Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction

Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question---correlation, predictability, predictive cost, observer synchronization, and the like---induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on. Tractably analyzing these dynamics relies on adapting the recently introduced meromorphic functional calculus, which specifies the spectral decomposition of functions of nondiagonalizable linear operators, even when the function poles and zeros coincide with the operator's spectrum. Along the way, we establish special properties of the projection operators that demonstrate how they capture the organization of subprocesses within a complex system. Circumventing the spurious infinities of alternative calculi, this leads in the sequel, Part II, to the first closed-form expressions for complexity measures, couched either in terms of the Drazin inverse (negative-one power of a singular operator) or the eigenvalues and projection operators of the appropriate transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht

Context-sensitive planning for autonomous vehicles

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (leaves 59-64).by David Vengerov.M.S

Identifying the recurrence of sleep apnea using a harmonic hidden Markov model

We propose to model time-varying periodic and oscillatory processes by means of a hidden Markov model where the states are defined through the spectral properties of a periodic regime. The number of states is unknown along with the relevant periodicities, the role and number of which may vary across states. We address this inference problem by a Bayesian nonparametric hidden Markov model assuming a sticky hierarchical Dirichlet process for the switching dynamics between different states while the periodicities characterizing each state are explored by means of a trans-dimensional Markov chain Monte Carlo sampling step. We develop the full Bayesian inference algorithm and illustrate the use of our proposed methodology for different simulation studies as well as an application related to respiratory research which focuses on the detection of apnea instances in human breathing traces

How incidents impact congestion on roadways: A queuing network approach

Motivated by the need for transportation infrastructure and incident management planning, we study traffic density under non-recurrent congestion. This paper provides an analytical solution approximating the stationary distribution of traffic density in roadways where deterioration of service occurs unpredictably. The proposed solution generalizes a queuing model discussed in the literature to long segments that are not space-homogeneous. We compare single and tandem queuing approaches to segments of different lengths and verify whether each model is appropriate. A single-queue approach works sufficiently well in segments with similar traffic behavior across space. In contrast, a tandem-queue approach more appropriately describes the density behavior for long segments with sections having distinct traffic characteristics. These models have a comparable fit to the ones generated using a lognormal distribution. However, they also have interpretable parameters, directly connecting the distribution of congestion to the dynamics of roadway behavior. The proposed models are general, adaptable, and tractable, thus being instrumental in infrastructure and incident management

Bayesian analysis of nonstationary periodic time series

Identifying the periodicities present in a cyclical process allows us to gain knowledge about the sources of variability that drive that phenomenon. For instance, respiratory traces obtained from a plethysmograph used on rodents in experimental sleep apnea research reveal many sudden changes in their periodic features as the rat spontaneously changes its breathing pattern during its sleep-wake activities. Similarly, human temperature, as measured by a wearable sensing device over several days at relatively high temporal resolution (e.g. 5 minutes), may be subject to a different periodic behaviour during the night when the individual transitions between different stages of sleep. While the theory and methods for analyzing the periodicities of time series data are relatively well-developed for the case of stationary time series, the task of modelling time series that undergo regime shifts in periodicity, amplitude and phase remains challenging because the timing of the changes and the relevant periodicities are usually unknown (both in value and number). This thesis introduces new methodologies for the automated analysis of non-stationary periodic time series. In the first part of this research, we present a novel Bayesian approach for analyzing time series data that exhibit regime shifts in periodicity, amplitude and phase, where we approximate the time series using a piece-wise oscillatory model with unknown periodicities, and our goal is to estimate the change-points while simultaneously identifying the changing periodicities in the data. Our proposed methodology is based on a trans-dimensional Markov chain Monte Carlo (MCMC) algorithm that simultaneously updates the change-points and the periodicities relevant to any segment between them. We show that the proposed methodology successfully identifies time changing oscillatory behaviour in two applications which are relevant to e-Health and sleep research, namely the occurrence of ultradian oscillations in human skin temperature during the time of night rest, and the characterization of instances of sleep apnea in plethysmographic respiratory traces. In addition to detecting temporal changes, it may also be of interest to recognize the recurrence of a relevant periodic pattern. In the second half of this thesis, we consider periodic phenomena, whose behaviour switches over time, as realizations of a hidden Markov model where the number of states is unknown along with the relevant periodicities, the role of which varies over the different states. Flexibility on the number of states is achieved by using Bayesian nonparametric techniques that address the stochastic switching dynamics of the time series via a hierarchical Dirichlet process that captures the temporal mode persistence of the hidden states. The variable dimensionality regarding the number of periodicities that characterizes the different regimes is addressed by developing an appropriate trans-dimensional MCMC sampler. We illustrate the use of our proposed approach in a case study relevant to respiratory research, namely the detection of recurring instances of sleep apnea in human respiratory traces

Interactions between gaussian processes and bayesian estimation

L’apprentissage (machine) de modèle et l’estimation d’état sont cruciaux pour interpréter les phénomènes sous-jacents à de nombreuses applications du monde réel. Toutefois, il est souvent difficile d’apprendre le modèle d’un système et de capturer les états latents, efficacement et avec précision, en raison du fait que la connaissance du monde est généralement incertaine. Au cours des dernières années, les approches d’estimation et de modélisation bayésiennes ont été extensivement étudiées afin que l’incertain soit réduit élégamment et de manière flexible. Dans la pratique cependant, différentes limitations au niveau de la modélisation et de l’estimation bayésiennes peuvent détériorer le pouvoir d’interprétation bayésienne. Ainsi, la performance de l’estimation est souvent limitée lorsque le modèle de système manque de souplesse ou/et est partiellement inconnu. De même, la performance de la modélisation est souvent restreinte lorsque l’estimateur Bayésien est inefficace. Inspiré par ces faits, nous proposons d’étudier dans cette thèse, les connections possibles entre modélisation bayésienne (via le processus gaussien) et l’estimation bayésienne (via le filtre de Kalman et les méthodes de Monte Carlo) et comment on pourrait améliorer l’une en utilisant l’autre. À cet effet, nous avons d’abord vu de plus près comment utiliser les processus gaussiens pour l’estimation bayésienne. Dans ce contexte, nous avons utilisé le processus gaussien comme un prior non-paramétrique des modèles et nous avons montré comment cela permettait d’améliorer l’efficacité et la précision de l’estimation bayésienne. Ensuite, nous nous somme intéressé au fait de savoir comment utiliser l’estimation bayésienne pour le processus gaussien. Dans ce cadre, nous avons utilisé différentes estimations bayésiennes comme le filtre de Kalman et les filtres particulaires en vue d’améliorer l’inférence au niveau du processus gaussien. Ceci nous a aussi permis de capturer différentes propriétés au niveau des données d’entrée. Finalement, on s’est intéressé aux interactions dynamiques entre estimation bayésienne et processus gaussien. On s’est en particulier penché sur comment l’estimation bayésienne et le processus gaussien peuvent ”travailler” de manière interactive et complémentaire de façon à améliorer à la fois le modèle et l’estimation. L’efficacité de nos approches, qui contribuent à la fois au processus gaussien et à l’estimation bayésienne, est montrée au travers d’une analyse mathématique rigoureuse et validée au moyen de différentes expérimentations reflétant des applications réelles.Model learning and state estimation are crucial to interpret the underlying phenomena in many real-world applications. However, it is often challenging to learn the system model and capture the latent states accurately and efficiently due to the fact that the knowledge of the world is highly uncertain. During the past years, Bayesian modeling and estimation approaches have been significantly investigated so that the uncertainty can be elegantly reduced in a flexible probabilistic manner. In practice, however, several drawbacks in both Bayesian modeling and estimation approaches deteriorate the power of Bayesian interpretation. On one hand, the estimation performance is often limited when the system model lacks in flexibility and/or is partially unknown. On the other hand, the modeling performance is often restricted when a Bayesian estimator is not efficient and/or accurate. Inspired by these facts, we propose Interactions Between Gaussian Processes and Bayesian Estimation where we investigate the novel connections between Bayesian model (Gaussian processes) and Bayesian estimator (Kalman filter and Monte Carlo methods) in different directions to address a number of potential difficulties in modeling and estimation tasks. Concretely, we first pay our attention to Gaussian Processes for Bayesian Estimation where a Gaussian process (GP) is used as an expressive nonparametric prior for system models to improve the accuracy and efficiency of Bayesian estimation. Then, we work on Bayesian Estimation for Gaussian Processes where a number of Bayesian estimation approaches, especially Kalman filter and particle filters, are used to speed up the inference efficiency of GP and also capture the distinct input-dependent data properties. Finally, we investigate Dynamical Interaction Between Gaussian Processes and Bayesian Estimation where GP modeling and Bayesian estimation work in a dynamically interactive manner so that GP learner and Bayesian estimator are positively complementary to improve the performance of both modeling and estimation. Through a number of mathematical analysis and experimental demonstrations, we show the effectiveness of our approaches which contribute to both GP and Bayesian estimation