Orthogonal Series Research Articles

Learning Novelty Detection Outside a Class of Random Curves with Application to COVID-19 Growth

Abstract Let a class of proper curves is specified by positive examples only. We aim to propose a learning novelty detection algorithm that decides whether a new curve is outside this class or not. In opposite to the majority of the literature, two sources of a curve variability are present, namely, the one inherent to curves from the proper class and observations errors’. Therefore, firstly a decision function is trained on historical data, and then, descriptors of each curve to be classified are learned from noisy observations.When the intrinsic variability is Gaussian, a decision threshold can be established from T 2 Hotelling distribution and tuned to more general cases. Expansion coefficients in a selected orthogonal series are taken as descriptors and an algorithm for their learning is proposed that follows nonparametric curve fitting approaches. Its fast version is derived for descriptors that are based on the cosine series. Additionally, the asymptotic normality of learned descriptors and the bound for the probability of their large deviations are proved. The influence of this bound on the decision threshold is also discussed.The proposed approach covers curves described as functional data projected onto a finite-dimensional subspace of a Hilbert space as well a shape sensitive description of curves, known as square-root velocity (SRV). It was tested both on synthetic data and on real-life observations of the COVID-19 growth curves.

Legendre orthogonal polynomial method in calculating reflection and transmission coefficients of fluid-loaded functionally gradient plates

Based on the Legendre orthogonal polynomial series expansion and the partial wave theory, a Legendre orthogonal polynomial method (LOPM) is proposed to calculate the reflection and transmission coefficients of plane waves at the liquid/solid interface of a liquid-loaded functionally gradient material (FGM) plate. The displacement solutions in FGM plate are fitted approximately by Legendre orthogonal polynomial series. The stresses and the governing differential equations of the FGM plate are derived. Based on the boundary conditions of the liquid/solid interface and the governing differential equations, the linearly independent equations are set up to calculate the reflection and transmission coefficients. Meanwhile, the expansion coefficients of the Legendre orthogonal polynomials are also obtained. The power function establishes the gradient model of the mechanical parameters along the thickness direction. The angular spectrums of reflection and transmission coefficients from LOPM are in well agreement with the calculation results from the transfer matrix method. By analyzing the convergence of the reflection coefficient spectrum, the critical value of the truncated order of Legendre orthogonal polynomials is determined. The mapping relationship between the FGM gradient models and the reflection coefficient angular spectrum, also the frequency spectrums and the displacement/stress distributions can be demonstrated simultaneously, which provides the theoretical fundamentals of the ultrasonic non-destructive testing for the mechanical properties of FGM and extends the application of the LOPM.

Modified spectral method for optimal estimation in linear continuous-time stochastic systems

The spectral method to solve estimation problems for linear continuous-time stochastic systems with polynomial measurements is presented. It is based on both the spectral form of mathematical description (the representation of deterministic functions and random processes by orthogonal series) and the particle filter. The main goal of this work is to implement the continuous-time particle filter without a time discretization. The proposed spectral method provides the possibility to solve estimation problems such as filtering, smoothing and prediction.

An Empirical Study of Moment Estimators for Quantile Approximation

We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against the state-of-the-art KLL quantile estimator on 14,072 real-world datasets drawn from the OpenML database. Our analysis highlights the effectiveness of variants of moment-based quantile approximation for highly space efficient summaries: their average performance using as few as five sample moments can approach the performance of a KLL sketch containing 500 elements. Experiments also illustrate the difficulty of applying the method reliably and showcases which moment-based approximations can be expected to fail or perform poorly.

Inequalities for Orthogonal Series and a Strengthening of the Carleman–Olevskii Theorem for Complete Orthonormal Systems

Inequalities for Orthogonal Series and a Strengthening of the Carleman–Olevskii Theorem for Complete Orthonormal Systems

Analysis of Lamb wave propagation in a functionally graded piezoelectric small-scale plate based on the modified couple stress theory

This work focuses on the propagation characteristics of Lamb-type waves in a functionally graded piezoelectric (FGP) small-scale plate. Based on the modified couple stress theory, Legendre orthogonal polynomial method (LOPM) is extended to perform the solving process. By introducing the rectangular window function and expanding the electric potential and displacement vectors to Legendre orthogonal polynomial series, the governing equations satisfying the electric-open and electric-short circuit boundary conditions are presented respectively. The solutions based on the global matrix method (GMM) are deduced to verify the correctness of this extended LOPM. Since there is no layered processing, the extended LOPM provides a more realistic analysis model for the FGP small-scale plate and has a high computational efficiency. Furthermore, the dispersion characteristics, stress and electromechanical coupling factor are investigated in details. Results show that the size and piezoelectric effect increase the phase velocity, and the piezoelectricity suppresses the influence of couple stress on the dispersion characteristics. The Lamb wave phase velocities of the first and second modes with different Ly always converge to two fixed values at different convergence speeds. The conclusions obtained can be applied to the design and dynamic response evaluation of composite small-scale structures.

Reliability analysis of horizontally loaded pile considering spatial variability of soil parameters

In the design and analysis of pile foundations, the horizontal load is an important factor that needs to be considered. The reliability analysis can take the uncertainty of various parameters into consideration and thus reduce the design deviations, especially the uncertainty of soil parameters. This paper proposes a reliability analysis method considering the spatial variability of soil for the horizontally loaded pile. Firstly, a parameter sensitivity analysis was performed by the Sobol sensitivity method, and the critical variables, including the undrained shear strength Cu, the soil strain ε50, which represents the soil strain when principal stress difference reaches half (50%) of the maximum value in triaxial test, and the horizontal load and applied moment, are identified and taken as the random variables. Moreover, the spatial variability of soil parameters (Cu and ε50) are simulated as one-dimensional random field through the orthogonal series expansion method. The failure probabilities of pile foundation under the failure modes of deflection and strength are evaluated by Monte Carlo simulation. Finally, the proposed method is verified through case study in which the influences of the spatial variability of soil parameters, the uncertainty of the horizontal load on the failure probability were investigated. The results show that the uncertainty of horizontal loads and Cu make significant effects on the reliability of pile foundations. The increase of correlation length of Cu and ε50 will lead to the increase of the mean value of pile response, which has different effects on the failure probability under different failure modes.

Orthogonal structure and orthogonal series in and on a double cone or a hyperboloid

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by hyperplanes at the two ends. On each domain a family of orthogonal polynomials, related to the Gegebauer polynomials, is studied and shown to share two characteristic properties of spherical harmonics: they are eigenfunctions of a second order linear differential operator with eigenvalues depending only on the polynomial degree, and they satisfy an addition formula that provides a closed form formula for the reproducing kernel of the orthogonal projection operator. The addition formula leads to a convolution structure, which provides a powerful tool for studying the Fourier orthogonal series on these domains. Furthermore, another family of orthogonal polynomials, related to the Hermite polynomials, is defined and shown to be the limit of the first family, and their properties are derived accordingly.

Fourier Method in Initial Boundary Value Problems for Regions with Curvilinear Boundaries

The algorithm of the generalized Fourier method associated with the use of orthogonal splines is presented on the example of an initial boundary value problem for a region with a curvilinear boundary. It is shown that the sequence of finite Fourier series formed by the method algorithm converges at each moment to the exact solution of the problem – an infinite Fourier series. The structure of these finite Fourier series is similar to that of partial sums of an infinite Fourier series. As the number of grid nodes increases in the area under consideration with a curvilinear boundary, the approximate eigenvalues and eigenfunctions of the boundary value problem converge to the exact eigenvalues and eigenfunctions, and the finite Fourier series approach the exact solution of the initial boundary value problem. The method provides arbitrarily accurate approximate analytical solutions to the problem, similar in structure to the exact solution, and therefore belongs to the group of analytical methods for constructing solutions in the form of orthogonal series. The obtained theoretical results are confirmed by the results of solving a test problem for which both the exact solution and analytical solutions of discrete problems for any number of grid nodes are known. The solution of test problem confirm the findings of the theoretical study of the convergence of the proposed method and the proposed algorithm of the method of separation of variables associated with orthogonal splines, yields the approximate analytical solutions of initial boundary value problem in the form of a finite Fourier series with any desired accuracy. For any number of grid nodes, the method leads to a generalized finite Fourier series which corresponds with high accuracy to the partial sum of the Fourier series of the exact solution of the problem.

Assessing the in vitro impact of ceftazidime on aztreonam/avibactam susceptibility testing for highly resistant MBL-producing Enterobacterales.

Aztreonam/avibactam is a combination agent that shows promise in treating infections caused by highly antibiotic-resistant MBL-producing Enterobacterales. This combination can be achieved by combining two FDA-approved drugs: ceftazidime/avibactam and aztreonam. It is unknown whether ceftazidime in the combination ceftazidime/aztreonam/avibactam has a synergistic or antagonistic effect on the in vitro activity of aztreonam/avibactam by significantly increasing or decreasing the MIC. To determine whether increasing ceftazidime concentrations affect the MICs of aztreonam/avibactam alone. A custom 8 × 8 chequerboard broth microdilution (BMD) panel was made using a digital dispenser (Hewlett-Packard, Corvallis, OR, USA). The panel included orthogonal 2-fold dilution series of aztreonam and ceftazidime ranging from 0.5 to 64 mg/L. Avibactam concentration was kept constant at 4 mg/L throughout the chequerboard. Thirty-seven Enterobacterales isolates from the CDC & FDA Antibiotic Resistance Isolate Bank or CDC's internal collection with intermediate or resistant interpretations to aztreonam and ceftazidime/avibactam were included for testing. All isolates harboured at least one of the following MBL genes: blaIMP, blaNDM or blaVIM. Regardless of the concentration of ceftazidime, aztreonam/avibactam with ceftazidime MICs for all 37 isolates were within one 2-fold doubling dilution of the aztreonam/avibactam MIC. Ceftazidime, in the combination ceftazidime/avibactam/aztreonam, did not affect the in vitro activity of aztreonam/avibactam in this sample of isolates. These findings can help assure clinical and public health laboratories that testing of aztreonam/avibactam by BMD can act as a reliable surrogate test when the combination of ceftazidime/avibactam and aztreonam is being considered for treatment of highly antibiotic-resistant MBL-producing Enterobacterales.

Spatio - temporal variations of sea surface temperature in coastal waters of Khanh Hoa province (South Viet Nam) during the period of 2010-2019

Sea Surface Temperature (SST) is considered to be the most important parameter in oceanography. SST is not only an important physical parameter for studying the exchange of water vapor and heat between sea surface and atmosphere but also provides an useful index for oceanographic studies such as ocean circulation, water mass, ocean front, upwelling current, seawater mixing and ocean ecological environment. Empirical orthogonal function (EOF) and time series analysis was used to assess spatio-temporal variability of sea surface temperature in coastal waters of Khanh Hoa province (south Viet Nam) from Multi-scale Ultra-high Resolution Sea Surface Temperature (MUR SST) data for a period of 10 years (2010-2019). The first 4 EOF modes of SST explained 99.80% of the spatio-temporal variance in total variabilities. The first EOF mode explained 98.10% of the total variation of SST, representing the distribution and variability of SST in the summer with a decreasing trend from north to south. The second EOF mode explained 1,10% of the total variation of SST, representing the variation of the SST distribution in winter and spring with an increasing trend from shore to offshore. SST had a seasonal variation, it was highest in summer and lowest in winter. Trend analysis of SST time series in the past 10 years showed that SST increased 0,03oC/year in the period 1/2010-12/2019. The trend and the speed of increase/decrease of SST varied with each different period. The results of the study are useful documents to support the management and exploitation of marine ecosystems.

Statistical Feature Extraction and System Identification Algorithms for Partial Discharge Signal Classification Using Laguerre Polynomial Expansion

In this paper, a novel algorithm for partial discharge (PD) pulse waveform analysis and separation based on orthogonal series expansion using Laguerre polynomials is proposed. A system identification technique is also developed to help increase the accuracy rate of the signal classification. Laguerre polynomial series expansions are applied on the PD pulses to effectively extract important information and well-discriminative features. Laguerre expansion coefficients inherently contain information about the amplitude (intensity) and the relaxation time of a dynamic system and this helps direct characterization of PD pulses. The proposed technique is applied to PD pulses acquired from a laboratory set of multiple PD sources. It is shown that this statistical method can classify partial discharge signals with high accuracies, even when the signals are visually indistinguishable. In this research, two methods are developed for system identification. One is based on a deterministic approach in the form of a recursive formula and the other one is a stochastic method based on a group Lasso methodology. The aim of system identification is to improve the classification accuracy by removing the effect of the measurement system from the observed PD waveforms. The accuracy rate of the proposed algorithm is evaluated using the signals that are captured from simultaneous artificial defects in high pressure SF <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</inf> and in low pressure air.

Free vibration analysis of functionally graded porous plates in contact with bounded fluid

This study investigates the free vibrations of functionally graded porous (FGP) plates in contact with fluid. As an illustrative problem, a rectangular FGP plate is considered as the vertical wall of a rigid tank containing liquid. The plate displacement field is approximated based on the first-order shear deformation theory (FSDT). Thus, the expressions of its strain and kinetic energy are obtained. The fluid is assumed to be incompressible, inviscid, and irrotational. Its velocity potential functions are obtained by applying suitable boundary and compatibility conditions. By using the Rayleigh–Ritz method with two-variable orthogonal polynomials series expansion, the natural frequencies of the FGP plate in contact with fluid are calculated. To validate the method, the results obtained for the natural frequencies are compared with those available in the literature for FGP plates without fluid and isotropic plates with fluid. Then, three different types of porosity distribution are considered to evaluate their effects on the natural frequencies of the plate. Besides, the effect are investigated of different parameters, such as the fluid depth, side-to-thickness ratio, length-to-width (aspect) ratio.

Remarks on the Uniform and Absolute Convergence of Orthogonal Series

Remarks on the Uniform and Absolute Convergence of Orthogonal Series

Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.

Small-Sample Comparison of the Gamma Kernel and the Orthogonal Series Methods of Density Estimation

Small-Sample Comparison of the Gamma Kernel and the Orthogonal Series Methods of Density Estimation

Estimation of a Time-Varying Parameter from Long Range Dependent Data

We consider a time series regression with a time-varying parameter and long-range dependent data. No restriction is placed on the behavior of the time-varying parameter, allowing for both smooth changes and abrupt breaks.The time-varying parameter is estimated by a nonlinear orthogonal series estimator which is shown to have mini max estimation error and no spurious jumps in the large sample limit.

قابلية جمع المتسلسلات المتعامدة بالطريقة المصفوفية المطلقة

ظهرت في مطلع القرن العشرين نظرية المتسلسلات المتعامدة، التي تتعامل مع جمل من الدوال المتعامدة كتعميم طبيعي لنظرية المتسلسلات. وقد تمت دراسة قابلية جمع المتسلسلات المتعامدة بالعديد من الطرائق مثل طرائق سيزارو وهولدر وريمان وبوريل وهاوسدورف ونيورلند ونيورلند المعممة والطريقة المصفوفية. كما تم في هذا البحث دراسة قابلية جمع المتسلسلات المتعامدة البسيطة بطرائق مختلفة نوعاً ما تسمى الطرائق المطلقة والتي تعتمد على الفروق بشكل عام. وذلك من خلال الاعتماد على بعض الأبحاث والمقالات والكتب العلمية في مجال قابلية جمع المتسلسلات. إن الطريقة المصفوفية المطلقة تعد من أهم طرائق قابلية الجمع المطلقة، حيث إن معظم طرائق قابلية الجمع المطلقة، ما هي إلا حالات خاصة من الطريقة المصفوفية المطلقة. ونوصي أخيراً بدراسة الحالة المضاعفة.

A time-variant uncertainty propagation analysis method based on a new technique for simulating non-Gaussian stochastic processes

In this paper, the time-variant uncertainty propagation analysis is defined to solve the output stochastic process of a time-variant function with uncertainty. And a time-variant uncertainty propagation analysis method is constructed with the combination of an extended orthogonal series expansion method (extended OSE) and sparse grid numerical integration (SGNI). The SGNI serving as a classical uncertainty propagation method is utilized here to solve the moments and autocorrelation function at discrete time points of the time-variant performance function. And the extended OSE is proposed to simulate the output stochastic process based on the results from SGNI. By extended OSE, a non-Gaussian stochastic process is represented as the sumof orthogonal time functions with random coefficients, and these coefficients can be directly obtained by discretization of the target process. For these coefficients are correlated and non-Gaussian, the correlated polynomial chaos expansion method (c-PCE) is presented to represent them in terms of correlated standard Gaussian variables, and then the principal component analysis (PCA) is adopted to transform them into independent ones with dimension reduction. Finally we can obtain an explicit expression to represent the non-Gaussian process whatever it is stationary or non-stationary. Three illustrative examples are used to verify the performance of the extended OSE. In addition, two engineering problems are investigated to demonstrate the effectiveness of the time-variant uncertainty propagation method.

Spectral method for solving linear Caputo fractional stochastic differential equations

We concern on the problem of finding the solution to the linear Caputo fractional stochastic differential equation with additive and multiplicative noise. It is proposed to apply the spectral method based on the spectral form of mathematical description. This method provides both an explicit form of the solution as the orthogonal series with random coefficients and a continuous-time approximation of this solution as the partial sum. Earlier, the spectral method has been applied for solving linear (non-fractional) stochastic differential equations. The proposed method is demonstrated on the modeling fractional Ornstein– Uhlenbeck process described by a linear Caputo fractional stochastic differential equation with additive noise.