Benet, Luis

Ferson, Scott

Forets, Marcelo

Gray, Ander

Schilling, Christian

English

We combine reachability analysis and probability bounds analysis, which allow for imprecisely known random variables (multivariate intervals or p-boxes) to be specified as the initial states of a dynamical system. In combination, the methods allow for the temporal evolution of p-boxes to be rigorously computed, and they give interval probabilities for formal verification problems, also called failure probability calculations in reliability analysis. The methodology places no constraints on the input probability distribution or p-box and can handle dependencies generally in the form of copulas

    Aalborg UniversitetRigorous time evolution of p-boxes in non-linear ODEsGray, Ander; Forets, Marcelo; Schilling, Christian; Benet, Luis; Ferson, ScottPublished in:Book of Extended Abstracts for the 32nd European Safety and Reliability ConferencePublication date:2022Document VersionAccepted author manuscript, peer reviewed versionLink to publication from Aalborg UniversityCitation for published version (APA):Gray, A., Forets, M., Schilling, C., Benet, L., & Ferson, S. (2022). Rigorous time evolution of p-boxes in non-linear ODEs. In Book of Extended Abstracts for the 32nd European Safety and Reliability Conference (pp. 154-155)General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            - Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            - You may not further distribute the material or use it for any profit-making activity or commercial gain            - You may freely distribute the URL identifying the publication in the public portal -Take down policyIf you believe that this document breaches copyright please contact us at vbn@aub.aau.dk providing details, and we will remove access tothe work immediately and investigate your claim.Downloaded from vbn.aau.dk on: November 19, 2022Rigorous time evolution of p-boxes in non-linear ODEsAnder GrayCulham Science Centre, United Kingdom Atomic Energy Authority. E-mail: ander.gray@ukaea.ukMarcelo ForetsUniversidad de la República, Montevideo, Uruguay. E-mail: mforets@fing.edu.uyChristian SchillingAalborg University, Aalborg, Denmark. E-mail: christianms@cs.aau.dkLuis BenetInstituto de Ciencias Fı́sicas, Universidad Nacional Autónoma de México. E-mail: benet@icf.unam.mxScott FersonInstitute for Risk and Uncertainty, University of Liverpool, UK. E-mail: scott.ferson@liverpool.ac.ukWe combine reachability analysis and probability bounds analysis, which allow for imprecisely known randomvariables (multivariate intervals or p-boxes) to be specified as the initial states of a dynamical system. In combination,the methods allow for the temporal evolution of p-boxes to be rigorously computed, and they give intervalprobabilities for formal verification problems, also called failure probability calculations in reliability analysis.The methodology places no constraints on the input probability distribution or p-box and can handle dependenciesgenerally in the form of copulas.Keywords: reachability analysis, probability bounds analysis, automatically verified computing, set-based methods,p-box, uncertainty propagation.1. Reachability AnalysisReachability analysis studies the rigorous timeevolution of sets in non-linear dynamical systems.Usually reachability problems are presented as aninterval initial value problem, where a boundedset of trajectories starting from an interval box isrigorously computed. Figure 1 shows an exampleof a two-dimensional set propagated through anon-linear dynamical system. The set-based dy-namics are solved with Taylor models (Benet et al.2019), which for specific time intervals [ti, ti+1]rigorously represent the possible system stateswith a taylor series plus an interval remainder. TheTaylor model reach sets are integrated throughtime using a verified Picard iteration, giving a rig-orous outer approximation of the set of trajectoriesfrom the initial interval. For verification problems(reliability analysis) the method evaluates threepossible probabilities: Pf = 0 (guaranteed to besafe), Pf = 1 (guaranteed to be unsafe), andtimefailuredomainFig. 1. Colours show time evolution of an intervalODE, and the grey set is a hypothetical elliptical failuredomain which just touches the reach sets. Producedusing ReachabilityAnalysis.jl (Bogomolov et al. 2019)the interval probability Pf = [0, 1] (unknownsafety). The last of these is the case for the greyfailure domain in Figure 1, and this imprecision isa drawback of the method.Extended abstract collection of the 32nd European Safety and Reliability Conference.Edited by Maria Chiara Leva, Edoardo Patelli, Luca Podofillini and Simon WilsonCopyright © 2022 by ESREL2022 Organizers.12 Gray, Forets, Schilling, Benet, Ferson2. Probability Bounds AnalysisWe extend reachability analysis to allow for p-boxes to be specified as inputs, giving rigor-ous bounds on failure probabilities. Probabilityboxes (Gray et al. 2022) represent a set of prob-ability distributions using interval bounds on cdfsand are a generalisation of both intervals anddistributions. A multivariate p-box can be con-structed with imprecise Sklar’s theorem (Monteset al. 2015) using N marginal p-boxes and a sin-gle n-copula to capture stochastic dependence. Arigorous outer approximation (a belief function)of a p-box can be constructed using a finite setof intervals X (focal elements) and probabilitymasses m. In the multivariate case the H-volumemay be used to assign the masses to the focalelements using the copula. The focal elementsmay then be propagated using interval analysisY = f(X), with the masses conserved by eachinterval m(X) = m(f(X)). The output p-box canbe reconstructed, and the interval failure probabil-ity computed on some domain U using the belief(lower) and plausibility (upper) measuresP(U) =∑Y⊆Um(Y ),P(U) =∑Y ∩U ̸=∅m(Y ).3. Imprecise Probabilistic ReachabilityWe first take the support of the multivariate p-box and perform a single reachability calculation,getting the Taylor model approximation using theentire input domain. Subsets of this input domain(focal elements) can be tightly propagated throughthe Taylor model using interval arithmetic. Thisallows us to perform the probability boundingcalculation as a supplementary extra to a singlereachability calculation if required, for example ifthe failure domain cannot be proven to be safe.Setting the inputs to Figure 1 as the beta dis-tributed p-boxes X1 ∼ beta([2, 3], [3, 4]) andX2 ∼ beta([7, 8], [2, 3]) defined by interval rangesfor the traditional beta parameters, and with acorrelation of ρ = −0.8 using a Gaussian copula,Figure 2 shows the time evolution of X2 between0 and 5 seconds. Further, it can be proven thattx2(t)cdfFig. 2. Time evolution of a p-box for X2 of the ODEfrom Figure 1 in t = [0, 5]s.the failure probability is in the interval Pf =[0, 0.00367578], with the interval width comingfrom the imprecision of the input p-boxes, andthe rigorous discretisation error of the p-box (only100 focal elements per dimension) and outer ap-proximation from the system dynamics.AcknowledgementThis work has been partly carried out within the frame-work of the EUROfusion Consortium, funded by theEuropean Union via the Euratom Research and Train-ing Programme (Grant Agreement No 101052200 —EUROfusion). This research was partly supported byDIREC - Digital Research Centre Denmark and the Vil-lum Investigator Grant S4OS. Luis Benet acknowledgessupport from the PAPIIT-UNAM project IG-101122ReferencesBenet, L., M. Forets, D. P. Sanders, and C. Schilling(2019). TaylorModels.jl: Taylor models in Julia andits application to validated solutions of ODEs. InSWIM.Bogomolov, S., M. Forets, G. Frehse, K. Potomkin, andC. Schilling (2019). JuliaReach: a toolbox for set-based reachability. In HSCC, pp. 39–44. ACM.Gray, A., S. Ferson, and E. Patelli (2022). Probability-BoundsAnalysis.jl: Arithmetic with sets of distribu-tions. Submitted to the Proceedings of JuliaCon.Montes, I., E. Miranda, R. Pelessoni, and P. Vicig(2015). Sklar’s theorem in an imprecise setting.Fuzzy Sets and Systems 278, 48–66.

Rigorous time evolution of p-boxes in non-linear ODEs

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Rigorous time evolution of p-boxes in non-linear ODEs

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