A behavior driven approach for sampling rare event situations for autonomous vehicles

Performance evaluation of urban autonomous vehicles requires a realistic model of the behavior of other road users in the environment. Learning such models from data involves collecting naturalistic data of real-world human behavior. In many cases, acquisition of this data can be prohibitively expensive or intrusive. Additionally, the available data often contain only typical behaviors and exclude behaviors that are classified as rare events. To evaluate the performance of AV in such situations, we develop a model of traffic behavior based on the theory of bounded rationality. Based on the experiments performed on a large naturalistic driving data, we show that the developed model can be applied to estimate probability of rare events, as well as to generate new traffic situations

Approximation of event probabilities in noisy cellular processes

Molecular noise, which arises from the randomness of the discrete events in the cell, significantly influences fundamental biological processes. Discrete-state continuous-time stochastic models (CTMC) can be used to describe such effects, but the calculation of the probabilities of certain events is computationally expensive. We present a comparison of two analysis approaches for CTMC. On one hand, we estimate the probabilities of interest using repeated Gillespie simulation and determine the statistical accuracy that we obtain. On the other hand, we apply a numerical reachability analysis that approximates the probability distributions of the system at several time instances. We use examples of cellular processes to demonstrate the superiority of the reachability analysis if accurate results are required

Numerical analysis of stochastic biochemical reaction networks

Numerical solution of the chemical master equation for stochastic reaction networks typically suffers from the state space explosion problem due to the curse of dimensionality and from stiffness due to multiple time scales. The dimension of the state space equals the number of molecular species involved in the reaction network and the size of the system of differential equations equals the number of states in the corresponding continuous-time Markov chain, which is usually enormously huge and often even infinite. Thus, efficient numerical solution approaches must be able to handle huge, possibly infinite and stiff systems of differential equations efficiently. In this thesis, we present efficient techniques for the numerical analysis of the biochemical reaction networks. We present an approximate numerical integration approach that combines a dynamical state space truncation procedure with efficient numerical integration schemes for systems of ordinary differential equations including adaptive step size selection based on local error estimates. We combine our dynamical state space truncation with the method of conditional moments, and present the implementation details and numerical results. We also incorporate ideas from importance sampling simulations into a non-simulative numerical method that approximates transient rare event probabilities based on a dynamical truncation of the state space. Finally, we present a maximum likelihood method for the estimation of the model parameters given noisy time series measurements of molecular counts. All approaches presented in this thesis are implemented as part of the tool STAR, which allows to model and simulate the biochemical reaction networks. The efficiency and accuracy is demonstrated by numerical examples.Numerische Lösungen der chemischen Master-Gleichung für stochastische Reaktionsnetzwerke leiden typischerweise an dem Zustandsraumexplosionsproblem aufgrund der hohen Dimensionalität und der Steifigkeit durch mehrfache Zeitskalen. Die Dimension des Zustandsraumes entspricht der Anzahl der molekularen Spezies von dem Reaktionsnetzwerk und die Größe des Systems von Differentialgleichungen entspricht der Anzahl der Zustände in der entsprechenden kontinuierlichen Markov-Kette, die in der Regel enorm gross und oft sogar unendlich gross ist. Daher müssen numerische Methoden in der Lage sein, riesige, eventuell unendlich grosse und steife Systeme von Differentialgleichungen effizient lösen zu können. In dieser Arbeit beschreiben wir effiziente Methoden für die numerische Analyse biochemischer Reaktionsnetzwerke. Wir betrachten einen inexakten numerischen Integrationsansatz, bei dem eine dynamische Zustandsraumbeschneidung und ein Verfahren mit einem effizienten numerischen Integrationsschema für Systeme von gewöhnlichen Differentialgleichungen benutzt werden. Wir kombinieren unsere dynamische Zustandsraumbeschneidungsmethode mit der Methode der bedingten Momente und beschreiben die Implementierungdetails und numerischen Ergebnisse. Wir benutzen auch Ideen des importance sampling für eine nicht-simulative numerische Methode, die basierend auf der Zustandsraumbeschneidung die Wahrscheinlichkeiten von seltenen Ereignissen berechnen kann. Schließlich beschreiben wir eine Maximum-Likelihood-Methode für die Schätzung der Modellparameter bei verrauschten Zeitreihenmessungen von molekularen Anzahlen. Alle in dieser Arbeit beschriebenen Ansätze sind in dem Software-Tool STAR implementiert, das erlaubt, biochemische Reaktionsnetzwerke zu modellieren und zu simulieren. Die Effizienz und die Genauigkeit werden durch numerische Beispiele gezeigt

Development and application of accurate molecular mechanics sampling methods : from atomic clusters to protein tetramers

In this PhD Thesis molecular systems off increasing size and complexity are investigated, using both standard sampling and advanced sampling methods. The implementation and validation of two of those rare events sampling methods is described, namely the SA-MC and PINS algorithm. The development and use of a toolkit for fitting force fields parameters (for the Lennard-Jones and Multipoles parameters), the Fitting Wizard, is presented. The stability of the Hæmoglobin tetramer is also investigated in solution using standard Molecular Dynamics. The two first Chapters introduce the necessary theoretical background, and are followed by the results sections containing the articles written during this PhD

Empirical Game Theoretic Models for Autonomous Driving: Methods and Applications

In recent years, there has been enormous public interest in autonomous vehicles (AV), with more than 80 billion dollars invested in self-driving car technology. However, for the foreseeable future, self-driving cars will interact with human driven vehicles and other human road users, such as pedestrians and cyclists. Therefore, in order to ensure safe operation of AVs, there is need for computational models of humans traffic behaviour that can be used for testing and verification of autonomous vehicles. Game theoretic models of human driving behaviour is a promising computational tool that can be used in many phases of AV development. However, traditional game theoretic models are typically built around the idea of rationality, i.e., selection of the most optimal action based on individual preferences. In reality, not only is it hard to infer diverse human preferences from observational data, but real-world traffic shows that humans rarely choose the most optimal action that a computational model suggests. The thesis makes a set of methodological contributions towards modelling sub-optimality in driving behaviour within a game theoretic framework. These include solution concepts that account for boundedly rational behaviour in hierarchical games, addressing challenges of bounded rationality in dynamic games, and estimation of multi-objective utility aggregation from observational data. Each of these contributions are evaluated based on a novel multi-agent traffic dataset. Building on the game theoretic models, the second part of the thesis demonstrates the application of the models by developing novel safety validation methodologies for testing AV planners. The first application is an automated generation of interpretable variations of lane change behaviour based on Quantal Best Response model. The proposed model is shown to be effective for generating both rare-event situations and to replicate the typical behaviour distribution observed in naturalistic data. The second application is safety validation of strategic planners in situations of dynamic occlusion. Using the concept of hypergames, in which different agents have different views of the game, the thesis develops a new safety surrogate metric, dynamic occlusion risk (DOR), that can be used to evaluate the risk associated with each action in situations of dynamic occlusion. The thesis concludes with a taxonomy of strategic interactions that maps complex design specific strategies in a game to a simpler taxonomy of traffic interactions. Regulations around what strategies an AV should execute in traffic can be developed over the simpler taxonomy, and a process of automated mapping can protect the proprietary design decisions of an AV manufacturer

Rare-Event Estimation and Calibration for Large-Scale Stochastic Simulation Models

Stochastic simulation has been widely applied in many domains. More recently, however, the rapid surge of sophisticated problems such as safety evaluation of intelligent systems has posed various challenges to conventional statistical methods. Motivated by these challenges, in this thesis, we develop novel methodologies with theoretical guarantees and numerical applications to tackle them from different perspectives. In particular, our works can be categorized into two areas: (1) rare-event estimation (Chapters 2 to 5) where we develop approaches to estimating the probabilities of rare events via simulation; (2) model calibration (Chapters 6 and 7) where we aim at calibrating the simulation model so that it is close to reality. In Chapter 2, we study rare-event simulation for a class of problems where the target hitting sets of interest are defined via modern machine learning tools such as neural networks and random forests. We investigate an importance sampling scheme that integrates the dominating point machinery in large deviations and sequential mixed integer programming to locate the underlying dominating points. We provide efficiency guarantees and numerical demonstration of our approach. In Chapter 3, we propose a new efficiency criterion for importance sampling, which we call probabilistic efficiency. Conventionally, an estimator is regarded as efficient if its relative error is sufficiently controlled. It is widely known that when a rare-event set contains multiple "important regions" encoded by the dominating points, importance sampling needs to account for all of them via mixing to achieve efficiency. We argue that the traditional analysis recipe could suffer from intrinsic looseness by using relative error as an efficiency criterion. Thus, we propose the new efficiency notion to tighten this gap. In particular, we show that under the standard Gartner-Ellis large deviations regime, an importance sampling that uses only the most significant dominating points is sufficient to attain this efficiency notion. In Chapter 4, we consider the estimation of rare-event probabilities using sample proportions output by crude Monte Carlo. Due to the recent surge of sophisticated rare-event problems, efficiency-guaranteed variance reduction may face implementation challenges, which motivate one to look at naive estimators. In this chapter we construct confidence intervals for the target probability using this naive estimator from various techniques, and then analyze their validity as well as tightness respectively quantified by the coverage probability and relative half-width. In Chapter 5, we propose the use of extreme value analysis, in particular the peak-over-threshold method which is popularly employed for extremal estimation of real datasets, in the simulation setting. More specifically, we view crude Monte Carlo samples as data to fit on a generalized Pareto distribution. We test this idea on several numerical examples. The results show that in the absence of efficient variance reduction schemes, it appears to offer potential benefits to enhance crude Monte Carlo estimates. In Chapter 6, we investigate a framework to develop calibration schemes in parametric settings, which satisfies rigorous frequentist statistical guarantees via a basic notion that we call eligibility set designed to bypass non-identifiability via a set-based estimation. We investigate a feature extraction-then-aggregation approach to construct these sets that target at multivariate outputs. We demonstrate our methodology on several numerical examples, including an application to calibration of a limit order book market simulator. In Chapter 7, we study a methodology to tackle the NASA Langley Uncertainty Quantification Challenge, a model calibration problem under both aleatory and epistemic uncertainties. Our methodology is based on an integration of distributionally robust optimization and importance sampling. The main computation machinery in this integrated methodology amounts to solving sampled linear programs. We present theoretical statistical guarantees of our approach via connections to nonparametric hypothesis testing, and numerical performances including parameter calibration and downstream decision and risk evaluation tasks