Bibliographies: 'Nonlinear Expectation' – Grafiati (2024)

Abstract:

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not subadditive in general, so it is beyond the scope of Peng’s framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time inconsistency, that is, the usual “tower property” fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step toward a new understanding of nonlinear expectations under probability distortion and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

Abstract:

The proposal of considering nonlinear principal component analysis as a kernel eigenvalue problem has provided an extremely powerful method of extracting nonlinear features for a number of classification and regression applications. Whereas the utilization of Mercer kernels makes the problem of computing principal components in, possibly, infinite-dimensional feature spaces tractable, there are still the attendant numerical problems of diagonalizing large matrices. In this contribution, we propose an expectation-maximization approach for performing kernel principal component analysis and show this to be a computationally efficient method, especially when the number of data points is large.

Abstract:

This thesis presents a comprehensive set of techniques for solving, simulating, analysing and controlling large scale, nonlinear, econometric models that contain rational expectations of future dated variables. These expectations are generally treated as model consistent, whereby the expectation is set to the deterministic projection of the model. Solutions to such models are distinguished from those of conventional models by the fact that they are not recursive in time. The outcome for the current period depends on the expected outcome for future periods as well as past periods. This property means that all of the basic numerical procedures need to be altered. We consider the following topics: solution algorithms for the two—point boundary value problem; terminal conditions, uniqueness and stability; experimental design and stochastic simulation; model forms, solution modes and historical tracking; control methods including optimal control. We find that suitable procedures allow us to undertake all of the experiments usually conducted with conventional models. Each topic is illustrated by application to three large scale models of the United Kingdom economy which contain rational expectations terms. Only one of these models is constructed following the new-classical paradigm and hence their comparative properties revealed by our experiments are of some interest.

Abstract:

This thesis deals with estimation of states and parameters in nonlinear and non-Gaussian dynamic systems. Sequential Monte Carlo methods are mainly used to this end. These methods rely on models of the underlying system, motivating some developments of the model concept. One of the main reasons for the interest in nonlinear estimation is that problems of this kind arise naturally in many important applications. Several applications of nonlinear estimation are studied. The models most commonly used for estimation are based on stochastic difference equations, referred to as state-space models. This thesis is mainly concerned with models of this kind. However, there will be a brief digression from this, in the treatment of the mathematically more intricate differential-algebraic equations. Here, the purpose is to write these equations in a form suitable for statistical signal processing. The nonlinear state estimation problem is addressed using sequential Monte Carlo methods, commonly referred to as particle methods. When there is a linear sub-structure inherent in the underlying model, this can be exploited by the powerful combination of the particle filter and the Kalman filter, presented by the marginalized particle filter. This algorithm is also known as the Rao-Blackwellized particle filter and it is thoroughly derived and explained in conjunction with a rather general class of mixed linear/nonlinear state-space models. Models of this type are often used in studying positioning and target tracking applications. This is illustrated using several examples from the automotive and the aircraft industry. Furthermore, the computational complexity of the marginalized particle filter is analyzed. The parameter estimation problem is addressed for a relatively general class of mixed linear/nonlinear state-space models. The expectation maximization algorithm is used to calculate parameter estimates from batch data. In devising this algorithm, the need to solve a nonlinear smoothing problem arises, which is handled using a particle smoother. The use of the marginalized particle filter for recursive parameterestimation is also investigated. The applications considered are the camera positioning problem arising from augmented reality and sensor fusion problems originating from automotive active safety systems. The use of vision measurements in the estimation problem is central to both applications. In augmented reality, the estimates of the camera’s position and orientation are imperative in the process of overlaying computer generated objects onto the live video stream. The objective in the sensor fusion problems arising in automotive safety systems is to provide information about the host vehicle and its surroundings, such as the position of other vehicles and the road geometry. Information of this kind is crucial for many systems, such as adaptive cruise control, collision avoidance and lane guidance.

Abstract:

Many common statistical techniques for modeling multidimensional dynamic data sets can be seen as variants of one (or multiple) underlying linear/nonlinear model(s). These statistical techniques fall into two broad categories of supervised and unsupervised learning. The emphasis of this dissertation is on unsupervised learning under multiple generative models. For linear models, this has been achieved by collective observations and derivations made by previous authors during the last few decades. Factor analysis, polynomial chaos expansion, principal component analysis, gaussian mixture clustering, vector quantization, and Kalman filter models can all be unified as some variations of unsupervised learning under a single basic linear generative model. Hidden Markov modeling (HMM), however, is categorized as an unsupervised learning under multiple linear/nonlinear generative models. This dissertation is primarily focused on hidden Markov models (HMMs).On the first half of this dissertation we study enhancements on the theory of hidden Markov modeling. These include three branches: 1) a robust as well as a closed-form parameter estimation solution to the expectation maximization (EM) process of HMMs for the case of elliptically symmetrical densities; 2) a two-step HMM, with a combined state sequence via an extended Viterbi algorithm for smoother state estimation; and 3) a duration-dependent HMM, for estimating the expected residency frequency on each state. Then, the second half of the dissertation studies three novel applications of these methods: 1) the applications of Markov switching models on the Bifurcation Theory in nonlinear dynamics; 2) a Game Theory application of HMM, based on fundamental theory of card counting and an example on the game of Baccarat; and 3) Trust modeling and the estimation of trustworthiness metrics in cyber security systems via Markov switching models.As a result of the duration dependent HMM, we achieved a better estimation for the expected duration of stay on each regime. Then by robust and closed form solution to the EM algorithm we achieved robustness against outliers in the training data set as well as higher computational efficiency in the maximization step of the EM algorithm. By means of the two-step HMM we achieved smoother probability estimation with higher likelihood than the standard HMM.
Ph. D.

Abstract:

In this paper we study parameter estimation via the Expectation Maximization (EM) algorithm for a continuous-time hidden Markov model with diffusion and point process observation. Inference problems of this type arise for instance in credit risk modelling. A key step in the application of the EM algorithm is the derivation of finite-dimensional filters for the quantities that are needed in the E-Step of the algorithm. In this context we obtain exact, unnormalized and robust filters, and we discuss their numerical implementation. Moreover, we propose several goodness-of-fit tests for hidden Markov models with Gaussian noise and point process observation. We run an extensive simulation study to test speed and accuracy of our methodology. The paper closes with an application to credit risk: we estimate the parameters of a hidden Markov model for credit quality where the observations consist of rating transitions and credit spreads for US corporations.

Abstract:

Dealing with missing data in spatio-temporal time series constitutes important branch of general missing data problem. Since the statistical properties of time-dependent data characterized by sequentiality of observations then any interruption of consecutiveness in time series will cause severe problems. In order to make reliable analyses in this case missing data must be handled cautiously without disturbing the series statistical properties, mainly as temporal and spatial dependencies.In this study we aimed to compare several imputation methods for the appropriate completion of missing values of the spatio-temporal meteorological time series. For this purpose, several missing imputation methods are assessed on their imputation performances for artificially created missing data in monthly total precipitation and monthly mean temperature series which are obtained from the climate stations of Turkish State Meteorological Service. Artificially created missing data are estimated by using six methods. Single Arithmetic Average (SAA), Normal Ratio (NR) and NR Weighted with Correlations (NRWC) are the three simple methods used in the study. On the other hand, we used two computational intensive methods for missing data imputation which are called Multi Layer Perceptron type Neural Network (MLPNN) and Monte Carlo Markov Chain based on Expectation-Maximization Algorithm (EM-MCMC). In addition to these, we propose a modification in the EM-MCMC method in which results of simple imputation methods are used as auxiliary variables. Beside the using accuracy measure based on squared errors we proposed Correlation Dimension (CD) technique for appropriate evaluation of imputation performances which is also important subject of Nonlinear Dynamic Time Series Analysis.

Abstract:

The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms.

Abstract:

Cette thèse s’intéresse au problème de l’identification de modèle thermique d’un bâtiment intelligent, dont les objets connectés pallient la non-mesure des grandeurs physiques d’intérêt. Un premier algorithme traite de l’estimation boucle ouverte du système de bâtiment exploité en boucle fermée. Cet algorithme est ensuite modifié pour intégrer l’incertitude de mesure des données. Nous suggérons ainsi une méthode en boucle fermée, non-intrusive car s’affranchissant de la nécessité de mesurer la température intérieure. Puis, nous revenons à des approches en boucle ouverte. Les différents algorithmes permettent respectivement de réduire le biais contenu dans la mesure de température extérieure par une sonde connectée, de remplacer le coûteux capteur de flux solaire par un capteur de température extérieure, et enfin d’utiliser la courbe de charge totale, et non désagrégée, en tirant profit de signaux On/Off des objets connectés
This thesis is devoted to the problem of the identification of a thermal model of a smart building, whose connected objects alleviate the lack of measurements of the physical quantities of interest. The first algorithm deals with the estimation of the open-loop building system, despite its actual exploitation in closed loop. This algorithm is then modified to account for the uncertainty of the data. We suggest a closedloop estimation of the building system as soon as the indoor temperature is not measured. Then, we return to open-loop approaches. The different algorithms enable respectively to reduce the possible bias contained in a connected outdoor air temperature sensor, to replace the costly solar flux sensor by another connected temperature sensor, and finally to directly use the total load curve, without disaggregation, by making the most of the On/Off signals of the connected objects

Abstract:

Cette thèse porte sur la modélisation mathématique de la dynamique du cancer ; elle se divise en deux projets de recherche.Dans le premier projet, nous estimons les paramètres de la limite déterministe d'un processus stochastique modélisant la dynamique du mélanome (cancer de la peau) traité par immunothérapie. L'estimation est réalisée à l'aide d'un modèle statistique non-linéaire à effets mixtes et l'algorithme SAEM, à partir des données réelles de taille tumorale mesurée au cours du temps chez plusieurs patients. Avec ce modèle mathématique qui ajuste bien les données, nous évaluons la probabilité de rechute du mélanome (à l'aide de l'algorithme Importance Splitting), et proposons une optimisation du protocole de traitement (doses et instants du traitement).Nous proposons dans le second projet, une méthode d'approximation de vraisemblance basée sur une approximation de l'algorithme Belief Propagation à l'aide de l'algorithme Expectation-Propagation, pour une approximation diffusion du modèle stochastique de mélanome observée chez un seul individu avec du bruit gaussien. Cette approximation diffusion (définie par une équation différentielle stochastique) n'ayant pas de solution analytique, nous faisons recours à une méthode d'Euler pour approcher sa solution (après avoir testé la méthode d'Euler sur le processus de diffusion d'Ornstein Uhlenbeck). Par ailleurs, nous utilisons une méthode d'approximation de moments pour faire face à la multidimensionnalité et la non-linéarité de notre modèle. A l'aide de la méthode d'approximation de vraisemblance, nous abordons l'estimation de paramètres dans des Modèles de Markov Cachés
This thesis is about mathematical modeling of cancer dynamics ; it is divided into two research projects.In the first project, we estimate the parameters of the deterministic limit of a stochastic process modeling the dynamics of melanoma (skin cancer) treated by immunotherapy. The estimation is carried out with a nonlinear mixed-effect statistical model and the SAEM algorithm, using real data of tumor size. With this mathematical model that fits the data well, we evaluate the relapse probability of melanoma (using the Importance Splitting algorithm), and we optimize the treatment protocol (doses and injection times).We propose in the second project, a likelihood approximation method based on an approximation of the Belief Propagation algorithm by the Expectation-Propagation algorithm, for a diffusion approximation of the melanoma stochastic model, noisily observed in a single individual. This diffusion approximation (defined by a stochastic differential equation) having no analytical solution, we approximate its solution by using an Euler method (after testing the Euler method on the Ornstein Uhlenbeck diffusion process). Moreover, a moment approximation method is used to manage the multidimensionality and the non-linearity of the melanoma mathematical model. With the likelihood approximation method, we tackle the problem of parameter estimation in Hidden Markov Models

Abstract:

This chapter presents theoretical foundations of main methods of solving rational expectations models with a special focus on perturbation approaches. First, it gives some insights into the solution methods for linear models. Second, it shows how to use the perturbation approach for solving nonlinear models. It then documents the limits of this approach. The perturbation approach, though the most common solution method in the macroeconomic literature, is inappropriate in contexts of large fluctuations (large shocks or regime switching) and of strong nonlinearities (e.g., occasionally binding constraints). The former case is illustrated by regime switching models. The latter case is illustrated by a study of existing methods for solving rational expectations models under the zero lower bound constraint, that is, the condition of non-negativity of the nominal interest rate. The chapter concludes with a presentation of global methods available when the perturbation approach fails in solving models.

Abstract:

Abstract This paper describes a generalized nonlinear model of active magnetic bearings, for arbitrary number, size and position of pole-pairs, including the coordinate coupling between the vertical and horizontal directions, the nonlinearility of the power amplifier and the effect of finite permeability of the ferromagnetic bearing material. As an example, a practical magnetic bearing with four pole-pairs is examined. The predicted force from the model is quite consistent with experimental results. For the PD controller implemented for a horizontal rotor, the expectation of the equilibrium of the rotor also agrees quite well with the measurement.

Abstract:

While addressing different load cases according to the IEC guidelines for offshore wind turbines, designers are required to estimate long-term extreme and fatigue loads; this is usually done by carrying out time-domain stochastic turbine response simulations. This involves simulation of the stochastic inflow wind field on the rotor plane, of irregular (random) waves on the support structure, and of the turbine response. Obtaining realistic response of the turbine depends, among other factors, on appropriate modeling of the incident wind and waves. The current practice for modeling waves on offshore wind turbines is limited to the representation of linear irregular waves. While such models are appropriate for deep waters, they are not accurate representations of waves in shallow waters where offshore wind turbines are most commonly sited. In shallow waters, waves are generally nonlinear in nature. It is, therefore, of interest to assess the influence of alternative wave models on the behavior of wind turbines (e.g., on the tower response) as well as on extrapolated long-term turbine loads. The expectation is that nonlinear (second-order) irregular waves can better describe waves in shallow waters. In this study, we investigate differences in turbine response statistics and in long-term load predictions that arise from the use of alternative wave models. We compute loads on the monopile support structure of a 5MW offshore wind turbine model for several representative environmental states where we focus on differences in estimates of the extreme tower bending moment at the mudline due to linear and nonlinear waves. Finally, we compare long-term load predictions using inverse reliability procedures with both the linear and nonlinear wave models. We present convergence criteria that may be used to establish accurate 20-year loads and discuss comparative influences of wind versus waves in long-term load prediction.

Abstract:

Abstract Precision and productivity are very important criteria for the evaluation of modular tool systems and require a thermally stable process with tolerances in the micrometer range. During the past decades there has been an increasing interest in compensating thermally induced errors. In this paper we investigate wheather a prediction of thermal displacement based on a nonlinear regression analysis is possible, namely using the alternating conditional expectation algorithm (ACE) introduced by Breiman and Friedman, 1985. The data we are analyzing were generated by two different finite element spindle models of modular tool systems. As the main result we find that the ACE-algorithm is a powerful tool to model the relation between temperatures and displacements. It could also be a promising approach to handle well-known hysteresis effects. Limitations of this study are the model restricted results, next our findings have to be validated on real data.