Fitting Jump Models

We describe a new framework for fitting jump models to a sequence of data. The key idea is to alternate between minimizing a loss function to fit multiple model parameters, and minimizing a discrete loss function to determine which set of model parameters is active at each data point. The framework is quite general and encompasses popular classes of models, such as hidden Markov models and piecewise affine models. The shape of the chosen loss functions to minimize determine the shape of the resulting jump model.Comment: Accepted for publication in Automatic

Time Consistent Policy in Markov Switching Models

In this paper we consider the quadratic optimal control problem with regime shifts and forward-looking agents. This extends the results of Zampolli (2003) who considered models without forward-looking expectations. Two algorithms are presented: The first algorithm computes the solution of a rational expectation model with random parameters or regime shifts. The second algorithm computes the time-consistent policy and the resulting Nash-Stackelberg equilibrium. The formulation of the problem is of general form and allows for model uncertainty and incorporation of policymaker’s judgement. We apply these methods to compute the optimal (non-linear) monetary policy in a small open economy subject to (symmetric or asymmetric) risks of change in some of its key parameters such as inflation inertia, degree of exchange rate pass-through, elasticity of aggregate demand to interest rate, etc.. We normally find that the time-consistent response to risk is more cautious. Furthermore, the optimal response is in some cases non-monotonic as a function of uncertainty. We also simulate the model under assumptions that the policymaker and the private sector hold the same beliefs over the probabilities of the structural change and different beliefs (as well as different assumptions about the knowledge of each other’s reaction function).monetary policy, regime switching, model uncertainty, time consistency

A Decomposition Approach to Multi-Agent Systems with Bernoulli Packet Loss

In this paper, we extend the decomposable systems framework to multi-agent systems with Bernoulli distributed packet loss with uniform probability. The proposed sufficient analysis conditions for mean-square stability and $H_2$-performance - which are expressed in the form of linear matrix inequalities - scale linearly with increased network size and thus allow to analyse even very large-scale multi-agent systems. A numerical example demonstrates the potential of the approach by application to a first-order consensus problem.Comment: 11 pages, 4 figure

Analysis of Embedded Controllers Subject to Computational Overruns

Microcontrollers have become an integral part of modern everyday embedded systems, such as smart bikes, cars, and drones. Typically, microcontrollers operate under real-time constraints, which require the timely execution of programs on the resource-constrained hardware. As embedded systems are becoming increasingly more complex, microcontrollers run the risk of violating their timing constraints, i.e., overrunning the program deadlines. Breaking these constraints can cause severe damage to both the embedded system and the humans interacting with the device. Therefore, it is crucial to analyse embedded systems properly to ensure that they do not pose any significant danger if the microcontroller overruns a few deadlines.However, there are very few tools available for assessing the safety and performance of embedded control systems when considering the implementation of the microcontroller. This thesis aims to fill this gap in the literature by presenting five papers on the analysis of embedded controllers subject to computational overruns. Details about the real-time operating system's implementation are included into the analysis, such as what happens to the controller's internal state representation when the timing constraints are violated. The contribution includes theoretical and computational tools for analysing the embedded system's stability, performance, and real-time properties.The embedded controller is analysed under three different types of timing violations: blackout events (when no control computation is completed during long periods), weakly-hard constraints (when the number of deadline overruns is constrained over a window), and stochastic overruns (when violations of timing constraints are governed by a probabilistic process). These scenarios are combined with different implementation policies to reduce the gap between the analysis and its practical applicability. The analyses are further validated with a comprehensive experimental campaign performed on both a set of physical processes and multiple simulations.In conclusion, the findings of this thesis reveal that the effect deadline overruns have on the embedded system heavily depends the implementation details and the system's dynamics. Additionally, the stability analysis of embedded controllers subject to deadline overruns is typically conservative, implying that additional insights can be gained by also analysing the system's performance

Large-scale games in large-scale systems

Many real-world problems modeled by stochastic games have huge state and/or action spaces, leading to the well-known curse of dimensionality. The complexity of the analysis of large-scale systems is dramatically reduced by exploiting mean field limit and dynamical system viewpoints. Under regularity assumptions and specific time-scaling techniques, the evolution of the mean field limit can be expressed in terms of deterministic or stochastic equation or inclusion (difference or differential). In this paper, we overview recent advances of large-scale games in large-scale systems. We focus in particular on population games, stochastic population games and mean field stochastic games. Considering long-term payoffs, we characterize the mean field systems using Bellman and Kolmogorov forward equations.Comment: 30 pages. Notes for the tutorial course on mean field stochastic games, March 201

Robust Multi-Object Tracking: A Labeled Random Finite Set Approach

The labeled random finite set based generalized multi-Bernoulli filter is a tractable analytic solution for the multi-object tracking problem. The robustness of this filter is dependent on certain knowledge regarding the multi-object system being available to the filter. This dissertation presents techniques for robust tracking, constructed upon the labeled random finite set framework, where complete information regarding the system is unavailable

Applications of hidden Markov models in financial modelling

This thesis was submitted for the degree of Doctor of Philosophy and was awarded by Brunel University.Various models driven by a hidden Markov chain in discrete or continuous time are developed to capture the stylised features of market variables whose levels or values constitute as the underliers of financial derivative contracts or investment portfolios. Since the parameters are switching regimes, the changes and developments in the economy as soon as they arise are readily reflected in these models. The change of probability measure technique and the EM algorithm are fundamental techniques utilised in the optimal parameter estimation. Recursive adaptive filters for the state of the Markov chain and other auxiliary processes related to the Markov chain are derived which in turn yield self-tuning dynamic financial models. A hidden Markov model (HMM)-based modelling set-up for commodity prices is developed and the predictability of the gold market under this setting is examined. An Ornstein-Uhlenbeck (OU) model with HMM parameters is proposed and under this set-up, we address two statistical inference issues: the sensitivity of the model to small changes in parameter estimates and the selection of the optimal number of states. The extended OU model is implemented on a data set of 30-day Canadian T-bill yields. An exponential of a Markov-switching OU process plus a compound Poisson process is put forward as a model for the evolution of electricity spot prices. Using a data set compiled by Nord Pool, we illustrate the vast improvements gained in incorporating regimes in the model. A multivariate HMM is employed as a framework in providing the solutions of two asset allocation problems; one involves the mean-variance utility function and the other entails the CVaR constraint. Finally, the valuation of credit default swaps highlights the important considerations necessitated by pricing in a regime-switching environment. Certain numerical schemes are applied to obtain approximations for the default probabilities and swap rates.Brunel Research Initiative and Enterprise Fund (BRIEF) and European Union (Marie Curie Fellowship

Statistical modelling and analysis of traffic: a dynamic approach

In both developed and emerging-economies, major cities continue to experience increasing traffic congestion. To address this issue, complex Traffic Management Systems (TMS) are employed in recent years to help manage traffic. These systems fuse traffic-surveillance-related information from a variety of sensors deployed across traffic networks. A TMS requires real-time information to make effective control decisions and to deliver trustworthy information to users, such as travel time, congestion level, etc. There are three fundamental inputs required by TMS, namely, traffic volume, vehicular speed, and traffic density. Using conventional traffic loop detectors one can directly measure flow and velocity. However, traffic density is more difficult to measure. The situation becomes more difficult for multi-lane motorways due to drivers lane-change behaviour. This research investigates statistical modelling and analysis of traffic flow. It contributes to the literature of transportation and traffic management and research in several aspects. First, it takes into account lane-changes in traffic modelling through incorporating a Markov chain model to describe the drivers lane-change behaviour. Secondly, the lane change probabilities between two adjacent lanes are not assumed to be fixed but rather they depend on the current traffic condition. A discrete choice model is used to capture drivers lane choice behaviour. The drivers choice probabilities are modelled by several traffic-condition related attributes such as vehicle time headway, traffic density and speed. This results in a highly nonlinear state equation for traffic density. To address the issue of high nonlinearity of the state space model, the EKF and UKF is used to estimate the traffic density recursively. In addition, a new transformation approach has been proposed to transform the observation equation from a nonlinear form to a linear one so that the potential approximation in the EKF & UKF can be avoided. Numerical studies have been conducted to investigate the performance of the developed method. The proposed method outperformed the existing methods for traffic density estimation in simulation studies. Furthermore, it is shown that the computational cost for updating the estimate of traffic densities for a multi-lane motorway is kept at a minimum so that online applications are feasible in practice. Consequently the traffic densities can be monitored and the relevant information can be fed into the traffic management system of interest