Error Estimates Research Articles

Advanced Ensemble Framework for Diabetes Outcome Forecasting

Abstract: Diabetes is one of the chronic diseases, which is increasing from year to year. Developing an automated system that can detect diabetes patients plays an important role in medical science. A stack classifier model is designed for the detection of diabetes by combining three base estimators such as Random Forest Classifier, LightGBM Classifier, and K-Nearest Neighbors Classifier, and Logistic Regression as meta-classifiers. The data preprocessing includes transforming categorical variables into numerical format. Each base learner is trained on the preprocessed data and predictions are made. In the meta learner stage, Logistic Regression is trained to make predictions based on the predictions of the base learners. The goal of the meta learner is to learn how to combine the predictions of the base learners to make a more accurate final prediction on whether the patient is diabetic or not. The use of a stacking classifier improves prediction accuracy compared to using a single classifier. The developed model gives an accuracy of 98%. The 5-fold cross validation is used to get a more robust estimation of generalization error. Thus, the developed model offers a means to enhance early detection and elevate the quality of care for diabetic patients.

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

Abstract We first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta -$scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta \le \frac 1 2$. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

Semi-active damping optimization of vibrational systems using the reduced basis method

In this article, we consider vibrational systems with semi-active damping that are described by a second-order model. In order to minimize the influence of external inputs to the system response, we are optimizing some damping values. As minimization criterion, we evaluate the energy response, that is the H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}_2$$\\end{document}-norm of the corresponding transfer function of the system. Computing the energy response includes solving Lyapunov equations for different damping parameters. Hence, the minimization process leads to high computational costs if the system is of large dimension. We present two techniques that reduce the optimization problem by applying the reduced basis method to the corresponding parametric Lyapunov equations. In the first method, we determine a reduced solution space on which the Lyapunov equations and hence the resulting energy response values are computed approximately in a reasonable time. The second method includes the reduced basis method in the minimization process. To evaluate the quality of the approximations, we introduce error estimators that evaluate the error in the controllability Gramians and the energy response. Finally, we illustrate the advantages of our methods by applying them to two different examples.

Exact Integral Formulas for False Discovery Rate and the Variance of False Discovery Proportion.

Multiple hypothesis testing is an integral component of data analysis for large-scale technologies such as proteomics, transcriptomics, or metabolomics, for which the false discovery rate (FDR) and positive FDR (pFDR) have been accepted as error estimation and control measures. The pFDR is the expectation of false discovery proportion (FDP), which refers to the ratio of the number of null hypotheses to that of all rejected hypotheses. In practice, the expectation of ratio is approximated by the ratio of expectation; however, the conditions for transforming the former into the latter have not been investigated. This work derives exact integral expressions for the expectation (pFDR) and variance of FDP. The widely used approximation (ratio of expectations) is shown to be a particular case (in the limit of a large sample size) of the integral formula for pFDR. A recurrence formula is provided to compute the pFDR for a predefined number of null hypotheses. The variance of FDP was approximated for a practical application in peptide identification using forward and reversed protein sequences. The simulations demonstrate that the integral expression exhibits better accuracy than the approximate formula in the case of a small number of hypotheses. For large sample sizes, the pFDRs obtained by the integral expression and approximation do not differ substantially. Applications to proteomics data sets are included.

Estimation and hypothesis test for varying coefficient single-index multiplicative models

ABSTRACT Estimation and hypothesis test for varying coefficient single-index multiplicative models are considered in this paper. To estimate an unknown single-index parameter, a profile product relative error estimation is proposed for the single-index parameter with a leave-one-component-out estimation method. A Wald-type test statistic is proposed to test a linear hypothesis test of the single-index. We employ the smoothly clipped absolute deviation penalty to simultaneously select variables and estimate regression coefficients. To study the model checking problem, we propose a variant of the integrated conditional moment test statistic by using a linear projection weighting function, and we also suggest a bootstrap procedure for calculating critical values. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real example is analysed for illustration.

A parareal exponential integrator finite element method for semilinear parabolic equations

AbstractIn this article, we present a parareal exponential finite element method, with the help of variational formulation and parareal framework, for solving semilinear parabolic equations in rectangular domains. The model equation is first discretized in space using the finite element method with continuous piecewise multilinear rectangular basis functions, producing the semi‐discrete system. We then discretize the temporal direction using the explicit exponential Runge–Kutta approach accompanied by the parareal framework, resulting in the fully‐discrete numerical scheme. To further improve computational speed, we design a fast solver for our method based on tensor product spectral decomposition and fast Fourier transform. Under certain regularity assumption, we successfully derive optimal error estimates for the proposed parallel‐based method with respect to ‐norm. Extensive numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the performance of our method.

Analysis and simulation of the (2 + 1)-dimensional Fisher’s reaction-diffusion equation with higher order finite element method

In this article, a finite element analysis of Fisher’s reaction-diffusion equation is performed, by using quadratic shape elements. The higher-order elements in the finite element method (FEM) are known to yield better results in the approximation of solutions. However, such elements are less explored in the literature. Therefore, quadratic shape functions are taken into consideration for the finite element analysis of Fisher’s reaction-diffusion equation. A priori error estimates for the semi-discrete solution are derived by applying the Galerkin approximation. For the fully discrete solution, a priori error estimates are derived using the Crank-Nicolson method and the predictor-corrector scheme. Also, the stability of the time-discrete scheme is analyzed by the energy technique. Third-order convergence for space discretization in L 2 ( Ω ) norm and second-order convergence in H 1 ( Ω ) norm are observed. Furthermore, second-order convergence for time discretization is also found. Finally, we validate our theoretical results by considering a variety of numerical examples in one and two-dimensional spaces, which also illustrate the reliability of the developed finite element algorithm. Through comparison tables, 3D plots, and convergence plots, the results of the current study are compared with the results from the existing literature.

A new approach of B-spline wavelets to solve fractional differential equations

This paper presents a groundbreaking method for solving the multi-order fractional differential (M-OFD), both linear and nonlinear, as well as fractional partial differential equations (FPDE)s. This approach involves constructing an operational matrix of fractional derivatives using linear B-spline (LB-S) wavelet functions with perfect subtlety. The new method has two crucial features. Firstly, it simplifies the problem by converting it into a set of algebraic equations, which is a significant advantage. This makes the method highly accurate and reliable. Secondly, it uses thresholding to dramatically reduce the computational workload in linear problems. This leads to lightning-fast and highly efficient problem-solving. The newly developed scheme underwent a thorough examination of its error estimates and convergence, revealing remarkable results in terms of accuracy and efficiency. The analysis provides a comprehensive understanding of the scheme’s performance, highlighting its potential as a dependable and effective method. Based on the findings, it is evident that the proposed method not only delivers exceptional precision but also operates with remarkable efficiency.

An Adaptive Discrete Integral Terminal Sliding Mode Control Method for a Two-Joint Manipulator

In response to the trajectory tracking control problem of manipulators under measurement disturbances, a novel multi-input multi-output discrete integral terminal sliding mode control scheme is proposed. Initially, this scheme establishes a dynamic model of a two-joint manipulator based on the Lagrangian dynamics analysis method. Subsequently, a discrete integral terminal sliding mode control law based on the dynamic model of the two joints is designed, incorporating delayed estimation of unknown disturbances and discretization errors in the manipulator system. To enhance the trajectory tracking accuracy of the control scheme and suppress the impact of sliding mode chattering on the manipulator system, an adaptive switching term is introduced into the discrete integral terminal sliding mode control law. The paper derives an adaptive discrete integral terminal sliding mode control scheme and provides stability proof for the proposed approach. Simulation experiments are conducted to compare the proposed adaptive discrete integral terminal sliding mode control scheme with classical discrete sliding mode control schemes and discrete integral terminal sliding mode control schemes. The simulation results demonstrate that the designed adaptive discrete integral terminal sliding mode control scheme maintains trajectory tracking errors within 0.004 radians for each joint of the manipulator, with minimal changes in control torque for each joint. The absolute integral of the control torque variations is calculated at 5.85×103, which is lower than other control schemes, thereby validating the effectiveness and superiority of the proposed approach.

A neural network inverse problem solution for a MIMO magnetic levitation actuator

A multiple-input and multiple-output planar actuator is proposed, which utilizes mechanically driven stator magnet arrays to levitate a permanent magnet mover. A state of levitation and actuation is obtained by mechanically altering the orientation of the stator magnets to control the forces and torques on the mover. A challenge for the design and control of the actuator is inverting the relationship between the force and stator magnet rotation angles, as there is no closed-form analytical solution. In this study, a feed-forward neural network is applied to model the forward relation between stator magnet angle input and a force and torque output to reduce the forward computation time for the design process and for error estimation in real-time applications. Additionally, the neural network is considered for inverting the solution for a motion profile sampled at 1000 Hz. The developed forward model is able to calculate the forces and torques on the mover a factor 10 faster than the equivalent charge or Fourier model with an absolute error of 3 mN and 0.1 mNm for the forces and torques, respectively, and a feed-forward neural network is able to accurately learn an inverse solution for small motion profiles.

Zero‐sum game‐based H∞ optimal finite‐time prescribed performance control for nonlinear multiagent systems with actuator faults

AbstractThis article proposes an optimal leader‐follower consensus control protocol with finite‐time prescribed performance for nonlinear multiagent systems suffering from the actuator fault based on the zero‐sum game framework. First, to guarantee the specified performance of the system within a finite time frame, a novel error transformation function (ETF) is constructed. The ETF proposed in this paper can adjust the mapping range of error transformation to achieve better error convergence performance, thereby enhancing the flexibility of the controller design. Then, to compensate for the negative impact of actuator fault, an error‐related item is separated from the optimal performance index function to design an online fault estimator, which can obtain a better error estimation accuracy. Furthermore, to lessen the need for difficult‐to‐verify persistent excitation signal in adaptive dynamic programming algorithm, both historical and real‐time data are applied to adjust the updating law of neural network weight. Moreover, it demonstrates that the signals within the closed‐loop system are bounded and the expected finite‐time prescribed performance can be ensured in the presence of faults. Finally, a numerical example of multiple single‐link robots manipulator system is executed to validate the efficacy of the developed scheme.

Convergence and error estimates of a penalization finite volume method for the compressible Navier–Stokes system

Abstract In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain. Consequently, in order to study convergence and error estimates of a numerical method domain-related discretization errors, the so-called variational crimes, need to be taken into account. In this paper we apply the penalty approach to control domain-related discretization errors. We embed the physical domain into a large enough cubed domain and study the convergence of a finite volume method for the corresponding domain-penalized problem. We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem. If a strong solution exists, the dissipative weak solution emanating from the same initial data coincides with the strong solution. In this case, we apply a novel tool of the relative energy and derive the error estimates between the numerical solution and the strong solution. Extensive numerical experiments that confirm theoretical results are presented.

Error estimate of a fully decoupled numerical scheme based on the Scalar Auxiliary Variable (SAV) method for the Boussinesq system

In this paper, we will investigate the unconditional stability and error estimate of the fully decoupled numerical scheme for the Boussinesq equations. The newly constructed numerical scheme is based on the pressure correction technique and the SAV method, in which all coupling terms and nonlinear terms are completely decoupled, that is, we only need to solve several linear constant-coefficient equations. We rigorously prove the unconditional stability and convergence of the time-marching scheme and discuss all the details of the algorithm implementation. Finally, we implement some numerical experiments to verify its stability and accuracy.

Complexity analysis of quasi continuous level Monte Carlo

Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest. Continuous level Monte Carlo methods allow naturally for samplewise adaptive mesh refinements, which are indicated by (goal-oriented) error estimators. The samplewise refinement levels are drawn in the estimator from an exponentially-distributed random variable. Unfortunately in practical examples this results in higher costs due to high variance in the samples. In this paper we propose a variant of continuous level Monte Carlo, where a quasi Monte Carlo sequence is utilized to ''sample'' the exponential random variable. We provide a complexity theorem for this novel estimator and show that this theoretically and practically results in a variance reduction of the whole estimator.

Simultaneous estimation of genotype error and uncalled deletion rates in whole genome sequence data.

Genotype data include errors that may influence conclusions reached by downstream statistical analyses. Previous studies have estimated genotype error rates from discrepancies in human pedigree data, such as Mendelian inconsistent genotypes or apparent phase violations. However, uncalled deletions, which generally have not been accounted for in these studies, can lead to biased error rate estimates. In this study, we propose a genotype error model that considers both genotype errors and uncalled deletions when calculating the likelihood of the observed genotypes in parent-offspring trios. Using simulations, we show that when there are uncalled deletions, our model produces genotype error rate estimates that are less biased than estimates from a model that does not account for these deletions. We applied our model to SNVs in 77 sequenced White British parent-offspring trios in the UK Biobank. We use the Akaike information criterion to show that our model fits the data better than a model that does not account for uncalled deletions. We estimate the genotype error rate at SNVs with minor allele frequency > 0.001 in these data to be [Formula: see text]. We estimate that 77% of the genotype errors at these markers are attributable to uncalled deletions [Formula: see text].

The Recovery and Re-Calibration of a 13-Month Aerosol Extinction Profiles Dataset from Searchlight Observations from New Mexico, after the 1963 Agung Eruption

The recovery and re-calibration of a dataset of vertical aerosol extinction profiles of the 1963/64 stratospheric aerosol layer measured by a searchlight at 32° N in New Mexico, US, is reported. The recovered dataset consists of 105 aerosol extinction profiles at 550 nm that cover the period from December 1963 to December 1964. It is a unique record of the portion of the aerosol cloud from the March 1963 Agung volcanic eruption that was transported into the Northern Hemisphere subtropics. The data-recovery methodology involved re-digitizing the 105 original aerosol extinction profiles from individual Figures within a research report, followed by the re-calibration. It involves inverting the original equation used to compute the aerosol extinction profile to retrieve the corresponding normalized detector response profile. The re-calibration of the original aerosol extinction profiles used Rayleigh extinction profiles calculated from local soundings. Rayleigh and aerosol slant transmission corrections are applied using the MODTRAN code in transmission mode. Also, a best-estimate aerosol phase function was calculated from observations and applied to the entire column. The tropospheric aerosol phase function from an AERONET station in the vicinity of the searchlight location was applied between 2.76 to 11.7 km. The stratospheric phase function, applied for a 12.2 to 35.2 km altitude range, is calculated from particle-size distributions measured by a high-altitude aircraft in the vicinity of the searchlight in early 1964. The original error estimate was updated considering unaccounted errors. Both the re-calibrated aerosol extinction profiles and the re-calibrated stratospheric aerosol optical depth magnitudes showed higher magnitudes than the original aerosol extinction profiles and the original stratospheric aerosol optical depth, respectively. However, the magnitudes of the re-calibrated variables show a reasonable agreement with other contemporary observations. The re-calibrated stratospheric aerosol optical depth demonstrated its consistency with the tropics-to-pole decreasing trend, associated with the major volcanic eruption stratospheric aerosol pattern when compared to the time-coincident stratospheric aerosol optical depth lidar observations at Lexington at 42° N.

Distributed formation control of multi-UAV systems using relative information

An adaptive distributed control strategy is designed for the formation of multiple unmanned aerial vehicle (UAV) groups by using only relative measurement information. A distributed estimator is first designed to get the estimate of the consensus error. Then, an adaptive distributed controller is proposed to realize the formation tracking with the relative measurements. Based on the position and velocity information, a control barrier function (CBF) algorithm is applied to achieve the UAV formation under safe position constraints. In addition, in order to break up and reorganize multiple UAV groups, the proposed controller is designed based on the time-varying network topology. It is proved that the closed-loop system is uniformly asymptotically stability. Simulation and experiment results indicate that the proposed strategy can achieve formation control of multiple UAVs under safety constraints based on the measurement of relative measurement information.

Discrete anisotropic curve shortening flow in higher codimension

Abstract We introduce a novel formulation for the evolution of parametric curves by anisotropic curve shortening flow in ${{\mathbb{R}}}^{d}$, $d\geq 2$. The reformulation hinges on a suitable manipulation of the parameterization’s tangential velocity, leading to a strictly parabolic differential equation. Moreover, the derived equation is in divergence form, giving rise to a natural variational numerical method. For a fully discrete finite element approximation based on piecewise linear elements we prove optimal error estimates. Numerical simulations confirm the theoretical results and demonstrate the practicality of the method.

Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization

In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

A stabilized SAV difference scheme and its accelerated solver for spatial fractional Cahn–Hilliard equations

A novel energy-stable scheme is proposed to solve the spatial fractional Cahn–Hilliard equations, using the idea of scalar auxiliary variable (SAV) approach and stabilization technique. Thanks to the stabilization technique, it is shown that larger temporal stepsizes can be applied in numerical simulations. Moreover, the proposed SAV finite difference scheme is non-coupled and linearly implicit, which can be efficiently solved by the preconditioned conjugate gradient (PCG) method with a sine transform based preconditioner. Optimal error estimates of the fully-discrete scheme are obtained rigorously. Numerical examples are given to confirm the theoretical results and show the higher efficiency of the proposed scheme than the previous SAV schemes.