-dimensional diffusions on finite time intervals. To begin with, we prove an existence result for the joint density by means of Malliavin calculus. Moreover, we derive expansions for the joint moments of the triplet (H,L,X) at time Delta w.r.t. Delta. Here, X stands for the underlying diffusion whereas H and L denote its running maximum and its running minimum, respectively. In a first approach that entirely relies on elementary estimates, such as Doob’s inequality and Cauchy-Schwarz’ inequality, we derive an expansion w.r.t. the square root of the time parameter Delta including powers of 2. A more sophisticated ansatz uses partial differential equation techniques to determine an expansion of the one-barrier hitting time probability for pinned diffusions. For an expansion of the transition density of diffusions is known, one obtains an overall expansion of the joint probability of (H,X) w.r.t. Delta. The developed distributional properties enable us to establish a theory for martingale estimating functions constructed from range-based data in a parameterized diffusion model. A small-Delta-optimality approach, that uses the approximated moments, yields a simplification of the relatively complicated estimating procedure and we obtain asymptotic optimality results when the sampling frequency Delta tends to 0. When it comes to estimating the drift coefficient the range-based method is not superior to the method relying on equidistant observations of the underlying diffusion alone. However, there is an enormous gain in efficiency at the estimation for the diffusion coefficient. Incorporating the maximum and the minimum into the analysis significantly lowers the asymptotic variance of the estimators for the parameter in this scenario.

Die vorliegende Arbeit befasst sich mit Innovationsdiffusionsmodellen und deren Anwendung in der Marketingpraxis. Sie hat zwei Ziele: Einen Überblick über existierende Innovationsmodelle zu schaffen und ein neues besseres Modell zu entwickeln. Es wird ein neuer Klassifizierungsansatz vorgeschlagen, mit dessen Hilfe ein strukturierter Überblick über die vorhandenen zahlreichen Innovationsdiffusionsmodelle möglich wird. Die Klassifizierung beruht auf den Annahmen in den Innovationsdiffusionsmodellen. Dies erlaubt im Gegensatz zu den bekannten Klassifizierungen (z.B. von Roberts/Lattin (2000)) die Bildung von disjunkten Modellklassen. Anhand der neuen Klassifizierung werden die prominenten Modelle, wie z.B. Bass-Modell (1969) bzw. Kalish-Modell (1985) eingeordnet und ihre Vor- und Nachteile aufgezeigt. Dieser Ansatz erleichtert eine Entscheidung für das beste zu verwendende Modell, wenn bekannt ist, welche Daten (Absatzdaten, Daten über Konsumenten etc.) zur Verfügung stehen und/oder welches Ziel (Absatzprognose, Preisbestimmung) verfolgt wird. Im zweiten Teil der Arbeit wird ein neues hybrides Innovationsdiffusionsmodell – das Information-Disicion-Evaluation-Modell (IED-Modell) – vorgestellt. Das IED-Modell besitzt zahlreiche Vorteile gegenüber existierenden Innovationsdiffusionsmodellen. Die Struktur des IED-Modells ist sehr allgemein so, dass das IED-Modell als eine Modellklasse bezeichnet werden könnte. Werden die Annahmen des IED-Modells genau definiert (z.B. über die Anzahl der Wettbewerbsprodukte usw.), erhält es eine explizite Form, die prominenten Innovationsdiffusionsmodellen ähnlich oder vollkommen identisch sein kann (für die Erstellung einer expliziten Form des IED-Modells siehe www.ied-modell.de). Ein solcher allgemeiner Modellierungsansatz des IED-Modells ist neu für die Innovationsdiffusionsforschung. Das IED-Modell und dessen Annahmen werden mittels Monte-Carlo-Simulationen analysiert. Beim empirischen Test an realen Daten wird das IED-Modell mit vier renommierten Innovationsdiffusionsmodellen verglichen. Laut diesem Vergleich ist das Anpassungsvermögen des IED-Modells im Durchschnitt besser als das der vier Vergleichsmodelle. Bei drei- und zehnmonatlichen Prognosen zeigte das IED-Modell eine sehr gute Vorhersagefähigkeit. / This work assesses innovation diffusion models and their application in marketing management. Its two principal aims are: (1) to give an overview of existing innovation diffusion models and (2) to develop a new and improved model. A new classification approach is proposed. The classification methodology bases on typical assumptions made in innovation diffusion models. Unlike prior classifications, e.g. Roberts/Lattin (2000), this approach allows for disjunctive classes. By means of this classification renowned models like Bass Model (1969) or Kalish Model (1985) are categorized, and their advantages and disadvantages are analyzed. This helps decide which model should be used depending on data availability (sales data, consumer data etc.) and the overall goal of a model investigation (sales forecast, pricing etc.). In the second part of this work the new hybrid innovation diffusion model – Innovation-Decision-Evaluation model (IED model) – is described. The model has several advantages compared with existing models. The structure of the IED Model is non-specific so that the IED model can be described as a distinct model class. When assumptions of the IED model are specified (e.g. number of competitive products) the model gets an explicit form which can be similar or even identical to other innovation diffusion models (for the design of an explicit model form see also www.ied-modell.de). Such a generalized modeling approach in IED modelling is new in innovation diffusion research. The IED model and its assumptions are analysed with Monte Carlo simulations. Its results are also empirically tested and compared with four renowned innovation diffusion models. The comparison reveals that the IED model has the best average fit and good forecast goodness.

Durant cette thèse, nous nous sommes intéressés aux outils permettant d'extraire les propriétés structurelles et temporelles de réseaux dynamiques ainsi que les caractéristiques de certains scénarios de diffusion pouvant s'opérer sur ces réseaux. Nous avons travaillé sur un jeu de données spécifiques, issu du projet MOSAR, qui comporte entre autre le réseau de proximité des personnes au cours du temps durant 6 mois à l'hôpital de Berk-sur-mer. Ce réseau est particulier dans le sens où il est constitué de trois dimensions: temporelle, structurelle par la répartition des personnes en services et fonctionnelle car chaque personne appartient à une catégorie socio-professionnelle. Pour chacune des dimensions, nous avons utilisé des outils existants en physique statistique ainsi qu'en théorie des graphes pour extraire des informations permettant de décrire certaines propriétés du réseau. Cela nous a permis de souligner le caractère très structuré de la répartition des contacts qui suit la répartition en services et mis en évidence les accointances entre certaines catégories professionnelles. Concernant la partie temporelle, nous avons mis en avant l'évolution périodique circadienne et hebdomadaire ainsi que les différences fondamentales entre l'évolution des interactions des patients et celle des personnels. Nous avons aussi présenté des outils permettant de comparer l'activité entre deux périodes données et de quantifier la similarité de ces périodes. Nous avons ensuite utilisé la technique de simulation pour extraire des propriétés de diffusion de ce réseau afin de donner quelques indices pour établir une politique de prévention. / In this thesis, we focus on tools whose aim is to extract structural and temporal properties of dynamic networks as well as diffusion characteristics which can occur on these networks. We work on specific data, from the European MOSAR project, including the network of individuals proximity from time to time during 6 months at the Brek-sur-Mer Hospital. The studied network is notable because of its three dimensions constitution : the structural one induced by the distribution of individuals into distinct services, the functional dimension due to the partition of individual into groups of socio-professional categories and the temporal dimension.For each dimension, we used tools well known from the areas of statistical physics as well as graphs theory in order to extract information which enable to describe the network properties. These methods underline the specific structure of the contacts distribution which follows the individuals distribution into services. We also highlight strong links within specific socio-professional categories. Regarding the temporal part, we extract circadian and weekly patterns and quantify the similarities of these activities. We also notice distinct behaviour within patients and staff evolution. In addition, we present tools to compare the network activity within two given periods. To finish, we use simulations techniques to extract diffusion properties of the network to find some clues in order to establish a prevention policy.

In this thesis, we consider stochastic systems which arise in different real-world application contexts. The first problem we consider is based on product adoption and pricing. A monopolist selling a product has to appropriately price the product over time in order to maximize the aggregated profit. The demand for a product is uncertain and is influenced by a number of factors, some of which are price, advertising, and product technology. We study the influence of price on the demand of a product and also how demand affects future prices. Our approach involves mathematically modelling the variation in demand as a function of price and current sales. We present a simulation-based algorithm for computing the optimal price path of a product for a given period of time. The algorithm we propose uses a smoothed-functional based performance gradient descent method to find a price sequence which maximizes the total profit over a planning horizon. The second system we consider is in the domain of sensor networks. A sensor network is a collection of autonomous nodes, each of which senses the environment. Sensor nodes use energy for sensing and communication related tasks. We consider the problem of finding optimal energy sharing policies that maximize the network performance of a system comprising of multiple sensor nodes and a single energy harvesting(EH) source. Nodes periodically sense a random field and generate data, which is stored in their respective data queues. The EH source harnesses energy from ambient energy sources and the generated energy is stored in a buffer. The nodes require energy for transmission of data and and they receive the energy for this purpose from the EH source. There is a need for efficiently sharing the stored energy in the EH source among the nodes in the system, in order to minimize average delay of data transmission over the long run. We formulate this problem in the framework of average cost infinite-horizon Markov Decision Processes[3],[7]and provide algorithms for the same.

In the area of Traffic Sign Recognition (TSR), deep learning models are trained to detect and classify images of traffic signs. The amount of data available to train these models is often limited, and collecting more data is time-consuming and expensive. A possible complement to traditional data acquisition, is to generate synthetic images with a generative machine learning model. This thesis investigates the use of denoising diffusion probabilistic models for generating synthetic data of one or multiple traffic sign classes, when providing different amount of real images for that class (classes). In the few-sample method, the number of images used was from 1 to 1000, and zero images were used in the zero-shot method. The results from the few-sample method show that combining synthetic images with real images when training a traffic sign classifier, increases the performance in 3 out of 6 investigated cases. The results indicate that the developed zero-shot method is useful if further refined, and potentially could enable generation of realistic images of signs not seen in the training data.

Generative modeling for computer vision has shown immense progress in the last few years, revolutionizing the way we perceive, understand, and manipulate visual data. This rapidly evolving field has witnessed advancements in image generation, 3D animation, and video prediction that unlock diverse applications across multiple fields including entertainment, design, healthcare, and education. As the demand for sophisticated computer vision systems continues to grow, this dissertation attempts to drive innovation in the field by exploring novel formulations of conditional generative models, and innovative applications in images, 3D animations, and video. Our research focuses on architectures that offer reversible transformations of noise and visual data, and the application of encoder-decoder architectures for generative tasks and 3D content manipulation. In all instances, we incorporate conditional information to enhance the synthesis of visual data, improving the efficiency of the generation process as well as the generated content. Prior successful generative techniques which are reversible between noise and data include normalizing flows and denoising diffusion models. The continuous variant of normalizing flows is powered by Neural Ordinary Differential Equations (Neural ODEs), and have shown some success in modeling the real image distribution. However, they often involve huge number of parameters, and high training time. Denoising diffusion models have recently gained huge popularity for their generalization capabilities especially in text-to-image applications. In this dissertation, we introduce the use of Neural ODEs to model video dynamics using an encoder-decoder architecture, demonstrating their ability to predict future video frames despite being trained solely to reconstruct current frames. In our next contribution, we propose a conditional variant of continuous normalizing flows that enables higher-resolution image generation based on lower-resolution input. This allows us to achieve comparable image quality to regular normalizing flows, while significantly reducing the number of parameters and training time. Our next contribution focuses on a flexible encoder-decoder architecture for accurate estimation and editing of full 3D human pose. We present a comprehensive pipeline that takes human images as input, automatically aligns a user-specified 3D human/non-human character with the pose of the human, and facilitates pose editing based on partial input information. We then proceed to use denoising diffusion models for image and video generation. Regular diffusion models involve the use of a Gaussian process to add noise to clean images. In our next contribution, we derive the relevant mathematical details for denoising diffusion models that use non-isotropic Gaussian processes, present non-isotropic noise, and show that the quality of generated images is comparable with the original formulation. In our final contribution, devise a novel framework building on denoising diffusion models that is capable of solving all three video tasks of prediction, generation, and interpolation. We perform ablation studies using this framework, and show state-of-the-art results on multiple datasets. Our contributions are published articles at peer-reviewed venues. Overall, our research aims to make a meaningful contribution to the pursuit of more efficient and flexible generative models, with the potential to shape the future of computer vision. / La modélisation générative pour la vision par ordinateur a connu d’immenses progrès ces dernières années, révolutionnant notre façon de percevoir, comprendre et manipuler les données visuelles. Ce domaine en constante évolution a connu des avancées dans la génération d’images, l’animation 3D et la prédiction vidéo, débloquant ainsi diverses applications dans plusieurs domaines tels que le divertissement, le design, la santé et l’éducation. Alors que la demande de systèmes de vision par ordinateur sophistiqués ne cesse de croître, cette thèse s’efforce de stimuler l’innovation dans le domaine en explorant de nouvelles formulations de modèles génératifs conditionnels et des applications innovantes dans les images, les animations 3D et la vidéo. Notre recherche se concentre sur des architectures offrant des transformations réversibles du bruit et des données visuelles, ainsi que sur l’application d’architectures encodeur-décodeur pour les tâches génératives et la manipulation de contenu 3D. Dans tous les cas, nous incorporons des informations conditionnelles pour améliorer la synthèse des données visuelles, améliorant ainsi l’efficacité du processus de génération ainsi que le contenu généré. Les techniques génératives antérieures qui sont réversibles entre le bruit et les données et qui ont connu un certain succès comprennent les flux de normalisation et les modèles de diffusion de débruitage. La variante continue des flux de normalisation est alimentée par les équations différentielles ordinaires neuronales (Neural ODEs) et a montré une certaine réussite dans la modélisation de la distribution d’images réelles. Cependant, elles impliquent souvent un grand nombre de paramètres et un temps d’entraînement élevé. Les modèles de diffusion de débruitage ont récemment gagné énormément en popularité en raison de leurs capacités de généralisation, notamment dans les applications de texte vers image. Dans cette thèse, nous introduisons l’utilisation des Neural ODEs pour modéliser la dynamique vidéo à l’aide d’une architecture encodeur-décodeur, démontrant leur capacité à prédire les images vidéo futures malgré le fait d’être entraînées uniquement à reconstruire les images actuelles. Dans notre prochaine contribution, nous proposons une variante conditionnelle des flux de normalisation continus qui permet une génération d’images à résolution supérieure à partir d’une entrée à résolution inférieure. Cela nous permet d’obtenir une qualité d’image comparable à celle des flux de normalisation réguliers, tout en réduisant considérablement le nombre de paramètres et le temps d’entraînement. Notre prochaine contribution se concentre sur une architecture encodeur-décodeur flexible pour l’estimation et l’édition précises de la pose humaine en 3D. Nous présentons un pipeline complet qui prend des images de personnes en entrée, aligne automatiquement un personnage 3D humain/non humain spécifié par l’utilisateur sur la pose de la personne, et facilite l’édition de la pose en fonction d’informations partielles. Nous utilisons ensuite des modèles de diffusion de débruitage pour la génération d’images et de vidéos. Les modèles de diffusion réguliers impliquent l’utilisation d’un processus gaussien pour ajouter du bruit aux images propres. Dans notre prochaine contribution, nous dérivons les détails mathématiques pertinents pour les modèles de diffusion de débruitage qui utilisent des processus gaussiens non isotropes, présentons du bruit non isotrope, et montrons que la qualité des images générées est comparable à la formulation d’origine. Dans notre dernière contribution, nous concevons un nouveau cadre basé sur les modèles de diffusion de débruitage, capable de résoudre les trois tâches vidéo de prédiction, de génération et d’interpolation. Nous réalisons des études d’ablation en utilisant ce cadre et montrons des résultats de pointe sur plusieurs ensembles de données. Nos contributions sont des articles publiés dans des revues à comité de lecture. Dans l’ensemble, notre recherche vise à apporter une contribution significative à la poursuite de modèles génératifs plus efficaces et flexibles, avec le potentiel de façonner l’avenir de la vision par ordinateur.

4D Imaging Radar is increasingly becoming a critical component in various industries due to beamforming technology and hardware advancements. However, it does not replace visual data in the form of 2D images captured by an RGB camera. Instead, 4D radar point clouds are a complementary data source that captures spatial information and velocity in a Doppler dimension that cannot be easily captured by a camera's view alone. Some discriminative features of the scene captured by the two sensors are hypothesized to have a shared representation. Therefore, a more interpretable visualization of the radar output can be obtained by learning a mapping from the empirical distribution of the radar to the distribution of images captured by the camera. To this end, the application of deep generative models to generate images conditioned on 4D radar data is explored. Two approaches that have become state-of-the-art in recent years are tested, generative adversarial networks and diffusion models. They are compared qualitatively through visual inspection and by two quantitative metrics: mean squared error and object detection count. It is found that it is easier to control the generative adversarial network's generative process through conditioning than in a diffusion process. In contrast, the diffusion model produces samples of higher quality and is more stable to train. Furthermore, their combination results in a hybrid sampling method, achieving the best results while simultaneously speeding up the diffusion process. / 4D bildradar får en alltmer betydande roll i olika industrier tack vare utveckling inom strålformningsteknik och hårdvara. Det ersätter dock inte visuell data i form av 2D-bilder som fångats av en RGB-kamera. Istället utgör 4D radar-punktmoln en kompletterande datakälla som representerar spatial information och hastighet i form av en Doppler-dimension. Det antas att vissa beskrivande egenskaper i den observerade miljön har en abstrakt representation som de två sensorerna delar. Därmed kan radar-datan visualiseras mer intuitivt genom att lära en transformation från fördelningen över radar-datan till fördelningen över bilderna. I detta syfte utforskas tillämpningen av djupa generativa modeller för bilder som är betingade av 4D radar-data. Två metoder som har blivit state-of-the-art de senaste åren testas: generativa antagonistiska nätverk och diffusionsmodeller. De jämförs kvalitativt genom visuell inspektion och med kvantitativa metriker: medelkvadratfelet och antalet korrekt detekterade objekt i den genererade bilden. Det konstateras att det är lättare att styra den generativa processen i generativa antagonistiska nätverk genom betingning än i en diffusionsprocess. Å andra sidan är diffusionsmodellen stabil att träna och producerar generellt bilder av högre kvalité. De bästa resultaten erhålls genom en hybrid: båda metoderna kombineras för att dra nytta av deras respektive styrkor. de identifierade begränsningarna i de enskilda modellerna och kurera datan för att jämföra hur dessa modeller skalar med större datamängder och mer variation.

<p dir="ltr">In Chapter 1, we introduce geometric motions from the general perspective of gradient flows. Here we develop the basic framework in which to pose the two main results of this thesis.</p><p dir="ltr">In Chapter 2, we examine the pinch-off phenomenon for a tubular surface moving by surface diffusion. We prove the existence of a one parameter family of pinching profiles obeying a long wavelength approximation of the dynamics.</p><p dir="ltr">In Chapter 3, we study a diffusion-based numerical scheme for curve shortening flow. We prove that the scheme is one time-step consistent.</p>

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.