Discrete Time Research Articles

A method for calculating fall risk parameters from discrete stride time series regardless of sensor placement

BackgroundTo complement traditional clinical fall risk assessments, research is oriented towards adding real-life gait-related fall risk parameters (FRP) using inertial sensors fixed to a specific body position. While fixing the sensor position can facilitate data processing, it can reduce user compliance. A newly proposed step detection method, Smartstep, has been proven to be robust against sensor position and real-life challenges. Moreover, FRP based on step variability calculated from stride times (Standard deviation (SD), Coefficient of Variance (Cov), fractal exponent, and sample entropy of stride duration) proved to be useful to prospectively predict the fall risk. Research questionsTo evaluate whether Smartstep is convenient for calculating FRP from different sensor placements. Methods29 elderly performed a 6-minute walking test with IMU placed on the waist and the wrist. FRP were computed from step-time estimated from Smartstep and compared to those obtained from foot-mounted inertial sensors: precision and recall of the step detection, Root mean square error (RMSE) and Intraclass Correlation Coefficient (ICC) of stride durations, and limits of agreement of FRP. ResultsThe step detection precision and recall were respectively 99.5% and 95.9% for the waist position, and 99.4% and 95.7% for the wrist position. The ICC and RMSE of stride duration were 0.91 and 54 ms respectively for both the waist and the hand position. The limits of agreement of Cov, SD, fractal exponent, and sample entropy of stride duration are respectively 2.15%, 25 ms, 0.3, 0.5 for the waist and 1.6%, 16 ms, 0.23, 0.4 for the hand. SignificanceRobust against the elderly's gait and different body locations, especially the wrist, this method can open doors toward ambulatory measurements of steps, and calculation of different discrete stride-related falling risk indicators.

Posterior Representations for Bayesian Context Trees: Sampling, Estimation and Convergence

We revisit the Bayesian Context Trees (BCT) modelling framework for discrete time series, which was recently found to be very effective in numerous tasks including model selection, estimation and prediction. A novel representation of the induced posterior distribution on model space is derived in terms of a simple branching process, and several consequences of this are explored in theory and in practice. First, it is shown that the branching process representation leads to a simple variable-dimensional Monte Carlo sampler for the joint posterior distribution on models and parameters, which can efficiently produce independent samples. This sampler is found to be more efficient than earlier MCMC samplers for the same tasks. Then, the branching process representation is used to establish the asymptotic consistency of the BCT posterior, including the derivation of an almost-sure convergence rate. Finally, an extensive study is carried out on the performance of the induced Bayesian entropy estimator. Its utility is illustrated through both simulation experiments and real-world applications, where it is found to outperform several state-of-the-art methods.

High-order Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation

Abstract We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis41, 1696–1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu $ is large, the error is $O(h^{k+1}+\varDelta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\varDelta t})+\varDelta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.

Population Dependent Two-Sex Branching Process with Random Mating and Overlapping Generations

Abstract This research focuses on stochastic modeling of the evolution over time of biological populations through branching processes. We introduce a new class of discrete time two-sex branching processes with random mating and overlapping generations. Mating and reproduction are considered to be influenced by the numbers of females and males existing in the population. This evolution over time with generational overlap is a novel research in two-sex branching process literature. We study general probabilistic properties and establish some limiting results of biological interest.

Sequential Optimal Trajectory Planning Scheme for Robotic Manipulators along Specified Path Based on Direct Collocation Method

Robotic manipulators play a pivotal role in modern intelligent manufacturing and unmanned production systems, often tasked with executing specific paths accurately. However, the input of the robotic manipulators is trajectory which is a path with time information. The resulting core technology is trajectory planning methods which are broadly classified into two categories: maximum velocity curve (MVC) methods and multiphase direct collocation (MPDC) methods. This paper concentrates on addressing challenges associated with the latter methods. In MPDC methods, the solving efficiency and accuracy are greatly influenced by the number of discretization nodes. When dealing with systems with complex dynamics, such as robotic manipulators, striking a balance between solving time and path discretization errors becomes crucial. We use a mesh refinement (MR) algorithm to find a suitable number of nodes under the premise of ensuring the path discretization error. So, the actual device can effectively implement the planned solutions. Nonetheless, the conventional application of the MR algorithm requires solving the original problem in each iteration; these processes are extremely time-consuming and may fail to solve when dealing with a complex dynamic system. As a result, we propose a sequential optimal trajectory planning scheme to solve the problem efficiently by dividing the original optimal control (OC) problem into two stages: path planning (PP) and trajectory planning (TP). In the PP stage, we employ a DC method based on arc length and an MR algorithm to identify key nodes along the specified path. This aims to minimize the approximation error introduced during discretization. In the TP stage, the identified key nodes serve as boundary conditions for an MPDC method based on time. This facilitates the generation of an optimal trajectory that maximizes motion performance, considering constant velocity in Cartesian space and dynamic constraints while keeping the path discretization error. Simulation and experiment are conducted with a six-axis robotic manipulator, ROCR6, and show significant potential for a wide range of applications in robotics.

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

Development of discrete opposition-based NSGA-III model for optimizing trade-off between discrete time, cost, and resource in construction projects

Development of discrete opposition-based NSGA-III model for optimizing trade-off between discrete time, cost, and resource in construction projects

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

Biased versus unbiased numerical methods for stochastic simulations

Approximate numerical methods are one of the most used strategies to extract information from many-interacting-agents systems. In particular, numerical approximations are of extended use to deal with epidemic, ecological and biological models, since unbiased methods like the Gillespie algorithm can become unpractical due to high CPU time usage required. However, the use of approximations has been debated and there is no clear consensus about whether unbiased methods or biased approach is the best option. In this work, we derive scaling relations for the errors in approximations based on binomial extractions. This finding allows us to build rules to compute the optimal values of both the discretization time and number of realizations needed to compute averages with the biased method with a target precision and minimum CPU-time usage. Furthermore, we also present another rule to discern whether the unbiased method or biased approach is more efficient. Ultimately, we will show that the choice of the method should depend on the desired precision for the estimation of averages.

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

On V-Geometric Ergodicity Markov Chains of the Two-Inertia Systems

This study employs the diffusion process to construct Markov chains for analyzing the common two-inertia systems used in industry. Two-inertia systems are prevalent in commonly used equipment, where the load is influenced by the coupling of external force and the drive shaft, leading to variations in the associated output states. Traditionally, the control of such systems is often guided by empirical rules. This paper examines the equilibrium distribution and convergence rate of the two-inertia system and develops a predictive model for its long-term operation. We explore the qualitative behavior of the load end at discrete time intervals. Our findings are applicable not only in control engineering, but also provide insights for small-scale models incorporating dual-system variables.

Degree evolution in a general growing network

We consider the preferential attachment model introduced by Deijfen and Lindholm (2009) in which, at every discrete time step: (i) either we add a vertex and connect it to an older vertex; or (ii) we add an edge between two random vertices; or (iii) we delete one edge. We show that, when the deletion probability equals 1/3, the expected degree of any given vertex grows logarithmically, thus correcting a statement made in Lindholm and Vallier (2011). Moreover we show that, when the deletion probability is strictly less than 1/3, then the function which scales the expected degree of a given vertex, identified in Lindholm and Vallier (2011), also guarantees almost sure convergence for the degree process of a given vertex.

Variational data assimilation with finite-element discretization for second-order parabolic interface equation

Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Methods for implementing integrated step-selection functions with incomplete data

Integrated step-selection analyses (iSSAs) are versatile and powerful frameworks for studying habitat and movement preferences of tracked animals. iSSAs utilize integrated step-selection functions (iSSFs) to model movements in discrete time, and thus, require animal location data that are regularly spaced in time. However, many real-world datasets are incomplete due to tracking devices failing to locate an individual at one or more scheduled times, leading to slight irregularities in the duration between consecutive animal locations. To address this issue, researchers typically only consider bursts of regular data (i.e., sequences of locations that are equally spaced in time), thereby reducing the number of observations used to model movement and habitat selection. We reassess this practice and explore four alternative approaches that account for temporal irregularity resulting from missing data. Using a simulation study, we compare these alternatives to a baseline approach where temporal irregularity is ignored and demonstrate the potential improvements in model performance that can be gained by leveraging these additional data. We also showcase these benefits using a case study on a spotted hyena (Crocuta crocuta).

A Net Present Value Analysis of Opportunity-Based Age Replacement Models in Discrete Time

Two important opportunistic age replacement models, under replacement first and last disciplines, are generalized in discrete time. The net present value (NPV) is applied to formulate the expected total costs. The priority of multiple replacement options is considered to classify the cost model with discounting into six cases. Since the NPV method accurately calculates the expected replacement costs over an infinite horizon in an unstable economic environment, we discuss some optimal opportunistic age replacement policies which minimize the expected total discounted costs over an infinite time horizon. Furthermore, we formulate a unified model under each discipline, merging six discrete time replacement models with probabilistic priority. Finally, a case study on optimal replacement first and last policies for pole air switches in a Japanese power company is presented.

An Empirical Study On Imaging Sensitivity Of SKA1-LOW Based on Point Source Sensitivity

The Square Kilometre Array (SKA) radio telescope will achieve unparalleled sensitivity and angular resolution, substantially progressing our research into the formation and development of the early universe. Sensitivity is one of the most critical telescope metrics, often used to determine the observing mode and observing time of a radio source. It is also used to determine imaging parameters to balance imaging sensitivity and spatial resolution during data processing. In this paper, to meet the needs of continuum imaging pipeline data processing for the SKA1-LOW telescope, we derive equations for point-source-sensitivity (PSS) calculations under multichannel, multipolarization, discrete frequency sampling, discrete time sampling, and different weighting schemes. After validating the correctness of the equation, we improved and refined the sensitivity calculation application in the Radio Astronomy Simulation, Calibration, and Imaging Library, and further calculated the variation of the PSS versus angular resolution for the full-scale SKA1-LOW with different weighting schemes, and provided SKA1-LOW broadband image performance as functions of angular scale for four frequencies. These results can provide not only a theoretical basis for the construction and commissioning of the SKA1-LOW telescope but also guidance for imaging processing in future scientific data processing.

Temporal approximation of stochastic evolution equations with irregular nonlinearities

In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong errorEk∞:=(Esupj∈{0,…,Nk}‖U(tj)-Uj‖Xp)1/p→0(k→0),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_{k}^{\\infty } {:}{=}\\Big (\\mathbb {E}\\sup _{j\\in \\{0, \\ldots , N_k\\}} \\Vert U(t_j) - U^j\\Vert _X^p\\Big )^{1/p} \\rightarrow 0\\quad (k \\rightarrow 0), \\end{aligned}$$\\end{document}where p∈[2,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p \\in [2,\\infty )$$\\end{document}, U is the mild solution, Uj\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U^j$$\\end{document} is obtained from a time discretisation scheme, k is the step size, and Nk=T/k\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_k = T/k$$\\end{document} for final time T>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T>0$$\\end{document}. This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong errorEk:=(supj∈{0,…,Nk}E‖U(tj)-Uj‖Xp)1/p,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extrm{E}_k {:}{=}\\bigg (\\sup _{j\\in \\{0,\\ldots ,N_k\\}}\\mathbb {E}\\Vert U(t_j) - U^{j}\\Vert _X^p\\bigg )^{1/p}, \\end{aligned}$$\\end{document}which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.

Modeling gas flow in low-permeability formations: An efficient combination of mixed finite elements and high order time integration schemes

Numerical simulation of gas flow in low permeability formations is a challenging task due to the high nonlinearity induced by (i) the compressibility of the gas, (ii) the Klinkenberg slippage effect and (iii) the Langmuir adsorption of the gas on the pore surface. Because of these nonlinearities, modeling gas flow in low permeability formations requires a great deal of computational effort. In this work, we develop an efficient numerical model using advanced spatial and temporal discretization methods for a simultaneous solution of the coupled equations of gas flow, cubic Peng-Robinson equation of state, slippage effect and Langmuir adsorption. The spatial discretization is performed with the lumped hybrid formulation of the mixed finite element method which is well adapted for fluid flow in heterogeneous porous media. The time integration is performed with high-order methods via the method of lines (MOL) which allows large time steps and efficient solution of the nonlinear system of equations. Numerical experiments, performed for gas extraction in a heterogeneous domain, point out the high efficiency of the new model since it can be until 10 times more efficient than the classical first-order time discretization method. Results of global sensitivity analysis show that the intrinsic gas-phase permeability and the Klinkenberg factor are the most influential parameters controlling the pumping rate in the case of gas extraction in a homogeneous domain.

Well‐posedness and error analysis of wave equations with Markovian switching

Compared to traditional partial differential equation modeling methods, Markov switching models can accurately capture the abrupt changes or jumps that complex systems often experience in the real world. In this paper, we propose a novel wave equation model with Markovian switching to represent complex systems with state‐jumping phenomena better, and the well‐posedness of the model is proved. In addition, a numerical method with non‐uniform grids is also proposed for the proposed model to simulate the data in realistic situations, which is based on the use of finite element discretization in space and central difference discretization in time. Finally, we conduct several experiments to analyze the errors and stability of the proposed model and the traditional model. The results show that the Markov switching model proposed in this paper has a smaller error than the traditional models while ensuring stability and can more accurately simulate the state jump phenomena of real‐world systems.