Special Cases

International audienceThis chapter reviews special cases of lower previsions, that are instrumental in practical applications. We emphasize their various advantages and drawbacks, as well as the kind of problems in which they can be the most useful

Imprecise Continuous-Time Markov Chains

Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMC's), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of "precise" continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of "precise" models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMC's and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points

Optimal state estimation and control of space systems under severe uncertainty

This thesis presents novel methods and algorithms for state estimation and optimal control under generalised models of uncertainty. Tracking, scheduling, conjunction assessment, as well as trajectory design and analysis, are typically carried out either considering the nominal scenario only or under assumptions and approximations of the underlying uncertainty to keep the computation tractable. However, neglecting uncertainty or not quantifying it properly may result in lengthy design iterations, mission failures, inaccurate estimation of the satellite state, and poorly assessed risk metrics. To overcome these challenges, this thesis proposes approaches to incorporate proper uncertainty treatment in state estimation, navigation and tracking, and trajectory design. First, epistemic uncertainty is introduced as a generalised model to describe partial probabilistic models, ignorance, scarce or conflicting information, and, overall, a larger umbrella of uncertainty structures. Then, new formulations for state estimation, optimal control, and scheduling under mixed aleatory and epistemic uncertainties are proposed to generalise and robustify their current deterministic or purely aleatory counterparts. Practical solution approaches are developed to numerically solve such problems efficiently. Specifically, a polynomial reinitialisation approach for efficient uncertainty propagation is developed to mitigate the stochastic dimensionality in multi-segment problems. For state estimation and navigation, two robust filtering approaches are presented: a generalisation of the particle filtering to epistemic uncertainty exploiting samples’ precomputations; a sequential filtering approach employing a combination of variational inference and importance sampling. For optimal control under uncertainty, direct shooting-like transcriptions with a tunable high-fidelity polynomial representation of the dynamical flow are developed. Uncertainty quantification, orbit determination, and navigation analysis are incorporated in the main optimisation loop to design trajectories that are simultaneously optimal and robust. The methods developed in this thesis are finally applied to a variety of novel test cases, ranging from LEO to deep-space missions, from trajectory design to space traffic management. The epistemic state estimation is employed in the robust estimation of debris’ conjunction analyses and incorporated in a robust Bayesian framework capable of autonomous decision-making. An optimisation-based scheduling method is presented to efficiently allocate resources to heterogeneous ground stations and fusing information coming from different sensors, and it is applied to the optimal tracking of a satellite in highly perturbed very-low Earth orbit, and a low-resource deep-space spacecraft. The optimal control methods are applied to the robust optimisation of an interplanetary low-thrust trajectory to Apophis, and to the robust redesign of a leg of the Europa Clipper tour with an initial infeasibility on the probability of impact with Jupiter’s moon.This thesis presents novel methods and algorithms for state estimation and optimal control under generalised models of uncertainty. Tracking, scheduling, conjunction assessment, as well as trajectory design and analysis, are typically carried out either considering the nominal scenario only or under assumptions and approximations of the underlying uncertainty to keep the computation tractable. However, neglecting uncertainty or not quantifying it properly may result in lengthy design iterations, mission failures, inaccurate estimation of the satellite state, and poorly assessed risk metrics. To overcome these challenges, this thesis proposes approaches to incorporate proper uncertainty treatment in state estimation, navigation and tracking, and trajectory design. First, epistemic uncertainty is introduced as a generalised model to describe partial probabilistic models, ignorance, scarce or conflicting information, and, overall, a larger umbrella of uncertainty structures. Then, new formulations for state estimation, optimal control, and scheduling under mixed aleatory and epistemic uncertainties are proposed to generalise and robustify their current deterministic or purely aleatory counterparts. Practical solution approaches are developed to numerically solve such problems efficiently. Specifically, a polynomial reinitialisation approach for efficient uncertainty propagation is developed to mitigate the stochastic dimensionality in multi-segment problems. For state estimation and navigation, two robust filtering approaches are presented: a generalisation of the particle filtering to epistemic uncertainty exploiting samples’ precomputations; a sequential filtering approach employing a combination of variational inference and importance sampling. For optimal control under uncertainty, direct shooting-like transcriptions with a tunable high-fidelity polynomial representation of the dynamical flow are developed. Uncertainty quantification, orbit determination, and navigation analysis are incorporated in the main optimisation loop to design trajectories that are simultaneously optimal and robust. The methods developed in this thesis are finally applied to a variety of novel test cases, ranging from LEO to deep-space missions, from trajectory design to space traffic management. The epistemic state estimation is employed in the robust estimation of debris’ conjunction analyses and incorporated in a robust Bayesian framework capable of autonomous decision-making. An optimisation-based scheduling method is presented to efficiently allocate resources to heterogeneous ground stations and fusing information coming from different sensors, and it is applied to the optimal tracking of a satellite in highly perturbed very-low Earth orbit, and a low-resource deep-space spacecraft. The optimal control methods are applied to the robust optimisation of an interplanetary low-thrust trajectory to Apophis, and to the robust redesign of a leg of the Europa Clipper tour with an initial infeasibility on the probability of impact with Jupiter’s moon

Valid and efficient imprecise-probabilistic inference with partial priors, III. Marginalization

As Basu (1977) writes, "Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics," but after more than 50 years since Basu wrote these words, the two mainstream schools of thought in statistics have yet to solve the problem. Fortunately, the two mainstream frameworks aren't the only options. This series of papers rigorously develops a new and very general inferential model (IM) framework for imprecise-probabilistic statistical inference that is provably valid and efficient, while simultaneously accommodating incomplete or partial prior information about the relevant unknowns when it's available. The present paper, Part III in the series, tackles the marginal inference problem. Part II showed that, for parametric models, the likelihood function naturally plays a central role and, here, when nuisance parameters are present, the same principles suggest that the profile likelihood is the key player. When the likelihood factors nicely, so that the interest and nuisance parameters are perfectly separated, the valid and efficient profile-based marginal IM solution is immediate. But even when the likelihood doesn't factor nicely, the same profile-based solution remains valid and leads to efficiency gains. This is demonstrated in several examples, including the famous Behrens--Fisher and gamma mean problems, where I claim the proposed IM solution is the best solution available. Remarkably, the same profiling-based construction offers validity guarantees in the prediction and non-parametric inference problems. Finally, I show how a broader view of this new IM construction can handle non-parametric inference on risk minimizers and makes a connection between non-parametric IMs and conformal prediction.Comment: Follow-up to arXiv:2211.14567. Feedback welcome at https://researchers.one/articles/23.09.0000

State Estimation for Distributed Systems with Stochastic and Set-membership Uncertainties

State estimation techniques for centralized, distributed, and decentralized systems are studied. An easy-to-implement state estimation concept is introduced that generalizes and combines basic principles of Kalman filter theory and ellipsoidal calculus. By means of this method, stochastic and set-membership uncertainties can be taken into consideration simultaneously. Different solutions for implementing these estimation algorithms in distributed networked systems are presented