Emulation of Poincaré return maps with Gaussian Kriging models

In this paper we investigate the use of Gaussian emulators to give an accurate and computationally fast method to approximate return maps, a tool used to study the dynamics of differential equations. One advantage of emulators over other approximation techniques is that they encode deterministic data exactly, so where values of the return map are known these are also outputs of the emulator output, another is that emulators allow us to simultaneously emulate a parameterized family of ODEs giving a tool to assess the behavior of perturbed systems. The methods introduced here are illustrated using two well-known dynamical systems: The Rossler equations, and the Billiard system. We show that the method can be used to look at return maps and discuss the further implications for full computation of differential equation outputs

Estimation of Stress-Strength model in the Generalized Linear Failure Rate Distribution

In this paper, we study the estimation of $R=P [Y < X ]$, also so-called the stress-strength model, when both $X$ and $Y$ are two independent random variables with the generalized linear failure rate distributions, under different assumptions about their parameters. We address the maximum likelihood estimator (MLE) of $R$ and the associated asymptotic confidence interval. In addition, we compute the MLE and the corresponding Bootstrap confidence interval when the sample sizes are small. The Bayes estimates of $R$ and the associated credible intervals are also investigated. An extensive computer simulation is implemented to compare the performances of the proposed estimators. Eventually, we briefly study the estimation of this model when the data obtained from both distributions are progressively type-II censored. We present the MLE and the corresponding confidence interval under three different progressive censoring schemes. We also analysis a set of real data for illustrative purpose.Comment: 31 pages, 2 figures, preprin

Estimation in causal graphical models

Pearl (2000), Spirtes et al (1993) and Lauritzen (2001) set up a new framework to encode the causal relationships between the random variables by a causal Bayesian network. The estimation of the conditional probabilities in a Bayesian network has received considerable attention by several investigators (e. g., Jordan (1998), Geiger and Heckerman (1997), Ileckerman et al (1995)), but, this issue has not been studied in a causal Bayesian network. In this thesis, we define the multicausal essential graph on the equivalence class of Bayesian networks in which each member of this class manifests a sort of strong type of invariance under (causal) manipulation called hypercausality. We then characterise the families of prior distributions on the parameters of the Bayesian networks which are consistent with hypercausality and show that their unmanipulated uncertain Bayesian networks must demonstrate the independence assumptions. As a result, such prior distributions satisfy a generalisation of the Geiger and lieckerman condition. In particular, when the corresponding essential graph is undirected, the mentioned class of prior distributions will reduce to the Hyper-Dirichlet family (see Chapter 6). In tile second part of this thesis, we will calculate certain local sensitivity measures and through them we are able to provide the solutions for the following questions: Is the network structure that is learned from data robust with respect to changes of the directionality of some specific arrows? Is the local conditional distributions associated with the specified node robust with respect to the changes to its prior distribution or with respect to the changes to the local conditional distribution of another node? Most importantly, is the posterior distribution associated with the parameters of any node robust with respect to the changes to the prior distribution associated with the parameters of one specific node? Finally, are the quantities mentioned above robust with respect to the changes in the independence assumptions described in Chapter 3? Most of the local sensitivity measures (particularly, local measures of the overall posteriors sensitivity), developed in the last decade, tend to diverge to infinity as the sample size becomes very large (Gustafson (1994) and Gustafson et al (1996)). This is in contrast to our knowledge that, starting from different priors, posteriors tend to agree as the data accumulate. Here we define a now class of metrics with more satisfactory asymptotic behaviour. The advantage of the corresponding local sensitivity measures is boundedness for large sample size

Probabilistic modeling of flood characterizations with parametric and minimum information pair-copula model

This paper highlights the usefulness of the minimum information and parametric pair-copula construction (PCC) to model the joint distribution of flood event properties. Both of these models outperform other standard multivariate copula in modeling multivariate flood data that exhibiting complex patterns of dependence, particularly in the tails. In particular, the minimum information pair-copula model shows greater flexibility and produces better approximation of the joint probability density and corresponding measures have capability for effective hazard assessments. The study demonstrates that any multivariate density can be approximated to any degree of desired precision using minimum information pair-copula model and can be practically used for probabilistic flood hazard assessment

Estimation in causal graphical models

Pearl (2000), Spirtes et al (1993) and Lauritzen (2001) set up a new framework to encode the causal relationships between the random variables by a causal Bayesian network. The estimation of the conditional probabilities in a Bayesian network has received considerable attention by several investigators (e. g., Jordan (1998), Geiger and Heckerman (1997), Ileckerman et al (1995)), but, this issue has not been studied in a causal Bayesian network. In this thesis, we define the multicausal essential graph on the equivalence class of Bayesian networks in which each member of this class manifests a sort of strong type of invariance under (causal) manipulation called hypercausality. We then characterise the families of prior distributions on the parameters of the Bayesian networks which are consistent with hypercausality and show that their unmanipulated uncertain Bayesian networks must demonstrate the independence assumptions. As a result, such prior distributions satisfy a generalisation of the Geiger and lieckerman condition. In particular, when the corresponding essential graph is undirected, the mentioned class of prior distributions will reduce to the Hyper-Dirichlet family (see Chapter 6). In tile second part of this thesis, we will calculate certain local sensitivity measures and through them we are able to provide the solutions for the following questions: Is the network structure that is learned from data robust with respect to changes of the directionality of some specific arrows? Is the local conditional distributions associated with the specified node robust with respect to the changes to its prior distribution or with respect to the changes to the local conditional distribution of another node? Most importantly, is the posterior distribution associated with the parameters of any node robust with respect to the changes to the prior distribution associated with the parameters of one specific node? Finally, are the quantities mentioned above robust with respect to the changes in the independence assumptions described in Chapter 3? Most of the local sensitivity measures (particularly, local measures of the overall posteriors sensitivity), developed in the last decade, tend to diverge to infinity as the sample size becomes very large (Gustafson (1994) and Gustafson et al (1996)). This is in contrast to our knowledge that, starting from different priors, posteriors tend to agree as the data accumulate. Here we define a now class of metrics with more satisfactory asymptotic behaviour. The advantage of the corresponding local sensitivity measures is boundedness for large sample size.EThOS - Electronic Theses Online ServiceIran. Vizārat-i ʻUlūm, Taḥqīqāt va Fanāvarī [Iran. Ministry of Science, Research and Technology] (VUTF)GBUnited Kingdo

Performance Boundary Identification for the Evaluation of Automated Vehicles using Gaussian Process Classification

Safety is an essential aspect in the facilitation of automated vehicle deployment. Current testing practices are not enough, and going beyond them leads to infeasible testing requirements, such as needing to drive billions of kilometres on public roads. Automated vehicles are exposed to an indefinite number of scenarios. Handling of the most challenging scenarios should be tested, which leads to the question of how such corner cases can be determined. We propose an approach to identify the performance boundary, where these corner cases are located, using Gaussian Process Classification. We also demonstrate the classification on an exemplary traffic jam approach scenario, showing that it is feasible and would lead to more efficient testing practices.Comment: 6 pages, 5 figures, accepted at 2019 IEEE Intelligent Transportation Systems Conference - ITSC 2019, Auckland, New Zealand, October 201

Probabilistic sensitivity analysis of optimized preventive maintenance strategies for deteriorating infrastructure assets

Efficient life-cycle management of civil infrastructure systems under continuous deterioration can be improved by studying the sensitivity of optimised preventive maintenance decisions with respect to changes in model parameters. Sensitivity analysis in maintenance optimisation problems is important because if the calculation of the cost of preventive maintenance strategies is not sufficiently robust, the use of the maintenance model can generate optimised maintenances strategies that are not cost-effective. Probabilistic sensitivity analysis methods (particularly variance based ones), only partially respond to this issue and their use is limited to evaluating the extent to which uncertainty in each input contributes to the overall output's variance. These methods do not take account of the decision-making problem in a straightforward manner. To address this issue, we use the concept of the Expected Value of Perfect Information (EVPI) to perform decision-informed sensitivity analysis: to identify the key parameters of the problem and quantify the value of learning about certain aspects of the life-cycle management of civil infrastructure system. This approach allows us to quantify the benefits of the maintenance strategies in terms of expected costs and in the light of accumulated information about the model parameters and aspects of the system, such as the ageing process. We use a Gamma process model to represent the uncertainty associated with asset deterioration, illustrating the use of EVPI to perform sensitivity analysis on the optimisation problem for age-based and condition-based preventive maintenance strategies. The evaluation of EVPI indices is computationally demanding and Markov Chain Monte Carlo techniques would not be helpful. To overcome this computational difficulty, we approximate the EVPI indices using Gaussian process emulators. The implications of the worked numerical examples discussed in the context of analytical efficiency and organisational learning