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Gaussian processes are a widely embraced technique for regression and classification due to their good prediction accuracy, analytical tractability and built-in capabilities for uncertainty quantification. However, they suffer from the curse of dimensionality whenever the number of variables increases. This challenge is generally addressed by assuming additional structure in theproblem, the preferred options being either additivity or low intrinsic dimensionality. Our contribution for high-dimensional Gaussian process modeling is to combine them with a multi-fidelity strategy, showcasing the advantages through experiments on synthetic functions and datasets.

We investigate the multiplicity model with m values of some test statistic independently drawn from a mixture of no effect (null) and positive effect (alternative), where we seek to identify, the alternative test results with a controlled error rate. We are interested in the case where the alternatives are rare. A number of multiple testing procedures filter the set of ordered p-values in order to eliminate the nulls. Such an approach can only work if the p-values originating from the alternatives form one or several identifiable clusters. The Benjamini and Hochberg (BH) method, for example, assumes that this cluster occurs in a small interval $(0,\Delta)$ and filters out all or most of the ordered p-values $p_{(r)}$ above a linear threshold $s \times r$. In repeated applications this filter controls the false discovery rate via the slope s. We propose a new adaptive filter that deletes the p-values from regions of uniform distribution. In cases where a single cluster remains, the p-values in an interval are declared alternatives, with the mid-point and the length of the interval chosen by controlling the data-dependent FDR at a desired level.

Using dominating sets to separate vertices of graphs is a well-studied problem in the larger domain of identification problems. In such problems, the objective is typically to separate any two vertices of a graph by their unique neighbourhoods in a suitably chosen dominating set of the graph. Such a dominating and separating set is often referred to as a \emph{code} in the literature. Depending on the types of dominating and separating sets used, various problems arise under various names in the literature. In this paper, we introduce a new problem in the same realm of identification problems whereby the code, called the \emph{open-separating dominating code}, or the \emph{OSD-code} for short, is a dominating set and uses open neighbourhoods for separating vertices. The paper studies the fundamental properties concerning the existence, hardness and minimality of OSD-codes. Due to the emergence of a close and yet difficult to establish relation of the OSD-codes with another well-studied code in the literature called the open locating dominating codes, or OLD-codes for short, we compare the two on various graph classes. Finally, we also provide an equivalent reformulation of the problem of finding OSD-codes of a graph as a covering problem in a suitable hypergraph and discuss the polyhedra associated with OSD-codes, again in relation to OLD-codes of some graph classes already studied in this context.

Heteroskedasticity testing in nonparametric regression is a classic statistical problem with important practical applications, yet fundamental limits are unknown. Adopting a minimax perspective, this article considers the testing problem in the context of an $\alpha$-H\"{o}lder mean and a $\beta$-H\"{o}lder variance function. For $\alpha > 0$ and $\beta \in (0, 1/2)$, the sharp minimax separation rate $n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$ is established. To achieve the minimax separation rate, a kernel-based statistic using first-order squared differences is developed. Notably, the statistic estimates a proxy rather than a natural quadratic functional (the squared distance between the variance function and its best $L^2$ approximation by a constant) suggested in previous work. The setting where no smoothness is assumed on the variance function is also studied; the variance profile across the design points can be arbitrary. Despite the lack of structure, consistent testing turns out to still be possible by using the Gaussian character of the noise, and the minimax rate is shown to be $n^{-4\alpha} + n^{-1/2}$. Exploiting noise information happens to be a fundamental necessity as consistent testing is impossible if nothing more than zero mean and unit variance is known about the noise distribution. Furthermore, in the setting where the variance function is $\beta$-H\"{o}lder but heteroskedasticity is measured only with respect to the design points, the minimax separation rate is shown to be $n^{-4\alpha} + n^{-\left((1/2) \vee (4\beta/(4\beta+1))\right)}$ when the noise is Gaussian and $n^{-4\alpha} + n^{-4\beta/(4\beta+1)} + n^{-2\beta}$ when the noise distribution is unknown.

Building prediction models from mass-spectrometry data is challenging due to the abundance of correlated features with varying degrees of zero-inflation, leading to a common interest in reducing the features to a concise predictor set with good predictive performance. In this study, we formally established and examined regularized regression approaches, designed to address zero-inflated and correlated predictors. In particular, we describe a novel two-stage regularized regression approach (ridge-garrote) explicitly modelling zero-inflated predictors using two component variables, comprising a ridge estimator in the first stage and subsequently applying a nonnegative garrote estimator in the second stage. We contrasted ridge-garrote with one-stage methods (ridge, lasso) and other two-stage regularized regression approaches (lasso-ridge, ridge-lasso) for zero-inflated predictors. We assessed the predictive performance and predictor selection properties of these methods in a comparative simulation study and a real-data case study to predict kidney function using peptidomic features derived from mass-spectrometry. In the simulation study, the predictive performance of all assessed approaches was comparable, yet the ridge-garrote approach consistently selected more parsimonious models compared to its competitors in most scenarios. While lasso-ridge achieved higher predictive accuracy than its competitors, it exhibited high variability in the number of selected predictors. Ridge-lasso exhibited slightly superior predictive accuracy than ridge-garrote but at the expense of selecting more noise predictors. Overall, ridge emerged as a favourable option when variable selection is not a primary concern, while ridge-garrote demonstrated notable practical utility in selecting a parsimonious set of predictors, with only minimal compromise in predictive accuracy.

A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.

This note presents a method that provides optimal monotone conditional error functions for a large class of adaptive two stage designs. The presented method builds on a previously developed general theory for optimal adaptive two stage designs where sample sizes are reassessed for a specific conditional power and the goal is to minimize the expected sample size. The previous theory can easily lead to a non-monotonous conditional error function which is highly undesirable for logical reasons and can harm type I error rate control for composite null hypotheses. The here presented method extends the existing theory by introducing intermediate monotonising steps that can easily be implemented.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

The integer autoregressive (INAR) model is one of the most commonly used models in nonnegative integer-valued time series analysis and is a counterpart to the traditional autoregressive model for continuous-valued time series. To guarantee the integer-valued nature, the binomial thinning operator or more generally the generalized Steutel and van Harn operator is used to define the INAR model. However, the distributions of the counting sequences used in the operators have been determined by the preference of analyst without statistical verification so far. In this paper, we propose a test based on the mean and variance relationships for distributions of counting sequences and a disturbance process to check if the operator is reasonable. We show that our proposed test has asymptotically correct size and is consistent. Numerical simulation is carried out to evaluate the finite sample performance of our test. As a real data application, we apply our test to the monthly number of anorexia cases in animals submitted to animal health laboratories in New Zealand and we conclude that binomial thinning operator is not appropriate.

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.