Podcasts about ddc:510

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

In regression models for ordinal response, each covariate can be equipped with either a simple, global effect or a more flexible and complex effect which is specific to the response categories. Instead of a priori assuming one of these effect types, as is done in the majority of the literature, we argue in this paper that effect type selection shall be data-based. For this purpose, we propose a novel and general penalty framework that allows for an automatic, data-driven selection between global and category-specific effects in all types of ordinal regression models. Optimality conditions and an estimation algorithm for the resulting penalized estimator are given. We show that our approach is asymptotically consistent in both effect type and variable selection and possesses the oracle property. A detailed application further illustrates the workings of our method and demonstrates the advantages of effect type selection on real data.

Regression models with functional responses and covariates constitute a powerful and increasingly important model class. However, regression with functional data poses well known and challenging problems of non-identifiability. This non-identifiability can manifest itself in arbitrarily large errors for coefficient surface estimates despite accurate predictions of the responses, thus invalidating substantial interpretations of the fitted models. We offer an accessible rephrasing of these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they are likely to occur in practice. Specifically, non-identifiability that persists under smoothness assumptions on the coefficient surface can occur if the functional covariate's empirical covariance has a kernel which overlaps that of the roughness penalty of the spline estimator. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate their practical consequences under varying assumptions about the data generating processes. A case study illustrates the practical significance of the problem. Based on theoretical considerations and our empirical evaluation, we provide immediately applicable diagnostics for lack of identifiability and give recommendations for avoiding estimation artifacts in practice.

The aim of this thesis is to provide a rigorous mathematical derivation of the Vlasov-Poisson equation and the Vlasov-Maxwell equations in the large N limit of interacting charged particles. We will extend a method previously proposed by Boers and Pickl to perform a mean field limit for the Vlasov-Poisson equation with the full Coulomb singularity and an N-dependent cut-off decreasing as $N^{-1/3 + epsilon}$. We will then discuss an alternative approach, deriving the Vlasov-Poisson equation as a combined mean field and point-particle limit of an N-particle Coulomb system of extended charges. Finally, we will combine both methods to prove a mean field limit for the relativistic Vlasov-Maxwell system in 3+1 dimensions. In each case, convergence of the empirical measures to solutions of the corresponding mean field equation can be shown for typical initial conditions. This implies, in particular, the propagation of chaos for the respective dynamics.

In this dissertation we investigate long-term interest rates, i.e. interest rates with maturity going to infinity, in the post-crisis interest rate market. Three different concepts of long-term interest rates are considered for this purpose: the long-term yield, the long-term simple rate, and the long-term swap rate. We analyze the properties as well as the interrelations of these long-term interest rates. In particular, we study the asymptotic behavior of the term structure of interest rates in some specific models. First, we compute the three long-term interest rates in the HJM framework with different stochastic drivers, namely Brownian motions, Lévy processes, and affine processes on the state space of positive semidefinite symmetric matrices. The HJM setting presents the advantage that the entire yield curve can be modeled directly. Furthermore, by considering increasingly more general classes of drivers, we were able to take into account the impact of different risk factors and their dependence structure on the long end of the yield curve. Finally, we study the long-term interest rates and especially the long-term swap rate in the Flesaker-Hughston model and the linear-rational methodology.

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.

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.

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 $dinN$, i.e. $|S_N^L|^2le L^{-gamma}$ in the thermodynamic limit $N/L^dto rho>0$ 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.

Tue, 21 Jul 2015 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/18590/ https://edoc.ub.uni-muenchen.de/18590/1/Jahn_Thomas.pdf Jahn, Thomas ddc:510, ddc:500, Fakultät für Mathematik, Informatik und Statistik 0

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common space-structure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R. The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions, the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact, the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included.

Regression models with functional covariates for functional responses constitute a powerful and increasingly important model class. However, regression with functional data poses challenging problems of non-identifiability. We describe these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they arise. Specifically, functional covariates whose empirical covariance has lower effective rank than the number of marginal basis function used to represent the coefficient surface can lead to unidentifiability. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate its practical consequences under varying assumptions about the data generating processes. Based on theoretical considerations and our empirical evaluation, we provide easily verifiable criteria for lack of identifiability and provide actionable advice for avoiding spurious estimation artifacts. Applicability of our strategy for mitigating non-identifiability is demonstrated in a case study on the Canadian Weather data set.

Tue, 2 Jun 2015 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/18333/ https://edoc.ub.uni-muenchen.de/18333/1/Riedl_Leonhard.pdf Riedl, Leonhard ddc:510, ddc:500, Fakultät für Mathe

Global tests are in demand whenever it is of interest to draw inferential conclusions about sets of variables as a whole. The present thesis attempts to develop such tests for the case of multivariate ordinal data in possibly high-dimensional set-ups, and has primarily been motivated by research questions that arise from data collected by means of the 'International Classification of Functioning, Disability and Health'. The thesis essentially comprises two parts. In the first part two tests are discussed, each of which addresses one specific problem in the classical two-group scenario. Since both are permutation tests, their validity relies on the condition that, under the null hypothesis, the joint distribution of the variables in the set to be tested is the same in both groups. Extensive simulation studies on the basis of the tests proposed suggest, however, that violations of this condition, from the purely practical viewpoint, do not automatically lead to invalid tests. Rather, two-sample permutation tests' failure appears to depend on numerous parameters, such as the proportion between group sizes, the number of variables in the set of interest and, importantly, the test statistic used. In the second part two further tests are developed which both can be used to test for association, if desired after adjustment for certain covariates, between a set of ordinally scaled covariates and an outcome variable within the range of generalized linear models. The first test rests upon explicit assumptions on the distances between the covariates' categories, and is shown to be a proper generalization of the traditional Cochran-Armitage test to higher dimensions, covariate-adjusted scenarios and generalized linear model-specific outcomes. The second test in turn parametrizes these distances and thus keeps them flexible. Based on the tests' power properties, practical recommendations are provided on when to favour one or the other, and connections with the permutation tests from the first part of the thesis are pointed out. For illustration of the methods developed, data from two studies based on the 'International Classification of Functioning, Disability and Health' are analyzed. The results promise vast potential of the proposed tests in this data context and beyond.

In 1992, Englert [Phys. Rev. A, 45;127--134] found a momentum energy functional for atoms and discussed the relation to the Thomas-Fermi functional (Lenz [Z. Phys., 77;713--721]). We place this model in a mathematical setting. Our results include a proof of existence and uniqueness of a minimizing momentum density for this momentum energy functional. Further, we investigate some properties of this minimizer, among them the connection with Euler's equation. We relate the minimizers of the Thomas-Fermi functional and the momentum energy functional found by Englert by explicit transforms. It turns out that in this way results well-known in the Thomas-Fermi model can be transferred directly to the model under consideration. In fact, we gain equivalence of the two functionals upon minimization. Finally, we consider momentum dependent perturbations. In particular, we show that the atomic momentum density converges to the minimizer of the momentum energy functional as the total nuclear charge becomes large in a certain sense. This thesis is based on joint work with Prof. Dr. Heinz Siedentop and the main contents will also appear in a joint article.

When being interested in administering the best of two treatments to an individual patient i, it is necessary to know the individual treatment effects (ITEs) of the considered subjects and the correlation between the possible responses (PRs) for two treatments. When data are generated in a parallel–group design RCT, it is not possible to determine the ITE for a single subject since we only observe two samples from the marginal distributions of these PRs and not the corresponding joint distribution due to the ’Fundamental Problem of Causal Inference’ [Holland, 1986, p. 947]. In this article, we present a counterfactual approach for estimating the joint distribution of two normally distributed responses to two treatments. This joint distribution can be estimated by assuming a normal joint distribution for the PRs and by using a normally distributed baseline biomarker which is defined to be functionally related to the sum of the ITE components. Such a functional relationship is plausible since a biomarker and the sum encode for the same information in a RCT, namely the variation between subjects. As a result of the interpretation of the biomarker as a proxy for the sum of ITE components, the estimation of the joint distribution is subjected to some constraints. These constraints can be framed in the context of linear regressions with regard to the proportions of variances in the responses explained and with regard to the residual variation. As a consequence, a new light is thrown on the presence of treatment–biomarker interactions. We applied our approach to a classical medical data example on exercise and heart rate.

Financial bubbles have been present in the history of financial markets from the early days up to the modern age. An asset is said to exhibit a bubble when its market value exceeds its fundamental valuation. Although this phenomenon has been thoroughly studied in the economic literature, a mathematical martingale theory of bubbles, based on an absence of arbitrage has only recently been developed. In this dissertation, we aim to further contribute to the developement of this theory. In the first part we construct a model that allows us to capture the birth of a financial bubble and to describe its behavior as an initial submartingale in the build-up phase, which then turns into a supermartingale in the collapse phase. To this purpose we construct a flow in the space of equivalent martingale measures and we study the shifting perception of the fundamental value of a given asset. In the second part of the dissertation, we study the formation of financial bubbles in the valuation of defaultable claims in a reduced-form setting. In our model a bubble is born due to investor heterogeneity. Furthermore, our study shows how changes in the dynamics of the defaultable claim's market price may lead to a different selection of the martingale measure used for pricing. In this way we are able to unify the classical martingale theory of bubbles with a constructive approach to the study of bubbles, based on the interactions between investors.

An efficient way to represent the domain knowledge is relational data, where information is recorded in form of relationships between entities. Relational data is becoming ubiquitous over the years for knowledge representation due to the fact that many real-word data is inherently interlinked. Some well-known examples of relational data are: the World Wide Web (WWW), a system of interlinked hypertext documents; the Linked Open Data (LOD) cloud of the Semantic Web, a collection of published data and their interlinks; and finally the Internet of Things (IoT), a network of physical objects with internal states and communications ability. Relational data has been addressed by many different machine learning approaches, the most promising ones are in the area of relational learning, which is the focus of this thesis. While conventional machine learning algorithms consider entities as being independent instances randomly sampled from some statistical distribution and being represented as data points in a vector space, relational learning takes into account the overall network environment when predicting the label of an entity, an attribute value of an entity or the existence of a relationship between entities. An important feature is that relational learning can exploit contextual information that is more distant in the relational network. As the volume and structural complexity of the relational data increase constantly in the era of Big Data, scalability and the modeling power become crucial for relational learning algorithms. Previous relational learning algorithms either provide an intuitive representation of the model, such as Inductive Logic Programming (ILP) and Markov Logic Networks (MLNs), or assume a set of latent variables to explain the observed data, such as the Infinite Hidden Relational Model (IHRM), the Infinite Relational Model (IRM) and factorization approaches. Models with intuitive representations often involve some form of structure learning which leads to scalability problems due to a typically large search space. Factorizations are among the best-performing approaches for large-scale relational learning since the algebraic computations can easily be parallelized and since they can exploit data sparsity. Previous factorization approaches exploit only patterns in the relational data itself and the focus of the thesis is to investigate how additional prior information (comprehensive information), either in form of unstructured data (e.g., texts) or structured patterns (e.g., in form of rules) can be considered in the factorization approaches. The goal is to enhance the predictive power of factorization approaches by involving prior knowledge for the learning, and on the other hand to reduce the model complexity for efficient learning. This thesis contains two main contributions: The first contribution presents a general and novel framework for predicting relationships in multirelational data using a set of matrices describing the various instantiated relations in the network. The instantiated relations, derived or learnt from prior knowledge, are integrated as entities' attributes or entity-pairs' attributes into different adjacency matrices for the learning. All the information available is then combined in an additive way. Efficient learning is achieved using an alternating least squares approach exploiting sparse matrix algebra and low-rank approximation. As an illustration, several algorithms are proposed to include information extraction, deductive reasoning and contextual information in matrix factorizations for the Semantic Web scenario and for recommendation systems. Experiments on various data sets are conducted for each proposed algorithm to show the improvement in predictive power by combining matrix factorizations with prior knowledge in a modular way. In contrast to a matrix, a 3-way tensor si a more natural representation for the multirelational data where entities are connected by different types of relations. A 3-way tensor is a three dimensional array which represents the multirelational data by using the first two dimensions for entities and using the third dimension for different types of relations. In the thesis, an analysis on the computational complexity of tensor models shows that the decomposition rank is key for the success of an efficient tensor decomposition algorithm, and that the factorization rank can be reduced by including observable patterns. Based on these theoretical considerations, a second contribution of this thesis develops a novel tensor decomposition approach - an Additive Relational Effects (ARE) model - which combines the strengths of factorization approaches and prior knowledge in an additive way to discover different relational effects from the relational data. As a result, ARE consists of a decomposition part which derives the strong relational leaning effects from a highly scalable tensor decomposition approach RESCAL and a Tucker 1 tensor which integrates the prior knowledge as instantiated relations. An efficient least squares approach is proposed to compute the combined model ARE. The additive model contains weights that reflect the degree of reliability of the prior knowledge, as evaluated by the data. Experiments on several benchmark data sets show that the inclusion of prior knowledge can lead to better performing models at a low tensor rank, with significant benefits for run-time and storage requirements. In particular, the results show that ARE outperforms state-of-the-art relational learning algorithms including intuitive models such as MRC, which is an approach based on Markov Logic with structure learning, factorization approaches such as Tucker, CP, Bayesian Clustered Tensor Factorization (BCTF), the Latent Factor Model (LFM), RESCAL, and other latent models such as the IRM. A final experiment on a Cora data set for paper topic classification shows the improvement of ARE over RESCAL in both predictive power and runtime performance, since ARE requires a significantly lower rank.

Forest-fire processes were first introduced in the physics literature as a toy model for self-organized criticality. The term self-organized criticality describes interacting particle systems which are governed by local interactions and are inherently driven towards a perpetual critical state. As in equilibrium statistical physics, the critical state is characterized by long-range correlations, power laws, fractal structures and self-similarity. We study several different forest-fire models, whose common features are the following: All models are continuous-time processes on the vertices of some graph. Every vertex can be "vacant" or "occupied by a tree". We start with some initial configuration. Then the process is governed by two competing random mechanisms: On the one hand, vertices become occupied according to rate 1 Poisson processes, independently of one another. On the other hand, occupied clusters are "set on fire" according to some predefined rule. In this case the entire cluster is instantaneously destroyed, i.e. all of its vertices become vacant. The self-organized critical behaviour of forest-fire models can only occur on infinite graphs such as planar lattices or infinite trees. However, in all relevant versions of forest-fire models, the destruction mechanism is a priori only well-defined for finite graphs. For this reason, one starts with a forest-fire model on finite subsets of an infinite graph and then takes the limit along increasing sequences of finite subsets to obtain a new forest-fire model on the infinite graph. In this thesis, we perform this kind of limit for two classes of forest-fire models and investigate the resulting limit processes.

Mon, 15 Dec 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/17923/ https://edoc.ub.uni-muenchen.de/17923/1/Huber_Josef_Georg.pdf Huber, Josef Georg ddc:510, ddc:500, Fakultät für Mathematik,

IT infrastructures can be quantitatively described by attributes, like performance or energy efficiency. Ever-changing user demands and economic attempts require varying short-term and long-term decisions regarding the alignment of an IT infrastructure and particularly its attributes to this dynamic surrounding. Potentially conflicting attribute goals and the central role of IT infrastructures presuppose decision making based upon reasoning, the process of forming inferences from facts or premises. The focus on specific IT infrastructure parts or a fixed (small) attribute set disqualify existing reasoning approaches for this intent, as they neither cover the (complex) interplay of all IT infrastructure components simultaneously, nor do they address inter- and intra-attribute correlations sufficiently. This thesis presents a process model for the integrated reasoning about quantitative IT infrastructure attributes. The process model’s main idea is to formalize the compilation of an individual reasoning function, a mathematical mapping of parametric influencing factors and modifications on an attribute vector. Compilation bases upon model integration to benefit from the multitude of existing specialized, elaborated, and well-established attribute models. The achieved reasoning function consumes an individual tuple of IT infrastructure components, attributes, and external influencing factors to expose a broad applicability. The process model formalizes a reasoning intent in three phases. First, reasoning goals and parameters are collected in a reasoning suite, and formalized in a reasoning function skeleton. Second, the skeleton is iteratively refined, guided by the reasoning suite. Third, the achieved reasoning function is employed for What-if analyses, optimization, or descriptive statistics to conduct the concrete reasoning. The process model provides five template classes that collectively formalize all phases in order to foster reproducibility and to reduce error-proneness. Process model validation is threefold. A controlled experiment reasons about a Raspberry Pi cluster’s performance and energy efficiency to illustrate feasibility. Besides, a requirements analysis on a world-class supercomputer and on the European-wide execution of hydro meteorology simulations as well as a related work examination disclose the process model’s level of innovation. Potential future work employs prepared automation capabilities, integrates human factors, and uses reasoning results for the automatic generation of modification recommendations.

Music has been the subject of formal approaches for a long time, ranging from Pythagoras’ elementary research on tonal systems to J. S. Bach’s elaborate formal composition techniques. Especially in the 20th century, much music was composed based on formal techniques: Algorithmic approaches for composing music were developed by composers like A. Schoenberg as well as in the scientific area. So far, a variety of mathematical techniques have been employed for composing music, e.g. probability models, artificial neural networks or constraint-based reasoning. In the recent time, interactive music systems have become popular: existing songs can be replayed with musical video games and original music can be interactively composed with easy-to-use applications running e.g. on mobile devices. However, applications which algorithmically generate music in real-time based on user interaction are mostly experimental and limited in either interactivity or musicality. There are many enjoyable applications but there are also many opportunities for improvements and novel approaches. The goal of this work is to provide a general and systematic approach for specifying and implementing interactive music systems. We introduce an algebraic framework for interactively composing music in real-time with a reasoning-technique called ‘soft constraints’: this technique allows modeling and solving a large range of problems and is suited particularly well for problems with soft and concurrent optimization goals. Our framework is based on well-known theories for music and soft constraints and allows specifying interactive music systems by declaratively defining ‘how the music should sound’ with respect to both user interaction and musical rules. Based on this core framework, we introduce an approach for interactively generating music similar to existing melodic material. With this approach, musical rules can be defined by playing notes (instead of writing code) in order to make interactively generated melodies comply with a certain musical style. We introduce an implementation of the algebraic framework in .NET and present several concrete applications: ‘The Planets’ is an application controlled by a table-based tangible interface where music can be interactively composed by arranging planet constellations. ‘Fluxus’ is an application geared towards musicians which allows training melodic material that can be used to define musical styles for applications geared towards non-musicians. Based on musical styles trained by the Fluxus sequencer, we introduce a general approach for transforming spatial movements to music and present two concrete applications: the first one is controlled by a touch display, the second one by a motion tracking system. At last, we investigate how interactive music systems can be used in the area of pervasive advertising in general and how our approach can be used to realize ‘interactive advertising jingles’.

This thesis contains results on singularity of nearcritical percolation scaling limits, on a rigidity estimate and on spontaneous rotational symmetry breaking. First it is shown that - on the triangular lattice - the laws of scaling limits of nearcritical percolation exploration paths with different parameters are singular with respect to each other. This generalises a result of Nolin and Werner, using a similar technique. As a corollary, the singularity can even be detected from an infinitesimal initial segment. Moreover, nearcritical scaling limits of exploration paths are mutually singular under scaling maps. Second full scaling limits of planar nearcritical percolation are investigated in the Quad-Crossing-Topology introduced by Schramm and Smirnov. It is shown that two nearcritical scaling limits with different parameters are singular with respect to each other. This result holds for percolation models on rather general lattices, including bond percolation on the square lattice and site percolation on the triangular lattice. Third a rigidity estimate for 1-forms with non-vanishing exterior derivative is proven. It generalises a theorem on geometric rigidity of Friesecke, James and Müller. Finally this estimate is used to prove a kind of spontaneous breaking of rotational symmetry for some models of crystals, which allow almost all kinds of defects, including unbounded defects as well as edge, screw and mixed dislocations, i.e. defects with Burgers vectors.

Tue, 11 Nov 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/18157/ https://edoc.ub.uni-muenchen.de/18157/1/Kugler_Johannes.pdf Kugler, Johannes ddc:510, ddc:500, Fakultät für Mathematik, Informatik und Statist

The mean prediction error of a classification or regression procedure can be estimated using resampling designs such as the cross-validation design. We decompose the variance of such an estimator associated with an arbitrary resampling procedure into a small linear combination of covariances between elementary estimators, each of which is a regular parameter as described in the theory of $U$-statistics. The enumerative combinatorics of the occurrence frequencies of these covariances govern the linear combination's coefficients and, therefore, the variance's large scale behavior. We study the variance of incomplete U-statistics associated with kernels which are partly but not entirely symmetric. This leads to asymptotic statements for the prediction error's estimator, under general non-empirical conditions on the resampling design. In particular, we show that the resampling based estimator of the average prediction error is asymptotically normally distributed under a general and easily verifiable condition. Likewise, we give a sufficient criterion for consistency. We thus develop a new approach to understanding small-variance designs as they have recently appeared in the literature. We exhibit the $U$-statistics which estimate these variances. We present a case from linear regression where the covariances between the elementary estimators can be computed analytically. We illustrate our theory by computing estimators of the studied quantities in an artificial data example.

We consider the mean prediction error of a classification or regression procedure as well as its cross-validation estimates, and investigate the variance of this estimate as a function of an arbitrary cross-validation design. We decompose this variance into a scalar product of coefficients and certain covariance expressions, such that the coefficients depend solely on the resampling design, and the covariances depend solely on the data's probability distribution. We rewrite this scalar product in such a form that the initially large number of summands can gradually be decreased down to three under the validity of a quadratic approximation to the core covariances. We show an analytical example in which this quadratic approximation holds true exactly. Moreover, in this example, we show that the leave-p-out estimator of the error depends on p only by means of a constant and can, therefore, be written in a much simpler form. Furthermore, there is an unbiased estimator of the variance of K-fold cross-validation, in contrast to a claim in the literature. As a consequence, we can show that Balanced Incomplete Block Designs have smaller variance than K-fold cross-validation. In a real data example from the UCI machine learning repository, this property can be confirmed. We finally show how to find Balanced Incomplete Block Designs in practice.

We consider the mean prediction error of a classification or regression procedure as well as its cross-validation estimates, and investigate the variance of this estimate as a function of an arbitrary cross-validation design. We decompose this variance into a scalar product of coefficients and certain covariance expressions, such that the coefficients depend solely on the resampling design, and the covariances depend solely on the data's probability distribution. We rewrite this scalar product in such a form that the initially large number of summands can gradually be decreased down to three under the validity of a quadratic approximation to the core covariances. We show an analytical example in which this quadratic approximation holds true exactly. Moreover, in this example, we show that the leave-p-out estimator of the error depends on p only by means of a constant and can, therefore, be written in a much simpler form. Furthermore, there is an unbiased estimator of the variance of K-fold cross-validation, in contrast to a claim in the literature. As a consequence, we can show that Balanced Incomplete Block Designs have smaller variance than K-fold cross-validation. In a real data example from the UCI machine learning repository, this property can be confirmed. We finally show how to find Balanced Incomplete Block Designs in practice.

Although each statistical unit on which measurements are taken is unique, typically there is not enough information available to account totally for its uniqueness. Therefore heterogeneity among units has to be limited by structural assumptions. One classical approach is to use random effects models which assume that heterogeneity can be described by distributional assumptions. However, inference may depend on the assumed mixing distribution and it is assumed that the random effects and the observed covariates are independent. An alternative considered here, are fixed effect models, which let each unit have its own parameter. They are quite flexible but suffer from the large number of parameters. The structural assumption made here is that there are clusters of units that share the same effects. It is shown how clusters can be identified by tailored regularized estimators. Moreover, it is shown that the regularized estimates compete well with estimates for the random effects model, even if the latter is the data generating model. They dominate if clusters are present.

Missing data is an important issue in almost all fields of quantitative research. A nonparametric procedure that has been shown to be useful is the nearest neighbor imputation method. We suggest a weighted nearest neighbor imputation method based on Lq-distances. The weighted method is shown to have smaller imputation error than available NN estimates. In addition we consider weighted neighbor imputation methods that use selected distances. The careful selection of distances that carry information on the missing values yields an imputation tool that outperforms competing nearest neighbor methods distinctly. Simulation studies show that the suggested weighted imputation with selection of distances provides the smallest imputation error, in particular when the number of predictors is large. In addition, the selected procedure is applied to real data from different fields.

Thu, 2 Oct 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/17574/ https://edoc.ub.uni-muenchen.de/17574/1/Hausmann_Steffen.pdf Hausmann, Steffen ddc:510, ddc:500, Fakultät für Mathematik, Informatik und Statistik 0

This thesis presents several non-parametric and parametric models for estimating dynamic dependence between financial time series and evaluates their ability to precisely estimate risk measures. Furthermore, the different dependence models are used to analyze the integration of emerging markets into the world economy. In order to analyze numerous dependence structures and to discover possible asymmetries, two distinct model classes are investigated: the multivariate GARCH and Copula models. On the theoretical side a new dynamic dependence structure for multivariate Archimedean Copulas is introduced which lifts the prevailing restriction to two dimensions and extends the multivariate dynamic Archimedean Copulas to more than two dimensions. On this basis a new mixture copula is presented using the newly invented multivariate dynamic dependence structure for the Archimedean Copulas and mixing it with multivariate elliptical copulas. Simultaneously a new process for modeling the time-varying weights of the mixture copula is introduced: this specification makes it possible to estimate various dependence structures within a single model. The empirical analysis of different portfolios shows that all equity portfolios and the bond portfolios of the emerging markets exhibit negative asymmetries, i.e. increasing dependence during market downturns. However, the portfolio consisting of the developed market bonds does not show any negative asymmetries. Overall, the analysis of the risk measures reveals that parametric models display portfolio risk more precisely than non-parametric models. However, no single parametric model dominates all other models for all portfolios and risk measures. The investigation of dependence between equity and bond portfolios of developed countries, proprietary, and secondary emerging markets reveals that secondary emerging markets are less integrated into the world economy than proprietary. Thus, secondary emerging markets are more suitable to diversify a portfolio consisting of developed equity or bond indices than proprietary

The topic of this thesis is a mathematical treatment of Anderson's orthogonality catastrophe. Named after P.W. Anderson, who studied the phenomenon in the late 1960s, the catastrophe is an intrinsic effect in Fermi gases. In his first work on the topic in [Phys. Rev. Lett. 18:1049--1051], Anderson studied a system of $N$ noninteracting fermions in three space dimensions and found the ground state to be asymptotically orthogonal to the ground state of the same system perturbed by a finite-range scattering potential. More precisely, let $Phi_L^N$ be the $N$-body ground state of the fermionic system in a $d$-dimensional box of length $L$,and let $Psi_L^N$ be the ground state of the corresponding system in the presence of the additional finite-range potential. Then the catastrophe brings about the asymptotic vanishing $$S_L^N := sim L^{-gamma/2}$$ of the overlap $S_L^N$ of the $N$-body ground states $Phi_L^N$ and $Psi_L^N$. The asymptotics is in the thermodynamic limit $Ltoinfty$ and $Ntoinfty$ with fixed density $N/L^dtovarrho > 0$. In [Commun. Math. Phys. 329:979--998], the overlap $S_L^N$ has been bounded from above with an asymptotic bound of the form $$abs{S_L^N}^2 lesssim L^{-tilde{gamma}}$$. The decay exponent $tilde{gamma}$ there corresponds to the one of Anderson in [Phys. Rev. Lett. 18:1049--1051]. Another publication by Anderson from the same year, [Phys. Rev. 164:352--359], contains the exact asymptotics with a bigger coefficient $gamma$. This thesis features a step towards the exact asymptotics. We prove a bound with a coefficient $gamma$ that corresponds in a certain sense to the one in [Phys. Rev. 164:352--359], and improves upon the one in [Commun. Math. Phys. 329:979--998]. We use the methods from [Commun. Math. Phys. 329:979--998], but treat every term in a series expansion of $ln S_L^N$, instead of only the first one. Treating the higher order terms introduces additional arguments since the trace expressions occurring are no longer necessarily nonnegative, which complicates some of the estimates. The main contents of this thesis will also be published in a forthcoming article co-authored with Martin Gebert, Peter Müller, and Peter Otte.

Generalized linear and additive models are very efficient regression tools but the selection of relevant terms becomes difficult if higher order interactions are needed. In contrast, tree-based methods also known as recursive partitioning are explicitly designed to model a specific form of interaction but with their focus on interaction tend to neglect the main effects. The method proposed here focusses on the main effects of categorical predictors by using tree type methods to obtain clusters. In particular when the predictor has many categories one wants to know which of the categories have to be distinguished with respect to their effect on the response. The tree-structured approach allows to detect clusters of categories that share the same effect while letting other variables, in particular metric variables, have a linear or additive effect on the response. An algorithm for the fitting is proposed and various stopping criteria are evaluated. The preferred stopping criterion is based on p-values representing a conditional inference procedure. In addition, stability of clusters are investigated and the relevance of variables is investigated by bootstrap methods. Several applications show the usefulness of tree-structured clustering and a small simulation study demonstrates that the fitting procedure works well.

To perform model selection in the context of multivariable regression, automated variable selection procedures such as backward elimination are commonly employed. However, these procedures are known to be highly unstable. Their stability can be investigated using bootstrap-based procedures: the idea is to perform model selection on a high number of bootstrap samples successively and to examine the obtained models, for instance in terms of the inclusion of specific predictor variables. However, from the literature such bootstrap-based procedures are known to yield misleading results in some cases. In this paper we aim to thoroughly investigate a particular important facet of these problems. More precisely, we assess the behaviour of regression models--with automated variable selection procedure based on the likelihood ratio test--fitted on bootstrap samples drawn with replacement and on subsamples drawn without replacement with respect to the number and type of included predictor variables. Our study includes both extensive simulations and a real data example from the NHANES study. The results indicate that models derived from bootstrap samples include more predictor variables than models fitted on original samples and that categorical predictor variables with many categories are preferentially selected over categorical predictor variables with fewer categories and over metric predictor variables. We conclude that using bootstrap samples to select variables for multivariable regression models may lead to overly complex models with a preferential selection of categorical predictor variables with many categories. We suggest the use of subsamples instead of bootstrap samples to bypass these drawbacks.

Mon, 21 Jul 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/17204/ https://edoc.ub.uni-muenchen.de/17204/1/Neofytidis_Christoforos.pdf Neofytidis, Christoforos ddc:510, ddc:500, Fakultät für Mathematik, Informatik und Stat

In this thesis, we investigate the question when a non-compact manifold can be quasi-isometric to a leaf in a foliation of a compact manifold. The point of departure is the result of Paul Schweitzer's that every non-compact manifold carries a Riemannian metric so that the resulting Riemannian manifold is not quasi-isometric to a leaf in a codimension one foliation of a compact manifold. We show that the coarse homology of these non-leaves is not finitely generated. This observation motivates the main question of this thesis: Does every leaf in a foliation of a compact manifold have finitely generated coarse homology? The answer to this question is a double negative: Firstly, we show that there exists a large class of two-dimensional leaves in codimension one foliations that have non-finitely generated coarse homology. Moreover, we improve Schweitzer's construction by showing that every Riemannian metric can be deformed to a codimension one non-leaf without affecting the coarse homology. In particular, we find non-leaves with trivial coarse homology. In order to answer these questions we develop computational tools for the coarse homology. Furthermore, we show that certain known criteria for manifolds to be a leaf are independent of one another and of the coarse homology.

In recent years, Hilbert's Programme has been resumed within the framework of constructive mathematics. This undertaking has already shown its feasability for a considerable part of commutative algebra. In particular, point-free methods have been playing a primary role, emerging as the appropriate language for expressing the interplay between real and ideal in mathematics. This dissertation is written within this tradition and has Sambin's notion of formal topology at its core. We start by developing general tools, in order to make this notion more immediate for algebraic application. We revise the Zariski spectrum as an inductively generated basic topology, and we analyse the constructive status of the corresponding principles of spatiality and reducibility. Through a series of examples, we show how the principle of spatiality is recurrent in the mathematical practice. The tools developed before are applied to specific problems in constructive algebra. In particular, we find an elementary characterization of the notion of codimension for ideals of a commutative ring, by means of which a constructive version of Krull's principal ideal theorem can be stated and proved. We prove a formal version of the projective Eisenbud-Evans-Storch theorem. Finally, guided by the algebraic intuition, we present an application in constructive domain theory, by proving a finite version of Kleene-Kreisel density theorem for non-flat information systems.

Wed, 9 Jul 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/17538/ https://edoc.ub.uni-muenchen.de/17538/1/Moest_Stephanie.pdf Möst, Stephanie ddc:510, ddc:500, Fakultät für Mathematik, Informatik und Statistik

When in a linear GMM model nuisance parameters are eliminated by multiplying the moment conditions by a projection matrix, the covariance matrix of the model, the inverse of which is typically used to construct an efficient GMM estimator, turns out to be singular and thus cannot be inverted. However, one can show that the generalized inverse can be used instead to produce an efficient estimator. Various other matrices in place of the projection matrix do the same job, i.e., they eliminate the nuisance parameters. The relations between those matrices with respect to the efficiency of the resulting estimators are investigated.

Quantitative analysis by means of discrete-state stochastic processes is hindered by the well-known phenomenon of state-space explosion, whereby the size of the state space may have an exponential growth with the number of objects in the model. When the stochastic process underlies a Markovian process algebra model, this problem may be alleviated by suitable notions of behavioural equivalence that induce lumping at the underlying continuous-time Markov chain, establishing an exact relation between a potentially much smaller aggregated chain and the original one. However, in the modelling of massively distributed computer systems, even aggregated chains may be still too large for efficient numerical analysis. Recently this problem has been addressed by fluid techniques, where the Markov chain is approximated by a system of ordinary differential equations (ODEs) whose size does not depend on the number of the objects in the model. The technique has been primarily applied in the case of massively replicated sequential processes with small local state space sizes. This thesis devises two different approaches that broaden the scope of applicability of efficient fluid approximations. Fluid lumpability applies in the case where objects are composites of simple objects, and aggregates the potentially massive, naively constructed ODE system into one whose size is independent from the number of composites in the model. Similarly to quasi and near lumpability, we introduce approximate fluid lumpability that covers ODE systems which can be aggregated after a small perturbation in the parameters. The technique of spatial aggregation, instead, applies to models whose objects perform a random walk on a two-dimensional lattice. Specifically, it is shown that the underlying ODE system, whose size is proportional to the number of the regions, converges to a system of partial differential equations of constant size as the number of regions goes to infinity. This allows for an efficient analysis of large-scale mobile models in continuous space like ad hoc networks and multi-agent systems.

The physical picture of particle behaviour that arises from experimental data is that it belongs to one of the following: Either the particle always stays near the origin or the particle comes from infinity, is scattered and escapes to infinity. In the quantum mechanical description these two categories of behaviour are associated with point spectrum and absolutely continuous spectrum respectively. The corresponding spectral measures are point measures or absolutely continuous measures. According to general results of the measure theory a measure can be decomposed in three parts: a pure point part, an absolutely continuous part and a singular continuous part. In contrast to the well-known particle behaviour of the other two types of spectrum, the singular continuous spectrum is more difficult to interpret. For the Schrödinger operator D.B. Pearson constructed an explicit class of potentials that give rise to purely singular continuous spectrum . This example allows the interpretation of the particle behaviour: The particle moves arbitrarily far away from the origin but it feels nevertheless the effect of the potential. Therefore it will recur infinitely often to the vicinity of the origin to run off infinitely often. The result for the Schröodinger operator leads to the question whether there can be found similar results in relativistic quantum mechanics. The aim of this paper is to construct for the first time an explicit potential for the Dirac operator that has purely singular continuous spectrum in (-infty,-1)cup (1,infty). The characteristic trait of this potential is that it consists of bumps whose distance is growing rapidly. This allows the particle to depart from the origin arbitrarily far. But the overall effect of the bumps will always lead the particle back to the origin.

In competing risks models one distinguishes between several distinct target events that end duration. Since the effects of covariates are specific to the target events, the model contains a large number of parameters even when the number of predictors is not very large. Therefore, reduction of the complexity of the model, in particular by deletion of all irrelevant predictors, is of major importance. A selection procedure is proposed that aims at selection of variables rather than parameters. It is based on penalization techniques and reduces the complexity of the model more efficiently than techniques that penalize parameters separately. An algorithm is proposed that yields stable estimates. We consider reduction of complexity by variable selection in two applications, the evolution of congressional careers of members of the US congress and the duration of unemployment.

Credit scoring models are the basis for financial institutions like retail and consumer credit banks. The purpose of the models is to evaluate the likelihood of credit applicants defaulting in order to decide whether to grant them credit. The area under the receiver operating characteristic (ROC) curve (AUC) is one of the most commonly used measures to evaluate predictive performance in credit scoring. The aim of this thesis is to benchmark different methods for building scoring models in order to maximize the AUC. While this measure is used to evaluate the predictive accuracy of the presented algorithms, the AUC is especially introduced as direct optimization criterion. The logistic regression model is the most widely used method for creating credit scorecards and classifying applicants into risk classes. Since this development process, based on the logit model, is standard in the retail banking practice, the predictive accuracy of this proceeding is used for benchmark reasons throughout this thesis. The AUC approach is a main task introduced within this work. Instead of using the maximum likelihood estimation, the AUC is considered as objective function to optimize it directly. The coefficients are estimated by calculating the AUC measure with Wilcoxon-Mann-Whitney and by using the Nelder-Mead algorithm for the optimization. The AUC optimization denotes a distribution-free approach, which is analyzed within a simulation study for investigating the theoretical considerations. It can be shown that the approach still works even if the underlying distribution is not logistic. In addition to the AUC approach and classical well-known methods like generalized additive models, new methods from statistics and machine learning are evaluated for the credit scoring case. Conditional inference trees, model-based recursive partitioning methods and random forests are presented as recursive partitioning algorithms. Boosting algorithms are also explored by additionally using the AUC as a loss function. The empirical evaluation is based on data from a German bank. From the application scoring, 26 attributes are included in the analysis. Besides the AUC, different performance measures are used for evaluating the predictive performance of scoring models. While classification trees cannot improve predictive accuracy for the current credit scoring case, the AUC approach and special boosting methods provide outperforming results compared to the robust classical scoring models regarding the predictive performance with the AUC measure.

Treatment efficacy in clinical trials is often assessed by time from treatment initiation to occurrence of a certain critical or beneficial event. In most cases the event of interest cannot be observed for all patients, as patients are only followed for a limited time or contact to patients is lost during their follow-up time. Therefore, certain methods were developed in the framework of the so called time-to-event or survival analysis, in order to obtain valid and consistent estimates in the presence of these "censored observations", using all available information. In classical event time analysis only one endpoint exists, as the death of a patient. As patients can die from different causes, in some clinical trials time to one out of two or more mutually exclusive types of event may be of interest. In many oncological studies, for example, time to cancer-specific death is considered as primary endpoint with deaths from other causes acting as so called competing risks. Different methods for data analysis in the competing risks framework were developed in recent years, which either focus on modelling the cause-specific or the subdistribution hazard rate or split the joint distribution of event times and event types into quantities, that can be estimated from observable data. In this work the analysis of event time data in the presence of competing risks is described, including the presentation and discussion of different regression approaches. A major topic of this work is the estimation of cause-specific and subdistribution hazard rates from a mixture model and a new approach using penalized B-splines (P-splines) for estimation of conditional hazard rates in a mixture model is proposed. In order to evaluate the behaviour of the new approach, a simulation study was conducted, using simulation techniques for competing risks data, which are described in detail in this work. The presented regression models were applied to data from a clinical cohort study investigating a risk stratification for cardiac mortality in patients, that survived a myocardial infarction. Finally, the use of the presented methods for event time analysis in the presence of competing risks and results obtained from the simulation study and the data analysis are discussed.

In this work, we derive the time-dependent Hartree(-Fock) equations as an effective dynamics for fermionic many-particle systems. Our main results are the first for a quantum mechanical mean-field dynamics for fermions; in previous works, the mean-field limit is usually either coupled to a semiclassical limit, or the interaction is scaled down so much, that the system behaves freely for large particle number N. We mainly consider systems with total kinetic energy bounded by const N and long-range interaction potentials, e.g., Coulomb interaction. Examples for such systems are large molecules or certain solid states. Our analysis also applies to attractive interactions, as, e.g., in fermionic stars. The fermionic Hartree(-Fock) equations are a standard tool to describe, e.g., excited states or chemical reactions of large molecules (like proteins). A deeper understanding of these equations as an approximation to the time evolution of a many body quantum system is thus highly relevant. We consider the fermionic Hatree equations (i.e., the Hartree-Fock equations without exchange term) in this work, since the exchange term is subleading in our setting. The main result is that the fermionic Hartree dynamics approximates the Schrödinger dynamics well for large N. This statement becomes exact in the thermodynamic limit N to infinity. We give explicit values for the rates of convergence. We prove two types of results. The first type is very general and concerns arbitrary free Hamiltonians (e.g., relativistic, non-relativistic, with external fields) and arbitrary interactions. The theorems give explicit conditions on the solutions to the fermonic Hartree equations under which a derivation of the mean-field dynamics succeeds. The second type of results scrutinizes situations where the conditions are fulfilled. These results are about non-relativistic free Hamiltonians with external fields, systems with total kinetic energy bounded by const N and with long-range interactions of the form x^(-s), with 0 < s < 6/5 (sometimes, for technical reasons, with a weaker or cut off singularity). We prove our main results by using a new method for deriving mean-field dynamics developed by Pickl in [Lett. Math. Phys., 97(2):151-164, 2011]. This method has been applied successfully in quantum mechanics for deriving the bosonic Hartree and Gross-Pitaevskii equations. Its application to fermions in this work is new. The method is based on a functional that "counts the number of particles outside the condensate", i.e., in the case of fermions, it measures those parts of the Schrödinger wave function that are not in the antisymmetric product of the Hartree states. We show that convergence of the functional to zero (which means that the mean-field equations approximate the dynamics well) is equivalent to convergence of the corresponding reduced one-particle density matrices in trace norm and in Hilbert-Schmidt norm. Finally, we show how also the recently treated semiclassical mean-field limits can be derived with this method.

Tue, 29 Apr 2014 12:00:00 +0100 https://edoc.ub.uni-muenchen.de/18656/ https://edoc.ub.uni-muenchen.de/18656/1/Sinibaldi_Jennifer.pdf Sinibaldi, Jennifer ddc:510, ddc:500, Fakultät für Mathematik, Informat

Regression models with lagged covariate effects are often used in biostatistical and geo- physical data analysis. In the difficult and all-important subject of earthquake research, strong long-lasting rainfall is assumed to be one of many complex trigger factors that lead to earthquakes. Geophysicists interpret the rain effect with an increase of pore pressure due to the infiltra- tion of rain water over a long time period. Therefore, a sensible statistical regression model examining the influence of rain on the number of earthquakes on day t has to contain rain information of day t and of preceding days t − 1 to t − L. In the first part of this thesis, the specific shape of lagged rain influence on the number of earthquakes is modeled. A novel penalty structure for interpretable and flexible estimates of lag coefficients based on spline representations is presented. The penalty structure enables smoothness of the resulting lag course and a shrinkage towards zero of the last lag coefficient via a ridge penalty. This additional ridge penalty offers an approach to another problem neglected in previous work. With the help of the additional ridge penalty, a suboptimal choice of the lag length L is no longer critical. We propose the use of longer lags, as our simulations indicate that superfluous coefficients are correctly estimated close to zero. We provide a user-friendly implementation of our flexible distributed lag (FDL) ap- proach, that can be used directly in the established R package mgcv for estimation of generalized additive models. This allows our approach to be immediately included in com- plex additive models for generalized responses even in hierarchical or longitudinal data settings, making use of established stable and well-tested algorithms. We demonstrate the performance and utility of the proposed flexible distributed lag model in a case study on (micro-) earthquake data from Mount Hochstaufen, Bavaria with focus on the specific shape of the lagged rain influence on the occurrence of earthquakes in different depths. The complex meteorological and geophysical data set was collected and provided by the Geophysical Observatory of the Ludwig-Maximilians University Munich. The benefit of flexible distributed lag modeling is shown in a detailed simulation study. In the second part of the thesis, the penalization concept is extended to lagged non- linear covariate influence. Here, we extend an approach of Gasparrini et al. (2010), that was up to now unpenalized. Detailed simulation studies illustrate again the benefits of the penalty structure. The flexible distributed lag nonlinear model is applied to data of the volcano Merapi in Indonesia, collected and provided by the Geophysical Observatory in Fürstenfeldbruck. In this data set, the specific shape of lagged rain influence on the occurrence of block and ash flows is examined.

This paper proposes a discrete-time hazard regression approach based on the relation between hazard rate models and excess over threshold models, which are frequently encountered in extreme value modelling. The proposed duration model employs a flexible link function and incorporates the grouped-duration analogue of the well-known Cox proportional hazards model and the proportional odds model as special cases. The theoretical setup of the model is motivated, and simulation results are reported to suggest that it performs well. The simulation results and an empirical analysis of US import durations also show that the choice of link function in discrete hazard models has important implications for the estimation results, and that severe biases in the results can be avoided when using a flexible link function as proposed in this study.

This thesis is concerned with quantum mechanical decay processes and their mathematical description. It consists out of three parts: In the first part we look at Laser induced ionization, whose mathematical description is often based on the so-called dipole approximation. Employing it essentially means to replace the Laser's vector potential $vec A(vec r,t)$ in the Hamiltonian by $vec A(0, t).$ Heuristically this is justified under usual experimental conditions, because the Laser varies only slowly in $vec r$ on atomic length scales. We make this heuristics rigorous by proving the dipole approximation in the limit in which the Laser's length scale becomes infinite compared to the atomic length scale. Our results apply to $N$-body Hamiltonians. In the second part we look at alpha decay as described by Skibsted (Comm. Math. Phys. 104, 1986) and show that Skibsted's model satisfies an energy-time uncertainty relation. Since there is no self-adjoint time operator, the uncertainty relation for energy and time can not be proven in the same way as the uncertainty relation for position and momentum. To define the time variance without a self-adjoint time operator, we will use the arrival time distribution obtained from the quantum current. Our proof of the energy-time uncertainty relation is then based on the quantitative scattering estimates that will be derived in the third part of the thesis and on a result from Skibsted. In addition to that, we will show that this uncertainty relation is different from the well known {it linewidth-lifetime relation}. The third part is about quantitative scattering estimates, which are of interest in their own right. For rotationally symmetric potentials having support in $[0,R_V]$ we will show that for $Rgeq R_V$, the time evolved wave function $e^{-iHt}psi$ satisfies begin{align}nonumber |1_R e^{-iHt}psi|_2^2leq c_1t^{-1}+c_2t^{-2}+c_3t^{-3}+c_4t^{-4} end{align} with explicit quantitative bounds on the constants $c_n$ in terms of the resonances of the $S$-Matrix. While such bounds on $|1_R e^{-iHt}psi|_2$ have been proven before, the quantitative estimates on the constants $c_n$ are new. These results are based on a detailed analysis of the $S$-matrix in the complex momentum plane, which in turn becomes possible by expressing the $S$-matrix in terms of the Jost function that can be factorized in a Hadamard product.