Fakultät für Mathematik, Informatik und Statistik - Digitale Hochschulschriften der LMU - Teil 02/02

Predicting the epidemiological effects of new vaccination programmes through mathematical-statistical transmission modelling is of increasing importance for the German Standing Committee on Vaccination. Such models commonly capture large populations utilizing a compartmental structure with its dynamics being governed by a system of ordinary differential equations (ODEs). Unfortunately, these ODE-based models are generally computationally expensive to solve, which poses a challenge for any statistical procedure inferring corresponding model parameters from disease surveillance data. Thus, in practice parameters are often fixed based on epidemiological knowledge hence ignoring uncertainty. A Bayesian inference framework incorporating this prior knowledge promises to be a more suitable approach allowing for additional parameter flexibility.

This thesis is concerned with statistical methods for performing Bayesian inference of ODE-based models. A posterior approximation approach based on a Gaussian distribution around the posterior mode through its respective observed Fisher information is presented. By employing a newly proposed method for adjusting the likelihood impact in terms of using a power posterior, the approximation procedure is able to account for the residual autocorrelation in the data given the model. As an alternative to this approximation approach, an adaptive Metropolis-Hastings algorithm is described which is geared towards an efficient posterior sampling in the case of a high-dimensional parameter space and considerable parameter collinearities. In order to identify relevant model components, Bayesian model selection criteria based on the marginal likelihood of the data are applied. The estimation of the marginal likelihood for each considered model is performed via a newly proposed approach which utilizes the available posterior sample obtained from the preceding Metropolis-Hastings algorithm.

Furthermore, the thesis contains an application of the presented methods by predicting the epidemiological effects of introducing rotavirus childhood vaccination in Germany. Again, an ODE-based compartmental model accounting for the most relevant transmission aspects of rotavirus is presented. After extending the model with vaccination mechanisms, it becomes possible to estimate the rotavirus vaccine effectiveness through routinely collected surveillance data. By employing the Bayesian framework, model predictions on the future epidemiological development assuming a high vaccination coverage rate incorporate uncertainty regarding both model structure and parameters. The forecast suggests that routine vaccination may cause a rotavirus incidence increase among older children and elderly, but drastically reduces the disease burden among the target group of young children, even beyond the expected direct vaccination effect by means of herd protection.

Altogether, this thesis provides a statistical perspective on the modelling of routine vaccination effects in order to assist decision making under uncertainty. The presented methodology is thereby easily applicable to other infectious diseases such as influenza.

Durch die immer größeren technischen Möglichkeiten mobiler Endgeräte sind die Voraussetzungen erfüllt, um diese zum mobilen Arbeiten oder zur Steuerung von industriellen Fertigungsprozessen einzusetzen. Aus Gründen der Informations- und Betriebssicherheit, sowie zur Umsetzung funktionaler Anforderungen, ist es aber vielfach erforderlich, die Verfügbarkeit von entsprechenden Zugriffsrechten auf Nutzer innerhalb autorisierter Zonen zu begrenzen. So kann z.B. das Auslesen kritischer Daten auf individuelle Büros oder die mobile Steuerung von Maschinen auf passende Orte innerhalb einer Fabrikhalle beschränkt werden. Dazu muss die Position des Nutzers ermittelt werden. Im realen Einsatz können Positionsschätzungen jedoch mit Fehlern in der Größe von autorisierten Zonen auftreten. Derzeit existieren noch keine Lösungen, welche diese Fehler in Autorisierungsentscheidungen berücksichtigen, um einhergehenden Schaden aus Falschentscheidungen zu minimieren. Ferner existieren derzeit keine Verfahren, um die Güteeigenschaften solcher Ortsbeschränkungen vor deren Ausbringung zu analysieren und zu entscheiden, ob ein gegebenes Positionierungssystem aufgrund der Größe seiner Positionsfehler geeignet ist.

In der vorliegenden Arbeit werden deshalb Lösungen zur Erfassung und Behandlung solcher Positionsfehler im Umfeld der standortbasierten Autorisierung vorgestellt. Hierzu wird zunächst ein Schätzverfahren für Positionsfehler in musterbasierten Positionierungsverfahren eingeführt, das aus den Charakteristika der durchgeführten Messungen eine Verteilung für den Standort des Nutzers ableitet. Um hieraus effizient die Aufenthaltswahrscheinlichkeit innerhalb einer autorisierten Zone zu bestimmen, wird ein Algorithmus vorgestellt, der basierend auf Vorberechnungen eine erhebliche Verbesserung der Laufzeit gegenüber der direkten Berechnung erlaubt. Erstmals wird eine umfassende Gegenüberstellung von existierenden standortbasierten Autorisierungsstrategien auf Basis der Entscheidungstheorie vorgestellt. Mit der risikobasierten Autorisierungsstrategie wird eine neue, aus entscheidungstheoretischer Sicht optimale Methodik eingeführt. Es werden Ansätze zur Erweiterung klassischer Zugriffskontrollmodelle durch Ortsbeschränkungen vorgestellt, welche bei ihrer Durchsetzung die Möglichkeit von Positionsfehlern und die Konsequenzen von Falschentscheidungen berücksichtigen. Zur Spezifikation autorisierter Zonen werden Eigenschaftsmodelle eingeführt, die, im Gegensatz zu herkömmlichen Polygonen, für jeden Ort die Wahrscheinlichkeit modellieren, dort eine geforderte Eigenschaft zu beobachten. Es werden ferner Methoden vorgestellt, um den Einfluss von Messausreißern auf Autorisierungsentscheidungen zu reduzieren. Ferner werden Analyseverfahren eingeführt, die für ein gegebenes Szenario eine qualitative und quantitative Bewertung der Eignung von Positionierungssystemen erlauben. Die quantitative Bewertung basiert auf dem entwickelten Konzept der Autorisierungsmodelle. Diese geben für jeden Standort die Wahrscheinlichkeit an, dort eine Positionsschätzung zu erhalten, die zur Autorisierung führt. Die qualitative Bewertung bietet erstmals ein binäres Kriterium, um für ein gegebenes Szenario eine konkrete Aussage bzgl. der Eignung eines Positionierungssystems treffen zu können. Die Einsetzbarkeit dieses Analyseverfahrens wird an einer Fallstudie verdeutlicht und zeigt die Notwendigkeit einer solchen Analyse bereits vor der Ausbringung von standortbasierter Autorisierung. Es wird gezeigt, dass für typische Positionierungssysteme durch die entwickelten risikobasierten Verfahren eine erhebliche Reduktion von Schaden aus Falschentscheidungen möglich ist und die Einsetzbarkeit der standortbasierten Autorisierung somit verbessert werden kann.

Large knowledge graphs increasingly add great value to various applications that require machines to recognize and understand queries and their semantics, as in search or question answering systems. These applications include Google search, Bing search, IBM’s Watson, but also smart mobile assistants as Apple’s Siri, Google Now or Microsoft’s Cortana. Popular knowledge graphs like DBpedia, YAGO or Freebase store a broad range of facts about the world, to a large extent derived from Wikipedia, currently the biggest web encyclopedia. In addition to these freely accessible open knowledge graphs, commercial ones have also evolved including the well-known Google Knowledge Graph or Microsoft’s Satori. Since incompleteness and veracity of knowledge graphs are known problems, the statistical modeling of knowledge graphs has increasingly gained attention in recent years. Some of the leading approaches are based on latent variable models which show both excellent predictive performance and scalability. Latent variable models learn embedding representations of domain entities and relations (representation learning). From these embeddings, priors for every possible fact in the knowledge graph are generated which can be exploited for data cleansing, completion or as prior knowledge to support triple extraction from unstructured textual data as successfully demonstrated by Google’s Knowledge-Vault project. However, large knowledge graphs impose constraints on the complexity of the latent embeddings learned by these models. For graphs with millions of entities and thousands of relation-types, latent variable models are required to exploit low dimensional embeddings for entities and relation-types to be tractable when applied to these graphs. The work described in this thesis extends the application of latent variable models for large knowledge graphs in three important dimensions. First, it is shown how the integration of ontological constraints on the domain and range of relation-types enables latent variable models to exploit latent embeddings of reduced complexity for modeling large knowledge graphs. The integration of this prior knowledge into the models leads to a substantial increase both in predictive performance and scalability with improvements of up to 77% in link-prediction tasks. Since manually designed domain and range constraints can be absent or fuzzy, we also propose and study an alternative approach based on a local closed-world assumption, which derives domain and range constraints from observed data without the need of prior knowledge extracted from the curated schema of the knowledge graph. We show that such an approach also leads to similar significant improvements in modeling quality. Further, we demonstrate that these two types of domain and range constraints are of general value to latent variable models by integrating and evaluating them on the current state of the art of latent variable models represented by RESCAL, Translational Embedding, and the neural network approach used by the recently proposed Google Knowledge Vault system. In the second part of the thesis it is shown that the just mentioned three approaches all perform well, but do not share many commonalities in the way they model knowledge graphs. These differences can be exploited in ensemble solutions which improve the predictive performance even further. The third part of the thesis concerns the efficient querying of the statistically modeled knowledge graphs. This thesis interprets statistically modeled knowledge graphs as probabilistic databases, where the latent variable models define a probability distribution for triples. From this perspective, link-prediction is equivalent to querying ground triples which is a standard functionality of the latent variable models. For more complex querying that involves e.g. joins and projections, the theory on probabilistic databases provides evaluation rules. In this thesis it is shown how the intrinsic features

With the widespread availability of wearable computers, equipped with sensors such as GPS or cameras, and with the ubiquitous presence of micro-blogging platforms, social media sites and digital marketplaces, data can be collected and shared on a massive scale. A necessary building block for taking advantage from this vast amount of information are efficient and effective similarity search algorithms that are able to find objects in a database which are similar to a query object. Due to the general applicability of similarity search over different data types and applications, the formalization of this concept and the development of strategies for evaluating similarity queries has evolved to an important field of research in the database community, spatio-temporal database community, and others, such as information retrieval and computer vision. This thesis concentrates on a special instance of similarity queries, namely k-Nearest Neighbor (kNN) Queries and their close relative, Reverse k-Nearest Neighbor (RkNN) Queries.

As a first contribution we provide an in-depth analysis of the RkNN join. While the problem of reverse nearest neighbor queries has received a vast amount of research interest, the problem of performing such queries in a bulk has not seen an in-depth analysis so far. We first formalize the RkNN join, identifying its monochromatic and bichromatic versions and their self-join variants. After pinpointing the monochromatic RkNN join as an important and interesting instance, we develop solutions for this class, including a self-pruning and a mutual pruning algorithm. We then evaluate these algorithms extensively on a variety of synthetic and real datasets.

From this starting point of similarity queries on certain data we shift our focus to uncertain data, addressing nearest neighbor queries in uncertain spatio-temporal databases. Starting from the traditional definition of nearest neighbor queries and a data model for uncertain spatio-temporal data, we develop efficient query mechanisms that consider temporal dependencies during query evaluation. We define intuitive query semantics, aiming not only at returning the objects closest to the query but also their probability of being a nearest neighbor. After theoretically evaluating these query predicates we develop efficient querying algorithms for the proposed query predicates. Given the findings of this research on nearest neighbor queries, we extend these results to reverse nearest neighbor queries.

Finally we address the problem of querying large datasets containing set-based objects, namely image databases, where images are represented by (multi-)sets of vectors and additional metadata describing the position of features in the image. We aim at reducing the number of kNN queries performed during query processing and evaluate a modified pipeline that aims at optimizing the query accuracy at a small number of kNN queries. Additionally, as feature representations in object recognition are moving more and more from the real-valued domain to the binary domain, we evaluate efficient indexing techniques for binary feature vectors.

In this thesis, we establish the scaling limit of several models of random trees and graphs, enlarging and completing the now long list of random structures that admit David Aldous' continuum random tree (CRT) as scaling limit. Our results answer important open questions, in particular the conjecture by Aldous for the scaling limit of random unlabelled unrooted trees. We also show that random graphs from subcritical graph classes admit the CRT as scaling limit, proving (in a strong from) a conjecture by Marc Noy and Michael Drmota, who conjectured a limit for the diameter of these graphs. Furthermore, we provide a new proof for results by Bénédicte Haas and Grégory Miermont regarding the scaling limits of random Pólya trees, extending their result to random Pólya trees with arbitrary vertex-degree restrictions.

After 2004, when it was possible for the first time to isolate graphene flakes, the interest in quantum mechanics of plain systems has been intensified significantly. In graphene, that is a single layer of carbon atoms aligned in a regular hexagonal structure, the generator of dynamics near the band edge is the massless Dirac operator in dimension two.

We investigate the spectrum of the two-dimensional massless Dirac operator H_D coupled to an external electro-magnetic field. More precisely, our focus lies on the characterisation of the spectrum σ(H_D) for field configurations that are generated by unbounded electric and magnetic potentials. We observe that the existence of gaps in σ(H_D) depends on the ratio V^2/B at infinity, which is a ratio of the electric potential V and the magnetic field B. In particular, a sharp bound on V^2/B is given, below which σ(H_D) is purely discrete. Further, we show that if the ratio V^2/B is unbounded at infinity, H_D has no spectral gaps for a huge class of fields B and potentials V . The latter statement leads to examples of two-dimensional massless Dirac operators with dense pure point spectrum. We extend the ideas, developed for H_D, to the classical Pauli (and the magnetic Schrödinger) operator in dimension two. It turns out that also such non-relativistic operators with a strong repulsive potential do admit criteria for spectral gaps in terms of B and V . Similarly as in the case of the Dirac operator, we show that those gaps do not occur in general if |V| is dominating B at infinity. It should be mentioned that this leads to a complete characterisation of the spectrum of certain Pauli (and Schrödinger) operators with very elementary, rotationally symmetric field configurations.

Considering for the Dirac operator H_D the regime of a growing ratio V^2/B, there happens a transition from pure point to continuous spectrum. A phenomenon that is particularly interesting from the dynamical point of view. Therefore, we address in a second part of the thesis the question under which spectral conditions ballistic wave package spreading in two-dimensional Dirac systems is possible. To be more explicit, we study the following problem: Do statements on the spectral type of H_D already suffice to decide whether the time mean of the expectation value

$$\frac{1}{T} \int_0^T \sps{\psi(t)}{|\bx|^2\psi(t)} \rd t $$

behaves like T^2? Here ψ(t) denotes the time evolution of a state ψ under the corresponding Dirac operator. We can answer that question affirmatively, at least for certain electro-magnetic fields with symmetry.

The goal of this thesis is a mathematical understanding of a phenomenon called Anderson's Orthogonality in the physics literature. Given two non-interacting fermionic systems which differ by an exterior short-range scattering potential, the scalar product of the corresponding ground states is predicted to decay algebraically in the thermodynamic limit. This decay is referred to as Anderson's orthogonality catastrophe in the physics literature and goes back to P.W.Anderson [Phys. Rev. Lett. 18:1049--1051] and is used to explain anomalies in the X-ray absorption spectrum of metals. We call this scalar product $S_N^L$, where $N$ refers to the particle number and $L$ to the diameter of the considered fermionic system. This decay was proven in the works [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512] for rather general pairs of Schrödinger operators in arbitrary dimension $d\in\N$, i.e. $|S_N^L|^2\le L^{-\gamma}$ in the thermodynamic limit $N/L^d\to \rho>0$ approaching a positive particle density. In the general case, the biggest found decay exponent is given by $\gamma=\frac 1 {\pi^2} \norm{\arcsin|T/2|}_{\text{HS}}$, where T refers to the scattering T-matrix.

In this thesis, we prove such upper bounds in more general situations than considered in both [Commun. Math. Phys. 329:979--998] and [arXiv:1407.2512]. Furthermore, we provide the first rigorous proof of the exact asymptotics Anderson predicted. We prove that in the $3$-dimensional toy model of a Dirac-$\delta$ perturbation that the exact decay exponent is given by $\zeta:= \delta^2/ \pi^2$. Here, $\delta$ refers to the s-wave scattering phase shift. In particular, this result shows that the previously found decay exponent $\gamma$ does not provide the correct asymptotics of $\S_N^L$ in general. Since the decay exponent is expressed in terms of scattering theory, these bounds depend on the existence of absolutely continuous spectrum of the underlying Schrödinger operators. We are able to deduce a different behavior in the contrary situation of Anderson localization. We prove the non-vanishing of the expectation value of the non-interacting many-body scalar product in the thermodynamic limit.

Apart from the behavior of the scalar product of the non-interacting ground states, we also study the asymptotics of the difference of the ground-state energies. We show that this difference converges in the thermodynamic limit to the integral of the spectral-shift function up to the Fermi energy. Furthermore, we quantify the error for models on the half-axis and show that higher order error terms depend on the particular thermodynamic limit chosen.

The present work deals with the idea of a multi-time wave function, i.e. a wave function with N space-time arguments for N particles.

Firstly, a careful derivation of the necessity of multi-time wave functions in relativistic quantum mechanics is given and a general formalism is developed.

Secondly, the physical meaning of multi-time wave functions is discussed in connection with realistic relativistic quantum theories, in particular the "Hypersurface Bohm-Dirac" model.

Thirdly, a first interacting model for multi-time wave functions of two Dirac particles in 1+1 space-time dimensions is constructed. Interaction is achieved by means of boundary conditions on configuration space-time, a mechanism closely related to zero-range physics. This is remarkable, as a restrictive consistency condition rules out various types of interaction and consequently no rigorous interacting model was known before.

Fourthly, the model is extended to more general types of interaction and to the N-particle case. Higher dimensions are also discussed.

Finally, the "Two-Body Dirac equations" of constraint theory are placed within the context of the multi-time formalism. In particular, the question of probability conservation is critically discussed, leading to further implications both for fundamental and applied questions.