Bayesian learning of models for estimating uncertainty in alert systems: application to air traffic conflict avoidance

Alert systems detect critical events which can happen in the short term. Uncertainties in data and in the models used for detection cause alert errors. In the case of air traffic control systems such as Short-Term Conflict Alert (STCA), uncertainty increases errors in alerts of separation loss. Statistical methods that are based on analytical assumptions can provide biased estimates of uncertainties. More accurate analysis can be achieved by using Bayesian Model Averaging, which provides estimates of the posterior probability distribution of a prediction. We propose a new approach to estimate the prediction uncertainty, which is based on observations that the uncertainty can be quantified by variance of predicted outcomes. In our approach, predictions for which variances of posterior probabilities are above a given threshold are assigned to be uncertain. To verify our approach we calculate a probability of alert based on the extrapolation of closest point of approach. Using Heathrow airport flight data we found that alerts are often generated under different conditions, variations in which lead to alert detection errors. Achieving 82.1% accuracy of modelling the STCA system, which is a necessary condition for evaluating the uncertainty in prediction, we found that the proposed method is capable of reducing the uncertain component. Comparison with a bootstrap aggregation method has demonstrated a significant reduction of uncertainty in predictions. Realistic estimates of uncertainties will open up new approaches to improving the performance of alert systems

Probabilistic reasoning and inference for systems biology

One of the important challenges in Systems Biology is reasoning and performing hypotheses testing in uncertain conditions, when available knowledge may be incomplete and the experimental data may contain substantial noise. In this thesis we develop methods of probabilistic reasoning and inference that operate consistently within an environment of uncertain knowledge and data. Mechanistic mathematical models are used to describe hypotheses about biological systems. We consider both deductive model based reasoning and model inference from data. The main contributions are a novel modelling approach using continuous time Markov chains that enables deductive derivation of model behaviours and their properties, and the application of Bayesian inferential methods to solve the inverse problem of model inference and comparison, given uncertain knowledge and noisy data. In the first part of the thesis, we consider both individual and population based techniques for modelling biochemical pathways using continuous time Markov chains, and demonstrate why the latter is the most appropriate. We illustrate a new approach, based on symbolic intervals of concentrations, with an example portion of the ERK signalling pathway. We demonstrate that the resulting model approximates the same dynamic system as traditionally defined using ordinary differential equations. The advantage of the new approach is quantitative logical analysis; we formulate a number of biologically significant queries in the temporal logic CSL and use probabilistic symbolic model checking to investigate their veracity. In the second part of the thesis, we consider the inverse problem of model inference and testing of alternative hypotheses, when models are defined by non-linear ordinary differential equations and the experimental data is noisy and sparse. We compare and evaluate a number of statistical techniques, and implement an effective Bayesian inferential framework for systems biology based on Markov chain Monte Carlo methods and estimation of marginal likelihoods by annealing-melting integration. We illustrate the framework with two case studies, one of which involves an open problem concerning the mediation of ERK phosphorylation in the ERK pathway

Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations

The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude. This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear. In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost. A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation

Development of a hybrid FE-SEA-experimental model

The vibro-acoustic response of complex structures with uncertain properties is a problem of great concern for modern industries. In recent years, much research has been devoted to the prediction of this response in the mid-frequency range where, because neither Finite element analysis nor statistical energy analysis are appropriate, a hybrid deterministic-statistical approach becomes a suitable solution. Despite its potential, the existence of systems with active components that are too complex to be modelled numerically can limit the application of the method. However, it may still be possible to measure the dynamical response of these structures experimentally. This paper is hence concerned with the possibility of integrating experimental data into a hybrid deterministic-statistical method. To explain the new methodology, two similar case studies, consisting of a deterministic source structure that is coupled to a statistical plate receiver using passive isolators, are used. For each case, the vibratory excitation, characterised using in-situ blocked force measurements, the source structure mobility, and the isolators stiffness are experimentally determined and inserted in the proposed hybrid model of the system. The paper explains the techniques used for obtaining the considered experimental data and the theoretical model proposed for describing the systems. To validate the proposed approach, the predicted vibration response of the receiver plate is compared to the one obtained by experimentally randomising the plate in both case studies. The results show that a good agreement is obtained, both for the ensemble average response of the receiver structure and for the ensemble variance of this response. Moreover, the upper con dence bounds predicted by the hybrid method enclose well the ensemble of experimental results. The cause of some narrow-band differences observed between the predicted response and the experimental measurements is finally discussed. It is therefore concluded that the capabilities of the hybrid deterministic-statistical method can be clearly enhanced through the incorporation of experimental data prescribing active sub-systems

Dynamical Modeling of NGC 6809: Selecting the best model using Bayesian Inference

The precise cosmological origin of globular clusters remains uncertain, a situation hampered by the struggle of observational approaches in conclusively identifying the presence, or not, of dark matter in these systems. In this paper, we address this question through an analysis of the particular case of NGC 6809. While previous studies have performed dynamical modeling of this globular cluster using a small number of available kinematic data, they did not perform appropriate statistical inference tests for the choice of best model description; such statistical inference for model selection is important since, in general, different models can result in significantly different inferred quantities. With the latest kinematic data, we use Bayesian inference tests for model selection and thus obtain the best fitting models, as well as mass and dynamic mass-to-light ratio estimates. For this, we introduce a new likelihood function that provides more constrained distributions for the defining parameters of dynamical models. Initially we consider models with a known distribution function, and then model the cluster using solutions of the spherically symmetric Jeans equation; this latter approach depends upon the mass density profile and anisotropy $\beta$ parameter. In order to find the best description for the cluster we compare these models by calculating their Bayesian evidence. We find smaller mass and dynamic mass-to-light ratio values than previous studies, with the best fitting Michie model for a constant mass-to-light ratio of $\Upsilon = 0.90^{+0.14}_{-0.14}$ and $M_{\text{dyn}}=6.10^{+0.51}_{-0.88} \times 10^4 M_{\odot}$. We exclude the significant presence of dark matter throughout the cluster, showing that no physically motivated distribution of dark matter can be present away from the cluster core.Comment: 12 pages, 10 figures, accepted for publication in MNRA

Robust H-infinity finite-horizon control for a class of stochastic nonlinear time-varying systems subject to sensor and actuator saturations

Copyright [2010] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This technical note addresses the robust H∞ finite-horizon output feedback control problem for a class of uncertain discrete stochastic nonlinear time-varying systems with both sensor and actuator saturations. In the system under investigation, all the system parameters are allowed to be time-varying, the parameter uncertainties are assumed to be of the polytopic type, and the stochastic nonlinearities are described by statistical means which can cover several classes of well-studied nonlinearities. The purpose of the problem addressed is to design an output feedback controller, over a given finite-horizon, such that the H∞ disturbance attenuation level is guaranteed for the nonlinear stochastic polytopic system in the presence of saturated sensor and actuator outputs. Sufficient conditions are first established for the robust H∞ performance through intensive stochastic analysis, and then a recursive linear matrix inequality (RLMI) approach is employed to design the desired output feedback controller achieving the prescribed H∞ disturbance rejection level. Simulation results demonstrate the effectiveness of the developed controller design scheme.This work was supported under Australian Research Council’s Discovery Projects funding scheme (project DP0880494) and by the German Science Foundation (DFG) within the priority programme 1305: Control Theory of Digitally Networked Dynamical Systems. Recommended by Associate Editor H. Ito

Ensemble energy average and energy flow relationships for nonstationary vibrating systems

This paper attempts to introduce a new point of view on energy analysis in structural dynamics with particular emphasis to its link with uncertainty and complexity. A linear, elastic system undergoing free vibrations, is considered. The system is subdivided into two subsystems and their respective energies together with the shared energy flow are analysed. First, the ensemble energy average of the two subsystems, assuming uncertain the natural frequencies, is investigated. It is shown how the energy averages follow a simple law when observing the long-term response of the system, obtained by a suitable asymptotic expansion. The second part of the analysis shows how the ensemble energy average of a set of random samples is representative even of the single case if the system is complex enough. The two previous points, combined, produce a result that applies to the energy sharing between two subsystems even independently of uncertainty: for complex systems, a simple energy sharing law is indeed stated. Moreover, in the case of absence of damping, a nonlinear relation between the energy flow and the energy (weighted) difference between the two subsystems is derived; on the other hand, when damping is present, this relationship becomes linear, including two terms: one is proportional to the energy (weighted) difference between the two subsystems, the other being proportional to its time derivative. Therefore, the approach suggests a way for deriving a general approach to energy sharing in vibration with results that, in some cases, are reminiscent of those met in Statistical Energy Analysis. Finally, computational experiments, performed on systems of increasing complexity, validate the theoretical results. (c) 2005 Elsevier Ltd. All rights reserved

Motion Planning of Uncertain Ordinary Differential Equation Systems

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and under-actuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as: system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it’s not accounted for in a given design. In this work uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and under-actuated systems, prescribes deterministic actuator inputs which yield uncertain state trajectories. The inverse dynamics formulation is the dual to the forward dynamic, and is only applicable to fully-actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to under-actuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories which yield uncertain unactuated states and actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case-studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space

Variance-constrained control for uncertain stochastic systems with missing measurements

Copyright [2005] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.In this paper, we are concerned with a new control problem for uncertain discrete-time stochastic systems with missing measurements. The parameter uncertainties are allowed to be norm-bounded and enter into the state matrix. The system measurements may be unavailable (i.e., missing data) at any sample time, and the probability of the occurrence of missing data is assumed to be known. The purpose of this problem is to design an output feedback controller such that, for all admissible parameter uncertainties and all possible incomplete observations, the system state of the closed-loop system is mean square bounded, and the steady-state variance of each state is not more than the individual prescribed upper bound. We show that the addressed problem can be solved by means of algebraic matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of some free parameters, which may be exploited to achieve further performance requirements. An illustrative numerical example is provided to demonstrate the usefulness and flexibility of the proposed design approach