Smooth Approximation Research Articles

On the robustness of methods to account for background bias in data assimilation to uncertainties in the bias estimates

AbstractFundamental to the theory of data assimilation is that the data (i.e., the observations and the background) provide an unbiased estimate of the true state. There are many situations when this assumption is known to be far from valid; and without bias correction (BC), significant biases will be present in the resulting analysis. Here, we compare two methods to account for biases in the background that do not require a change to the data assimilation algorithm: explicit BC and covariance inflation (CI). When the background bias is known perfectly it is clear that the BC method outperforms the CI method, in that it can completely remove the effect of the background bias whereas the CI method can only reduce it. However, the background bias can only be estimated when unbiased observations are available. A lack of unbiased observations means that the estimate of the background bias will always be subject to sample errors and structural errors due to poor assumptions about how the bias varies in space and time. Given these difficulties in estimating the background bias, the robustness of the two methods in producing an unbiased analysis is studied within an idealised linear system. It is found that the CI method is much less sensitive to errors in the background bias estimate and that a smooth estimate of the bias is crucial to the success of the BC method. However, the CI method is more sensitive to uncorrected biases in the observations.

FFT-Based Probability Density Imaging of Euler Solutions.

When using traditional Euler deconvolution optimization strategies, it is difficult to distinguish between anomalies and their corresponding Euler tails (those solutions are often distributed outside the anomaly source, forming "tail"-shaped spurious solutions, i.e., misplaced Euler solutions, which must be removed or marked) with only the structural index. The nonparametric estimation method based on the normalized B-spline probability density (BSS) is used to separate the Euler solution clusters and mark different anomaly sources according to the similarity and density characteristics of the Euler solutions. For display purposes, the BSS needs to map the samples onto the estimation grid at the points where density will be estimated in order to obtain the probability density distribution. However, if the size of the samples or the estimation grid is too large, this process can lead to high levels of memory consumption and excessive computation times. To address this issue, a fast linear binning approximation algorithm is introduced in the BSS to speed up the computation process and save time. Subsequently, the sample data are quickly projected onto the estimation grid to facilitate the discrete convolution between the grid and the density function using a fast Fourier transform. A method involving multivariate B-spline probability density estimation based on the FFT (BSSFFT), in conjunction with fast linear binning appropriation, is proposed in this paper. The results of two random normal distributions show the correctness of the BSS and BSSFFT algorithms, which is verified via a comparison with the true probability density function (pdf) and Gaussian kernel smoothing estimation algorithms. Then, the Euler solutions of the two synthetic models are analyzed using the BSS and BSSFFT algorithms. The results are consistent with their theoretical values, which verify their correctness regarding Euler solutions. Finally, the BSSFFT is applied to Bishop 5X data, and the numerical results show that the comprehensive analysis of the 3D probability density distributions using the BSSFFT algorithm, derived from the Euler solution subset of x0,y0,z0, can effectively separate and locate adjacent anomaly sources, demonstrating strong adaptability.

Bridging the Domain Gap in Scene Flow Estimation via Hierarchical Smoothness Refinement

This article introduces SmoothFlowNet3D, an innovative encoder-decoder architecture specifically designed for bridging the domain gap in scene flow estimation. To achieve this goal, SmoothFlowNet3D divides the scene flow estimation task into two stages: initial scene flow estimation and smoothness refinement. Specifically, SmoothFlowNet3D comprises a hierarchical encoder that extracts multi-scale point cloud features from two consecutive frames, along with a hierarchical decoder responsible for predicting the initial scene flow and further refining it to achieve smoother estimation. To generate the initial scene flow, a cross-frame nearest-neighbor search operation is performed between the features extracted from two consecutive frames, resulting in forward and backward flow embeddings. These embeddings are then combined to form the bidirectional flow embedding, serving as input for predicting the initial scene flow. Additionally, a flow smoothing module based on the self-attention mechanism is proposed to predict the smoothing error and facilitate the refinement of the initial scene flow for more accurate and smoother estimation results. Extensive experiments demonstrate that the proposed SmoothFlowNet3D approach achieves state-of-the-art performance on both synthetic datasets and real LiDAR point clouds, confirming its effectiveness in enhancing scene flow smoothness.

A nonconvex sparse recovery method for DOA estimation based on the trimmed lasso

Sparse direction-of-arrival (DOA) estimation methods can be formulated as a group-sparse optimization problem. Meanwhile, sparse recovery methods based on nonconvex penalty terms have been a hot topic in recent years due to their several appealing properties. Herein, this paper studies a new nonconvex regularized approach called the trimmed lasso for DOA estimation. We define the penalty term of the trimmed lasso in the multiple measurement vector model by ℓ2,1-norm. First, we use the smooth approximation function to change the nonconvex objective function to the convex weighted problem. Next, we derive sparse recovery guarantees based on the extended Restricted Isometry Property and regularization parameter for the trimmed lasso in the multiple measurement vector problem. Our proposed method can control the desired level of sparsity of estimators exactly and give a more precise solution to the DOA estimation problem. Numerical simulations show that our proposed method overperforms traditional approaches, which is more close to the Cramer-Rao bound.

TENSOR REGRESSION FOR INCOMPLETE OBSERVATIONS WITH APPLICATION TO LONGITUDINAL STUDIES.

Multivariate longitudinal data are frequently encountered in practice such as in our motivating longitudinal microbiome study. It is of general interest to associate such high-dimensional, longitudinal measures with some univariate continuous outcome. However, incomplete observations are common in a regular study design, as not all samples are measured at every time point, giving rise to the so-called blockwise missing values. Such missing structure imposes significant challenges for association analysis and defies many existing methods that require complete samples. In this paper we propose to represent multivariate longitudinal data as a three-way tensor array (i.e., sample-by-feature-by-time) and exploit a parsimonious scalar-on-tensor regression model for association analysis. We develop a regularized covariance-based estimation procedure that effectively leverages all available observations without imputation. The method achieves variable selection and smooth estimation of time-varying effects. The application to the motivating microbiome study reveals interesting links between the preterm infant's gut microbiome dynamics and their neurodevelopment. Additional numerical studies on synthetic data and a longitudinal aging study further demonstrate the efficacy of the proposed method.

Diffusion tensor imaging from past to present: A bibliometric analysis with trending topics and global productivity during 1980–2022

AbstractDiffusion tensor imaging (DTI) provides increasing opportunities for clinicians. As bibliometric studies define trends of novel modality applications, this study on DTI analyzed global productivity, most prolific institutions, and research areas. Articles obtained from the Web of Science (WoS) database were analyzed with the VOSviewer program, biblioshiny library in the R studio software bibliometrix package, exponential smoothing estimator, and with the SPSS (Version: 22.0) statistical software. In descending order, the United States, China, and Germany were the top 3 countries publishing DTI research. In descending order, the top 3 most productive institutions were California, Harvard and Johns Hopkins University. Trend topics include traumatic brain injury, tract‐based spatial statistics, fractional anisotropy, prognosis, machine learning, autism, white matter integrity and microstructure and Parkinson's disease. According to the estimations, 345 DTI articles will be published in 2025 (95% CI: 254–437). This study indicates the ever‐growing employment of DTI in clinical and basic sciences.

A conjecture of Zagier and the value distribution of quantum modular forms

In his influential paper on quantum modular forms, Zagier developed a conjectural framework describing the behavior of certain quantum knot invariants under the action of the modular group on their arguments. More precisely, when J_{K,0} denotes the colored Jones polynomial of a knot K , Zagier’s modularity conjecture describes the asymptotics of the quotient J_{K,0}(e^{2 \pi i \gamma(x)}) / J_{K,0}(e^{2 \pi i x}) as x \to \infty along rationals with bounded denominators, where \gamma \in \mathrm{SL}(2,\mathbb{Z}) . This problem is most accessible for the figure-eight knot 4_{1} , where the colored Jones polynomial has a simple explicit expression in terms of the q -Pochhammer symbol. Zagier also conjectured that the function h(x) = \log (J_{4_1,0}(e^{2 \pi i x}) / J_{4_1,0}(e^{2 \pi i /x})) can be extended to a function on \mathbb{R} which is continuous at irrationals. In the present paper, we prove Zagier’s continuity conjecture for all irrationals for which the sequence of partial quotients in the continued fraction expansion is unbounded. In particular, the continuity conjecture holds almost everywhere on the real line. We also establish a smooth approximation of h , uniform over all rationals, in accordance with the modularity conjecture. As an application, we find the limit distribution (after a suitable centering and rescaling) of \log J_{4_1,0}(e^{2 \pi i x}) , when x ranges over all reduced rationals in (0,1) with denominator at most N , as N \to \infty , thereby confirming a conjecture of Bettin and Drappeau.

Uniform convergence of the one-dimensional cubic Schrödinger equation

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in H s ( R ) ( s > 1 6 ) which is an alternative proof of Theorem 1.1 ( n = 1 , p = 3 ) of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Data fusion based on short-term memory Kalman filtering using intermittent-displacement and acceleration signal with a time-varying bias

Accurate displacement information plays a vital role in various applications from structural health monitoring to damage detection in physical systems. Due to practical reasons, displacement sensing is often performed using sensors that provide less accuracy or a low sampling rate. This study proposes an algorithm for displacement signal estimation that fuses high sampling rate acceleration data (which consists of unknown time-varying bias) with low sampling rate displacement data. In contrast to the present literature, the state vector in the proposed state-space formulation consists of short-term memory of the past realizations of displacement and velocity, and bias terms which allow (i) online estimation of the displacements by avoiding multi-rate Kalman filtering, and (ii) estimate time-varying bias in the measured acceleration signal. The state equations comprise the finite difference approximations of the time derivative of velocity and displacement which act as a moving average filter to provide smooth estimates, thus preventing the need for offline smoothing procedures. By inclusion of bias terms in the state vector, the drifts in displacement and velocity caused due to time-varying bias in measured acceleration are easily eliminated. The proposed method was found to provide accurate estimates of displacement, velocity, and time-varying bias in measured acceleration signals when tested on different types of bias, including but not limited to constant, linear, and parabolic variations. Additional tests on three oscillators subjected to three earthquake ground motions showed significantly better state estimates for stiff oscillators and improved estimates for flexible oscillators. Experimental validation on a base-excited two-story frame further affirms the capability of the method to estimate the hidden bias in the measured acceleration and provide an accurate estimate of displacement.

Towards optimal model evaluation: enhancing active testing with actively improved estimators

With rapid advancements in machine learning and statistical models, ensuring the reliability of these models through accurate evaluation has become imperative. Traditional evaluation methods often rely on fully labeled test data, a requirement that is becoming increasingly impractical due to the growing size of datasets. In this work, we address this issue by extending existing work on active testing (AT) methods which are designed to sequentially sample and label data for evaluating pre-trained models. We propose two novel estimators: the Actively Improved Levelled Unbiased Risk (AILUR) and the Actively Improved Inverse Probability Weighting (AIIPW) estimators which are derived from nonparametric smoothing estimation. In addition, a model recalibration process is designed for the AIIPW estimator to optimize the sampling probability within the AT framework. We evaluate the proposed estimators on four real-world datasets and demonstrate that they consistently outperform existing AT methods. Our study also shows that the proposed methods are robust to changes in subsample sizes, and effective at reducing labeling costs.

Equivalence between constrained optimal smoothing and Bayesian estimation

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) by adding convex constraints to the problem. Through a sequence of approximating Hilbertian subspaces and a discretised model, we prove that the Maximum a posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalisation of the paper Kimeldorf and Wahba (1970), where it is proved that the optimal smoothing solution is the mean of the posterior distribution. Synthetic and real data studies confirm the correspondence established in this paper.

A variational model for cartoon-texture decomposition of a color image

Image decomposition is one of the most important tasks in image processing. In this paper, we develop a novel saturation-value total variation model for color image decomposition. The approach is to propose a variational model containing an energy functional to derive the cartoon and texture decomposition in the saturation-value color space so that the color can be kept in the resulting cartoon component. The proposed idea is different from existing color image decomposition model where color information usually exists in both cartoon and texture components. In the variational model, saturation-value total variation is incorporated for regularizing the cartoon component of a color image, meanwhile, L1 norm is considered for modeling the texture component in the saturation-value color space. Theoretically, we develop a smooth approximation approach to study the existence of the solution of the proposed optimization problem. We formulate an effective and efficient algorithm to solve the proposed minimization problem based on the framework of alternating direction method of multipliers. Numerical examples are presented to demonstrate that the performance of the proposed model is better than that of other testing methods for several testing color images.

The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses

AbstractWe study the Langevin dynamics for spherical ‐spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder‐dependent initialization and on exponential time‐scales.

Smooth Information Criterion for Regularized Estimation of Item Response Models

Item response theory (IRT) models are frequently used to analyze multivariate categorical data from questionnaires or cognitive test data. In order to reduce the model complexity in item response models, regularized estimation is now widely applied, adding a nondifferentiable penalty function like the LASSO or the SCAD penalty to the log-likelihood function in the optimization function. In most applications, regularized estimation repeatedly estimates the IRT model on a grid of regularization parameters λ. The final model is selected for the parameter that minimizes the Akaike or Bayesian information criterion (AIC or BIC). In recent work, it has been proposed to directly minimize a smooth approximation of the AIC or the BIC for regularized estimation. This approach circumvents the repeated estimation of the IRT model. To this end, the computation time is substantially reduced. The adequacy of the new approach is demonstrated by three simulation studies focusing on regularized estimation for IRT models with differential item functioning, multidimensional IRT models with cross-loadings, and the mixed Rasch/two-parameter logistic IRT model. It was found from the simulation studies that the computationally less demanding direct optimization based on the smooth variants of AIC and BIC had comparable or improved performance compared to the ordinarily employed repeated regularized estimation based on AIC or BIC.

Smooth approximation of Lipschitz domains, weak curvatures and isocapacitary estimates

We provide a novel approach to approximate bounded Lipschitz domains via a sequence of smooth, bounded domains. The flexibility of our method allows either inner or outer approximations of Lipschitz domains which also possess weakly defined curvatures, namely, domains whose boundary can be locally described as the graph of a function belonging to the Sobolev space W2,q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,q}$$\\end{document} for some q≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\ge 1$$\\end{document}. The sequences of approximating sets is also characterized by uniform isocapacitary estimates with respect to the initial domain Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}.

Partition of unity method and smooth approximation

Partition of unity method and smooth approximation

Privacy-Preserving Robust Consensus for Distributed Microgrid Control Applications

Consensus-based distributed control has been proposed for coordinating distributed energy resources (DERs) in microgrids (MGs). As one key component, distributed average observers are used to estimate the average of a group of reference signals (e.g., voltage, current, or power). State-of-the-art distributed average observers could lead to loss of privacy due to information exchange on the communication channels. The DERs' reference signals, which contains private information, could be inferred by an eavesdropper. In this paper, a privacy-preserving distributed average observer is proposed, which is based on robust consensus and uses the state decomposition method to preserve privacy. Compared to the existing methods, the proposed observer does not require the knowledge of the reference signal's derivative and gives accurate and smooth estimation, and is thus applicable for MG distributed control applications. A detailed analysis regarding the convergence and privacy properties of the proposed observer is presented. The proposed observer is implemented on hardware controllers and validated in the context of distributed MG control applications through hardware-in-the-loop (HIL) tests.

Decision algorithm for educational institute selection with spherical fuzzy heronian mean operators and Aczel-Alsina triangular norm

This article presents a novel study of spherical fuzzy sets (SFSs), a more comprehensive framework of intuitionistic fuzzy sets and picture fuzzy sets. The SFS allows the decision-makers (DMs) to cope with complicated and insufficient information during the aggregation process. The Heronian mean (HrM) model theory is also utilized to express correlation among different input arguments or characteristics. Recently, the theory of Aczel Alsina triangular norms gained a lot of attention from various research scholars and has many capabilities to provide smooth approximations during decision analysis. In this article, we developed some appropriate operations of Aczel Alsina t-norms and t-conorms in light of spherical fuzzy (SF) information. We develop new mathematical ways to look at SF data to keep clarity and sufficient information. These are the SF Aczel Alsina Heronian mean (SFAAHrM) and SF Aczel Alsina weighted Heronian mean (SFAAWHrM) operators. Furthermore, we also present a list of new strategies based on Aczel Alsina operations, such as SF Aczel Alsina geometric Heronian mean (SFAAGHrM) and SF Aczel Alsina weighted geometric Heronian mean (SFAAWGHrM) operators. Some notable properties are also characterized to show the validity and effectiveness of our derived mathematical approaches. Considering our derived strategies, an algorithm for the multiple attribute decision-making (MADM) problem is established to resolve complicated real-life applications. A numerical example presents the compatibility of derived approaches and provides a solid mechanism to improve the performance of educational institutes. A comparison technique is also demonstrated to show the applicability and consistency of diagnosed approaches by contrasting the findings of pioneered approaches with existing methodologies.

A Symmetric Kernel Smoothing Estimation of the Time-Varying Coefficient for Medical Costs

In longitudinal studies, subjects are repeatedly observed at a set of distinct time points until the terminal event time. The time-varying coefficient model extends the parametric method and captures the dynamic trajectories of time-dependent covariate effects, thus enabling it to describe the potential relationship between the longitudinal variable and the observed time points. In this study, we propose a novel approach to the estimation of medical costs using a symmetric kernel smoothing method in the time-varying coefficient joint model. A smooth function of medical costs is derived by weighting the values of longitudinal data at all distinct observed time points via the combination of the kernel method and the inverse probability weighting method. For the simulation study, we first set up the true functions of time-varying coefficients; we then generated random samples for covariates and censored survival times. Subsequently, the longitudinal data of response variables could be produced. Further, numerical simulation experiments were conducted by using the proposed method and applying R code to the generated data. The estimated results for the parameters and non-parametric functions were compared with different settings. The numerical results illustrate that as the sample size increases, the bias and model-based standard errors decrease, and the performance improves with larger sample sizes. The estimates of functions in the model almost coincide with the true functions, as shown in the figures of the simulation study. Furthermore, the consistency of the obtained estimator is demonstrated via theoretical analysis, and a numerical simulation is performed to illustrate the performance of the proposed estimators. The proposed model is applied to a real-world data set acquired from a multicenter automatic defibrillator implantation trial (MADIT).

On the growth of Sobolev norm for the cubic NLS on two dimensional product space

We obtain polynomial bounds on the growth in time of Sobolev norm of solutions to the cubic defocussing nonlinear Schrödinger equation on two dimensional product space. We also give the angular improved bilinear Strichartz estimates for frequency localized functions, which estimates are used for enhancement of a smoothing estimates. Such upper bounds for the growth of Sobolev norms measure the transfer of energy from low to high modes as time grows on.