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We introduce the boolean convolution for probability measures on the unit circle. Roughly speaking, it describes the distribution of the product of two boolean independent unitary random variables. We find an analogue of the characteristic…
In a variety of scientific applications we wish to characterize a physical system using measurements or observations. This often requires us to solve an inverse problem, which usually has non-unique solutions so uncertainty must be…
By use of window functions, time-frequency analysis tools like Short Time Fourier Transform overcome a shortcoming of the Fourier Transform and enable us to study the time- frequency characteristics of signals which exhibit transient os-…
Stochastic motion of particles in a highly unstable potential generates a number of diverging trajectories leading to undefined statistical moments of the particle position. This makes experiments challenging and breaks down a standard…
The aim of this paper is to prove new uncertainty principles for an integral operator $\tt$ with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Faris's local uncertainty principle…
Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…
The notion of wavelets is defined. It is briefly described {\it what} are wavelets, {\it how} to use them, {\it when} we do need them, {\it why} they are preferred and {\it where} they have been applied. Then one proceeds to the…
This book chapter introduces the principles and practical applications of uncertainty quantification in machine learning. It explains how to identify and distinguish between different types of uncertainty and presents methods for…
Visualization and analysis of multivariate data and their uncertainty are top research challenges in data visualization. Constructing fiber surfaces is a popular technique for multivariate data visualization that generalizes the idea of…
Starting from considerations about meaning and subsequent use of asymmetric uncertainty intervals of experimental results, we review the issue of uncertainty propagation. We show that, using a probabilistic approach (the so-called Bayesian…
The modeling and uncertainty quantification of closed curves is an important problem in the field of shape analysis, and can have significant ramifications for subsequent statistical tasks. Many of these tasks involve collections of closed…
We propose a wavelet-based approach to construct consistent estimators of the pointwise H\"older exponent of a multifractional Brownian motion, in the case where this underlying process is not directly observed. The relative merits of our…
The aim of this paper is to prove an uncertainty principle for the representation of a vector in two bases. Our result extends previously known qualitative uncertainty principles into quantitative estimates. We then show how to transfer…
Droplets moving in a microfluidic loop device exhibit both periodic and chaotic behaviors based on the inlet droplet spacing. We propose that the periodic behavior is an outcome of a dispersed phase conservation principle. This conservation…
In this paper we study a family of nonlinear (conditional) expectations that can be understood as a stochastic process with uncertain parameters. We develop a general framework which can be seen as a version of the martingale problem method…
Bayesian inference is a popular approach to calibrating uncertainties, but it can underpredict such uncertainties when model misspecification is present, impacting its reliability to inform decision making. Recently, the statistics and…
Many systems arising in biological applications are subject to periodic forcing. In these systems the forcing parameter is not only time-varying but also known to have a periodic structure. We present an approach to estimating periodic,…
This paper is devoted to establish an invariance principle where the limit process is a multifractional Gaussian process with a multifractional function which takes its values in $(1/2,1)$. Some properties, such as regularity and local…
The notion of asymptotic unpredictability was recently introduced in (Commun. Nonlinear Sci. Numer. Simul. 134, 108029, 2024) for semiflows. Likewise unpredictable trajectories, asymptotically unpredictable ones are also capable of…
Local volatility is an important quantity in option pricing, portfolio hedging, and risk management. It is not directly observable from the market; hence calibrations of local volatility models are necessary using observable market data.…