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The purpose of this paper is to propose an improved optimization methodology of information granulation‐based fuzzy radial basis function neural networks (IG‐FRBFNN). In the IG‐FRBFNN, the membership functions of the premise part of fuzzy rules are determined by means of fuzzy c‐means (FCM) clustering. Also, high‐order polynomial is considered as the consequent part of fuzzy rules which represent input‐output relation characteristic of sub‐space and weighted least squares learning is used to estimate the coefficients of polynomial. Since the performance of IG‐RBFNN is affected by some parameters such as a specific subset of input variables, the fuzzification coefficient of FCM, the number of rules and the order of polynomial of consequent part of fuzzy rules, we need the structural as well as parametric optimization of the network. The proposed model is demonstrated with the use of two kinds of examples such as nonlinear function approximation problem and Mackey‐Glass time‐series data.

The type of polynomial of each fuzzy rule is determined by selection algorithm by considering the local error as performance index. In addition, the combined local error is introduced as a performance index considered by two kinds of parameters such as the polynomial type of each rule and the number of polynomial coefficients of each rule. Besides this, other structural and parametric factors of the IG‐FRBFNN are optimized to minimize the global error of model by means of the hierarchical fair competition‐based parallel genetic algorithm.

The performance of the proposed model is illustrated with the aid of two examples. The proposed optimization method leads to an accurate and highly interpretable fuzzy model.

The proposed hybrid optimization methodology is interesting for designing an accurate and highly interpretable fuzzy model. Hybrid optimization algorithm comes in the form of the combination of the combined local error and the global error.

The purpose of this paper is to consider the concept of Fuzzy Radial Basis Function Neural Networks with Information Granulation (IG‐FRBFNN) and their optimization realized by means of the Multiobjective Particle Swarm Optimization (MOPSO).

In fuzzy modeling, complexity, interpretability (or simplicity) as well as accuracy of the obtained model are essential design criteria. Since the performance of the IG‐RBFNN model is directly affected by some parameters, such as the fuzzification coefficient used in the FCM, the number of rules and the orders of the polynomials in the consequent parts of the rules, the authors carry out both structural as well as parametric optimization of the network. A multi‐objective Particle Swarm Optimization using Crowding Distance (MOPSO‐CD) as well as O/WLS learning‐based optimization are exploited to carry out the structural and parametric optimization of the model, respectively, while the optimization is of multiobjective character as it is aimed at the simultaneous minimization of complexity and maximization of accuracy.

The performance of the proposed model is illustrated with the aid of three examples. The proposed optimization method leads to an accurate and highly interpretable fuzzy model.

A MOPSO‐CD as well as O/WLS learning‐based optimization are exploited, respectively, to carry out the structural and parametric optimization of the model. As a result, the proposed methodology is interesting for designing an accurate and highly interpretable fuzzy model.

The purpose of this paper is to overcome the application limitations of other multi-variable regression based on polynomials due to the huge computation room and time cost.

In this paper, based on the idea of feature selection and cascaded regression, two strategies including Laguerre polynomials and manifolds optimization are proposed to enhance the accuracy of multi-variable regression. Laguerre polynomials were combined with the genetic algorithm to enhance the capacity of polynomials approximation and the manifolds optimization method was introduced to solve the co-related optimization problem.

Two multi-variable Laguerre polynomials regression methods are designed. Firstly, Laguerre polynomials are combined with feature selection method. Secondly, manifolds component analysis is adopted in cascaded Laguerre polynomials regression method. Two methods are brought to enhance the accuracy of multi-variable regression method.

With the increasing number of variables in regression problem, the stable accuracy performance might not be kept by using manifold-based optimization method. Moreover, the methods mentioned in this paper are not suitable for the classification problem.

Experiments are conducted on three types of datasets to evaluate the performance of the proposed regression methods. The best accuracy was achieved by the combination of cascade, manifold optimization and Chebyshev polynomials, which implies that the manifolds optimization has stronger contribution than the genetic algorithm and Laguerre polynomials.

The purpose of this paper is to propose a new acceleration/deceleration (acc/dec) algorithm for motion profiles. The motion efficiency, flexibility of the motion profiles and the residual vibration of the movement are discussed in this paper.

A dynamics model is developed to assess the residual vibration of these two kinds of motion profile. And a Simulink model is created to assess the motion efficiency and flexibility of the motion profiles with the proposed acc/dec algorithm.

Considering the flexibility of trigonometric motion profiles and the higher motion efficiency of S-curve motion profiles, the authors add the polynomial parts into the jerk profile of the cosine function acc/dec algorithm to hold the jerk when it reaches the maximum so that the motion efficiency can increase and decrease residual vibration at the same time. And the cyclical parameter k shows the decisive factor for the flexibility of trigonometric motion profiles.

Comparing with the traditional motion profiles, the proposed motion profiles have higher motion efficiency and excite less residual vibration. The acc/dec algorithm proposed in this paper is useful for the present motion control and servo system.

This chapter aims at shedding light upon how transforming or detrending a series can substantially impact predictions of mixed causal-noncausal (MAR) models, namely dynamic processes that depend not only on their lags but also on their leads. MAR models have been successfully implemented on commodity prices as they allow to generate nonlinear features such as locally explosive episodes (denoted here as bubbles) in a strictly stationary setting. The authors consider multiple detrending methods and investigate, using Monte Carlo simulations, to what extent they preserve the bubble patterns observed in the raw data. MAR models relies on the dynamics observed in the series alone and does not require economical background to construct a structural model, which can sometimes be intricate to specify or which may lack parsimony. The authors investigate oil prices and estimate probabilities of crashes before and during the first 2020 wave of the COVID-19 pandemic. The authors consider three different mechanical detrending methods and compare them to a detrending performed using the level of strategic petroleum reserves.

In this chapter we demonstrate the construction of inverse test confidence intervals for the turning-points in estimated nonlinear relationships by the use of the marginal or first derivative function. First, we outline the inverse test confidence interval approach. Then we examine the relationship between the traditional confidence intervals based on the Wald test for the turning-points for a cubic, a quartic, and fractional polynomials estimated via regression analysis and the inverse test intervals. We show that the confidence interval plots of the marginal function can be used to estimate confidence intervals for the turning-points that are equivalent to the inverse test. We also provide a method for the interpretation of the confidence intervals for the second derivative function to draw inferences for the characteristics of the turning-point.

This method is applied to the examination of the turning-points found when estimating a quartic and a fractional polynomial from data used for the estimation of an Environmental Kuznets Curve. The Stata do files used to generate these examples are listed in Appendix A along with the data.

The purpose of this paper is to propose methods to improve the least square error polynomial fitting of multi-input nonlinear sources that suffer from strong correlating inputs.

The polynomial fitting is improved by amplitude normalization, reducing the order of the model, utilizing Chebychev polynomials and finally perturbing the correlating controlling voltage spectra. The fitting process is estimated by the reliability figure and the condition number.

It is shown in the paper that perturbing one of the controlling voltages reduces the correlation to a large extend especially in the cross-terms of the multi-input polynomials. Chebychev polynomials reduce the correlation between the higher-order spectra derived from the same input signal, but cannot break the correlation between correlating input and output voltages.

Optimal perturbations are sought in a separate optimization loop, which slows down the fitting process. This is due to the fact that each nonlinear source that suffers from the correlation needs a different perturbation.

The perturbation, harmonic balance run and refitting of an individual nonlinear source inside a device model is new and original way to characterize and fit polynomial models.

The purpose of this study is to propose an isogeometric analysis (IGA) approach for solving the Reynolds equation in textured piston ring cylinder liner (PRCL) contacts.

The texture region is accurately and conveniently expressed by non-uniform rational B-splines (NURBS) besides hydrodynamic pressure and the oil film density ratio is represented in this mathematical form. A quadratic programming method combined with a Lagrange multiplier method is developed to address the cavitation issue.

The comparison with the results solved by an analytical method has verified the effectiveness of the proposed approach. In the study of the PRCL contact with two-dimensional circular dimple textures, the solution of the IGA approach shows high smoothness and accuracy, and it well satisfies the complementarity condition in the case of cavitation presence.

This paper proposes an IGA approach for solving the Reynolds equation in textured PRCL contacts. Its novelty is reflected in three aspects. First, NURBS functions are simultaneously used to express the solution domain, texture shape, hydrodynamic pressure and oil density ratio. Second, the streamline upwind/Petrov–Galerkin method is adopted to create a weak form for the Reynolds equation that takes the oil density ratio as a first-order unknown variable. Third, a quadratic programming approach is developed to impose the complementarity conditions between the hydrodynamic pressure and the oil density ratio.

The global methods of generalized differential quadrature (GDQ) and generalized integral quadrature (GIQ) are applied to solve three‐dimensional, incompressible, laminar boundary layer equations. The streamwise and crosswise velocity components are taken as the dependent variables. The normal velocity is obtained by integrating the continuity equation along the normal direction where the integral is approximated by GIQ approach with high order of accuracy. All the spatial derivatives are discretized by a GDQ scheme. After spatial discretization, the resultant ordinary differential equations are solved by the 4‐stage Runge—Katta scheme. Application of GDQ—GIQ approach to a test problem demonstrated that accurate numerical results can be obtained using just a few grid points.

A novel method for modelling permanent magnets is investigated based on numerical approximations with rational functions. This study aims to introduce the AAA algorithm and other recently developed, cutting-edge mathematical tools, which provide outstandingly fast and accurate numerical computation of potentials and vector fields.

First, the AAA algorithm is briefly introduced along with its main variants and other advanced mathematical tools involved in the modelling. Then, the analysis of a circular Halbach array with a one-pole pair is carried out by means of the AAA-least squares method, focusing on vector potential and flux density in the bore and validating results by means of classic finite element software. Finally, the investigation is completed by a finite difference analysis.

AAA methods for field analysis prove to be strikingly fast and accurate. Results are in excellent agreement with those provided by the finite element model, and the very good agreement with those from finite differences suggests future improvements. They are also easy programming; the MATLAB code is less than 200 lines. This indicates they can provide an effective tool for rapid analysis.

AAA methods in magnetostatics are novel, but their extension to analogous physical problems seems straightforward. Being a meshless method, it is unlikely that local non-linearities can be considered. An aspect of particular interest, left for future research, is the capability of handling inhomogeneous domains, i.e. solving general interface problems.

The authors use cutting-edge mathematical tools for the modelling of complex physical objects in magnetostatics.