Asymptotic Error Research Articles

Hermite interpolation with retractions on manifolds

AbstractInterpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. A novel procedure relying on the general concept of retractions is proposed to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. The well-posedness of the method is analyzed by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. A classical result on the asymptotic interpolation error of Hermite interpolation is extended to the manifold setting. Finally numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns illustrate these results and the effectiveness of the method.

A note on potential gains in precision of radiation risk estimates from joint analysis.

In estimating radiation-related risk of cancer and other diseases based on the RERF Life Span Study (LSS), joint analyses can be performed where multiple health outcome endpoints are combined in the same model, allowing some parameters to be estimated in common among all endpoints with possible increase in precision of radiation risk and other model parameter estimates. Using as a basis excess relative risk (ERR) and excess absolute risk (EAR) models of the type commonly used in analysis of LSS data at RERF, we use maximum likelihood theory to compute the asymptotic relative standard error of endpoint-specific radiation effect and other parameter estimates using joint analyses as compared to traditional independent analysis. We show that some gains in precision of endpoint-specific radiation risk parameter estimates can be achieved by sharing effect modifier and other model parameters, but only small or negligible gains in precision are achieved for endpoint-specific background modifying or effect modifying parameters when other model parameters are shared. The magnitude of the precision gain for radiation risk estimates depends on the number of endpoints, the baseline incidence rate of the endpoint, and the type of model being used.

Confidence intervals estimator of the kinetic parameters: do its reliability depend on the assembling method of the oxygen uptakes?

Gas exchange data acquired repeatedly under the same exercise conditions are assembled together to improve the kinetic parameters of breath-by-breath oxygen uptake. The latter are provided by the non-linear regression procedure, together with the corresponding estimators of the width of the Confidence Intervals (i.e., the Asymptotic Standard Errors; ASEs). We tested, for two different assembling procedures, whether the range of values identified by the ASE actually correspond to the 95% Confidence Interval. Ten O2 uptake responses were acquired on 10 healthy volunteers performing a square-wave moderate-intensity exercise. Kinetic parameters were estimated running the non-linear regression with a mono-exponential model on an increasingly greater number of responses (Nr, from 1 to 10), assembled together using the "stacking" and the "1-s-bins" procedures. Kinetic values obtained assembling together the 10 repetitions were assumed as "true" values. The time constant was not affected by Nr or by the assembling procedure (ANOVA; p>0.54 and p>0.16, respectively). The corresponding ASE decreased according to Nr (ANOVA; p=0.000), being significantly smaller for the "1-s-bins" procedure compared to the "stacking" one (ANOVA; p<0.001). Excluding 20s at the start of the fitting window, the range of values identified with the ASE provided by the "1-s-bins" and the "stacking" procedures included the "true" value in 85% and in 95% of cases, respectively. The "stacking" procedure should be preferred since it yielded ASEs for the time constant that provided a range of values satisfying the statistical meaning of the width of the Confidence Intervals, at the given degree of probability.

UWB RCS calculations of smooth targets via the multi-band beam summation method

We present an ultra-wide-band beam-summation (UWB-BS) algorithm for calculating the radar cross-section (RCS) of large smooth targets whose asymptotic numerical complexity is of O ( 1 ) . In this approach, the incident field is expanded into a sparse set of iso-diffracting Gaussian beams (ID-GB), tiling the phase space (PS) of beam origins and directions. The field of each ID-GB is then tracked through reflections from the scatterer until the beam emerges from the target zone and propagates to the far zone using a closed-form near-to-far (N2F) zone transformation, and finally, the beam contributions are aggregated to obtain the scattered field. The algorithm can accommodate a multi-octave frequency band by dividing it into octave-wide sub-bands and introducing a UWB-BS scheme such that the PS beam lattices in the higher sub-bands are obtained by spatial-spectral upsampling those of the lower bands. The PS summation of ID-GBs is a priori localized and involves a roughly constant relatively small number of contributing beams in all the sub-bands, leading to an O ( 1 ) (frequency-independent) asymptotic computational complexity. Furthermore, with the ID parametrization, the beam axes and their propagation and scattering parameters are frequency-independent and, therefore, need to be calculated only once and then reused for all frequencies, thus greatly reducing the numerical cost of computing the RCS at multiple frequencies. The paper describes all aspects of the algorithm, including the choice of the beam parameters for asymptotic error convergence and the utilization of localization for algorithmic efficacy.

Approximation of operators on real line

There are exponential operators which are defined on the whole real line, namely, the Ismail–May operators , connected to , and the Gauss–Weierstrass operators , connected with 1. Here we consider the composition of these two operators and observe that the composition operator is not of exponential type operator. We find moment‐generating functions and moments of these operators and estimate some point‐wise convergence of these operators using the characteristic function. We also estimate quantitative Voronovskaya type asymptotic formula and error estimation. Also, the difference of the composition operators with its components and is estimated in terms of modulus of continuity. Finally, we provide some graphs to see visualize the convergence by considering the function .

Comparing frequentist and Bayesian uncertainty quantification in two‐step constitutive model calibration

AbstractThe calibration of constitutive models using experimental observations is a well‐established task in computational solid mechanics. Especially for complex constitutive models, multistep calibrations are regularly applied. While this approach is advantageous for reducing the complexity of the calibration problem, the quantification of parameter uncertainties becomes more challenging. In this study, we perform two‐step statistical parameter inference for an elasto‐plastic constitutive model utilizing a frequentist approach and Bayesian inference. The uncertainties are quantified by different methods, in particular, we compare an asymptotic normality analysis and Gaussian error propagation in the frequentist setting with sampling‐based Bayesian inference. Although frequentist and Bayesian parameter inference differ significantly in the way parameters are calibrated, both methods yield similar estimates. Additionally, the uncertainties quantified by asymptotic normality considerations and Gaussian error propagation are quite similar. However, the uncertainties of the elasticity parameters in the frequentist approach are found to be significantly smaller compared to Bayesian inference, due to the underlying approximations in the frequentist setting. Thus, uncertainties can be estimated in both settings, but it is crucial to be aware of the inherent assumptions and simplifications when interpreting the results.

Distributed learning for kernel mode–based regression

AbstractWe propose a parametric kernel mode–based regression built on the mode value, which provides robust and efficient estimators for datasets containing outliers or heavy‐tailed distributions. To address the challenges posed by massive datasets, we integrate this regression method with distributed statistical learning techniques, which greatly reduces the required amount of primary memory and simultaneously accommodates heterogeneity in the estimation process. By approximating the local kernel objective function with a least squares format, we are able to preserve compact statistics for each worker machine, facilitating the reconstruction of estimates for the entire dataset with minimal asymptotic approximation error. Additionally, we explore shrinkage estimation through local quadratic approximation, showcasing that the resulting estimator possesses the oracle property through an adaptive LASSO approach. The finite‐sample performance of the developed method is illustrated using simulations and real data analysis.

Numerical approximations of a lattice Boltzmann scheme with a family of partial differential equations

In this contribution, we address the numerical solutions of high-order asymptotic equivalent partial differential equations with the results of a lattice Boltzmann scheme for an inhomogeneous advection problem in one spatial dimension. We first derive a family of equivalent partial differential equations at various orders, and we compare the lattice Boltzmann experimental results with a spectral approximation of the differential equations. For an unsteady situation, we show that the initialization scheme at a sufficiently high order of the microscopic moments plays a crucial role to observe an asymptotic error consistent with the order of approximation. For a stationary long-time limit, we observe that the measured asymptotic error converges with a reduced order of precision compared to the one suggested by asymptotic analysis.

Uncertainty of Line-of-sight Velocity Measurements of Faint Stars from Low- and Medium-resolution Optical Spectra

Massively multiplexed spectrographs will soon gather large statistical samples of stellar spectra. The accurate estimation of uncertainties on derived parameters, such as the line-of-sight velocity v los, especially for spectra with low signal-to-noise ratios (S/Ns), is paramount. We generated an ensemble of simulated optical spectra of stars as if they were observed with low- and medium-resolution fiber-fed instruments on an 8 m class telescope, similar to the Subaru Prime Focus Spectrograph, and determined v los by fitting stellar templates to the simulated spectra. We compared the empirical errors of the derived parameters—calculated from an ensemble of simulations—to the asymptotic errors determined from the Fisher matrix, as well as from Monte Carlo sampling of the posterior probability. We confirm that the uncertainty of v los scales with the inverse square root of the S/N, but also show how this scaling breaks down at low S/N and analyze the error and bias caused by template mismatch. We outline a computationally optimized algorithm to fit multiexposure data and provide a mathematical model of stellar spectrum fitting that maximizes the so called significance, which allows for calculating the error from the Fisher matrix analytically. We also introduce the effective line count, and provide a scaling relation to estimate the errors of v los measurements based on stellar type. Our analysis covers a range of stellar types with parameters that are typical of the Galactic outer disk and halo, together with analogs of stars in M31 and in satellite dwarf spheroidal galaxies around the Milky Way.

High dimensional discriminant analysis under weak sparsity

. In this work, we consider the discriminant analysis under the weak sparsity where many entries of the parameters are nearly zero. We develop a unified LASSO-typed framework to estimate the parameters for sparse discriminant analysis and derive the general non asymptotic error bound under the weak sparsity condition. As applications, we revisit the sparse linear discriminant analysis and sparse quadratic discriminant analysis. We establish the consistency of the estimators and also the misclassification error rate. These results extend the sparse discriminant analysis methods to weak sparsity setting and refine the existing theoretical results.

Asymptotic Error Analysis for the Discrete Iterated Galerkin Solution of Urysohn Integral Equations with Green's Kernels

Asymptotic Error Analysis for the Discrete Iterated Galerkin Solution of Urysohn Integral Equations with Green's Kernels

Error performance and capacity of subcarrier BPSK modulated THz wireless systems with pointing errors.

Terahertz (THz) technology is poised to play a pivotal role in the future sixth-generation (6 G) wireless systems, offering substantial benefits with its extremely high bandwidth and rapid transmission speeds. This paper explores subcarrier-modulated THz wireless systems, analyzing their performance under the log-normal and gamma-gamma models. It provides a detailed examination of how composite turbulence and pointing errors impact the system performance. Closed-form solutions for the bit error rate and outage probability are derived for various conditions of turbulence and pointing errors. Furthermore, the paper delves into the asymptotic error performance, presenting numerical results that confirm the accuracy of the derived solutions across different propagation distances, turbulence strengths, and pointing error scenarios. The study offers critical insights into how different conditions affect the system performance, which is essential for characterizing subcarrier THz links in forthcoming 6 G deployments.

Asymptotic generalization errors in the online learning of random feature models

Deep neural networks are widely used prediction algorithms whose performance often improves as the number of weights increases, leading to over-parametrization. We consider a two-layered neural network whose first layer is frozen while the last layer is trainable, known as the random feature model. We study over-parametrization in the context of a student-teacher framework by deriving a set of differential equations for the learning dynamics. For any finite ratio of hidden layer size and input dimension, the student cannot generalize perfectly, and we compute the non-zero asymptotic generalization error. Only when the student's hidden layer size is exponentially larger than the input dimension, an approach to perfect generalization is possible. Published by the American Physical Society 2024

Introducing Ph.D. students to asymptotic inference for two‐stage M‐estimators: Easing analytic and coding demands via the use of numerical derivatives

AbstractApplications of two‐stage M‐estimators (2SMEs) abound in empirical economics. Asymptotic theory for 2SMEs (correct formulation of the asymptotic standard errors [ASE]) has been available for decades. Nevertheless, due to the daunting nature of the requisite matrix formulations, when conducting statistical inference based on two‐stage estimates, applied researchers often implement bootstrapping methods or ignore the two‐stage nature of the estimator and report the uncorrected second‐stage outputs from packaged statistical software. In the present paper, we offer teachers of econometrics a pedagogical approach for introducing Ph.D. students to asymptotic inference for 2SMEs, with a view toward easier software implementation and empirical application. We seek to demonstrate to students (and their teachers) that the analytic and coding demands for calculating correct ASEs for the 2SME need not be burdensome (or prohibitive). The main instructional (and practical) innovation that we offer in this regard is our suggested use of numerical derivative (ND) software for calculating the most challenging components of the ASE formulations. An exercise demonstrates to the student that, by implementing ND software, one can overcome the analytic and coding impediments to conducting inference based on 2SMEs, without abandoning rigor.

Accelerated Gradient Approach For Deep Neural Network-Based Adaptive Control of Unknown Nonlinear Systems.

Recent connections in the adaptive control literature to continuous-time analogs of Nesterov's accelerated gradient method have led to the development of new real-time adaptation laws based on accelerated gradient methods. However, previous results assume that the system's uncertainties are linear-in-the-parameters (LIP). To compensate for non-LIP uncertainties, our preliminary results developed a neural network (NN)-based accelerated gradient adaptive controller to achieve trajectory tracking for nonlinear systems; however, the development and analysis only considered single-hidden-layer NNs. In this article, a generalized deep NN (DNN) architecture with an arbitrary number of hidden layers is considered, and a new DNN-based accelerated gradient adaptation scheme is developed to generate estimates of all the DNN weights in real-time. A nonsmooth Lyapunov-based analysis is used to guarantee the developed accelerated gradient-based DNN adaptation design achieves global asymptotic tracking error convergence for general nonlinear control affine systems subject to unknown (non-LIP) drift dynamics and exogenous disturbances. A comprehensive set of simulation studies are conducted on a two-state nonlinear system, a robotic manipulator, and a complex 20-D nonlinear system to demonstrate the improved performance of the developed method. Our simulation studies demonstrate enhanced tracking and function approximation performance from both DNN architectures and accelerated gradient adaptation.

Discrete-time Euler-smoothing methods for time-varying convex constrained optimization

In this paper, we propose two kinds of prediction–correction methods, discrete-time Euler-smoothing methods, for solving the general time-varying convex constrained optimization problem. By using the complementarity function and smoothing technique, the Karush–Kuhn–Tucker (KKT) trajectory of the problem can be reformulated as a system of smooth equations and considering the dynamics of the system of smooth equations in a discrete constant time-sampling periods scheme. The proposed methods mainly consist of the prediction step derived by Euler approximation and the correction step generated by one or multiple Newton iterations. Under some reasonable conditions, the approximate solution trajectories produced by the methods achieve the asymptotic error bound behaving as O(h4) with respect to optimal trajectory both in the primal and dual space as the smoothing parameter approaches to zero, where h is the sampling period. Finally, the experimental results show that our methods are efficient, and the steady-state tracking errors are much smaller than the other prediction–correction methods under the same conditions.

Hybrid selection combining scheme for correlated cooperative communication over [formula omitted]-[formula omitted] fading channels

Due to poor (limited) scattering, a correlation among channels is inevitable, resulting in performance degradation of wireless communication systems. In this paper, we propose a hybrid selection combining (SC), also known as switched scaled – SC for correlated cooperative communication over κ-μ fading channel. By using paired error analysis, we derive the end-to-end symbol error probability of this system over bivariate κ-μ channel fading with M-ary phase shift keying (MPSK) modulation. In addition, we obtain significant parameters of hybrid SS-SC, such as the optimum scaling factor and optimum threshold by using the second derivative test. For every signal to noise ratio (SNR), we obtain different optimum thresholds and scaling factors. Our analytical results were found to overlap significantly with those of the simulation results. They were also compared with the variants of SC and switched diversity techniques. Our study demonstrates that the proposed hybrid technique performs better than the other SC and switched diversity schemes over correlated and uncorrelated fading channels. Furthermore, we derive an asymptotic symbol error probability for the proposed system, and the asymptotic results are in agreement with the analytical results at higher SNRs. Our study also shows that the diversity order of the system depends on the mutually exclusive events of the proposed technique.

Age-Related Constraints in the Visuomotor Plasticity of Postural Control as Revealed by a Whole-Body Mirror Learning Task.

Whether visuomotor plasticity of postural control is a trainable feature in older age remains an open question despite the wealth of visually guided exercise games promising to improve balance skill. We asked how aging affects adaptation and learning of a visual feedback (VF) reversal during visually guided weight shifting and whether this skill is modulated by explicit knowledge. Twenty-four older (71.43 ± 2.54years) and 24 young (24.04 ± 0.93years) participants were exposed to a 180° VF reversal while tracking a horizontally moving target by voluntarily weight shifting between two force platforms. An explicit strategy was available to half of the participants with detailed instruction to counter the VF rotation. Individual error data were fitted to an exponential function to assess adaptation. Fewer older (12/24) than younger (21/24) participants adapted to the VF reversal, displaying error curves that fitted the exponential function. Older adults who adapted to the VF reversal (responders, n = 12) reached an asymptote in performance in the same weight shifting cycle and displayed a similar mean asymptotic error compared with young participants. Young but not older responders exhibited an aftereffect when the VF reversal was removed. Instruction did not influence spatial error modulations regardless of age. The large individual variations within the older adults' group during early adaptation suggest age-specific limitations in using explicit cognitive strategies when older adults are exposed to an abrupt mirror feedback reversal that requires a change in weight shifting direction during whole-body postural tracking.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Asymptotic Errors in the Superconvergence of Discontinuous Galerkin Methods Based on Upwind-Biased Fluxes for 1D Linear Hyperbolic Equations

In this paper, we study the superconvergence of the semi-discrete discontinuous Galerkin (DG) method for linear hyperbolic equations in one spatial dimension. The asymptotic errors in cell averages, downwind point values, and the postprocessed solution are derived for the initial discretization by Gaussian projection (for periodic boundary condition) or Cao projection Cao et al. (SIAM J. Numer. Anal. 5, 2555–2573 (2014)) (for Dirichlet boundary condition). We proved that the error constant in the superconvergence of order 2k+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2k+1$$\\end{document} for DG methods based on upwind-biased fluxes depends on the parity of the order k. The asymptotic errors are demonstrated by various numerical experiments for scalar and vector hyperbolic equations.