Orthogonal Expansion Research Articles

Побудова і аналіз універсальних двовимірних розподілів, які мають обме-жену область варіювання у вигляді прямокутника

When solving parametric reliability problems, one often has to construct distributions of statistical data to find the probability of containment in the operability region. This paper considers the problem of 2D statistical ensemble fitting. The use of a 2D normal distribution in statistical data description is not always justified because statistical ensembles rather frequently (at the level of marginal components and a stochastic relationship between them) have properties different from the normal case. From a practical standpoint, it is desirable for researchers to describe 2D statistical ensembles with the use of universal distributions, which allow one to cover a wide range of source data using a single analytical form. In the process of fitting, account should be made of bounded ranges of random variables. The paper considers who universal distribution construction methods, which are based on 1D orthogonal Jacobi polynomial expansions. In these distributions, the random variable range is a rectangle. In the first method, a 2D distribution is constructed using a direct expansion in the 1D Jacobi polynomials. A 2D Jacobi distribution function and regression lines are obtained, and methods to fit it are considered. In theory, a distribution obtained in this way can be used, up to the fourth order inclusive, for marginal and even reduced moments different from the normal case. However, its real capabilities are limited to values of reduced moments (1D and even) that differ from the normal case only very slightly. Otherwise, the probability surface may enter negative ranges with the occurrence of multiple modes. The second way to construct a 2D distribution is to use a normal copula and 1D Jacobi distributions as components. The resulting 2D distribution allows one to deal with 1D distributions different from the normal case and linear correlation. This approach is justified because, according to research data, it is a linear stochastic relationship that relates a significant part of 2D statistical ensembles, and marginal distributions deviate from the normal case. Regression lines of a distribution of this kind are obtained, and it is shown that they are curved because marginal distributions differ from the normal one. The paper considers the practical example of fitting a 2D ensemble of characteristics of a liquid-propellant rocket engine some components of which are related via a linear stochastic relationship (the parameters that characterize a nonlinear stochastic relationship proved to be insignificant) and have 1D distributions different from the normal one. The fitted and observed frequencies are in rather good agreement. It is shown that a distribution based on a normal copula is more universal, and it is recommended for practical calculations.

An analytical model for thermoelastic damping in laminated microring resonators

Predicting thermoelastic damping (TED) is crucial in the design of micro-resonators with high quality factors. Until now, some analytical models have been developed to evaluate the TED effect on microrings which are widely used in many micro-resonators. However, most previous works are limited to microrings with single homogeneous material. This paper develops an analytical model for investigating the TED in the laminated microrings. The temperature fields along the thickness and circumferential directions in the laminated microring are presented using the generalized orthogonal expansion technique. The analytical model for the TED is obtained by computing the dissipated energy and the maximum elastic energy in each layer. The present analytical model for evaluating the TED is validated by comparing with the previously published result and the finite element method (FEM). The effects of the thickness and the property of the metallic layer on the TED characteristics of two- and three-layered microrings are analyzed. The present model can be used to optimize the design of the microring resonators with high quality factors.

Research on High-frequency stock price prediction based on Chebyshev-Stacking and Weighted LSTM neural network [1

To solve the problem of high-frequency stock price prediction, this paper proposed a prediction model based on Chebyshev-Stacking and a weighted LSTM neural network. The proposed method extracts the function characteristic information of the high-frequency stock price series through Chebyshev orthogonal polynomial basis expansion. Considering that the potential model structure between each component of the function feature vector and the residual sequence predicted by the LSTM neural network is unknown and there is a certain noise, this paper used the Stacking framework to enhance the data and weighed the bias and variance of the prediction model. In addition, since the number of predictor variable periods of the LSTM neural network is a hyperparameter, the model averaging method based on distance covariance is used for optimization. The results of actual data analysis show that the proposed method is significantly better than the original LSTM neural network in terms of mean square error, absolute error, and relative error. By selecting the different number of training sets, the robustness of the improved model is verified. Finally, the proposed method can also be used in practical applications such as daily average temperature prediction, missile trajectory prediction, and real-time monitoring of atmospheric environment quality.

A hierarchical Bayesian framework embedded with an improved orthogonal series expansion for Gaussian processes and fields identification

A new hierarchical Bayesian framework (HBM) is proposed for identification of Gaussian processes or fields, which are usually used for simulating uncertainty in temporal variability of loads or spatial variability of material properties. An improved orthogonal series expansion (iOSE) is embedded into the proposed framework by simulating the Gaussian process or field through correlated Gaussian variables, and then HBM is applied to quantify their uncertainty. Hyper parameters to be identified are set to be the mean value and standard deviation vectors of these Gaussian variables, as well as the parameters in autocorrelation function (ACF) of the Gaussian process or field which are used to replace correlation coefficients of correlated Gaussian variables for reducing the number of hyper parameters. With the identified hyper parameters, a simulation model of the Gaussian process or field can be obtained based on the iOSE expression. In addition, model class selection is introduced to select the optimal number of orthogonal functions and integral points involved in iOSE as well as select the category of ACF among several alternative models, known to influence the simulated expression and accuracy. Studies conducted on two dynamic examples verify the effectiveness of proposed framework.

A High frequency stock price prediction model based on Boosting and information entropy weighted LSTM neural network

With the continuous development of data acquisition technology, high-frequency time series forecasting, such as stock prices, has become a hot issue. This paper proposes a LSTM neural network based on Boosting and information entropy weighting. The proposed method extracts the functional features of high-frequency time series by using orthogonal polynomial expansion, and uses the Boosting frame to recursively fit the residual predicted by LSTM neural network. Considering that the dimension of the prediction variable of LSTM neural network is a super parameter, we propose a model average method based on information entropy weighting, which theoretically balances the variance and bias of the prediction model. In addition, the basic model in the Boosting framework can be arbitrary, which greatly expands the applicability of the proposed method. The results of real data analyses show that the proposed method can effectively improve the prediction accuracy of the original LSTM neural network and has robustness. Finally, the proposed method can be further applied to the real-time monitoring of trace elements, the monitoring of road traffic flow and the prediction of daily average temperature curve in environmental science.

On Orthogonal Polynomials and Finite Moment Problem

Background: This paper is an improvement of a previous work on the problem recovering a function or probability density function from a finite number of its geometric moments, [1]. The previous worked solved the problem with the help of the B-Spline theory which is a great approach as long as the resulting linear system is not very large. In this work, two solution algorithms based on the approximate representation of the target probability distribution function via an orthogonal expansion are provided. One primary application of this theory is the reconstruction of the Particle Size Distribution (PSD), occurring in chemical engineering applications. Another application of this theory is the reconstruction of the Radon transform of an image at an unknown angle using the moments of the transform at known angles which leads to the reconstruction of the image form limited data. Objective: The aim is to recover a probability density function from a finite number of its geometric moments. Methods: The tool is the orthogonal expansion approach. The Shifted-Legendre Polynomials and the Chebyshev Polynomials as bases for the orthogonal expansion are used in this study. Results: A high degree of accuracy has been obtained in recovering a function without facing a possible ill-conditioned linear system, which is the case with many typical approaches of solving the problem. In fact, for a normalized template function f on the interval [0, 1], and a reconstructed function ; the reconstruction accuracy is measured in two domains. One is the moment domain, in which the error (difference between the moments of f and the moments of ) is zero. The other measure is the standard difference in the norm -space ||f- || which can be ≈ 10-6 or less. Conclusion: This paper discusses the problem of recovering a function from a finite number of its geometric moments for the PSD application. Linear transformations were used, as needed, so that the function is supported on the unit interval [0, 1], or on [0, α] for some choice of α. This transformation forces the sequence of moments to vanish. Then, an orthogonal expansion of the Scaled Shifted Legendre Polynomials, as well as the Chebyshev Polynomials, are developed. The result shows good accuracy in recovering different types of synthetic functions. It is believed that up to fifteen moments, this approach is safe and reliable.

Two-Dimensional Conjugated Mass Transfer of Carbon Dioxide Absorption in a Hollow-Fiber Gas-Liquid Membrane Contactor.

The absorption efficiencies of CO2 in hollow-fiber membrane contactors using an ethanolamine (MEA) solvent under both concurrent- and countercurrent-flow operations were investigated theoretically and experimentally. Two-dimensional mathematical modeling was developed by Happel’s free surface model, and the resultant partial differential equations were solved analytically using the separated variables method with the use of an orthogonal expansion technique. A simplified expression of Sherwood number variations was reported by employing the relevant operations conditions and expressed in terms of the computed eigenvalues for predicting concentration distribution and absorption efficiency. It is emphasized that, in comparing various fiber packing configurations, both theoretical predictions and experimental results should be compared to find the absorption flux increment accomplished by the CO2/N2 stream passing through the fiber cells under the same mass flow rate. The value of the present mathematical treatment is evident to propose a simplified expression of the averaged Sherwood number variations, and provides the predictions of the absorption flux, absorption efficiency, average Sherwood number with the absorbent Graetz number, inlet CO2 concentration, and absorbent flow rates as parameters. The availability of such concise expressions, as developed directly from the analytical formulations, is the value of the present study. The experiments of the CO2 absorption using MEA with alumina (Al2O3) hollow fiber membranes are also set up to confirm the accuracy of the theoretical predictions. The accuracy derivations between the experimental results and theoretical predictions for concurrent- and countercurrent-flow operations are and , respectively. The operations of the hollow-fiber membrane contactor implementing N = 7 fiber cells and N = 19 fiber cells offer an inexpensive method of improving absorption efficiency by increasing fiber numbers with consideration of device performance.

A Matlab Toolbox for Extended Dynamic Mode Decomposition Based on Orthogonal Polynomials and p-q Quasi-Norm Order Reduction

Extended Dynamic Mode Decomposition (EDMD) allows an approximation of the Koopman operator to be derived in the form of a truncated (finite dimensional) linear operator in a lifted space of (nonlinear) observable functions. EDMD can operate in a purely data-driven way using either data generated by a numerical simulator of arbitrary complexity or actual experimental data. An important question at this stage is the selection of basis functions to construct the observable functions, which in turn is determinant of the sparsity and efficiency of the approximation. In this study, attention is focused on orthogonal polynomial expansions and an order-reduction procedure called p-q quasi-norm reduction. The objective of this article is to present a Matlab library to automate the computation of the EDMD based on the above-mentioned tools and to illustrate the performance of this library with a few representative examples.

A Stochastic Switched Optimal Control Approach to Formation Mission Design for Commercial Aircraft

This article studies the formation mission design problem for commercial aircraft in the presence of uncertainties. Specifically, it considers uncertainties in the departure times of the aircraft and in the fuel burn savings for the trailing aircraft. Given several commercial flights, the problem consists in arranging them in formation or solo flights and finding the trajectories that minimize the expected value of the DOC of the flights. The formation mission design problem is formulated as an optimal control problem of a stochastic switched dynamical system and solved using nonintrusive gPC-based stochastic collocation. The stochastic collocation method converts the SSOCP into an augmented deterministic switched optimal control problem. With this approach, a small number of sample points of the random parameters are used to jointly solve particular instances of the switched optimal control problem. The obtained solutions are then expressed as orthogonal polynomial expansions in terms of the random parameters using these sample points. This technique allows statistical and global sensitivity analysis of the stochastic solutions to be conducted at a low computational cost. The aim of this article is to establish if, in the presence of uncertainties, a formation mission is beneficial with respect to solo flight in terms of the expected value of the direct operating costs. Several numerical experiments have been conducted in which uncertainties on the departure times and on the fuel saving during formation flight have been considered. The obtained results demonstrate that benefits can be achieved even in the presence of these uncertainties.

Contactless Measurement of Current and Mutual Inductance in Wireless Power Transfer System Based on Sandwich Structure

The mutual inductance between the transmitter and receiver sides is critical for the efficiency of a wireless power transfer (WPT) system. This article proposes a novel online mutual inductance measurement method based on a local detection coil with a sandwich structure. The mutual inductance and the receiver current are obtained by measuring the induced voltage of the sandwich coil and the output voltage of the inverter, with contactless measurement from the receiver side. Using the ferrite substrate as the skeleton of the sandwich detection coil not only increases the differential mutual inductance between the receiver coil and the sandwich coil but also maintains the differential mutual inductance almost constant when horizontal misalignment occurs. Moreover, the differential mutual inductance is used to calculate the mutual inductance between the transmitter and receiver coils and the orthogonal eigenvalue expansion method is exploited to analyze the sandwich detection coil in a finite region. Then, the factors affecting the differential mutual inductance are analyzed and optimized, and a stable differential mutual inductance is obtained by formulating a standard coil design procedure. Finally, a prototype is built to verify the proposed method. The experimental results show that the differential mutual inductance remains almost constant when the horizontal misalignment reaches 50% of the outer radius of the transmitter/receiver coil, and the maximum calculated error of the detected mutual inductance is 2.18%.

APPROXIMATION BY HAAR WAVELET AND THEIR ORDER OF APPROXIMATION IN DIFFERENT SPACES

Wavelet analysis uses as a new class of orthogonal expansion in with regularity-approximation properties. In this study, we approximate any general function in by Haar wavelet in different smoothness spaces such as Lebesgue space, Lipschitz continuous space, Sobolev and Besov spaces.

Evaluation of Different Methods for Voltage Sag Source Detection

This paper compares and evaluates three different methods for voltage sag source detection. First method (method I) is based on the assumption, that the energy flow at the monitoring point increases during downstream events and decreases during upstream events. Second and third method (methods II and III) are both based on the assumption that currents measured at the monitoring point increase during downstream events and decrease during upstream events. The slope of a current-voltage trajectory is investigated in method II, while a real current component is observed within method III. Both current-based methods (II and III) require fundamental harmonic components of sampled voltages and currents, which are extracted using discrete orthogonal series expansion, such as Fourier or Walsh. Algorithms of this type are especially appropriate for studying steady-state and periodically repeating conditions. Voltage sags are, on the contrary, transient disturbance events. Thus, usage of the discussed algorithms may not be appropriate. Furthermore, criteria within methods II and III are checked for each phase individually. In the cases of asymmetrical voltage sags exact interpretation of the obtained results, therefore, might not be possible. Method I is, on the contrary, based on instantaneous values of line voltages and currents, while three-phase criterion is used. An exact interpretation of the results obtained by this method is, therefore, also possible in cases of asymmetrical voltage sags. All the discussed methods for voltage sag source detection have been tested by applying extensive simulations and field tests. The results for ground faults, asymmetrical voltage sags, upstream events and motor starting have been analyzed in order to evaluate all the discussed methods. The obtained results show that all discussed methods are very successful in cases of heavy motor starting and other symmetrical voltage sags. In cases of asymmetrical voltage sags the methods II and III do not work well, especially for those originating from the upstream side, while the method I is not successful only in particular cases of voltage sags due to upstream ground faults. Based on the performed evaluation it can be concluded, that further development is still needed to increase the degree of confidence in the discussed methods.

Optimal One-Pass Nonparametric Estimation Under Memory Constraint

For nonparametric regression in the streaming setting, where data constantly flow in and require real-time analysis, a main challenge is that data are cleared from the computer system once processed due to limited computer memory and storage. We tackle the challenge by proposing a novel one-pass estimator based on penalized orthogonal basis expansions and developing a general framework to study the interplay between statistical efficiency and memory consumption of estimators. We show that, the proposed estimator is statistically optimal under memory constraint, and has asymptotically minimal memory footprints among all one-pass estimators of the same estimation quality. Numerical studies demonstrate that the proposed one-pass estimator is nearly as efficient as its nonstreaming counterpart that has access to all historical data.

Dynamic behavior of an inhomogeneous piezoelectric/piezomagnetic half space with a circular ring structure under SH wave

In this article, the dynamic characteristics of a circular ring structure in an inhomogeneous piezoelectric/piezomagnetic half spaces subjected to anti-plane shear wave is studied. The material parameters are assumed to have an exponential function distribution along the coordinate axis, and the Helmholtz equation includes variable coefficients due to inhomogeneity. First, by introducing new variables, the Helmholtz equation is transformed into standard form. Next, on the basis of boundary conditions, integral equations are setup and solved by applying orthogonal function expansion technique and effective truncation. Results clarified the influence on the dynamic stress concentration factor, electric field intensity concentration factor and magnetic field intensity concentration factor under proper conditions. Besides, the analytical solutions are compared with the finite element solutions to verify the accuracy of the conclusions in this article.

Numerical scheme for Erdélyi–Kober fractional diffusion equation using Galerkin–Hermite method

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi–Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion, and grey Brownian motion. To obtain a convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi–Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singularity in the time term appearing in the main equation, the proposed method converges slower than first order. Finally, we provide a numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

Maximal functions and multiplier theorem for Fourier orthogonal series

Under the assumption that orthogonal polynomials of several variables admit an addition formula, we can define a convolution structure and use it to study the Fourier orthogonal expansions on a space of homogeneous type. We define a maximal function via the convolution structure induced by the addition formula and use it to establish a Marcinkiewicz multiplier theorem. For the space of homogeneous type defined by a family of weight functions on conic domains, we show that the maximal function is bounded by the Hardy-Littlewood maximal function so that the multiplier theorem holds on conic domains.

Torsional deformation of an infinite elastic solid weakened by a penny-shaped crack in the presence of surface elasticity and Dugdale plastic zone

We consider the torsional deformation of an infinite three-dimensional isotropic elastic solid weakened by a penny-shaped crack whose boundary is enhanced by the incorporation of surface elasticity. Furthermore, we assume the presence of a Dugdale-type plastic zone around the crack front. The ensuing elastoplastic problem gives rise to a non-standard mixed boundary value problem which is then reduced to a hypersingular integral equation upon application of the Hankel transform. The hypersingular integral equation is subsequently solved numerically using orthogonal expansions revealing the effects of surface elasticity on the size of the plastic zone and crack-face torsional displacements. Our results demonstrate the effect of surface elasticity on the width of the plastic zone thus providing a significant shielding effect while increasing the load-bearing capacity of the solid in the vicinity of the defect. Our conclusions will be of particular interest when the presence of surface elasticity along a crack boundary is used in enhanced continuum-based models to study fracture at micro- and nano-scales, for example, in the prediction of fracture and failure in micro- and nano-structures.

Goodness-of-fit tests in proportional hazards models with random effects.

This paper deals with testing the functional form of the covariate effects in a Cox proportional hazards model with random effects. We assume that the responses are clustered and incomplete due to right censoring. The estimation of the model under the null (parametric covariate effect) and the alternative (nonparametric effect) is performed using the full marginal likelihood. Under the alternative, the nonparametric covariate effects are estimated using orthogonal expansions. The test statistic is the likelihood ratio statistic, and its distribution is approximated using a bootstrap method. The performance of the proposed testing procedure is studied through simulations. The method is also applied on two real data sets one from biomedical research and one from veterinarymedicine.

Cesàro Means of Weighted Orthogonal Expansions on Regular Domains

In this paper, we investigate Cesàro means for the weighted orthogonal polynomial expansions on spheres with weights being invariant under a general finite reflection group on Rd. Our theorems extend previous results only for specific reflection groups. Precisely, we consider the weight function hκ(x):=∏ν∈R+|x,ν|κν,x∈Rd on the unit sphere; the upper estimates of the Cesàro kernels and Cesàro means are obtained and used to prove the convergence of the Cesàro (C,δ) means in the weighted Lp space for δ above the corresponding index. We also establish similar results for the corresponding estimates on the unit ball and the simplex.

3D Morphology of Open Clusters in the Solar Neighborhood with Gaia EDR 3. II. Hierarchical Star Formation Revealed by Spatial and Kinematic Substructures

We identify members of 65 open clusters in the solar neighborhood using the machine-learning algorithm StarGO based on Gaia EDR3 data. After adding members of 20 clusters from previous studies we obtain 85 clusters, and study their morphology and kinematics. We classify the substructures outside the tidal radius into four categories: filamentary (f1) and fractal (f2) for clusters <100 Myr, and halo (h) and tidal tail (t) for clusters >100 Myr. The kinematical substructures of f1-type clusters are elongated; these resemble the disrupted cluster Group X. Kinematic tails are distinct in t-type clusters, especially Pleiades. We identify 29 hierarchical groups in four young regions (Alessi 20, IC 348, LP 2373, LP 2442); 10 among these are new. The hierarchical groups form filament networks. Two regions (Alessi 20, LP 2373) exhibit global orthogonal expansion (stellar motion perpendicular to the filament), which might cause complete dispersal. Infalling-like flows (stellar motion along the filament) are found in UBC 31 and related hierarchical groups in the IC 348 region. Stellar groups in the LP 2442 region (LP 2442 gp 1–5) are spatially well mixed but kinematically coherent. A merging process might be ongoing in the LP 2442 subgroups. For younger systems (≲30 Myr), the mean axis ratio, cluster mass, and half-mass–radius tend to increase with age values. These correlations between structural parameters may imply two dynamical processes occurring in the hierarchical formation scenario in young stellar groups: (1) filament dissolution and (2) subgroup mergers.