Higher Order Differences Research Articles

Directed Network Comparison Using Motifs.

Analyzing and characterizing the differences between networks is a fundamental and challenging problem in network science. Most previous network comparison methods that rely on topological properties have been restricted to measuring differences between two undirected networks. However, many networks, such as biological networks, social networks, and transportation networks, exhibit inherent directionality and higher-order attributes that should not be ignored when comparing networks. Therefore, we propose a motif-based directed network comparison method that captures local, global, and higher-order differences between two directed networks. Specifically, we first construct a motif distribution vector for each node, which captures the information of a node's involvement in different directed motifs. Then, the dissimilarity between two directed networks is defined on the basis of a matrix, which is composed of the motif distribution vector of every node and the Jensen-Shannon divergence. The performance of our method is evaluated via the comparison of six real directed networks with their null models, as well as their perturbed networks based on edge perturbation. Our method is superior to the state-of-the-art baselines and is robust with different parameter settings.

A novel high accuracy finite-difference time-domain method

The finite-difference time-domain (FDTD) method is widely used for numerical simulations of electromagnetic waves and acoustic waves. It is known, however, that the Courant condition is restricted in higher dimensions and with higher order differences in space. Although it is possible to relax the Courant condition by utilizing the third-degree difference in space, there remains a large anisotropy in the numerical dispersion at large Courant numbers. This study aims to reduce the anisotropy in the numerical dispersion and relax the Courant condition simultaneously. A new third-degree difference operator including the Laplacian is introduced to the time-development equations of FDTD(2,4) with second- and fourth-order accuracies. The present numerical simulations have demonstrated that numerical oscillations due to the anisotropic dispersion relation are reduced with the new operator.Graphical

Single-Loop Robust Model Predictive Speed Regulation of PMSM Based on Exogenous Signal Preview

Speed regulation problem of permanent magnet synchronous motor (PMSM) with overcurrent protection is addressed in this paper by using a single-loop robust model predictive control (MPC) approach. The unique feature of this new disturbance rejection MPC method is the utilization of the preview information of lumped exogenous signal. Different from most of the existing disturbance observer-based MPC approaches, the reference inputs, the lumped disturbances in PMSM, and their higher-order differences are explicitly considered as a part of this new concept of lumped exogenous signal by exploiting output regulation theory. Preview of the possible future behaviors of this exogenous signal is conducted based upon the development of two generalized proportional integral observers (GPIOs). An accuracy enhanced prediction model is obtained by exploiting the estimation and preview of the exogenous signal during the prediction horizon. Speed regulation problem of the PMSM is transformed into an error stabilization problem by means of regulator equations and preview information where the faster time-varying exogenous signal is considered. The overcurrent protection problem is also taken into account throughout the receding horizon optimization process. Experimental studies show that the proposed method achieves effective functionality in overcurrent protection, and much better dynamic performance against time-varying exogenous inputs.

Unravelling brain connectivity patterns in body dysmorphic disorder during decision-making on visual illusions: A graph theoretical approach

Body dysmorphic disorder (BDD) is characterized by an excessive preoccupation with perceived defects in physical appearance, and is associated with compulsive checking. Visual illusions are illusory or distorted subjective perceptions of visual stimuli, which are induced by specific visual cues or contexts. While previous research has investigated visual processing in BDD, the decision-making processes involved in visual illusion processing remain unknown. The current study addressed this gap by investigating the brain connectivity patterns of BDD patients during decision-making about visual illusions. Thirty-six adults - 18 BDD (9 female) and 18 healthy controls (10 female) - viewed 39 visual illusions while their EEG was recorded. For each image, participants were asked to indicate (1) whether they perceived the illusory features of the images; and (2) their degree of confidence in their response. Our results did not uncover group-level differences in susceptibility to visual illusions, supporting the idea that higher-order differences, as opposed to lower-level visual impairments, can account for the visual processing differences that have previously been reported in BDD. However, the BDD group had lower confidence ratings when they reported illusory percepts, reflecting increased feelings of doubt. At the neural level, individuals with BDD showed greater theta band connectivity while making decisions about the visual illusions, likely reflecting higher intolerance to uncertainty and thus increased performance monitoring. Finally, control participants showed increased left-to-right and front-to-back directed connectivity in the alpha band, which may suggest more efficient top-down modulation of sensory areas in control participants compared to individuals with BDD. Overall, our findings are consistent with the idea that higher-order disruptions in BDD are associated with increased performance monitoring during decision-making, which may be related to constant mental rechecking of responses.

On Polynomials Orthogonal with Respect to an Inner Product Involving Higher-Order Differences: The Meixner Case

In this contribution we consider sequences of monic polynomials orthogonal with respect to the Sobolev-type inner product f,g=⟨uM,fg⟩+λTjf(α)Tjg(α), where uM is the Meixner linear operator, λ∈R+, j∈N, α≤0, and T is the forward difference operator Δ or the backward difference operator ∇. Moreover, we derive an explicit representation for these polynomials. The ladder operators associated with these polynomials are obtained, and the linear difference equation of the second order is also given. In addition, for these polynomials, we derive a (2j+3)-term recurrence relation. Finally, we find the Mehler–Heine type formula for the particular case α=0.

Higher order differences on arbitrary discrete time scales and related generating functions

In this paper we uncover some fundamental relationships involving the weights one needs to calculate the n-th difference of a function on an arbitrary discrete time scale. One of several interesting results we obtain is a formula that allows one to calculate the value of any desired finite difference coefficient directly from the graininess function for the time scale under consideration. This foundational work beautifully combines analysis, algebra, and combinatorics to obtain this and other interesting results. Some of the other interesting results include (i) for a fixed n, the n-th finite difference coefficients sum to 0 and (ii) there are nice and useful generating functions that encode various sequences of finite difference coefficients. Throughout we show the results coincide with well-known results in the special cases where the time scales are either quantum time scales or time scales with constant graininess.

Phase Unwrapping of Coarsely Sampled Maps Using Higher-Order Methods

We consider the problem of 2-D phase unwrapping that of reconstructing the absolute phase (up to a constant) given a noisy <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\pi $ </tex-math></inline-formula> -wrapped phase map as input. In particular, we present an algorithm that excels at unwrapping low-resolution phase maps, that is, maps obtained by coarsely sampling a field of view. Our key observation is that the magnitudes of higher-order differences on phase maps are typically much lower than corresponding first-order differences. We develop this insight into formulating “higher-order Itoh conditions.” Using this, we build an optimization-based framework that leverages higher-order information to estimate the first-order derivatives of the unwrapped phase with the aid of appropriate total variation and irrotationality-based regularizers. The first-order derivatives are then integrated using a minimum spanning tree (MST) approach to produce the unwrapped phase map. We compare the performance on synthetic terrain maps and real-world data obtained from interferometric synthetic aperture radar (InSAR) with other contemporary algorithms to demonstrate superior performance on key metrics such as absolute error, Feature SIMilarity (FSIM), and image sharpness for low-resolution phase maps.

Closing the species gap: Translational approaches to studying sensory processing differences relevant for autism spectrum disorder.

The study of sensory phenotypes has great potential for increasing research translation between species, a necessity to decipher the neural mechanisms that contribute to higher-order differences in neurological conditions such as autism spectrum disorder (ASD). Over the past decade, despite separate advances in our understanding of the structural and functional differences within the brain of autistic and non-autistic individuals and in rodent models for ASD, researchers have had difficulty translating the findings in murine species to humans, mostly due to incompatibility in experimental methodologies used to screen for ASD phenotypes. Focusing on sensory phenotypes offers an avenue to close the species gap because sensory pathways are highly conserved across species and are affected by the same risk-factors as the higher-order brain areas mostly responsible for the diagnostic criteria for ASD. By first reviewing how sensory processing has been studied to date, we direct our focus to electrophysiological and behavioral techniques that can be used to study sensory phenotypes consistently across species. Using auditory sensory phenotypes as a template, we seek to improve the accessibility of translational methods by providing a framework for collecting cohesive data in both rodents and humans. Specifically, evoked-potentials, acoustic startle paradigms, and psychophysical detection/discrimination paradigms can be created and implemented in a coordinated and systematic fashion across species. Through careful protocol design and collaboration, sensory processing phenotypes can be harnessed to bridge the gap that exists between preclinical animal studies and human testing, so that mutually held questions in autism research can be answered. LAY SUMMARY: It has always been difficult to relate results from animal research to humans. We try to close this gap by studying changes in sensory processing using careful protocol design and collaboration between clinicians and researchers. Sensory pathways are comparable between animals and humans, and are affected in the same way as the rest of the brain in ASD. Using changes in hearing as a template, we point the field in an innovative direction by providing a framework for collecting cohesive data in rodents and humans.

Concentration Inequalities for Bounded Functionals via Log-Sobolev-Type Inequalities

In this paper, we prove multilevel concentration inequalities for bounded functionals f = f(X_1, ldots , X_n) of random variables X_1, ldots , X_n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k-tensors of higher order differences of f. We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes f(X) = sup _{g in {mathcal {F}}} {|g(X)|} and suprema of homogeneous chaos in bounded random variables in the Banach space case f(X) = sup _{t} {Vert sum _{i_1 ne ldots ne i_d} t_{i_1 ldots i_d} X_{i_1} cdots X_{i_d}Vert }_{{mathcal {B}}}. The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U-statistics with bounded kernels h and for the number of triangles in an exponential random graph model.

Condition monitoring of electric-cam mechanisms based on Model-of-Signals of the drive current higher-order differences

Condition monitoring of electric motor driven mechanisms is of great importance in industrial machines. The knowledge of the actual health state of such components permits to address maintenance policies which results in better exploitation of their actual operational life span and consequently in maintenance cost reduction. In this paper, we exploit the way electric cams are implemented on the vast majority of PLC/Motion controllers to develop a suitable condition monitoring procedure. This technique relies on computing the higher-order differences of the current absorbed by slave motors to get signals that do not depend on a priori knowledge of the cam trajectory and of the mechanism nominal model. Subsequently, we will use these data in the Model-of-Signals framework, to gather information on the mechanism’s health condition, which in turn can be used to perform predictive maintenance policies. The differenced signal is modelled as an ARMA process and the model capabilities in condition monitoring are then shown in simulation and experimental application. Besides, this framework allows exploiting the edge-computing capabilities of the machinery controllers by implementing recursive estimation algorithms.

Digital humanities methods as a gateway to inter and transdisciplinarity

ABSTRACT Advances in digital methods not only allow the application of quantitative methods to new problems but also offer new possibilities for collaborative qualitative research. This permits us to look again at the ‘two cultures’ of scientific and humanities research as an interdisciplinary continuum of methods rather than as opposites. DInformed by teaching digital research methods courses in history, politics, international relations and digital humanities at undergraduate and graduate level over the past 30 years, this paper draws on Unsworth’s idea of ‘Scholarly primitives’ sand the ideas of Willard McCarthy to suggests that a range of methods provide the basic ingredients on which a variety of disciplinary perspectives are built. Thus an integrated perspective can flow from tools and methods upwards to epistemology and ontology in research-based learning. This allows us to equip our students to use a range of common research tools which may be applied across disciplines, while at the same time making them more aware of higher order differences in disciplinary perspectives.

Fixed Points of Meromorphic Functions and Their Higher Order Differences and Shifts

Abstract In this paper, we investigate the relationships between fixed points of meromorphic functions, and their higher order differences and shifts, and generalize the case of fixed points into the more general case for first order difference and shift. Concretely, some estimates on the order and the exponents of convergence of special points of meromorphic functions and their differences and shifts are obtained.

Penalized estimation of flexible hidden Markov models for time series of counts

We propose an effectively nonparametric approach to fitting hidden Markov models to time series of counts, where the state-dependent distributions are estimated in a completely data-driven way without the need to specify a parametric family of distributions. To avoid overfitting, a roughness penalty based on higher-order differences between adjacent count probabilities is added to the likelihood, which is demonstrated to produce smooth state-dependent probability mass functions. The feasibility of the suggested approach is assessed in simulation experiments, and further illustrated in two real-data applications, where we model the distributions of (i) major earthquake counts and (ii) acceleration counts of an oceanic whitetip shark (Carcharhinus longimanus) over time. The proposed methodology is implemented in the accompanying R package countHMM, which is available on CRAN.

Method of Refinement by Higher Order Differences for 3D Poisson Equation with Nonlocal Boundary Conditions

Method of Refinement by Higher Order Differences for 3D Poisson Equation with Nonlocal Boundary Conditions

Commentary: Visual Fixation in Human Newborns Correlates with Extensive White Matter Networks and Predicts Long-Term Neurocognitive Development.

Commentary: Visual Fixation in Human Newborns Correlates with Extensive White Matter Networks and Predicts Long-Term Neurocognitive Development.

WITHDRAWN: Method of corrections by higher order differences for one nonlocal boundary-value problem

Abstract We consider the Bitsadze–Samarskii type nonlocal boundary value problem for Poisson equation in a unit square, which is solved by a difference scheme of second-order accuracy. Using this approximate solution, we correct the right-hand side of the difference scheme. It is shown that the solution of the corrected scheme converges at the rate O ( | h | s ) in the discrete L 2 -norm provided that the solution of the original problem belongs to the Sobolev space with exponent s ∈ [ 2 , 4 ] .

Compatible convergence estimates in the method of refinement by higher-order differences

We consider the Dirichlet problem for an elliptic equation with constant coefficients, which is solved by a difference scheme of second-order accuracy. By using the approximate solution, we correct the right-hand side of the difference scheme. We show that the solution of the corrected scheme is convergent at the rate O(|h|m) in the discrete L2-norm provided that the solution of the original problem belongs to the Sobolev space with exponent m ∈ [2, 4].

The bistability of higher-order differences of discrete periodic signals

A theorem on higher-order absolute differences taken from successive terms of bounded sequences is proved. This theorem establishes the property of bistability of such difference series and suggests a method for converting periodic discrete-time signals into the binary digital form based only on the computation of absolute differences.

Robust estimation of dynamic fixed-effects panel data models

This paper extends an existing outlier-robust estimator of linear dynamic panel data models with fixed effects, which is based on the median ratio of two consecutive pairs of first-order differenced data. To improve its precision and robustness properties, a general procedure based on higher-order pairwise differences and their ratios is designed. The asymptotic distribution of this class of estimators is derived. Further, the breakdown point properties are obtained under contamination by independent additive outliers and by the patches of additive outliers, and are used to select the pairwise differences that do not compromise the robustness properties of the procedure. The proposed estimator is additionally compared with existing methods by means of Monte Carlo simulations.

Rating scales as predictors--the old question of scale level and some answers.

Rating scales as predictors in regression models are typically treated as metrically scaled variables or, alternatively, are coded in dummy variables. The first approach implies a scale level that is not justified, the latter approach results in a large number of parameters to be estimated. Therefore, when rating scales are dummy-coded, applications are often restricted to the use of a few predictors. The penalization approach advocated here takes the scale level serious by using only the ordering of categories but is shown to work in the high dimensional case. We consider the proper modeling of rating scales as predictors and selection procedures by using penalization methods that are tailored to ordinal predictors. In addition to the selection of predictors, the clustering of categories is investigated. Existing methodology is extended to the wider class of generalized linear models. Moreover, higher order differences that allow shrinkage towards a polynomial as well as monotonicity constraints and alternative penalties are introduced. The proposed penalization approaches are illustrated by use of the Motivational States Questionnaire.