Quantile regression is a field with steadily growing importance in statistical modeling. It is a complementary method to linear regression, since computing a range of conditional quantile functions provides a more accurate modelling of the stochastic relationship among variables, especially in the tails. We introduce a novel nonrestrictive and highly flexible nonparametric quantile regression approach based on C- and D-vine copulas. Vine copulas allow for separate modeling of marginal distributions and the dependence structure in the data, and can be expressed through a graph theoretical model given by a sequence of trees. This way we obtain a quantile regression model, that overcomes typical issues of quantile regression such as quantile crossings or collinearity, the need for transformations and interactions of variables. Our approach incorporates a two-step ahead ordering of variables, by maximizing the conditional log-likelihood of the tree sequence, while taking into account the next two tree levels. We show that the nonparametric conditional quantile estimator is consistent. The performance of the proposed methods is evaluated in both low- and high-dimensional settings using simulated and real world data. The results support the superior prediction ability of the proposed models.

In this paper we develop methodology for testing relevant hypotheses in a tuning-free way. Our main focus is on functional time series, but extensions to other settings are also discussed. Instead of testing for exact equality, for example for the equality of two mean functions from two independent time series, we propose to test a \textit{relevant} deviation under the null hypothesis. In the two sample problem this means that an $L^2$-distance between the two mean functions is smaller than a pre-specified threshold. For such hypotheses self-normalization, which was introduced by Shao (2010) and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses and demonstrate the particular advantages of this approach in the the comparisons of eigenvalues and eigenfunctions.

An extension of the D-vine based forward regression procedure to a R-vine forward regression is proposed. In this extension any R-vine structure can be taken into account. Moreover, a new heuristic is proposed to determine which R-vine structure is the most appropriate to model the conditional distribution of the response variable given the covariates. It is shown in the simulation that the performance of the heuristic is comparable to the D-vine based approach. Furthermore, it is explained how to extend the heuristic into a situation when more than one response variable are of interest. Finally, the proposed R-vine regression is applied to perform a stress analysis on the manufacturing sector which shows its impact on the whole economy.

Reference: Zhu, Kurowicka and Nane. https://doi.org/10.1016/j.csda.2020.107091

Many mathematical models of physical processes contain uncertainties due to incomplete models or measurement errors and lack of knowledge associated with the model inputs. We consider processes which are formulated in terms of classical partial differential equations (PDEs). The challenge and novelty is that the PDEs contain random coefficient functions, e.g., some transformations of Gaussian random fields. Random PDEs are much more flexible and can model more complex situations compared to classical PDEs with deterministic coefficients. However, each sample of a PDE-based model is extremely expensive. To alleviate the high costs the numerical analysis community has developed so-called multilevel estimators which work with a hierarchy of PDE models with different resolution and cost. We review the basic idea of multilevel estimators and discuss our own recent contributions:

i) a multilevel best linear unbiased estimator to approximate the expectation of a scalar output quantity of interest associated with a random PDE [1, 2],

ii) a multilevel sequential Monte Carlo method for Bayesian inverse problems [3],

iii) a multilevel sequential importance method to estimate the probability of rare events [4].

[1] D. Schaden, E. Ullmann: On multilevel best linear unbiased estimators. SIAM/ASA J. Uncert. Quantif. 8(2), pp. 601-635, 2020

[2] D. Schaden, E. Ullmann: Asymptotic analysis of multilevel best linear unbiased estimators, arXiv:2012.03658

[3] J. Latz, I. Papaioannou, E. Ullmann: Multilevel Sequential² Monte Carlo for Bayesian Inverse Problems. J. Comput. Phys., 368, pp. 154-178, 2018

[4] F. Wagner, J. Latz, I. Papaioannou, E. Ullmann: Multilevel sequential importance sampling for rare event estimation. SIAM J. Sci. Comput. 42(4), pp. A2062–A2087, 2020

We propose a coefficient of conditional dependence between two random variables $Y$ and $Z$, given a set of other variables $X_1, \cdots , X_p$, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in $[0, 1]$, where the limit is 0 if and only if $Y$ and $Z$ are conditionally independent given $X_1, \cdots , X_p$, and is 1 if and only if Y is equal to a measurable function of $Z$ given $X_1, \cdots , X_p$. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial $R^2$ statistic for measuring conditional dependence by regression. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.

Many driving factors of physical systems are latent or unobserved. Thus, understanding such systems and producing robust predictions crucially relies on accounting for the influence of the latent structure. I will discuss methodological and theoretical advances in two important problems in latent-variable modeling. The first problem focuses on developing false discovery methods for latent-variable models that are parameterized by low-rank matrices, where the traditional perspective on false discovery control is ill-suited due to the non-discrete nature of the underlying decision spaces. To overcome this challenge, I will present a geometric reformulation of the notion of a discovery as well as a specific algorithm to control false discoveries in these settings. The second problem aims to estimate causal relations among a collection of observed variables with latent effects. Given access to data arising from perturbations (interventions), I will introduce a regularized maximum-likelihood framework that provably identifies the underlying causal structure and improves robustness to distributional changes. Throughout, I will explore the utility of the proposed methodologies for real-world applications such as water resource management.