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41	Range-based parameter estimation in diffusion models / statistical concepts and analytical foundations Henkel, Hartmuth 04 October 2010 (has links) 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. Wertebereich in Diffusionsmodellen Martingalschätzfunktionen Small-Delta-Optimalität Range in diffusion models Range-based parameter estimation Martingale estimating functions Small-Delta-Optimality 510 Mathematik 27 Mathematik SK 830 ddc:510
42	Eine neue Klasse hybrider Innovationsdiffusionsmodelle / ein theoretischer Vergleich mit existierenden Ansätzen und eine Analyse mit Simulationen und Realdaten Grishchenko, Yulia 18 September 2007 (has links) 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. Marketing Innovationsdiffusion Innovationsdiffusionsmodelle Absatzprognose Preisbestimmung Monte-Carlo-Simulation Information-Evaluation-Decision-Modell Innovation diffusion innovation diffusion models marketing sales forecast pricing Information-Evaluation-Decision-Model 330 Wirtschaft 17 Wirtschaft QP 210 ddc:330
43	Scene Reconstruction From 4D Radar Data with GAN and Diffusion : A Hybrid Method Combining GAN and Diffusion for Generating Video Frames from 4D Radar Data / Scenrekonstruktion från 4D-radardata med GAN och Diffusion : En Hybridmetod för Generation av Bilder och Video från 4D-radardata med GAN och Diffusionsmodeller Djadkin, Alexandr January 2023 (has links) 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. Deep generative models Generative adversarial networks Diffusion models GAN DGM 4D imaging radar Djupa generativa modeller Generativa antagonistiska nätverk Diffusionsmodeller GAN DGM 4D-bildradar Computer Sciences Datavetenskap (datalogi) Computer and Information Sciences Data- och informationsvetenskap
44	Asymptotic Analysis of Models for Geometric Motions Gavin Ainsley Glenn (17958005) 13 February 2024 (has links) <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> Numerical analysis Partial differential equations partial differential equations (PDE) motion by mean curvature Mean Curvature Flow surface diffusion models gradient flows diffusion generated motion codimension-2 pinch-off dynamics consistency estimates heat equation axisymmetric surface diffusion Geometric PDEs one-parameter family singularity formation Numerical analysis. convergence rate Differential equations. Ordinary Differential Equations (ODE)
45	Highway Development Decision-Making Under Uncertainty: Analysis, Critique and Advancement El-Khatib, Mayar January 2010 (has links) 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. highway development decision making under uncertainty real options jump processes infinitely divisible distributions Lévy processes measure theory characteristic function characteristic exponent Lévy jump measure probability measure cádlág stochastic process Wiener process Poisson process compound Poisson process finite activity models jump diffusion models Merton model Gaussian jump process Kou model double exponential jump process leptokurtic volatility smile infinite activity models pure jumps process generalized hyperbolic model modified Bessel function negative inverse Gaussian model geometric Brownian motion model log-normal distribution model normality assumption heavy-tailed distribution asymmetric distribution quantile-quantile plot Rydberg algorithm parameter calibration moments cummulants sum of squares Monte Carlo simulation least-squares regression backward dynamic programming discrete-time Markov chain land price traffic demand highway service quality index data collection traffic volume Annual Average Daily Traffic Ontario Ministry of Transportation Land Registry Office Freedom of Information Act seasonality value of transportation types of transportation systems importance of highway systems cost of wrong decisions monetary cost private sector cost human cost land expropriation environmental cost irreversibility cost factor time cost factor political cost factor opportunity cost opportunity cost of wrong decisions economic cost of wrong decisions economic cost of mis-estimation economic profit of wrong decisions cost of inaction price of making right decisions scope of decisions rehabilitation decision land acquisition decision expansion decision uncertainty modeling land acquisition cost expropriation cost construction cost material cost Montréal-Mirabel International Airport Pickering Airport 407 Express Toll Route Highway 407 Statistics
46	Highway Development Decision-Making Under Uncertainty: Analysis, Critique and Advancement El-Khatib, Mayar January 2010 (has links) 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. highway development decision making under uncertainty real options jump processes infinitely divisible distributions Lévy processes measure theory characteristic function characteristic exponent Lévy jump measure probability measure cádlág stochastic process Wiener process Poisson process compound Poisson process finite activity models jump diffusion models Merton model Gaussian jump process Kou model double exponential jump process leptokurtic volatility smile infinite activity models pure jumps process generalized hyperbolic model modified Bessel function negative inverse Gaussian model geometric Brownian motion model log-normal distribution model normality assumption heavy-tailed distribution asymmetric distribution quantile-quantile plot Rydberg algorithm parameter calibration moments cummulants sum of squares Monte Carlo simulation least-squares regression backward dynamic programming discrete-time Markov chain land price traffic demand highway service quality index data collection traffic volume Annual Average Daily Traffic Ontario Ministry of Transportation Land Registry Office Freedom of Information Act seasonality value of transportation types of transportation systems importance of highway systems cost of wrong decisions monetary cost private sector cost human cost land expropriation environmental cost irreversibility cost factor time cost factor political cost factor opportunity cost opportunity cost of wrong decisions economic cost of wrong decisions economic cost of mis-estimation economic profit of wrong decisions cost of inaction price of making right decisions scope of decisions rehabilitation decision land acquisition decision expansion decision uncertainty modeling land acquisition cost expropriation cost construction cost material cost Montréal-Mirabel International Airport Pickering Airport 407 Express Toll Route Highway 407 Statistics

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