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The computation of a similarity measure for genomic data is a standard tool in computational genetics. The principal components of such matrices are routinely used to correct for biases due to confounding by population stratification, for instance in linear regressions. However, the calculation of both a similarity matrix and its singular value decomposition (SVD) are computationally intensive. The contribution of this article is threefold. First, we demonstrate that the calculation of three matrices (called the covariance matrix, the weighted Jaccard matrix, and the genomic relationship matrix) can be reformulated in a unified way which allows for the application of a randomized SVD algorithm, which is faster than the traditional computation. The fast SVD algorithm we present is adapted from an existing randomized SVD algorithm and ensures that all computations are carried out in sparse matrix algebra. The algorithm only assumes that row-wise and column-wise subtraction and multiplication of a vector with a sparse matrix is available, an operation that is efficiently implemented in common sparse matrix packages. An exception is the so-called Jaccard matrix, which does not have a structure applicable for the fast SVD algorithm. Second, an approximate Jaccard matrix is introduced to which the fast SVD computation is applicable. Third, we establish guaranteed theoretical bounds on the accuracy (in L 2 norm and angle) between the principal components of the Jaccard matrix and the ones of our proposed approximation, thus putting the proposed Jaccard approximation on a solid mathematical foundation, and derive the theoretical runtime of our algorithm. We illustrate that the approximation error is low in practice and empirically verify the theoretical runtime scalings on both simulated data and data of the 1000 Genome Project. [ABSTRACT FROM AUTHOR]

The objective of the present study is to evaluate the psychometric properties of the Scale of Attitudes toward Gender Equality in Football in the Context of Schools (SAGEFS) in the international context. This sample was formed by N = 6,101 students. The study was conducted by applying the SAGEFS. The model of the three factors correlated in the complete sample and for each country was correlated using AFC. The structural model was appraised by employing eight indices: the relative Chi-squared index; the goodness of fit index and its adjusted formula; the normal fit index; the comparative fit index; the standardized Quadratic Mean; and the Quadratic Mean Error of Approximation. To conclude, the results evidence the presence of psychometric properties that are indispensable for the measurement of attitudes toward gender equality in the context of schools. [ABSTRACT FROM AUTHOR]

Estimates of PM2.5 levels are crucial for monitoring air quality and studying the epidemiological impact of air quality on the population. Currently, the most precise measurements of PM2.5 are obtained from ground stations, resulting in limited spatial coverage. In this study, we consider satellite-based PM2.5 retrieval, which involves conversion of high-resolution satellite retrieval of Aerosol Optical Depth (AOD) into high-resolution PM2.5 retrieval. To improve the accuracy of the AOD to PM2.5 conversion, we employ the machine learning based post-process correction to correct the AOD-to-PM conversion ratio derived from Modern-Era Retrospective analysis for Research and Applications, Version 2 (MERRA-2) reanalysis model data. The post-process correction approach utilizes a fusion and downscaling of satellite observation and retrieval data, MERRA-2 reanalysis data, various high resolution geographical indicators, meteorological data and ground station observations for learning a predictor for the approximation error in the AOD to PM2.5 conversion ratio. The corrected conversion ratio is then applied to estimate PM2.5 levels given the high-resolution satellite AOD retrieval data derived from Sentinel-3 observations. Our model produces PM2.5 estimates with a spatial resolution of 100 meters at satellite overpass times. Additionally, we have incorporated an ensemble of neural networks to provide error envelopes for machine learning related uncertainty in the PM2.5 estimates. [ABSTRACT FROM AUTHOR]

This paper provides a convergence and stability analysis of the incremental value iteration algorithm under the influence of various errors. Incremental control is firstly used to linearize the continuous-time nonlinear system, recursive least squares (RLS) identification is then introduced to identify the incremental model online. Based on the incremental model, the value iteration algorithm is used to design an optimal adaptive controller, with an analytical optimal control law. Moreover, the convergence of the developed incremental value iteration algorithm is proved. The stability of the controller is analyzed using Lyapunov stability theory. Finally, a flight control simulation verifies the robustness of the controller to various initial conditions, as well as adaptation to actuator faults. [ABSTRACT FROM AUTHOR]

This work presents the development of a state-input affine differential neural network that approximates a class of nonlinear systems with uncertain dynamics and perturbations. The feasibility of using differential neural networks is based on the approximation capabilities of artificial neural networks, assuring that the approximation error of the right-hand side of the system with uncertain dynamics is bounded. The dynamics of the free parameters of the differential neural network is obtained using the Lyapunov's second stability method, which provides the convergence of the identification error to an invariant set around the origin. The approximation capabilities of the proposed identifier are compared with a classical differential neural network, aiming to identify the dynamics of a Stewart platform, which is used as testing example. The results show that the cumulative norm of identification's error is smaller using the state-input affine differential neural network, which demonstrates better identification outcomes with the proposed modified identifier. An additional advantage of the state-input affine differential neural network is that it has a quasi-linear structure that could help formulate a controller for several systems, which is a further research interest. • Learning laws for State-input affine differential neural network (SIADNN) identifier. • Design based on the second stability method of Lyapunov. • The novel DNN identifier has more applications than the traditional DNN identifier. • Evaluation of performing the non-parametric identification of a Stewart platform. [ABSTRACT FROM AUTHOR]

When preparing a pure state with a quantum circuit, there is an unavoidable approximation error due to the compilation error in fault-tolerant implementation. A recently proposed approach called probabilistic state synthesis, where the circuit is probabilistically sampled, is able to reduce the approximation error compared to conventional deterministic synthesis. In this paper, we demonstrate that the optimal probabilistic synthesis quadratically reduces the approximation error. Moreover, we show that a deterministic synthesis algorithm can be efficiently converted into a probabilistic one that achieves this quadratic error reduction. We also numerically demonstrate how this conversion reduces the T-count and analytically prove that this conversion halves an information-theoretic lower bound on the circuit size. In order to derive these results, we prove general theorems about the optimal convex approximation of a quantum state. Furthermore, we demonstrate that this theorem can be used to analyze an entanglement measure. [ABSTRACT FROM AUTHOR]

In this paper, the scope of discrete asymptotic homogenization employing voxel (cartesian) mesh discretization is expanded to estimate high fidelity effective properties of any periodic heterogeneous media with arbitrary Bravais's lattice symmetry, including those with non-orthogonal periodic bases. A framework was developed in Python with a proposed fast–nearest neighbour algorithm to accurately estimate the periodic boundary conditions of the discretized representative volume element of the lattice unit cell. Convergence studies are performed, and numerical errors caused by both voxel meshing and periodic boundary condition approximation processes are discussed in detail. It is found that the numerical error in periodicity approximation is cyclically dependent on the number of divisions performed during the meshing process and, thus, is minimized with a refined voxel mesh. Validation studies are performed by comparing the elastic properties of 2D hexagon lattices with orthogonal and non-orthogonal bases. The developed methodology was also applied to derive the effective properties of several lattice topologies, and variation of their anisotropic macroscopic properties with relative densities is presented as material selection charts. [ABSTRACT FROM AUTHOR]

The success of a multi-kilometre drive by a solar-powered rover at the lunar south pole depends upon careful planning in space and time due to highly dynamic solar illumination conditions. An additional challenge is that the rover may be subject to random faults that can temporarily delay long-range traverses. The majority of existing global spatiotemporal planners assume a deterministic rover-environment model and do not account for random faults. In this paper, we consider a random fault profile with a known, average spatial fault rate. We introduce a methodology to compute recovery policies that maximize the probability of survival of a solar-powered rover from different start states. A recovery policy defines a set of recourse actions to reach a safe location with sufficient battery energy remaining, given the local solar illumination conditions. We solve a stochastic reach-avoid problem using dynamic programming to find an optimal recovery policy. Our focus, in part, is on the implications of state space discretization, which is required in practical implementations. We propose a modified dynamic programming algorithm that conservatively accounts for approximation errors. To demonstrate the benefits of our approach, we compare against existing methods in scenarios where a solar-powered rover seeks to safely exit from permanently shadowed regions in the Cabeus area at the lunar south pole. We also highlight the relevance of our methodology for mission formulation and trade safety analysis by comparing different rover mobility models in simulated recovery drives from the LCROSS impact region. • The safety of a solar-powered rover depends on its location, energy and time of day. • Maximally safe behaviours depend on when and where delays occur during a traverse. • The probability of survival of a rover varies non-monotonically with time. • Min–max dynamic programming can conservatively account for the temporal risk ambiguity. [ABSTRACT FROM AUTHOR]

Ship operators typically depend on the visual method for draft reading, which may lead to errors or approximations, and it further introduces draft survey calculation errors and cannot provide continuous updates. Ensuring accurate real-time ship draft measurement becomes crucial for enhancing navigational safety, optimizing vessel performance, and achieving precise cargo and consumable measurements. With the current approach towards automation and remote operation of ships, the need to rely on the accuracy of the measurements provided by the sensors has increased. Although different sensor types are gradually being adopted for draft measurement, they encounter challenges in the demanding marine environment which may result in noisy and inaccurate readings. This paper aims to estimate the true draft of a ship in different conditions from noisy sensor measurements using the Kalman filter algorithm. The purpose of the algorithm is to reduce uncertainty in draft measurement that is generated from inaccuracies in the sensor or from the dynamic marine environment. The paper involves designing the Kalman Filter algorithm for draft measurement to work within the different conditions the ship may experience. Simulating different situations and analyzing the result, the application of the filter shows the advantage in real-time draft measurement in both static and dynamic conditions. [ABSTRACT FROM AUTHOR]

It has been shown that neural networks are vulnerable to adversarial attacks: adding well-crafted, imperceptible perturbations to their input can modify their output. Adversarial training is one of the most effective approaches to training robust models against such attacks. Unfortunately, this method is much slower than vanilla training of neural networks since it needs to construct adversarial examples for the entire training data at every iteration. By leveraging the theory of coreset selection, we show how selecting a small subset of training data provides a principled approach to reducing the time complexity of robust training. To this end, we first provide convergence guarantees for adversarial coreset selection. In particular, we show that the convergence bound is directly related to how well our coresets can approximate the gradient computed over the entire training data. Motivated by our theoretical analysis, we propose using this gradient approximation error as our adversarial coreset selection objective to reduce the training set size effectively. Once built, we run adversarial training over this subset of the training data. Unlike existing methods, our approach can be adapted to a wide variety of training objectives, including TRADES, ℓ p -PGD, and Perceptual Adversarial Training. We conduct extensive experiments to demonstrate that our approach speeds up adversarial training by 2–3 times while experiencing a slight degradation in the clean and robust accuracy. [ABSTRACT FROM AUTHOR]

Spiking neural networks (SNNs) become popular choices for processing spatiotemporal input data and enabling low‐power event‐driven spike computation on neuromorphic processors. However, direct SNN training algorithms are not well compatible with error back‐propagation process, while indirect conversion algorithms based on artificial neural networks (ANNs) are usually accuracy–lossy due to various approximation errors. Both of them suffer from lower accuracies compared with their reference ANNs and need lots of time steps to achieve stable performance in deep architectures. In this article, a novel conversion framework is presented for deep SNNs with negative‐spike dynamics, which takes a quantization constraint and spike compensation technique into consideration during ANN‐to‐SNN conversion, and a truly lossless accuracy performance with their ANN counterparts is obtained. The converted SNNs can retain full advantages of simple leaky‐integrate‐and‐fire spiking neurons and are very suited for hardware implementation. In the experimental results, it is shown that converted spiking LeNet on MNIST/FashionMNIST and VGG‐Net on CIFAR‐10 dataset yield the state‐of‐the‐art classification accuracies with quite shortened computing time steps and much fewer synaptic operations. [ABSTRACT FROM AUTHOR]

In this paper, we provide a new framework for studying continued fractions (CFs) by means of the backwards continued fraction (BCF). We develop an approximation theory for BCFs based on taking expansions of a fixed length, show the correspondence between continued fractions and their BCFs counterpart, and illustrate a rich approximation theory for continued fractions based off the methods of the approximation theory for the backwards case. In particular, we construct explicit functions that are sharp bounds for the BCF or CF error infinitely often over any BCF or CF cylinder set, and work out the details to pass seamlessly between the BCF and CF expansion of any real number. [ABSTRACT FROM AUTHOR]

The detection of elongated structures like lines or edges is an essential component in semantic image analysis. Classical approaches that rely on significant image gradients quickly reach their limits when the structure is context-dependent, amorphous, or not directly visible. This study introduces a principled mathematical description of elongated structures with various origins and shapes. Among others, it serves as an expressive operational description of target functions that can be well approximated by Convolutional Neural Networks. The nominal position of a curve and its positional uncertainty are encoded as a heatmap by convolving the curve distribution with a filter function. We propose a low-error approximation to the expensive numerical integration by evaluating a distance-dependent function, enabling a lightweight implementation with linear time complexity. We analyze the method's numerical approximation error and behavior for different curve types and signal-to-noise levels. Application to surgical 2D and 3D data, semantic boundary detection, skeletonization, and other related tasks demonstrate the method's versatility at low errors. [ABSTRACT FROM AUTHOR]

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorization rank of a non-negative matrix have been proposed. One of them is the cophenetic correlation coefficient ( c c c ), widely used in data science to evaluate the number of clusters in a hierarchical clustering. In previous work, it was shown that c c c performs better than other methods for rank selection in non-negative matrix factorization (NMF) when the underlying structure of the matrix consists of orthogonal clusters. In this article, we show that using the ratio of c c c to the approximation error significantly improves the accuracy of the rank selection. We also propose a new criterion, c o n c o r d a n c e , which, like c c c , benefits from the stochastic nature of NMF; its accuracy is also improved by using its ratio-to-error form. Using real and simulated data, we show that c o n c o r d a n c e , with a CUSUM-based automatic detection algorithm for its original or ratio-to-error forms, significantly outperforms c c c . It is important to note that the new criterion works for a broader class of matrices, where the underlying clusters are not assumed to be orthogonal. [ABSTRACT FROM AUTHOR]

Digital differentiators enable the computation of the derivative of a continuous-time signal at discrete time instances, and they are used in many signal processing applications. This paper derives a unified filter order estimate for digital differentiators that are realized with linear-phase finite-length impulse response filters and designed in the minimax sense. The estimate is useful at the high-level system design when assessing the implementation complexity and it enables fewer designs when finding the minimal filter order required to satisfy a prescribed tolerable approximation error. The proposed unified estimate covers both wideband and lowpass differentiators of integer degrees up to ten. Furthermore, degree-individual filter order estimates are derived which improve and extend previous results. The performance of both the unified and degree-individual order estimates is evaluated through simulation examples and compared with previous estimates. [ABSTRACT FROM AUTHOR]

As we know, feedforward neural networks (FNNs) with sigmoidal activation function are universal approximators. Theoretically, for any continuous function defined on a compact set, there exists an FNN such that the FNN can approximate the function with arbitrary accuracy, which constitutes a theoretical guarantee that FNN can be used as an efficient learning machine. This paper addresses the construction and approximation for FNNs. We construct an FNN with sigmoidal activation function and estimate its approximation error. In particular, an inverse theorem of the approximation is established, which implies the equivalence characterization theorem of the approximation and reveals the relationship between the topological structure of the FNN and its approximation ability. As keys in this study, the concepts of modulus of continuity of function, K-functional, and their relationship are utilized, and two Bernstein-type inequalities are established. [ABSTRACT FROM AUTHOR]

We show that Bayesian posteriors concentrate on the outcome distributions that approximately minimize the Kullback–Leibler divergence from the empirical distribution, uniformly over sample paths, even when the prior does not have full support. This generalizes Diaconis and Freedman's (1990) uniform convergence result to, e.g., priors that have finite support, are constrained by independence assumptions, or have a parametric form that cannot match some probability distributions. The concentration result lets us provide a rate of convergence for Berk's (1966) result on the limiting behavior of posterior beliefs when the prior is misspecified. We provide a bound on approximation errors in "anticipated‐utility" models, and extend our analysis to outcomes that are perceived to follow a Markov process. [ABSTRACT FROM AUTHOR]

The main issue we are studying in this paper is that of aggregation maps, which refers to the process of combining various input values into a single output. We apply aggregated special maps on Mittag-Leffler-type functions in one parameter to get diverse approximation errors for fractionalorder systems in Hilfer sense using an optimal method. Indeed, making use of various well-known special functions that are initially chosen, we establish a new class of matrix-valued fuzzy controllers to evaluate maximal stability and minimal error. An example is given to illustrate the numerical results by charts and tables. [ABSTRACT FROM AUTHOR]

Folklore suggests that policy gradient can be more robust to misspecification than its relative, approximate policy iteration. This paper studies the case of state-aggregated representations, in which the state space is partitioned and either the policy or value function approximation is held constant over partitions. This paper shows a policy gradient method converges to a policy whose regret per period is bounded by ϵ, the largest difference between two elements of the state-action value function belonging to a common partition. With the same representation, both approximate policy iteration and approximate value iteration can produce policies whose per-period regret scales as ϵ / (1 − γ) , where γ is a discount factor. Faced with inherent approximation error, methods that locally optimize the true decision objective can be far more robust. This paper was accepted by Hamid Nazerzadeh, data science. Supplemental Material: Data are available at https://doi.org/10.1287/mnsc.2023.4788. [ABSTRACT FROM AUTHOR]

A new parameter estimation method for a UAV braking system with unknown friction parameters is suggested. The unknown part containing friction is separated from the coupling system. The law of parameter updating is driven by the estimation error extracted by the auxiliary filter. A controller is developed to guarantee the convergence of the parameter estimation and tracking errors simultaneously. Furthermore, a designed performance index function enables the system to track the desired slip rate optimally. The optimal value of the performance function is updated by a single critic neural network (NN) to obtain comprehensive optimization of the control energy consumption, dynamic tracking error, and filtering error during braking. In addition, the unknown bounded interference and neural network approximation errors are compensated by robust integral terms. Simulation results are presented to confirm the effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

This paper studies stochastic games on large graphs and their graphon limits. We propose a new formulation of graphon games based on a single typical player's label-state distribution. In contrast, other recently proposed models of graphon games work directly with a continuum of players, which involves serious measure-theoretic technicalities. In fact, by viewing the label as a component of the state process, we show in our formulation that graphon games are a special case of mean field games, albeit with certain inevitable degeneracies and discontinuities that make most existing results on mean field games inapplicable. Nonetheless, we prove the existence of Markovian graphon equilibria under fairly general assumptions as well as uniqueness under a monotonicity condition. Most importantly, we show how our notion of graphon equilibrium can be used to construct approximate equilibria for large finite games set on any (weighted, directed) graph that converges in cut norm. The lack of players' exchangeability necessitates a careful definition of approximate equilibrium, allowing heterogeneity among the players' approximation errors, and we show how various regularity properties of the model inputs and underlying graphon lead naturally to different strengths of approximation. Funding: D. Lacker was partially supported by the Air Force Office of Scientific Research [Grant FA9550-19-1-0291] and the National Science Foundation [Award DMS-2045328]. [ABSTRACT FROM AUTHOR]

Linear fixed-point equations in Hilbert spaces arise in a variety of settings, including reinforcement learning, and computational methods for solving differential and integral equations. We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. First, we prove an instance-dependent upper bound on the mean-squared error for a linear stochastic approximation scheme that exploits Polyak–Ruppert averaging. This bound consists of two terms: an approximation error term with an instance-dependent approximation factor and a statistical error term that captures the instance-specific complexity of the noise when projected onto the low-dimensional subspace. Using information-theoretic methods, we also establish lower bounds showing that both of these terms cannot be improved, again in an instance-dependent sense. A concrete consequence of our characterization is that the optimal approximation factor in this problem can be much larger than a universal constant. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation, establishing their optimality. Funding: This work was partially supported by grants from the Office of Naval Research [Grant DOD-ONR-N00014-18-1-2640] and the National Science Foundation (NSF) [NSF-IIS Grant 1909365, NSF-DMS Grant 2015454, and NSF-CCF Grant 1955450] to M. J. Wainwright. Part of this work was performed when A. Pananjady was visiting the Simons Institute for the Theory of Computing, where he was supported by a Swiss Re Research Fellowship. Supplemental Material: The online supplementary file is available at https://doi.org/10.1287/moor.2022.1341. [ABSTRACT FROM AUTHOR]

In this paper, for partially observed Markov decision problems (POMDPs), we provide the convergence of a Q learning algorithm for control policies using a finite history of past observations and control actions, and consequentially, we establish near optimality of such limit Q functions under explicit filter stability conditions. We present explicit error bounds relating the approximation error to the length of the finite history window. We establish the convergence of such Q learning iterations under mild ergodicity assumptions on the state process during the exploration phase. We further show that the limit fixed point equation gives an optimal solution for an approximate belief Markov decision problem (MDP). We then provide bounds on the performance of the policy obtained using the limit Q values compared with the performance of the optimal policy for the POMDP, in which we also present explicit conditions using recent results on filter stability in controlled POMDPs. Whereas there exist many experimental results, (i) the rigorous asymptotic convergence (to an approximate MDP value function) for such finite memory Q learning algorithms and (ii) the near optimality with an explicit rate of convergence (in the memory size) under filter stability are results that are new to the literature to our knowledge. Funding: This research was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada. [ABSTRACT FROM AUTHOR]

As massive MIMO is a key technology in the future sixth generation (6G), the large-scale antenna arrays are widely considered in direction-of-arrival (DOA) estimation for they can provide larger aperture and higher estimation resolution. However, the conventional fully digital architecture requires one radio-frequency (RF) chain per antenna, and this is challenging for the high hardware costs and much more power consumption caused by the large number of RF chains. Therefore, an overlapped subarray (OSA) architecture-based hybrid massive MIMO array is proposed to reduce the hardware costs, and it can also have better DOA estimation accuracy compared to non-overlapped subarray (NOSA) architecture. The simulation results also show that the accuracy of the proposed OSA architecture has 6 ∘ advantage over the NOSA architecture with signal-to-noise ratio (SNR) at 10 dB. In addition, to improve the DOA estimation resolution, a deep learning (DL)-based estimator is proposed by combining convolution denoise autoencoder (CDAE) and deep neural network (DNN), where CDAE can remove the approximation error of sample covariance matrix (SCM) and DNN is used to perform high-resolution DOA estimation. From the simulation results, CDAE-DNN can achieve the accuracy lower bound at SNR = - 8 dB and the number of snapshots N = 100 , this means it has better performance in poor communication situation and can save more software resources compared to conventional estimators. [ABSTRACT FROM AUTHOR]

In this paper it is shown, that if a possibly inhomogeneous Markov chain with continuous time and finite state space is weakly ergodic and all the entries of its intensity matrix are locally integrable, then, using available results from the perturbation theory, its time-dependent probability characteristics can be approximately obtained from another Markov chain, having piecewise constant intensities and the same state space. The approximation error (the taxicab distance between the state probability distributions) is provided. It is shown how the Cauchy operator and the state probability distribution for an arbitrary initial condition can be calculated. The findings are illustrated with the numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, a Szasz-Favard-Mirakyan operator is exploited for chaos synchronization of a master-slave system. The universal approximation feature asset lets in Szasz-Favard-Mirakyan operators for approximating uncertainties, including un-modeled dynamics and disturbances. The above subject is considered in detail in this investigation. It is confirmed that the synchronization/approximation errors are uniformly bounded and stable if the Szasz-Favard-Mirakyan operators are applied as the regressors. Additionally, it has been presumed that the synchronization's error rate is not available, and an observer will be designed for its estimation. The Duffing–Holmes oscillator is examined as the computer-generated chaotic framework with the target of examining the functioning of the recommended synchronization observer-based controller. The outcomes are compared with an effective approximation-based control strategy. Unlike Chebyshev Neural Network approximators in which the system's inputs states are needed to define the regressor vector and approximate uncertainties, the suggested Szasz-Favard-Mirakyan operators–based strategy is independent of the system states for constructing the regressor vector. [ABSTRACT FROM AUTHOR]

In sampling theory, it is a popular trend to use auxiliary information to obtain more efficient estimators for the population parameters to increase the precision of the estimator. Estimators obtained using auxiliary information are supposed to be more efficient than the estimators obtained without using auxiliary information. The ratio, regression, product and difference methods take advantage of the auxiliary information at the estimation stage. Therefore, this study considered a generalized class of log-type estimators of finite population mean based on correlation coefficient as the proposed estimator for estimating the population mean of the study variable. It has been shown that the generalized class of log-type estimators has lesser mean square errors (MSEs) under the optimum values of the characterizing scalar as compared to some of the commonly used related estimators available in the literature. Further, an extension of the proposed generalized class of log-type estimators using multiple auxiliary variables such as coefficient of variation (CX), coefficient of kurtosis (b2(x)), and correlation coefficient (ρxy). have also base initiated in this dissertation. The expressions for the properties of the proposed family of estimators, that is; Bias and Mean Square Error (MSE), were derived to the first degree of approximation. We also obtained the optimum Mean Square Error (MSEopt.), and theoretical comparisons were made with the related existing estimators in literature. Following theoretical comparisons, it was demonstrated that the proposed family of estimators was more efficient than various related existing estimators compared with, under the obtained conditions. [ABSTRACT FROM AUTHOR]

Reinforcement learning (RL) is explored for motor control of a novel pneumatic-driven soft robot modeled after continuum media with a varying density. This model complies with closed-form Lagrangian dynamics, which fulfills the fundamental structural property of passivity, among others. Then, the question arises of how to synthesize a passivity-based RL model to control the unknown continuum soft robot dynamics to exploit its input–output energy properties advantageously throughout a reward-based neural network controller. Thus, we propose a continuous-time Actor–Critic scheme for tracking tasks of the continuum 3D soft robot subject to Lipschitz disturbances. A reward-based temporal difference leads to learning with a novel discontinuous adaptive mechanism of Critic neural weights. Finally, the reward and integral of the Bellman error approximation reinforce the adaptive mechanism of Actor neural weights. Closed-loop stability is guaranteed in the sense of Lyapunov, which leads to local exponential convergence of tracking errors based on integral sliding modes. Notably, it is assumed that dynamics are unknown, yet the control is continuous and robust. A representative simulation study shows the effectiveness of our proposal for tracking tasks. [ABSTRACT FROM AUTHOR]

The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment.

In a 3 D bounded and C 1 , α -smooth domain Ω, α ∈ (0 , 1) , we distribute a cluster of nanoparticles enjoying moderately contrasting relative permittivity and permeability which can be anisotropic. We show that the effective permittivity and permeability generated by such cluster is explicitly characterized by the corresponding electric and magnetic polarization tensors of the fixed shape. The error of the approximation of the scattered fields corresponding to the cluster and the effective medium is inversely proportional to the dilution parameter c r : = δ a , where a is the maximum diameter of the nanoparticles and δ is the minimum distance between them. The constant of the proportionality is given in terms of a-priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on their permittivity and permeability parameters, upper bound on the dilution parameter c r , the used incident frequency and the domain Ω). A key point of the analysis is to show that the Foldy-Lax field appearing in the meso-scale approximation, derived in [17] , is a discrete form of a (continuous) system of Lippmann-Schwinger equations with a related effective permittivity and permeability contrasts. To derive this, we prove that the Lippmann-Schwinger operator, for the Maxwell system, is invertible in the Hölder spaces. As a by-product, this shows a Hölder regularity property of the electromagnetic fields up to the boundary of the inhomogeneity. [ABSTRACT FROM AUTHOR]

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions. [ABSTRACT FROM AUTHOR]

Off-policy learning, where the goal is to learn about a policy of interest while following a different behavior policy, constitutes an important class of reinforcement learning problems. It is well-known that emphatic temporal-difference (TD) learning is a pioneering off-policy reinforcement learning method involving the use of the followon trace. Although the gradient emphasis learning (GEM) algorithm has recently been proposed to fix the problems of unbounded variance and large emphasis approximation error introduced by the followon trace from the perspective of stochastic approximation. This approach, however, is limited to a single gradient-TD2-style update instead of considering the update rules of other GTD algorithms. Overall, it remains an open question on how to better learn the emphasis for off-policy learning. In this paper, we rethink GEM and investigate introducing a novel two-time-scale algorithm called TD emphasis learning with gradient correction (TDEC) to learn the true emphasis. Further, we regularize the update to the secondary learning process of TDEC and obtain our final TD emphasis learning with regularized correction (TDERC) algorithm. We then apply the emphasis estimated by the proposed emphasis learning algorithms to the value estimation gradient and the policy gradient, respectively, yielding the corresponding emphatic TD variants for off-policy evaluation and actor-critic algorithms for off-policy control. Finally, we empirically demonstrate the advantage of the proposed algorithms on a small domain as well as challenging Mujoco robot simulation tasks. Taken together, we hope that our work can provide new insights into the development of a better alternative in the family of off-policy emphatic algorithms. [ABSTRACT FROM AUTHOR]

The experiences of close relationships-revised (ECR-R) is a widely used 36-item self-report measurement for measuring adult attachment. However, various short versions of the ECR-R have been developed and tested psychometrically. Given the cultural impact, a short version of the Thai ECR-R should be derived from the existing Thai version of the ECR-R. This study aimed to develop a 10-item version of the ECR-R that demonstrates comparable psychometric properties to the previous Thai version and the 18-item ECR-R. This study included four studies with a total of 1,322 participants. In study 1, 434 adults in a nonclinical setting were used for the development of the 10-item Thai ECR-R and tested in an independent sample. Studies 2, 3, and 4 were conducted on 312 adults in the clinical setting, 227 older adults in the nonclinical, and 123 older adults in clinical settings. The Cronbach alphas and corrected correlations between the ECR-R-18 and the ECR-R-10 in each study were calculated. Confirmatory factor analysis of the first-order two-factor solution model with fit statistics was examined with each sample. Correlations of the ECR-R-18 and the ECR-R-10 with other measurements were presented and compared. Known-group validity and measurement invariance test were also examined. The Cronbach alphas of the ECR-R-10 among all samples were acceptable, ranging between.77 and.85 for avoidance subscales and between.82 and.86 for anxiety subscales. The corrected correlation between the ECR-R-18 and ECR-R-10 was between.61 (p <.001) and.82 (p <.001). The values of the comparative fit index and Tucker-Lewis index for the model of ECR-R-10 were between.903 and.985, whereas the root-mean-square error of approximation was between.082 and.036, indicating that the model fits were acceptable. The ECR-R-10 was related to the measurements with a similar construct; however, no difference in the magnitude of correlation was observed between ECR-R-18 and ECR-R-10. Known group validity was established. Measurement invariance was successfully established across different age and gender groups, although it was only partially achieved with respect to clinical status. The ECR-R-10 provided equal or superior psychometric properties to the ECR-R-18 across age groups and settings. As it is a briefer scale, the ECR-R-10 can be practically used in general and clinical samples to reduce the burden of assessment, especially with older adults. Further investigation is needed to test the scale's temporal stability. [ABSTRACT FROM AUTHOR]

Vibration-based health diagnostics technique has shown great potentials to enhance the safety and reliability for many industrial rotary machinery. The emerging sparse representation classification (SRC) paradigm provides a promising tool for intelligent machinery health diagnostics. However, traditional SRC approaches neglect the useful priori information in rotary machinery vibration data, limiting the reconstruction ability and thus restricting their diagnostic accuracy. To address this issue, we present a novel sparsity assisted intelligent recognition (SAIR) methodology for vibration-based machinery health diagnostics. SAIR is constituted by two stages for dictionary design and intelligent recognition. In the dictionary design stage, SAIR exploits the useful priori information of the prediction shift-invariance property, to design class-specific dictionaries via an overlapping segmentation strategy. Additionally, this dictionary design strategy can leverage those local and nonlocal features within data segments. In the intelligent recognition stage, SAIR implements health state recognition by means of the sparsity-based health diagnostic strategy according to minimal sparse approximation errors. Finally, the feasibility and advantage of SAIR have been comprehensively evaluated for planetary gearbox health diagnostics, indicating that SAIR yields an overall diagnostic accuracy of 99.72%. Besides, comparative studies demonstrate that SAIR outperforms the advanced mainstream methods with better diagnostic accuracy and lower computation costs for vibration-based machinery health diagnostics. [ABSTRACT FROM AUTHOR]

In this article, q -Chlodowsky operators are utilized for synchronizing a chaotic master-slave system. The universal approximation property enables q -Chlodowsky operators to approximate uncertainties, which consist of disturbances and unmodeled dynamics. It is verified that uniformly ultimately bounded stability of the tracking/approximation errors can be assured if the q -Chlodowsky operators are utilized as regressors and the polynomial coefficients are tuned by the adaptive laws calculated in the stability analysis. Moreover, it has been assumed that the rate of synchronization error is not at hand and a filter-based strategy has been adopted instead of designing an observer. The Duffing-Holmes oscillator is studied and the results are also compared with three powerful approximation-based and robust control methods. Unlike neural network/fuzzy estimators in which system states are necessary to define the regressor vector, the proposed uncertainty estimator does not require them in the regressor vector. Furthermore, in comparison with radial basis functions neural networks (RBFNNs), we are involved in fewer adaptive coefficients in the regressor vector of the proposed uncertainty estimator. As a result, the tedious and time-consuming procedure of trial and error for tuning the uncertainty estimator is eliminated. [ABSTRACT FROM AUTHOR]

We consider a new approach to approximate stability analysis for a tri-additive functional inequality and to obtain the optimal approximation for permuting tri-derivations and tri-homomorphisms in unital matrix algebras via the vector-valued alternative fixed-point theorem, which is a popular technique of proving the stability of functional equations. We also present a small list of aggregation functions on the classical, well-known special functions to investigate the best approximation error estimates using a different concept of perturbation stability. [ABSTRACT FROM AUTHOR]

Gaussian processes (GPs) are a popular technique for global metamodeling applications. Their objective in such settings is to establish an efficient and globally accurate approximation of the response surface of computationally expensive simulation models. When developing such GPs, the design of (simulation) experiments (DoE) plays an important role in reducing the required number of model runs for obtaining accurate approximations. Sequential (adaptive) selection of experiments can provide significant advantages, especially when the response surface is characterized by localized nonlinearities. Such adaptive DoE strategies for global metamodeling applications typically focus on minimizing the predictive GP variance, representing an exploration strategy, while recent developments have additionally considered the reduction of the GP bias obtained through cross validation, representing an exploitation strategy. While significant focus has been placed on the definition of appropriate adaptive DoE criteria, computational challenges still exist that limit the widespread adoption of adaptive DoE techniques—for example, related to the additional computational demand for identifying the optimal new experiment(s) or the necessity to establish proper schemes to combine exploration and exploitation strategies. To address these specific challenges, this research investigates two new adaptive DoE formulations. The first one focuses on the approximation of the popular integrated mean square error (IMSE) DoE criterion. The computationally demanding GP predictive variance update (after addition of each candidate experiment), required in the original IMSE formulation, is replaced by an approximation based on the current predictive variance and the domain of influence that surrounds each new experiment. The approximation is established through a parametric formulation that leverages the GP kernel to describe the aforementioned domain, with characteristics that are progressively calibrated across the GP training stages, to minimize the discrepancy between the actual and the approximated IMSE. The second formulation establishes a multicriteria search for simultaneously identifying multiple Pareto optimal experiments that balance exploration and exploitation objectives, replacing conventional strategies that establish a weighted combination of these objectives to promote a single DoE selection criterion. [ABSTRACT FROM AUTHOR]

Corner rounding methods have been widely developed to pursue the smooth motions of machine tools. However, most corner rounding methods, which adopt the double inscribed transitions, still remain an inherent issue of retaining large curvatures of transition curves. Even for those double circumscribed transitions-based methods with relatively small curvatures, they also constrain excessively the transition lengths and are limited to a low-order continuity, deteriorating the feedrate and jerk of machine tools. For addressing these problems, a C3 continuous double circumscribed corner rounding (DCCR) method is proposed for five-axis linear tool path. In this method, the C3 continuous double circumscribed B-splines are specially designed to round the corners of tool position and tool orientation, whose transition lengths are analytically determined by jointly constraining the approximation errors, overlaps elimination, and parameter synchronization. Moreover,the excessive constrains of transition lengths imposed by traditional methods are alleviated by fully considering the effects of overlaps and parameter synchronization, and the jerk of rotary axes is also limited with a high-order continuity. Compared to the existing double inscribed corner rounding (DICR) and DCCR methods, experiment results demonstrate that our method can improve further the feedrate while limiting the jerk of machine tools. [ABSTRACT FROM AUTHOR]

Uncompensated motion errors can seriously affect the imaging quality of synthetic aperture sonars (SASs). In the existing line-by-line motion compensation (MOCO) algorithms for wide-beam multiple-receiver SAS systems, the approximate form of the range history error usually introduces a significant approximation error, and the residual two-dimensional (2D) range cell migration (RCM) caused by aperture-dependent motion errors is not corrected, resulting in the severe defocus of the image. In this paper, in the presence of translational and rotational errors in a multiple-receiver SAS system, the exact range history error concerning the five-degree-of-freedom (DOF) motion errors of the sway, heave, yaw, pitch, and roll under the non-stop-hop-stop case is derived. Based on this, a two-stage subaperture MOCO algorithm for wide-beam multiple-receiver SAS systems is proposed. We decompose the range history error into the beam-center term (BCT) and the residual spatial-variant term (RSVT) to compensate successively. In the first stage, the time delay and phase error caused by the BCT are compensated receiver-by-receiver through interpolation and phase multiplication in the azimuth-time domain. In the second stage, the data of a single pulse are regarded as a subaperture, and the RSVT is compensated in the subaperture range-Doppler (RD) domain. We divide the range into several blocks to correct RCM caused by the RSVT in the subaperture RD domain, and the phase error caused by the RSVT is compensated by phase multiplication. After compensation, the wide-beam RD algorithm is used for imaging. Simulated and real-data experiments verify the superiority and robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we first design a residual-type error estimator for a modified weak Galerkin (MWG) method of 2D H (curl) elliptic problems. Then, we show that the indicator is reliable with respect to the approximation error measured in terms of a natural energy norm, the main ingredient of our approach is to translate the error between weak solution and MWG solution into the conforming part and nonconforming part. We also prove the efficiency of the error estimator by using standard bubble functions. Finally, we provide several experiments to verify the performance of the error estimator within both uniform meshes and adaptive meshes. [ABSTRACT FROM AUTHOR]

The paper presents a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach is relatively simple and effective in application, and it takes advantage of some unique properties of the DG method. The error function is constructed in an enriched approximation space, utilizing the hierarchical nature of the basis functions. Among many versions of the DG method, the most popular one is based on the interior penalty approach. However, in this paper a DG method with finite difference (DGFD) is utilized, where the continuity of the approximate solution is enforced by finite difference conditions applied on the mesh skeleton. In the DG methods arbitrarily shaped finite elements can be used, so in this paper the meshes with polygonal finite elements are considered, including quadrilateral and triangular elements. Some benchmark examples are presented, in which Poisson's and linear elasticity problems are considered. The examples use various mesh densities and approximation orders to evaluate the errors. The error estimation maps, generated for the discussed tests, indicate a good correlation with the exact errors. In the last example, the error approximation concept is applied for an adaptive hp mesh refinement. [ABSTRACT FROM AUTHOR]

In this paper, the composite control of position and orientation of the quadrotor for tracking purposes are investigated. The quadrotor is an underactuated nonlinear system. So, based on the introduced virtual inputs the quadrotor is transformed into an actuated system. External disturbances and different sources of uncertainty such as input saturation are considered in the dynamic model of the quadrotor. Also, after transform implementation, some complicated terms appear which are considered as other sources of uncertainty. To guarantee the stabilization and tracking goals of the quadrotor in the perturbed model, the composite adaptive fuzzy PID-Like controller is proposed. This controller consists of three parts where the innovation of the paper appears in all terms. The main term is an adaptive fuzzy PID-like controller which able to approximate the coefficients of PID based on the decentralized fuzzy logic. Then, a H∞ compensator based on a modified Riccati-like equation is presented; which can attenuate the fuzzy approximation errors. Also, as the last term of the composite controller, the modified disturbance observer is designed that estimates the uncertainty and disturbance. The experimental result shows that the proposed method leads to resilient stabilization and tracking in comparison with conventional methods. [ABSTRACT FROM AUTHOR]

In this paper, we apply the well-known aggregation mappings on Mittag-Leffler-type functions to investigating new approximation error estimates of a W -Hilfer fractional differential equation, by a different concept of Ulam-type stability in both bounded and unbounded domains. [ABSTRACT FROM AUTHOR]

Let F be a real valued generalized biaxisymmetric potential (GBASP) in Lβ on SR, the open sphere of radius R about the origin. In this paper we have obtained the necessary and sufficient conditions on the rate of decrease of a sequence of best harmonic polynomial approximates to F such that F is harmonically continues as an entire function GBASP and determine their (p, q)- order and generalized (p, q)-type with respect to proximate order ρ(r). [ABSTRACT FROM AUTHOR]

In order to estimate the statistical phase of underwater propagation for coherent acoustic systems, a calculation of background structure functions, based on wavespeeds, provides a wavefront distortion metric that excludes the phase due to focusing. In 1890, Gouy measured a phase change through the focus that differs for spherical vs cylindrical convergence by a factor of 2 from π/2. Statistics for the locations of the probable focus waist can use structure functions based on the measured sound speed profiles. Phase estimates are susceptible to errors in phase estimation unless results consider the Gouy phase while passing through focus waists. These foci may counteract or average with other errors of estimation, including scattering, type of spreading (cylindrical vs spherical foci), and reflections from the surface and bottom. Oceans have a sufficient range for the spatial derivatives of sound speed profiles to contribute persistently to foci that we show have coherence time and cross-sectional spatial (imaging) coherence. As an alternative, specular point theory can estimate deterministic pressure solutions. Convergence of these ray models that violate the stationary phase requires methods such as catastrophe theory to estimate the focal effects. Sommerfeld had an early comprehensive ray theory description that includes the Gouy phase. Using Flatté's data, the foci are clearly time and length coherent. Statistically, the mutual coherence function for the Atlantic, Caribbean, and Gulf of Mexico regions shows that even distorted and out-of-phase convergence has full spatial coherence. [ABSTRACT FROM AUTHOR]

We present an open-source system that can optimize compressed trajectory representations for large fleets of vessels. We take into account the type of each vessel in order to choose a suitable configuration that can yield improved trajectory synopses, both in terms of approximation error and compression ratio. We employ a genetic algorithm that converges to a fine-tuned configuration per vessel type without any hyper-parameter tuning. These configurations can provide synopses that retain less than 10% of the original points with less than 20m approximation error in a real world dataset; in another dataset with 90% less samples than the previous one, the synopses retain 20% of the points and achieve less than 80m error. Additionally the level of compression can be chosen by the user, by setting the desired approximation error. Our system also supports incremental optimization by training in data batches, and therefore continuously improves performance. Furthermore, we employ a composite event recognition engine to efficiently detect complex maritime activities, such as ship-to-ship transfer and loitering; thanks to the synopses generated by the genetic algorithm instead of the raw trajectories, we make the recognition process faster while also maintaining the same level of recognition accuracy. Our extensive empirical study demonstrates the effectiveness of our system over large, real-world datasets. [ABSTRACT FROM AUTHOR]

The paper refers to interdisciplinary research in the areas of hierarchical cluster analysis of big data and ordering of primary data to detect objects in a color or in a grayscale image. To perform this on a limited domain of multidimensional data, an NP-hard problem of calculation of close to optimal piecewise constant data approximations with the smallest possible standard deviations or total squared errors (approximation errors) is solved. The solution is achieved by revisiting, modernizing, and combining classical Ward's clustering, split/merge, and K-means methods. The concepts of objects, images, and their elements (superpixels) are formalized as structures that are distinguishable from each other. The results of structuring and ordering the image data are presented to the user in two ways, as tabulated approximations of the image showing the available object hierarchies. For not only theoretical reasoning, but also for practical implementation, reversible calculations with pixel sets are performed easily, as with individual pixels in terms of Sleator–Tarjan Dynamic trees and cyclic graphs forming an Algebraic Multi-Layer Network (AMN). The detailing of the latter significantly distinguishes this paper from our prior works. The establishment of the invariance of detected objects with respect to changing the context of the image and its transformation into grayscale is also new. [ABSTRACT FROM AUTHOR]

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The new iterative method of approximating the boundary trajectory of a short-focus electron beam propagating in a free drift mode in a low-pressure ionized gas under the condition of compensation of the space charge of electrons is considered and discussed in the article. To solve the given approximation task, the root-polynomial functions of the fourth and fifth order were applied, the main features of which are the ravine character and the presence of one global minimum. As an initial approach to solving the approximation problem, the values of the polynomial coefficients are calculated by solving the interpolation problem. After this, the approximation task is solved iteratively. All necessary polynomial coefficients are calculated multiple times, taking into account the values of the function and its derivative at the reference points. The final values of polynomial coefficients of high-order root-polynomial functions are calculated using the dichotomy method. The article also provides examples of the applying fourth-order and fifth-order root-polynomial functions to approximate sets of numerical data that correspond to the description of ravine functions. The obtained theoretical results are interesting and important for the experts who study the physics of electron beams and design modern industrial electron beam technological equipment. [ABSTRACT FROM AUTHOR]

The strong approximation of a function is a useful tool to analyze the convergence of its Fourier series. It is based on the summability techniques. However, unlike matrix summability methods, it uses non-linear methods to derive an auxiliary sequence using approximation errors generated by the series under analysis. In this paper, we give some direct results on the strong means of Fourier series of functions in generalized Hölder and Zygmund spaces. To elaborate its use, we deduce some corollaries. [ABSTRACT FROM AUTHOR]