Pointwise construction of Lippschitz aggregation operators with specific properties

This paper describes an approach to pointwise construction of general aggregation operators, based on monotone Lipschitz approximation. The aggregation operators are constructed from a set of desired values at certain points, or from empirically collected data. It establishes tight upper and lower bounds on Lipschitz aggregation operators with a number of different properties, as well as the optimal aggregation operator, consistent with the given values. We consider conjunctive, disjunctive and idempotent n-ary aggregation operators; p-stable aggregation operators; various choices of the neutral element and annihilator; diagonal, opposite diagonal and marginal sections; bipolar and double aggregation operators. In all cases we provide either explicit formulas or deterministic numerical procedures to determine the bounds. The findings of this paper are useful for construction of aggregation operators with specified properties, especially using interpolation schemata.<br /

Dependence modeling with applications in financial econometrics

The amount of data available in banking, finance and economics steadily increases due to the ongoing technological progress and the continuing digitalization. A key element of many econometric models for analyzing this data are methods for assessing dependencies, cross-sectionally as well as intertemporally. For this reason, the thesis is centered around statistical and econometric methods for dependence modeling with applications in financial econometrics. The first part of this cumulative dissertation consists of three contributions. The first contribution provides a thorough explanation of the partial copula. It is a natural generalization of the partial correlation coefficient and several of its properties are investigated. In the second contribution, a different multivariate generalization of the partial correlation, the partial vine copula (PVC), is introduced. The PVC is a specific simplified vine copula (SVC) consisting of bivariate higher-order partial copulas, which are copula-based generalizations of sequentially computed partial correlations. Several properties of the PVC are presented and it is shown that, if SVCs are considered as approximations of multivariate distributions, the PVC has a special role as it is the limit of stepwise estimators. The third contribution introduces statistical tests for the simplifying assumption with a special focus on high-dimensional vine copulas. We propose a computationally feasible test for the simplifying assumption in high-dimensions, which is successfully applied to data sets with up to 49 dimensions. The novel test procedure is based on a decision tree which is used to identify the possibly strongest violation of the simplifying assumption. The asymptotic distribution of the test statistic is derived under consideration of estimation uncertainty in the copula parameters. The finite sample performance is analyzed in an extensive simulation study and the results show that the power of the test only slightly decreases in the dimensionality of the test problem. In the second part of the dissertation, the assessment of risk measures is studied with a special focus on the financial return data used for estimation. It is shown that the choice of the sampling scheme can greatly affect the results of risk assessment procedures if the assessment frequency and forecasting horizon are fixed. Specifically, we study sequences of variance estimates and show that they exhibit spurious seasonality, if the assessment frequency is higher than the sampling frequency of non-overlapping return data. The root cause of spurious seasonality is identified by deriving the theoretical autocorrelation function of sequences of variance estimates under general assumptions. To overcome spurious seasonality, alternative variance estimators based on overlapping return data are suggested. The third part of the dissertation is about state space methods for systems with lagged states in the measurement equation. Recently, a low-dimensional modified Kalman filter and smoother for such systems was proposed in the literature. Special attention is paid to the modified Kalman smoother, for which it is shown that the suggested smoother in general does not minimize the mean squared error (MSE). The correct MSE-minimizing modified Kalman smoother is derived and computationally more efficient smoothing algorithms are discussed. Finally, a comparison of the competing smoothers with regards to the MSE is performed

Regularized Optimal Transport and the Rot Mover's Distance

This paper presents a unified framework for smooth convex regularization of discrete optimal transport problems. In this context, the regularized optimal transport turns out to be equivalent to a matrix nearness problem with respect to Bregman divergences. Our framework thus naturally generalizes a previously proposed regularization based on the Boltzmann-Shannon entropy related to the Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We call the regularized optimal transport distance the rot mover's distance in reference to the classical earth mover's distance. We develop two generic schemes that we respectively call the alternate scaling algorithm and the non-negative alternate scaling algorithm, to compute efficiently the regularized optimal plans depending on whether the domain of the regularizer lies within the non-negative orthant or not. These schemes are based on Dykstra's algorithm with alternate Bregman projections, and further exploit the Newton-Raphson method when applied to separable divergences. We enhance the separable case with a sparse extension to deal with high data dimensions. We also instantiate our proposed framework and discuss the inherent specificities for well-known regularizers and statistical divergences in the machine learning and information geometry communities. Finally, we demonstrate the merits of our methods with experiments using synthetic data to illustrate the effect of different regularizers and penalties on the solutions, as well as real-world data for a pattern recognition application to audio scene classification

Nonparametric inference of interaction laws in systems of agents from trajectory data

Inferring the laws of interaction between particles and agents in complex dynamical systems from observational data is a fundamental challenge in a wide variety of disciplines. We propose a non-parametric statistical learning approach to estimate the governing laws of distance-based interactions, with no reference or assumption about their analytical form, from data consisting trajectories of interacting agents. We demonstrate the effectiveness of our learning approach both by providing theoretical guarantees, and by testing the approach on a variety of prototypical systems in various disciplines. These systems include homogeneous and heterogeneous agents systems, ranging from particle systems in fundamental physics to agent-based systems modeling opinion dynamics under the social influence, prey-predator dynamics, flocking and swarming, and phototaxis in cell dynamics

Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows

Copula-based stochastic modelling of evapotranspiration time series conditioned on rainfall as design tool in water resources management

In the last few decades, the frequency and intensity of water-related disasters, also called climate-related disasters, e.g. floods, storms, heat waves and droughts, has gone up considerably at both global and regional scales, causing significant damage to many societies and ecosystems. Understanding the behavior and frequency of these disasters is extremely important, not only for reducing their damages but also for the management of water resources. These disasters can often be characterized by multiple dependent variables and therefore require a flexible multivariate approach for studying such phenomena. In this study, we focus on copulas, which are multivariate functions that describe the dependence structure between stochastic variables, independently of their marginal behaviors. The study aimed at different potential applications of copulas in hydrology, such as a multivariate frequency analysis and a copula-based approach for assessing a rainfall model. And further, a stochastic copula-based evapotranspiration generator was developed. As an application, the potential impacts of climate change on river discharge was investigated partly based the latter generator