Quantified real constraint solving using modal intervals with applications to control

Les restriccions reals quantificades (QRC) formen un formalisme matemàtic utilitzat per modelar un gran nombre de problemes físics dins els quals intervenen sistemes d'equacions no-lineals sobre variables reals, algunes de les quals podent ésser quantificades. Els QRCs apareixen en nombrosos contextos, com l'Enginyeria de Control o la Biologia. La resolució de QRCs és un domini de recerca molt actiu dins el qual es proposen dos enfocaments diferents: l'eliminació simbòlica de quantificadors i els mètodes aproximatius. Tot i això, la resolució de problemes de grans dimensions i del cas general, resten encara problemes oberts. Aquesta tesi proposa una nova metodologia aproximativa basada en l'Anàlisi Intervalar Modal, una teoria matemàtica que permet resoldre problemes en els quals intervenen quantificadors lògics sobre variables reals. Finalment, dues aplicacions a l'Enginyeria de Control són presentades. La primera fa referència al problema de detecció de fallades i la segona consisteix en un controlador per a un vaixell a vela.

Bootstrap approaches for estimation and confidence intervals of long term memory processes

In this work, we investigate an alternative bootstrap approach based on a result of Ramsey [F.L. Ramsey, Characterization of the partial autocorrelation function, Ann. Statist. 2 (1974), pp. 1296-1301] and on the Durbin-Levinson algorithm to obtain a surrogate series from linear Gaussian processes with long range dependence. We compare this bootstrap method with other existing procedures in a wide Monte Carlo experiment by estimating, parametrically and semi-parametrically, the memory parameter d. We consider Gaussian and non-Gaussian processes to prove the robustness of the method to deviations from normality. The approach is also useful to estimate confidence intervals for the memory parameter d by improving the coverage level of the interval.

Maintenance optimisation of a multi-state series-parallel system considering economic dependence and state-dependent inspection intervals

This paper presents a maintenance optimisation method for a multi-state series-parallel system considering economic dependence and state-dependent inspection intervals. The objective function considered in the paper is the average revenue per unit time calculated based on the semi-regenerative theory and the universal generating function (UGF). A new algorithm using the stochastic ordering is also developed in this paper to reduce the search space of maintenance strategies and to enhance the efficiency of optimisation algorithms. A numerical simulation is presented in the study to evaluate the efficiency of the proposed maintenance strategy and optimisation algorithms. The simulation result reveals that maintenance strategies with opportunistic maintenance and state-dependent inspection intervals are more cost-effective when the influence of economic dependence and inspection cost is significant. The study further demonstrates that the optimisation algorithm proposed in this paper has higher computational efficiency than the commonly employed heuristic algorithms.

Critical timescales and time intervals for coupled linear processes

In 1991, McNabb introduced the concept of mean action time (MAT) as a finite measure of the time required for a diffusive process to effectively reach steady state. Although this concept was initially adopted by others within the Australian and New Zealand applied mathematics community, it appears to have had little use outside this region until very recently, when in 2010 Berezhkovskii and coworkers rediscovered the concept of MAT in their study of morphogen gradient formation. All previous work in this area has been limited to studying single–species differential equations, such as the linear advection–diffusion–reaction equation. Here we generalise the concept of MAT by showing how the theory can be applied to coupled linear processes. We begin by studying coupled ordinary differential equations and extend our approach to coupled partial differential equations. Our new results have broad applications including the analysis of models describing coupled chemical decay and cell differentiation processes, amongst others.

Beyond point forecasting : evaluation of alternative prediction intervals for tourist arrivals

This paper evaluates the performances of prediction intervals generated from alternative time series models, in the context of tourism forecasting. The forecasting methods considered include the autoregressive (AR) model, the AR model using the bias-corrected bootstrap, seasonal ARIMA models, innovations state space models for exponential smoothing, and Harvey’s structural time series models. We use thirteen monthly time series for the number of tourist arrivals to Hong Kong and Australia. The mean coverage rates and widths of the alternative prediction intervals are evaluated in an empirical setting. It is found that all models produce satisfactory prediction intervals, except for the autoregressive model. In particular, those based on the biascorrected bootstrap perform best in general, providing tight intervals with accurate coverage rates, especially when the forecast horizon is long.

Exact parametric confidence intervals for Bland-Altman limits of agreement

Purpose The previous literature on Bland-Altman analysis only describes approximate methods for calculating confidence intervals for 95% Limits of Agreement (LoAs). This paper describes exact methods for calculating such confidence intervals, based on the assumption that differences in measurement pairs are normally distributed. Methods Two basic situations are considered for calculating LoA confidence intervals: the first where LoAs are considered individually (i.e. using one-sided tolerance factors for a normal distribution); and the second, where LoAs are considered as a pair (i.e. using two-sided tolerance factors for a normal distribution). Equations underlying the calculation of exact confidence limits are briefly outlined. Results To assist in determining confidence intervals for LoAs (considered individually and as a pair) tables of coefficients have been included for degrees of freedom between 1 and 1000. Numerical examples, showing the use of the tables for calculating confidence limits for Bland-Altman LoAs, have been provided. Conclusions Exact confidence intervals for LoAs can differ considerably from Bland and Altman’s approximate method, especially for sample sizes that are not large. There are better, more precise methods for calculating confidence intervals for LoAs than Bland and Altman’s approximate method, although even an approximate calculation of confidence intervals for LoAs is likely to be better than none at all. Reporting confidence limits for LoAs considered as a pair is appropriate for most situations, however there may be circumstances where it is appropriate to report confidence limits for LoAs considered individually.

Methods for constructing uncertainty intervals for queries of Bayesian nets

In this paper the issue of finding uncertainty intervals for queries in a Bayesian Network is reconsidered. The investigation focuses on Bayesian Nets with discrete nodes and finite populations. An earlier asymptotic approach is compared with a simulation-based approach, together with further alternatives, one based on a single sample of the Bayesian Net of a particular finite population size, and another which uses expected population sizes together with exact probabilities. We conclude that a query of a Bayesian Net should be expressed as a probability embedded in an uncertainty interval. Based on an investigation of two Bayesian Net structures, the preferred method is the simulation method. However, both the single sample method and the expected sample size methods may be useful and are simpler to compute. Any method at all is more useful than none, when assessing a Bayesian Net under development, or when drawing conclusions from an ‘expert’ system.

An extravariation model for improving confidence intervals of population size estimates from removal data

We propose a new model for estimating the size of a population from successive catches taken during a removal experiment. The data from these experiments often have excessive variation, known as overdispersion, as compared with that predicted by the multinomial model. The new model allows catchability to vary randomly among samplings, which accounts for overdispersion. When the catchability is assumed to have a beta distribution, the likelihood function, which is refered to as beta-multinomial, is derived, and hence the maximum likelihood estimates can be evaluated. Simulations show that in the presence of extravariation in the data, the confidence intervals have been substantially underestimated in previous models (Leslie-DeLury, Moran) and that the new model provides more reliable confidence intervals. The performance of these methods was also demonstrated using two real data sets: one with overdispersion, from smallmouth bass (Micropterus dolomieu), and the other without overdispersion, from rat (Rattus rattus).

Asymptotic behaviour of point processes with Markov-dependent intervals

This splitting techniques for MARKOV chains developed by NUMMELIN (1978a) and ATHREYA and NEY (1978b) are used to derive an imbedded renewal process in WOLD's point process with MARKOV-correlated intervals. This leads to a simple proof of renewal theorems for such processes. In particular, a key renewal theorem is proved, from which analogues to both BLACKWELL's and BREIMAN's forms of the renewal theorem can be deduced.

Precise measurement of hyperfine intervals using avoided crossing of dressed states

We demonstrate a technique for precisely measuring hyperfine intervals in alkali atoms. The atoms form a three-level system in the presence of a strong control laser and a weak probe laser. The dressed states created by the control laser show significant linewidth reduction. We have developed a technique for Doppler-free spectroscopy that enables the separation between the dressed states to be measured with high accuracy even in room temperature atoms. The states go through an avoided crossing as the detuning of the control laser is changed from positive to negative. By studying the separation as a function of detuning, the center of the level-crossing diagram is determined with high precision, which yields the hyperfine interval. Using room temperature Rb vapor, we obtain a precision of 44 kHz. This is a significant improvement over the current precision of similar to1 MHz.