A mathematical framework for new fault detection schemes in nonlinear stochastic continuous-time dynamical systems

n this work, a mathematical unifying framework for designing new fault detection schemes in nonlinear stochastic continuous-time dynamical systems is developed. These schemes are based on a stochastic process, called the residual, which reflects the system behavior and whose changes are to be detected. A quickest detection scheme for the residual is proposed, which is based on the computed likelihood ratios for time-varying statistical changes in the Ornstein–Uhlenbeck process. Several expressions are provided, depending on a priori knowledge of the fault, which can be employed in a proposed CUSUM-type approximated scheme. This general setting gathers different existing fault detection schemes within a unifying framework, and allows for the definition of new ones. A comparative simulation example illustrates the behavior of the proposed schemes.

Approximate Bayesian techniques for inference in stochastic dynamical systems

This thesis is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variant of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here two new extended frameworks are derived and presented that are based on basis function expansions and local polynomial approximations of a recently proposed variational Bayesian algorithm. It is shown that the new extensions converge to the original variational algorithm and can be used for state estimation (smoothing). However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new methods are numerically validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, for which the exact likelihood can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz '63 (3-dimensional model). The algorithms are also applied to the 40 dimensional stochastic Lorenz '96 system. In this investigation these new approaches are compared with a variety of other well known methods such as the ensemble Kalman filter / smoother, a hybrid Monte Carlo sampler, the dual unscented Kalman filter (for jointly estimating the systems states and model parameters) and full weak-constraint 4D-Var. Empirical analysis of their asymptotic behaviour as a function of observation density or length of time window increases is provided.

Estimating parameters in stochastic systems:a variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein–Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.

Strong Consistency of the Conditional Least Squares Estimator for a Nonstationary Process. Example of the Garch Model

2000 Mathematics Subject Classification: Primary: 62M10, 62J02, 62F12, 62M05, 62P05, 62P10; secondary: 60G46, 60F15.

A Stochastic Control Approach to a Parabolic Equation, Reciprocal Processes

2000 Mathematics Subject Classi cation: 49L60, 60J60, 93E20.

A Generalized Quasi-Likelihood Estimator for Nonstationary Stochastic Processes−Asymptotic Properties and Examples

2010 Mathematics Subject Classification: 62F12, 62M05, 62M09, 62M10, 60G42.

Role of noise in population dynamics cycles

Noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase-forgetting quasicycles by converting damped into sustained oscillations. This function of noise becomes evident when considering Langevin equations whose deterministic part yields only damped oscillations. We formulate here a consistent and systematic approach to population dynamics, leading to a Fokker-Planck equation and the associate Langevin equations in accordance with this conceptual framework, founded on stochastic lattice-gas models that describe spatially structured predator-prey systems. Langevin equations in the population densities and predator-prey pair density are derived in two stages. First, a birth-and-death stochastic process in the space of prey and predator numbers and predator-prey pair number is obtained by a contraction method that reduces the degrees of freedom. Second, a van Kampen expansion in the inverse of system size is then performed to get the Fokker-Planck equation. We also study the time correlation function, the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence.

A hybrid commodity price-forecasting model applied to the sugar-alcohol sector

Accurate price forecasting for agricultural commodities can have significant decision-making implications for suppliers, especially those of biofuels, where the agriculture and energy sectors intersect. Environmental pressures and high oil prices affect demand for biofuels and have reignited the discussion about effects on food prices. Suppliers in the sugar-alcohol sector need to decide the ideal proportion of ethanol and sugar to optimise their financial strategy. Prices can be affected by exogenous factors, such as exchange rates and interest rates, as well as non-observable variables like the convenience yield, which is related to supply shortages. The literature generally uses two approaches: artificial neural networks (ANNs), which are recognised as being in the forefront of exogenous-variable analysis, and stochastic models such as the Kalman filter, which is able to account for non-observable variables. This article proposes a hybrid model for forecasting the prices of agricultural commodities that is built upon both approaches and is applied to forecast the price of sugar. The Kalman filter considers the structure of the stochastic process that describes the evolution of prices. Neural networks allow variables that can impact asset prices in an indirect, nonlinear way, what cannot be incorporated easily into traditional econometric models.

A framework for valuing derivative securities

This paper develops a general framework for valuing a wide range of derivative securities. Rather than focusing on the stochastic process of the underlying security and developing an instantaneously-riskless hedge portfolio, we focus on the terminal distribution of the underlying security. This enables the derivative security to be valued as the weighted sum of a number of component pieces. The component pieces are simply the different payoffs that the security generates in different states of the world, and they are weighted by the probability of the particular state of the world occurring. A full set of derivations is provided. To illustrate its use, the valuation framework is applied to plain-vanilla call and put options, as well as a range of derivatives including caps, floors, collars, supershares, and digital options.

Testing the relationship between morphological and molecular rates of change along phylogenies

Molecular evolution has been considered to be essentially a stochastic process, little influenced by the pace of phenotypic change. This assumption was challenged by a study that demonstrated an association between rates of morphological and molecular change estimated for total-evidence phylogenies, a finding that led some researchers to challenge molecular date estimates of major evolutionary radiations. Here we show that Omland's (1997) result is probably due to methodological bias, particularly phylogenetic nonindependence, rather than being indicative of an underlying evolutionary phenomenon. We apply three new methods specifically designed to overcome phylogenetic bias to 13 published phylogenetic datasets for vertebrate taxa, each of which includes both morphological characters and DNA sequence data. We find no evidence of an association between rates of molecular and morphological rates of change.

On the relation between the fractional Brownian motion and the fractional derivatives

Physics Letters A, vol. 372; Issue 7

Valuing an offshore exploration project through real options analysis

This thesis applied real options analysis to the valuation of an offshore oil exploration project, taking into consideration the several options typically faced by the management team of these projects. The real options process is developed under technical and price uncertainties, where it is considered that the mean reversion stochastic process is more adequate to describe the movement of oil price throught time. The valuation is realized to two case scenarios, being the first a simplified approach to develop the intuition of the used concepts, and the later a more complete cases that is resolved using both the binomial and trinomial processes to describe oil price movement. Real options methodology demonstrated to be capable of assessing and valuing the projects options, and of overcoming common capital budgeting methodologies flexibility limitation. The added value of the application of real options is evident, but so is the method's increased complexity, which adversely influence its widespread implementation.

Real-time monitoring of progression towards renal failure in primary care patients

First published online: December 16, 2014.

Adaptive numerical simulation of reaction - diffusion systems

Fluidized beds, granulation, heat and mass transfer, calcium dynamics, stochastic process, finite element methods, Rosenbrock methods, multigrid methods, parallelization

Consumers' behavior and the Bertrand paradox: an ACE approach

We analyze the classical Bertrand model when consumers exhibit some strategic behavior in deciding from which seller they will buy. We use two related but different tools. Both consider a probabilistic learning (or evolutionary) mechanism, and in the two of them consumers' behavior in uences the competition between the sellers. The results obtained show that, in general, developing some sort of loyalty is a good strategy for the buyers as it works in their best interest. First, we consider a learning procedure described by a deterministic dynamic system and, using strong simplifying assumptions, we can produce a description of the process behavior. Second, we use nite automata to represent the strategies played by the agents and an adaptive process based on genetic algorithms to simulate the stochastic process of learning. By doing so we can relax some of the strong assumptions used in the rst approach and still obtain the same basic results. It is suggested that the limitations of the rst approach (analytical) provide a good motivation for the second approach (Agent-Based). Indeed, although both approaches address the same problem, the use of Agent-Based computational techniques allows us to relax hypothesis and overcome the limitations of the analytical approach.