Dissertations / Theses on the topic 'Nonlinear Expectation'

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.

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.

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.

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.

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.

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.

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

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

Orientadores: Victor Hugo Lachos Dávila, Filidor Edilfonso Vilca Labra
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
Made available in DSpace on 2018-08-16T04:06:26Z (GMT). No. of bitstreams: 1 MedinaGaray_AldoWilliam_M.pdf: 1389516 bytes, checksum: 2763869ea52e11ede3c860714ea0e75e (MD5) Previous issue date: 2010
Resumo: Neste trabalho estudamos alguns aspectos de estimação e diagnóstico de influência global e local de modelos não lineares sob a classe de distribuição misturas da escala skew-normal, baseado na metodologia proposta por Cook (1986) e Poon & Poon (1999). Os modelos não lineares heteroscedásticos também são discutidos. Esta nova classe de modelos constitui uma generalização robusta dos modelos de regressão não linear simétricos, que têm como membros particulares distribuições com caudas pesadas, tais como skew-t, skew-slash, skew-normal contaminada, entre outras. A estimação dos parâmetros será obtida via o algoritmo EM proposto por Dempster et al. (1977). Estudos de testes de hipóteses são considerados utilizando as estatísticas de escore e da razão de verossimilhança, para testar a homogeneidade do parâmetro de escala. Propriedades das estatísticas do teste são investigadas através de simulações de Monte Carlo. Exemplos numéricos considerando dados reais e simulados são apresentados para ilustrar a metodologia desenvolvida
Abstrac: In this work, we studied some aspects of estimation and diagnostics on the global and local influence in nonlinear models under the class of scale mixtures of the skewnormal (SMSN) distribution, based on the methodology proposed by Cook (1986) e Poon & Poon (1999). Heteroscedastic nonlinear models are also discussed. This new class of models are a robust generalization of non-linear regression symmetrical models, which have as members individual distributions with heavy tails, such as skew-t, skew-slash, and skew-contaminated normal, among others. The parameter estimation will be obtained with the EM algorithm proposed by Dempster et al. (1977). Studies testing hypotheses are considered using the score statistics and the likelihood ratio test to test the homogeneity of scale parameter. Properties of test statistics are investigated through Monte Carlo simulations. Numerical examples considering real and simulated data are presented to illustrate the methodology
Mestrado
Métodos Estatísticos
Mestre em Estatística

Depuis la publication de l'ouvrage de Choquet (1955), la théorie d'espérance non linéaire a attiré avec grand intérêt des chercheurs pour ses applications potentielles dans les problèmes d'incertitude, les mesures de risque et le super-hedging en finance. Shige Peng a construit une sorte d'espérance entièrement non linéaire dynamiquement cohérente par l'approche des EDP. Un cas important d'espérance non linéaire cohérente en temps est la G-espérance, dans laquelle le processus canonique correspondant (B_{t})_{t≥0} est appelé G-mouvement brownien et joue un rôle analogue au processus de Wiener classique. L'objectif de cette thèse est d'étudier, dans le cadre de la G-espérance, certaines équations différentielles stochastiques rétrogrades (G-EDSR) à croissance quadratique avec applications aux problèmes de maximisation d'utilité robuste avec incertitude sur les modèles, certaines équations différentielles stochastiques (G-EDS) réfléchies et équations différentielles stochastiques rétrogrades réfléchies avec générateurs lipschitziens. On considère d'abord des G-EDSRs à croissance quadratique. Dans le Chapitre 2 nous fournissons un resultat d'existence et unicité pour des G-EDSRs à croissance quadratique. D'une part, nous établissons des estimations a priori en appliquant le théorème de type Girsanov, d'où l'on en déduit l'unicité. D'autre part, pour prouver l'existence de solutions, nous avons d'abord construit des solutions pour des G-EDSRs discretes en résolvant des EDPs non-linéaires correspondantes, puis des solutions pour les G-EDSRs quadratiques générales dans les espaces de Banach. Dans le Chapitre 3 nous appliquons les G-EDSRs quadratiques aux problèmes de maximisation d'utilité robuste. Nous donnons une caratérisation de la fonction valeur et une stratégie optimale pour les fonctions d'utilité exponentielle, puissance et logarithmique. Dans le Chapitre 4, nous traitons des G-EDSs réfléchies multidimensionnelles. Nous examinons d'abord la méthode de pénalisation pour résoudre des problèmes de Skorokhod déterministes dans des domaines non convexes et établissons des estimations pour des fonctions α-Hölder continues. A l'aide de ces résultats obtenus pour des problèmes déterministes, nous définissons le G-mouvement Brownien réfléchi et prouvons son existence et son unicité dans un espace de Banach. Ensuite, nous prouvons l'existence et l'unicité de solution pour les G-EDSRs multidimensionnelles réfléchies via un argument de point fixe. Dans le Chapitre 5, nous étudions l'existence et l'unicité pour les équations différentielles stochastiques rétrogrades réfléchies dirigées par un G-mouvement brownien lorsque la barrière S est un processus de G-Itô
Since the publication of Choquet's (1955) book, the theory of nonlinear expectation has attracted great interest from researchers for its potential applications in uncertainty problems, risk measures and super-hedging in finance. Shige Peng has constructed a kind of fully nonlinear expectation dynamically coherent by the PDE approach. An important case of time-consistent nonlinear expectation is G-expectation, in which the corresponding canonical process (B_{t})_{t≥0} is called G-Brownian motion and plays a similar role to the classical Wiener process. The objective of this thesis is to study, in the framework of the G-expectation, some backward stochastic differential equations (G-BSDE) under a quadratic growth condition on their coefficients with applications to robust utility maximization problems with uncertainty on models, Reflected stochastic differential equations (reflected G-SDE) and reflected backward stochastic differential equations with Lipschitz coefficients (reflected G-BSDE). We first consider G-BSDE with quadratic growth. In Chapter 2 we provide a result of existence and uniqueness for quadratic G-BSDEs. On the one hand, we establish a priori estimates by applying the Girsanov-type theorem, from which we deduce the uniqueness. On the other hand, to prove the existence of solutions, we first constructed solutions for discrete G-BSDEs by solving corresponding nonlinear PDEs, then solutions for the general quadratic G-BSDEs in the spaces of Banach. In Chapter 3 we apply quadratic G-BSDE to robust utility maximization problems. We give a characterization of the value function and an optimal strategy for exponential, power and logarithmic utility functions. In Chapter 4, we discuss multidimensional reflected G-SDE. We first examine the penalization method to solve deterministic Skorokhod problems in non-convex domains and establish estimates for continuous α-Hölder functions. Using these results for deterministic problems, we define the reflected G-Brownian motion and prove its existence and its uniqueness in a Banach space. Then we prove the existence and uniqueness of the solution for the multidimensional reflected G-SDE via a fixed point argument. In Chapter 5, we study the existence and uniqueness of the reflected backward stochastic differential equations driven by a G-Brownian motion when the obstacle S is a G-Itô process

by Long Mei.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2005.
Includes bibliographical references (leaves 73-77).
Abstracts in English and Chinese.
Acknowledgements --- p.iv
Abstract --- p.v
Table of Contents --- p.vii
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Nonlinear Factor Analysis Model --- p.1
Chapter 1.2 --- Main Objectives --- p.2
Chapter 1.2.1 --- Investigation of the performance of the ML approach with MCECM algorithm in NFA model --- p.2
Chapter 1.2.2 --- Investigation of the Robustness of the ML approach with MCECM algorithm --- p.3
Chapter 1.3 --- Structure of the Thesis --- p.3
Chapter 2 --- Theoretical Background of the MCECM Algorithm --- p.5
Chapter 2.1 --- Introduction of the EM algorithm --- p.5
Chapter 2.2 --- Monte Carlo integration --- p.7
Chapter 2.3 --- Markov Chains --- p.7
Chapter 2.4 --- The Metropolis-Hastings algorithm --- p.8
Chapter 3 --- Maximum Likelihood Estimation of a Nonlinear Factor Analysis Model --- p.10
Chapter 3.1 --- MCECM Algorithm --- p.10
Chapter 3.1.1 --- Motivation of Using MCECM algorithm --- p.11
Chapter 3.1.2 --- Introduction of the Realization of the MCECM algorithm --- p.12
Chapter 3.1.3 --- Implementation of the E-step via the MH Algorithm --- p.13
Chapter 3.1.4 --- Maximization Step --- p.15
Chapter 3.2 --- Monitoring Convergence of MCECM --- p.17
Chapter 3.2.1 --- Bridge Sampling Method --- p.17
Chapter 3.2.2 --- Average Batch Mean Method --- p.18
Chapter 4 --- Simulation Studies --- p.20
Chapter 4.1 --- The First Simulation Study with the Normal Distribution --- p.20
Chapter 4.1.1 --- Model Specification --- p.20
Chapter 4.1.2 --- The Selection of System Parameters --- p.22
Chapter 4.1.3 --- Monitoring the Convergence --- p.22
Chapter 4.1.4 --- Simulation Results for the ML Estimates --- p.25
Chapter 4.2 --- The Second Simulation Study with the Normal Distribution --- p.34
Chapter 4.2.1 --- Model Specification --- p.34
Chapter 4.2.2 --- Monitoring the Convergence --- p.35
Chapter 4.2.3 --- Simulation Results for the ML Estimates --- p.38
Chapter 4.3 --- The Third Simulation Study on Robustness --- p.47
Chapter 4.3.1 --- Model Specification --- p.47
Chapter 4.3.2 --- Monitoring the Convergence --- p.48
Chapter 4.3.3 --- Simulation Results for the ML Estimates --- p.51
Chapter 4.4 --- The Fourth Simulation Study on Robustness --- p.59
Chapter 4.4.1 --- Model Specification --- p.59
Chapter 4.4.2 --- Monitoring the Convergence --- p.59
Chapter 4.4.3 --- Simulation Results for the ML Estimates --- p.62
Chapter 5 --- Conclusion --- p.71
Bibliography --- p.73

In this thesis appropriate statistical methods to overcome two types of problems that occur during parameter estimation in chemical engineering systems are studied. The first problem is having too many parameters to estimate from limited available data, assuming that the model structure is correct, while the second problem involves estimating unmeasured disturbances, assuming that enough data are available for parameter estimation. In the first part of this thesis, a model is developed to predict rates of undesirable reactions during the finishing stage of nylon 66 production. This model has too many parameters to estimate (56 unknown parameters) and not having enough data to reliably estimating all of the parameters. Statistical techniques are used to determine that 43 of 56 parameters should be estimated. The proposed model matches the data well. In the second part of this thesis, techniques are proposed for estimating parameters in Stochastic Differential Equations (SDEs). SDEs are fundamental dynamic models that take into account process disturbances and model mismatch. Three new approximate maximum likelihood methods are developed for estimating parameters in SDE models. First, an Approximate Expectation Maximization (AEM) algorithm is developed for estimating model parameters and process disturbance intensities when measurement noise variance is known. Then, a Fully-Laplace Approximation Expectation Maximization (FLAEM) algorithm is proposed for simultaneous estimation of model parameters, process disturbance intensities and measurement noise variances in nonlinear SDEs. Finally, a Laplace Approximation Maximum Likelihood Estimation (LAMLE) algorithm is developed for estimating measurement noise variances along with model parameters and disturbance intensities in nonlinear SDEs. The effectiveness of the proposed algorithms is compared with a maximum-likelihood based method. For the CSTR examples studied, the proposed algorithms provide more accurate estimates for the parameters. Additionally, it is shown that the performance of LAMLE is superior to the performance of FLAEM. SDE models and associated parameter estimates obtained using the proposed techniques will help engineers who implement on-line state estimation and process monitoring schemes.
Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2013-12-23 15:12:35.738

Programme in Advanced Mathematics of Finance, University of the Witwatersrand, Johannesburg.
Peng introduced a typical ltration consistent nonlinear expectation, called a g-expectation in [40]. It satis es all properties of the classical mathematical expectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic di erential equation (BSDE) within the classical framework of It^o's calculus, with terminal condition given at some xed time T. In addition, this g-expectation is uniquely speci ed by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the speci cation of this function. Martingales, super- and submartingales have been de ned in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical nancial world have also been of great interest. g-Expectations have been applied to the pricing of contingent claims in the nancial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any nancial position. They also give an indication as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cash ows, we want to work with a multi-period, dynamic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. The Choquet expectation, introduced by Gustave Choquet, is another nonlinear expectation. An interesting question which is examined, is whether there are incidences when the g-expectation and the Choquet expectation coincide.