Essays in Statistics

This thesis is comprised of several contributions to the field of mathematical statistics, particularly with regards to computational issues of Bayesian statistics and functional data analysis. The first two chapters are concerned with computational Bayesian approaches that allow one to generate samples from an approximation to the posterior distribution in settings where the likelihood function of some statistical model of interest is unknown. This has led to a class of Approximate Bayesian Computation (ABC) methods whose performance depends on the ability to effectively summarize the information content of the data sample by a lower-dimensional vector of summary statistics. Ideally, these statistics are sufficient for the parameter of interest. However, it is difficult to establish sufficiency in a straightforward way if the likelihood of the model is unavailable. In Chapter 1 we propose an indirect approach to select sufficient summary statistics for ABC methods that borrows its intuition from the indirect estimation literature in econometrics. More precisely, we introduce an auxiliary statistical model that is large enough as to contain the structural model of interest. Summary statistics are then identified in this auxiliary model and mapped to the structural model of interest. We show sufficiency of these statistics for Indirect ABC methods based on parameter estimates (ABC-IP), likelihood functions (ABC-IL) and scores (ABC-IS) of the auxiliary model. A detailed simulation study investigates the performance of each proposal and compares it to a traditional, moment-based ABC approach. Particularly, the ABC-IL and ABC-IS algorithms are shown to perform better than both standard ABC and the ABC-IP methods. In Chapter 2 we extend the notion of Indirect ABC methods by proposing an efficient way of weighting the individual entries of the vector of summary statistics obtained from the score-based Indirect ABC approach (ABC-IS). In particular, the weighting matrix is given by the inverse of the asymptotic covariance matrix of the score vector of the auxiliary model and allows us to appropriately assess the distance between the true posterior distribution and the approximation based on the ABC-IS method. We illustrate the performance gain in a simulation study. An empirical application then implements the weighted ABC-IS method to the problem of estimating a continuous-time stochastic volatility model based on non-Gaussian Ornstein-Uhlenbeck processes. We show how a suitable auxiliary model can be constructed and confirm estimation results from concurring Bayesian estimation approaches suggested in the literature. In Chapter 3 we consider the problem of sampling from high-dimensional probability distributions that exhibit multiple, well-separated modes. Such distributions arise frequently, for instance, in the Bayesian estimation of macroeconomic DSGE models. Standard Markov Chain Monte Carlo (MCMC) methods, such as the Metropolis-Hastings algorithm, are prone to get trapped in local neighborhoods of the target distribution thus severely limiting the use of these methods in more complex models. We suggest the use of a Sequential Markov Chain Monte Carlo approach to overcome these difficulties and investigate its finite sample properties. The results show that Sequential MCMC methods clearly outperform standard MCMC approaches in a multimodal setting and can recover both the location as well as the mixture weights in a 12-dimensional mixture model. Moreover, we provide a detailed comparison of the effects different choices of tuning parameters have on the approximation to the true sampling distribution. These results can serve as valuable guidelines when applying this method to more complex economic models, such as the (Bayesian) estimation of Dynamic Stochastic General Equilibrium models. Chapters 4 and 5 study the statistical problem of prediction from a functional perspective. In many statistical applications, data is becoming available at ever increasing frequencies and it has thus become natural to think of discrete observations as realizations of a continuous function, say over the course of one day. However, as functions are generally speaking infinite-dimensional objects, the statistical analysis of such functional data is intrinsically different from standard multivariate techniques. In Chapter 4 we consider prediction in functional additive models of first-order autoregressive type for a time series of functional observations. This is a generalization of functional linear models that are commonly considered in the literature and has two advantages to be applied in a functional time series setting. First, it allows us to introduce a very general notion of time dependencies for functional data in this modeling framework. Particularly, it is rooted at the correlation structure of functional principal component scores and even allows for long memory behavior in the score series across the time dimension. Second, prediction in this modeling framework is straightforwardly implemented as it only concerns conditional means of scalar random variables and we suggest a k-nearest neighbors classification scheme. The theoretical contributions of this paper are twofold. In a first step, we verify the applicability of the functional principal components analysis under our notion of time dependence and obtain precise rates of convergence for the mean function and the covariance operator associated with the observed sample of functions. In a second step, we derive precise rates of convergence of the mean squared error for the proposed predictor, taking into account both the effect of truncating the infinite series expansion at some finite integer L as well as the effect of estimating the covariance operator and associated eigenelements based on a sample of N curves. In Chapter 5 we investigate the performance of functional models in a forecasting study of ground-level ozone-concentration surfaces over the geographical domain of Germany. Our perspective thus differs from the literature on spatially distributed functional processes (which are considered to be (univariate) functions of time that show spatial dependence) in that we consider smooth surfaces defined over some spatial domain that are sampled consecutively over time. In particular, we treat discrete observations that are sampled both over a spatial domain and over time as noisy realizations of some time series of smooth bivariate functions. In a first step we therefore discuss how smooth functions can be reconstructed from such noisy measurements through a finite element spline smoother that is defined over some triangulation of the spatial domain. In a second step we consider two forecasting approaches to functional time series. The first one is a functional linear model of first-order auto-regressive type, whereas the second considers the non-parametric extension to functional additive models discussed in Chapter 4. Both approaches are applied to predicting ground-level ozone concentration measured over the spatial domain of Germany and are shown to yield similar predictions

Semantics and Verification of UML Activity Diagrams for Workflow Modelling

This thesis defines a formal semantics for UML activity diagrams that is suitable for workflow modelling. The semantics allows verification of functional requirements using model checking. Since a workflow specification prescribes how a workflow system behaves, the semantics is defined and motivated in terms of workflow systems. As workflow systems are reactive and coordinate activities, the defined semantics reflects these aspects. In fact, two formal semantics are defined, which are completely different. Both semantics are defined directly in terms of activity diagrams and not by a mapping of activity diagrams to some existing formal notation. The requirements-level semantics, based on the Statemate semantics of statecharts, assumes that workflow systems are infinitely fast w.r.t. their environment and react immediately to input events (this assumption is called the perfect synchrony hypothesis). The implementation-level semantics, based on the UML semantics of statecharts, does not make this assumption. Due to the perfect synchrony hypothesis, the requirements-level semantics is unrealistic, but easy to use for verification. On the other hand, the implementation-level semantics is realistic, but difficult to use for verification. A class of activity diagrams and a class of functional requirements is identified for which the outcome of the verification does not depend upon the particular semantics being used, i.e., both semantics give the same result. For such activity diagrams and such functional requirements, the requirements-level semantics is as realistic as the implementation-level semantics, even though the requirements-level semantics makes the perfect synchrony hypothesis. The requirements-level semantics has been implemented in a verification tool. The tool interfaces with a model checker by translating an activity diagram into an input for a model checker according to the requirements-level semantics. The model checker checks the desired functional requirement against the input model. If the model checker returns a counterexample, the tool translates this counterexample back into the activity diagram by highlighting a path corresponding to the counterexample. The tool supports verification of workflow models that have event-driven behaviour, data, real time, and loops. Only model checkers supporting strong fairness model checking turn out to be useful. The feasibility of the approach is demonstrated by using the tool to verify some real-life workflow models

Targeted Fused Ridge Estimation of Inverse Covariance Matrices from Multiple High-Dimensional Data Classes

We consider the problem of jointly estimating multiple inverse covariance matrices from high-dimensional data consisting of distinct classes. An $\ell_2$-penalized maximum likelihood approach is employed. The suggested approach is flexible and generic, incorporating several other $\ell_2$-penalized estimators as special cases. In addition, the approach allows specification of target matrices through which prior knowledge may be incorporated and which can stabilize the estimation procedure in high-dimensional settings. The result is a targeted fused ridge estimator that is of use when the precision matrices of the constituent classes are believed to chiefly share the same structure while potentially differing in a number of locations of interest. It has many applications in (multi)factorial study designs. We focus on the graphical interpretation of precision matrices with the proposed estimator then serving as a basis for integrative or meta-analytic Gaussian graphical modeling. Situations are considered in which the classes are defined by data sets and subtypes of diseases. The performance of the proposed estimator in the graphical modeling setting is assessed through extensive simulation experiments. Its practical usability is illustrated by the differential network modeling of 12 large-scale gene expression data sets of diffuse large B-cell lymphoma subtypes. The estimator and its related procedures are incorporated into the R-package rags2ridges.Comment: 52 pages, 11 figure

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency

An approach to computer-based knowledge representation for the business environment using empirical modelling

The motivation for the thesis arises from the difficulties experienced by business people who are non-programmers with the inflexibilities of conventional packages and tools for model-making. After a review of current business software an argument is made for the need for a new computing paradigm that would offer more support for the way that people actually experience their business activities. The Empirical Mod- elling (EM) approach is introduced as a broad theoretical and practical paradigm for computing that can be viewed as a far-reaching generali- sation of the spreadsheet concept. The concepts and principles of EM emphasise the experiential pro- cesses underlying familiar abstractions and by which we come to iden- tify reliable components in everyday life and, in particular, business activities. The emphasis on experience and on interaction leads to the new claim that EM environments offer a framework for combining propositional, experiential and tacit knowledge in a way that is more accessible and supportive of cognitive processes than conventional computer-based modelling. It is proposed that such environments offer an alternative kind of knowledge representation. Turning to the imple- mentation and development of systems, the difficulties inherent in con- ventional methods are discussed and then the practical aspects of EM, and its potential for system building, are outlined. Finally, a more detailed study is made of Decision Support Systems and the ways in which the EM focus on experience, and knowledge through interaction, can contribute to the representation of qualitative aspects of business activities and their use in a more human-centred, but computer-supported, process of decision making. Illustrations of the practical application of EM methods to the requirements of a deci- sion support environment are given by means of extracts from a num- ber of existing EM models

Optimal slip control for tractors with feedback of drive torque

Traction efficiency of tractors barely reaches 50 % in field operations. On the other hand, modern trends in agriculture show growth of the global tractor markets and at the same time increased demands for greenhouse gas emission reduction as well as energy efficiency due to increasing fuel costs. Engine power of farm tractors is growing at 1.8 kW per year reaching today about 500 kW for the highest traction class machines. The problem of effective use of energy has become crucial. Existing slip control approaches for tractors do not fulfil this requirement due to fixed reference set-point. The present work suggests an optimal control scheme based on set-point optimization and on assessment of soil conditions, namely, wheel-ground parameter identification using fuzzy-logic-assisted adaptive unscented Kalman filter.:List of figures VIII List of tables IX Keywords XI List of abbreviations XII List of mathematical symbols XIII Indices XV 1 Introduction 1 1.1 Problem description and challenges 1 1.1.1 Development of agricultural industry 1 1.1.2 Power flows and energy efficiency of a farm tractor 2 1.2 Motivation 9 1.3 Purpose and approach 12 1.3.1 Purpose and goals 12 1.3.2 Brief description of methodology 14 1.3.2.1 Drive torque feedback 14 1.3.2.2 Measurement signals 15 1.3.2.3 Identification of traction parameters 15 1.3.2.4 Definition of optimal slip 15 1.4 Outline 16 2 State of the art in traction management and parameter estimation 17 2.1 Slip control for farm tractors 17 2.2 Acquisition of drive torque feedback 23 2.3 Tire-ground parameter estimation 25 2.3.1 Kalman filter 25 2.3.2 Extended Kalman filter 27 2.3.3 Unscented Kalman filter 27 2.3.4 Adaptation algorithms for Kalman filter 29 3 Modelling vehicle dynamics for traction control 31 3.1 Tire-soil interaction 31 3.1.1 Forces in wheel-ground contact 32 3.1.1.1 Vertical force 32 3.1.1.2 Tire-ground surface geometry 34 3.1.2 Longitudinal force 36 3.1.3 Zero-slip condition 37 3.1.3.1 Soil shear stress 38 3.1.3.2 Rolling resistance 39 3.2 Vehicle body and wheels 40 3.2.1 Short description of Multi-Body-Simulation 40 3.2.2 Vehicle body and wheel models 42 3.2.3 Wheel structure 43 3.3 Stochastic input signals 45 3.3.1 Influence of trend and low-frequency components 47 3.3.2 Modelling stochastic signals 49 3.4 Further components and general view of tractor model 53 3.4.1 Generator, intermediate circuit, electrical motors and braking resistor 53 3.4.2 Diesel engine 55 4 Identification of traction parameters 56 4.1 Description of identification approaches 56 4.2 Vehicle model 58 4.2.1 Vehicle longitudinal dynamics 58 4.2.2 Wheel rotational dynamics 59 4.2.3 Tire dynamic rolling radius and inner rolling resistance coefficient 60 4.2.4 Whole model 61 4.3 Static methods of parameter identification 63 4.4 Adaptation mechanism of the unscented Kalman filter 63 4.5 Fuzzy supervisor for the adaptive unscented Kalman filter 66 4.5.1 Structure of the fuzzy supervisor 67 4.5.2 Stability analysis of the adaptive unscented Kalman filter with the fuzzy supervisor 69 5 Optimal slip control 73 5.1 Approaches for slip control by means of traction control system 73 5.1.1 Feedback compensation law 73 5.1.2 Sliding mode control 74 5.1.3 Funnel control 77 5.1.4 Lyapunov-Candidate-Function-based control, other approaches and choice of algorithm 78 5.2 General description of optimal slip control algorithm 79 5.3 Estimation of traction force characteristic curves 82 5.4 Optimal slip set-point computation 85 6 Verification of identification and optimal slip control systems 91 6.1 Simulation results 91 6.1.1 Identification of traction parameters 91 6.1.1.1 Comparison of extended Kalman filter and unscented Kalman filter 92 6.1.1.2 Comparison of ordinary and adaptive unscented Kalman filters 96 6.1.1.3 Comparison of the adaptive unscented Kalman filter with the fuzzy supervisor and static methods 99 6.1.1.4 Description of soil conditions 100 6.1.1.5 Identification of traction parameters under changing soil conditions 101 6.1.2 Approximation of characteristic curves 102 6.1.3 Slip control with reference of 10% 103 6.1.4 Comparison of operating with fixed and optimal slip reference 104 6.2 Experimental verification 108 6.2.1 Setup and description of the experiments 108 6.2.2 Virtual slip control without load machine 109 6.2.3 Virtual slip control with load machine 113 7 Summary, conclusions and future challenges 122 7.1 Summary of results and discussion 122 7.2 Contributions of the dissertation 123 7.3 Future challenges 123 Bibliography 125 A Measurement systems 137 A.1 Measurement of vehicle velocity 137 A.2 Measurement of wheel speed 138 A.3 Measurement or estimation of wheel vertical load 139 A.4 Measurement of draft force 140 A.5 Further possible measurement systems 141 B Basic probability theoretical notions 142 B.1 Brief description of the theory of stochastic processes 142 B.2 Properties of stochastic signals 144 B.3 Bayesian filtering 145 C Modelling stochastic draft force and field microprofile 147 D Approximation of kappa-curves 152 E Simulation parameters 15