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We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are…

A simple proof of the convergence of the variational regularization, with the regularization parameter, chosen by the discrepancy principle, is given for linear operators under suitable assumptions. It is shown that the discrepancy…

Data-driven sequential decision has found a wide range of applications in modern operations management, such as dynamic pricing, inventory control, and assortment optimization. Most existing research on data-driven sequential decision…

The classical uncertainty principle of harmonic analysis states that a nontrivial function and its Fourier transform cannot both be sharply localized. It plays an important role in signal processing and physics. This paper generalizes the…

Reliable predictive uncertainty estimation plays an important role in enabling the deployment of neural networks to safety-critical settings. A popular approach for estimating the predictive uncertainty of neural networks is to define a…

Over the last decade, nonparametric methods have gained increasing attention for modeling complex data structures due to their flexibility and minimal structural assumptions. In this paper, we study a general multivariate nonparametric…

We consider the problem of filtering an unseen Markov chain from noisy observations, in the presence of uncertainty regarding the parameters of the processes involved. Using the theory of nonlinear expectations, we describe the uncertainty…

We propose a novel definition of Shapley values with uncertain value functions based on first principles using probability theory. Such uncertain value functions can arise in the context of explainable machine learning as a result of…

This article presents a theoretical study of uncertainty functionals on general measurable spaces. These functionals are fundamental in experimental design and global sensitivity analysis, where they are used to quantify variability and…

Many mathematical models utilize limit processes. Continuous functions and the calculus, differential equations and topology, all are based on limits and continuity. However, when we perform measurements and computations, we can achieve…

A theory of measurement uncertainty is presented, which, since it is based exclusively on the Bayesian approach and on the subjective concept of conditional probability, is applicable in the most general cases. The recent International…

We develop a new paradigm for finding bifurcations of solutions of nonlinear problems, which is based on the detection of extreme values of new type of variational functional associated with the considering problem. The variational…

Many real-world systems modeled using partial differential equations (PDEs) involve unknown parameters that must be estimated from limited, noisy system observations. While typically assumed to be constants, some of these unobserved…

The use of quantiles to obtain insights about multivariate data is addressed. It is argued that incisive insights can be obtained by considering directional quantiles, the quantiles of projections. Directional quantile envelopes are…

Functional data analysis is ubiquitous in most areas of sciences and engineering. Several paradigms are proposed to deal with the dimensionality problem which is inherent to this type of data. Sparseness, penalization, thresholding, among…

We introduce directional regularity, a new definition of anisotropy for multivariate functional data. Instead of taking the conventional view, which determines anisotropy as a notion of smoothness along a dimension, directional regularity…

The pressure function is a fundamental object in various areas of mathematics. Its regularity is studied to derive insights into phase transitions in certain physical systems or to determine the Hausdorff dimension of self-affine sets. In…

We provide constructive necessary and sufficient conditions for a family of periodic wavelets to be a Parseval wavelet frame. The criterion generalizes unitary and oblique extension principles. The case of one wavelet generator and…

The examples of rhythmical signals with variable period are considered. The definition of periodic function with the variable period is given as a model of such signals. The examples of such functions are given and their variable periods…

Time series classification is a task that aims at classifying chronological data. It is used in a diverse range of domains such as meteorology, medicine and physics. In the last decade, many algorithms have been built to perform this task…