Search results for "Mathematics and Statistics"
showing 10 items of 11 documents
Measuring High-Order Interactions in Rhythmic Processes Through Multivariate Spectral Information Decomposition
2021
Many complex systems in physics, biology and engineering are modeled as dynamical networks and described using multivariate time series analysis. Recent developments have shown that the emergent dynamics of a network system are significantly affected by interactions involving multiple network nodes which cannot be described using pairwise links. While these higher-order interactions can be probed using information-theoretic measures, a rigorous framework to describe them in the frequency domain is still lacking. This work presents an approach for the spectral decomposition of multivariate information measures, capable of identifying higher-order synergistic and redundant interactions betwee…
On the interpretability and computational reliability of frequency-domain Granger causality
2017
This Correspondence article is a comment which directly relates to the paper “A study of problems encountered in Granger causality analysis from a neuroscience perspective” (Stokes and Purdon, 2017). We agree that interpretation issues of Granger causality (GC) in neuroscience exist, partially due to the historically unfortunate use of the name “causality”, as described in previous literature. On the other hand, we think that Stokes and Purdon use a formulation of GC which is outdated (albeit still used) and do not fully account for the potential of the different frequency-domain versions of GC; in doing so, their paper dismisses GC measures based on a suboptimal use of them. Furthermore, s…
Multiscale Information Decomposition: Exact Computation for Multivariate Gaussian Processes
2017
Exploiting the theory of state space models, we derive the exact expressions of the information transfer, as well as redundant and synergistic transfer, for coupled Gaussian processes observed at multiple temporal scales. All of the terms, constituting the frameworks known as interaction information decomposition and partial information decomposition, can thus be analytically obtained for different time scales from the parameters of the VAR model that fits the processes. We report the application of the proposed methodology firstly to benchmark Gaussian systems, showing that this class of systems may generate patterns of information decomposition characterized by prevalently redundant or sy…
Extending the spectral decomposition of Granger causality to include instantaneous influences: application to the control mechanisms of heart rate va…
2021
Assessing Granger causality (GC) intended as the influence, in terms of reduction of variance of surprise, that a driver variable exerts on a given target, requires a suitable treatment of ‘instantaneous’ effects, i.e. influences due to interactions whose time scale is much faster than the time resolution of the measurements, due to unobserved confounders or insufficient sampling rate that cannot be increased because the mechanism of generation of the variable is inherently slow (e.g. the heartbeat). We exploit a recently proposed framework for the estimation of causal influences in the spectral domain and include instantaneous interactions in the modelling, thus obtaining (i) a novel index…
Synergistic Information Transfer in the Global System of Financial Markets.
2020
Uncovering dynamic information flow between stock market indices has been the topic of several studies which exploited the notion of transfer entropy or Granger causality, its linear version. The output of the transfer entropy approach is a directed weighted graph measuring the information about the future state of each target provided by the knowledge of the state of each driving stock market index. In order to go beyond the pairwise description of the information flow, thus looking at higher order informational circuits, here we apply the partial information decomposition to triplets consisting of a pair of driving markets (belonging to America or Europe) and a target market in Asia. Our …
A collaboratively-derived research agenda for e-assessment in undergraduate mathematics
2022
AbstractThis paper describes the collaborative development of an agenda for research on e-assessment in undergraduate mathematics. We built on an established approach to develop the agenda from the contributions of 22 mathematics education researchers, university teachers and learning technologists interested in this topic. The resulting set of 55 research questions are grouped into 5 broad themes: errors and feedback, student interactions with e-assessment, design and implementation choices, affordances offered by e-assessment tools, and mathematical skills. This agenda gives a framework for a programme of research aligned with practical concerns that will contribute to both theoretical an…
Multivariate correlation measures reveal structure and strength of brain–body physiological networks at rest and during mental stress
2021
In this work, we extend to the multivariate case the classical correlation analysis used in the field of network physiology to probe dynamic interactions between organ systems in the human body. To this end, we define different correlation-based measures of the multivariate interaction (MI) within and between the brain and body subnetworks of the human physiological network, represented, respectively, by the time series of delta, theta, alpha, and beta electroencephalographic (EEG) wave amplitudes, and of heart rate, respiration amplitude, and pulse arrival time (PAT) variability. MI is computed: (i) considering all variables in the two subnetworks to evaluate overall brain–body interaction…
The second Weyl coefficient for a first-order system
2020
For a scalar elliptic self-adjoint operator on a compact manifold without boundary we have two-term asymptotics for the number of eigenvalues between 0 and λ when λ → ∞, under an additional dynamical condition. (See [3, Theorem 3.5] for an early result in this direction.) In the case of an elliptic system of first order, the existence of two-term asymptotics was also established quite early and as in the scalar case Fourier integral operators have been the crucial tool. The complete computation of the coefficient of the second term was obtained only in the 2013 paper [2]. In the present paper we simplify that calculation. The main observation is that with the existence of two-term asymptoti…
Multiscale Granger causality
2017
In the study of complex physical and biological systems represented by multivariate stochastic processes, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. While methods to assess the dynamic complexity of individual processes at different time scales are well-established, multiscale analysis of directed interactions has never been formalized theoretically, and empirical evaluations are complicated by practical issues such as filtering and downsampling. Here we extend the very popular measure of Granger causality (GC), a prominent tool for assessing directed lagged interactions between joint processes, to quantify information transfer a…
Wardowski conditions to the coincidence problem
2015
In this article we first discuss the existence and uniqueness of a solution for the coincidence problem: Find p ∈ X such that Tp = Sp, where X is a nonempty set, Y is a complete metric space, and T, S:X → Y are two mappings satisfying a Wardowski type condition of contractivity. Later on, we will state the convergence of the Picard-Juncgk iteration process to the above coincidence problem as well as a rate of convergence for this iteration scheme. Finally, we shall apply our results to study the existence and uniqueness of a solution as well as the convergence of the Picard-Juncgk iteration process toward the solution of a second order differential equation. Ministerio de Economía y Competi…