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• The paper provides a clusterwise regression model for interval-valued variables. • It uses a k-means type algorithm and the center and range non-linear regression. • It is able to select the best pair of center and range (linear/ non-inear) functions. • Applications on synthetic and real data sets corroborate the proposed method. Interval-valued variables are required in data analysis since this type of data represents either the uncertainty existing in an error measurement or the natural variability of the data. Currently, methods and algorithms which aim to manage interval-valued data are very much required. Hence, this paper presents a center and range clusterwise nonlinear regression algorithm for interval-valued data. The proposed algorithm combines a k-means type algorithm with the center and range linear and nonlinear regression methods for interval-valued data, with the aim to identify both the partition of the data and the relevant regression models fitted on the center and range of the intervals simultaneously, one for each cluster. The proposed method is able to automatically select the best pair of center and range (linear and/or nonlinear) functions according to optimization criteria. A simulation study with synthetic data sets with the purpose of assessing the parameter estimation and the prediction performance of the proposed algorithm was undertaken. Finally, applications on real data sets were performed and the prediction accuracy of the proposed method was compared to the linear case. The results obtained showed that the proposed method performed well on both synthetic and real data sets. [ABSTRACT FROM AUTHOR]

The description of a physical problem through a model necessarily involves the introduction of parameters. Hence, one wishes to have a solution of the problem that is a function of all these parameters: a parametric solution. However, the construction of such parametric solutions exhibiting localization in space is only ensured by costly and time-consuming tests, which can be both numerical or experimental. Numerical methodologies used classically imply enormous computational efforts for exploring the design space. Therefore, parametric solutions obtained using advanced nonlinear regressions are an essential tool to address this challenge. However, classical regression techniques, even the most advanced ones, can lead to non physical interpolation in some fields such as fluid dynamics, where the solution localizes in different regions depending on the problem parameters choice. In this context, Optimal Transport (OT) offers a mathematical approach to measure distances and interpolate between general objects in a, sometimes, more physical way than the classical interpolation approach. Thus, OT has become fundamental in some fields such as statistics or computer vision, and it is being increasingly used in fields such as computational mechanics. However, the OT problem is usually computationally costly to solve and not adapted to be accessed in an online manner. Therefore, the aim of this paper is combining advanced nonlinear regressions with Optimal Transport in order to implement a parametric real-time model based on OT. To this purpose, a parametric model is built offline relying on Model Order Reduction and OT, leading to a real-time interpolation tool following Optimal Transport theory. Such a tool is of major interest in design processes, but also within the digital twin rationale. [ABSTRACT FROM AUTHOR]

High-precision short-term inbound passenger flow prediction is of great significance to the daily crowd management and line rescheduling in urban rail systems. Although current models have been applied to prediction, most methods need optimization to meet refined passenger flow management demand. In order to better predict the passenger flow, a novel Wave-LSTM model, based on long short-term memory network (LSTM) and wavelet, is introduced in this paper. In an empirical study with practical passenger flow data of Dongzhimen Station in the Beijing Subway system, the hybrid model exhibited more effective performance in terms of prediction accuracy than the existing algorithms, e.g., autoregressive integrated moving average (ARIMA), nonlinear regression (NAR), and traditional LSTM model. The study illustrates that our newly adopted model is a promising approach for predicting high-precision short-term inbound passenger flow. [ABSTRACT FROM AUTHOR]

In this paper we present the zero-adjusted reparameterized Birnbaum–Saunders regression model. This new model generalizes at least seven reparameterized Birnbaum–Saunders regression models. Finally, an application to real data shows the potential of the model. [ABSTRACT FROM AUTHOR]

This paper focuses on parameter estimation problems for non-uniformly sampled Hammerstein nonlinear systems. By combining the lifting technique and state space transformation, we derive a nonlinear regression identification model with different input and output updating rates. Furthermore, the unmeasurable state vector is estimated by Kalman filter, and by using the hierarchical identification principle, we develop a hierarchical recursive least squares algorithm for estimating the unknown parameters of the identification model. Finally, illustrative examples are given to indicate that the proposed algorithm is effective. [ABSTRACT FROM AUTHOR]

The main aim of this study is to introduce an efficient method based on the Chelyshkov polynomials and least squares support vector regression (LS-SVR) for solving a class of nonlinear stochastic differential equations (SDEs) by variable fractional Brownian motion (VFBm). The derivative operational matrix and variable-order fractional integral operator of Chelyshkov polynomials (ChPs) are obtained. These operators, the standard Brownian motion with help of the Gauss–Legendre quadrature are applied for generating VFBm. We apply the Chelyshkov polynomials kernel and the collocation LS-SVR method for training the network. Then, the formulation of the scheme gives rise to an optimization problem. Finally, the classical optimization and Newton's iterative scheme are used to train this problem. Moreover, we discuss convergence and error analysis of mentioned scheme. In the end, to reveal the superiority and efficiency of current paper, some test problems are applied. • We combined Chelyshkov polynomials and least squares support vector regression. • We solved stochastic differential equations by variable fractional Brownian motion. • We presented a new method for generating variable fractional Brownian motion. • We convert stochastic differential equations into an optimization problem. • Convergence of the proposed scheme is discussed. [ABSTRACT FROM AUTHOR]

The permutationally invariant combined hyperbolic inverse power representation (CHIPR) polynomial expansion in terms of hyperbolic secant basis proves to be an efficient model in describing the multi state coupled potential energy surface (MSCPES) and this fact was tested on the O 2 H + triatomic ion and its diatomic sub units upon dissociation with an unmatchable accuracy of fitting in comparison with the previous works (below 1 cm − 1 in the case of diatomic function and below 300 cm − 1 in the case of triatomic function). In general, this model is capable of modeling A 3 (H 3 , O 3 etc.) and ABC (OCS, HCO + etc.) triatomic system in addition to the A 2 B (O 2 H +) system. This paper presents a Mathematica notebook (M-CHIPR.nb file) to perform the fitting of di and triatomic potentials in a user friendly manner with proper explanations written along side wherever required in the.nb file supplied. The advantage of this code is that the computing clusters are not required to use it and can be run in normal pc with good processor and memory, apart from its numerical accuracy and the system independent consistency over the similar programs written in other languages. Program title: M-CHIPR CPC Library link to program files: https://doi.org/10.17632/4mbydkzyts.1 Licensing provisions: GPLv3 Programming language: Wolfram Mathematica version12 or higher Nature of problem: Finding an multi-variable Mathematica combined hyperbolic inverse power representation (M-CHIPR) function to describe a multi state coupled potential energy surface for a typical diatomic and triatomic molecule via extensive nonlinear regression method involving hundreds of optimizable variables with minimal fitting error with respect to the raw data points. Solution method: Default Mathematica implementation of nonlinear least square fitting of ab initio points to obtain a well optimized model function having ability to accurately predicting the data behavior. Additional comments including restrictions and unusual features: The entire work was tested on Apple Mac pc with Intel i7 processor having 4 cores. The version of Mathematica is 12 and therefore, the users are advised to test the supplied notebook with the same version and beyond to avoid incompatibility issues. [ABSTRACT FROM AUTHOR]

The issue of quality-related fault detection is a hot research topic in the process monitoring community in the recent five years. Several modifications based on partial least squares (PLS) have been proposed to solve the relevant problems for linear systems. For the systems with nonlinear characteristics, some modified algorithms based on kernel partial least squares (KPLS) have also been designed very recently. However, most of the existing methods suffer from the defect that their performances are not stable when the fault intensity increases. More importantly, there is no way yet to solve the linear and nonlinear problems in a uniform algorithm structure, which is very important for simplifying the design steps of fault detection systems. This paper aims to propose such approaches based on principal component regression (PCR) and kernel principal component regression (KPCR). Such that, relevant problems in linear and nonlinear systems can be solved in the same way. Two literature examples are used to test the performance of the proposed approaches. [ABSTRACT FROM AUTHOR]

• Adulthood obesity in Spain is modeled using official surveys from 1987 to 2017. • A compartmental model based on Itȳ stochastic differential equations is considered. • Excess weight is treated as an infectious disease that spreads through contact. • The parameters are estimated based on moment methods and maximum likelihood. • Comparison with a recent paper that used nonlinear regression and Bayesian inference. Obesity is growing riskily in developed and developing countries. This should pose major concerns for the countries, not only from the health point, but also from the economic perspective. Our case study relies on the excess weight dynamics in Spain. The Spanish National Health Survey (ENSE) 2017 collects the percentage of overweight and obese adults in Spain from the year 1987 to 2017. A recent contribution proposed a nonautonomous compartmental system of ordinary differential equations to calibrate the incidence of excess weight in the Spanish adulthood population. Essentially, three principles were followed: the total adulthood population is time-dependent, the subpopulations interact homogeneously along the country, and excess weight plays the role of an infectious disease that is transmitted through contact by social pressure. Accounting for both data and model errors, frequentist nonlinear regression and Bayesian inference were conducted. The methods agreed well in terms of fit, prediction, bands and sensitivity analysis. In the present paper, the deterministic compartmental system of ordinary differential equations is randomized in a different manner, by employing Itô-type stochastic differential equations. The derivatives of the compartments are perturbed by Gaussian white noise-type pure errors that have a rough and unpredictable structure. From the Euler-Maruyama discretization, several strategies are utilized for estimating the parameters, based on the moments method and maximum likelihood estimation. Comparison is performed numerically by assessing the fit to the data. [ABSTRACT FROM AUTHOR]

Abstract: Knowing an upper bound on the number of optimal design points greatly simplifies the search for an optimal design. Carathéodory’s Theorem is commonly used to identify an upper bound. However, the upper bound from Carathéodory’s Theorem is relatively loose when there are three or more parameters in the model. In this paper, an alternative approach of finding a sharper upper bound for classical optimality criteria is proposed. Examples are given to demonstrate how to use the new approach. [Copyright &y& Elsevier]

Abstract: Non-linear regression based on reproducing kernel Hilbert space (RKHS) has recently become very popular in fitting high-dimensional data. The RKHS formulation provides an automatic dimension reduction of the covariates. This is particularly helpful when the number of covariates () far exceed the number of data points. In this paper, we introduce a Bayesian nonlinear multivariate regression model for high-dimensional problems. Our model is suitable when we have multiple correlated observed response corresponding to same set of covariates. We introduce a robust Bayesian support vector regression model based on a multivariate version of Vapnik’s -insensitive loss function. The likelihood corresponding to the multivariate Vapnik’s -insensitive loss function is constructed as a scale mixture of truncated normal and gamma distribution. The regression function is constructed using the finite representation of a function in the reproducing kernel Hilbert space (RKHS). The kernel parameter is estimated adaptively by assigning a prior on it and using the Markov chain Monte Carlo (MCMC) techniques for computation. Practical applications of our model are demonstrated via applications in near-infrared (NIR) spectroscopy and simulation studies. Our Bayesian kernel models are highly accurate in predicting composition of materials based on its near infrared (NIR) spectroscopy signature. We have compared our method with popularly used methodologies in NIR spectroscopy, like partial least square (PLS), principal component regression (PCA), support vector machine (SVM), Gaussian process regression (GPR), and random forest (RF). In all the simulation and real case studies, our multivariate Bayesian RKHS regression model outperforms the standard methods by a substantially large margin. The implementation of our models based on MCMC is fairly fast and straight forward. [Copyright &y& Elsevier]

Abstract: This paper introduces a tree-based model that combines aspects of classification and regression trees (CART) and smooth transition regression (STR). The model is called the STR-tree. The main idea relies on specifying a parametric nonlinear model through a tree-growing procedure. The resulting model can be analyzed as a smooth transition regression with multiple regimes. Decisions about splits are entirely based on a sequence of Lagrange multiplier (LM) tests of hypotheses. An alternative specification strategy based on a 10-fold cross-validation is also discussed and a Monte Carlo experiment is carried out to evaluate the performance of the proposed methodology in comparison with standard techniques. The STR-tree model outperforms CART when the correct selection of the architecture of simulated trees is discussed. Furthermore, the LM test seems to be a promising alternative to 10-fold cross-validation. Function approximation is also analyzed. When put into proof with real and simulated data sets, the STR-tree model has a superior predictive ability than CART. [Copyright &y& Elsevier]

Abstract: Demidenko [2000. Is this the least squares estimate? Biometrika 87, 437–452] has established the relationship between the curvature of nonlinear regression and the local convexity of a sum of squares: the Hessian matrix is positive definite if the sum of squares is less than the minimum squared radius of the full curvature. In this paper, we continue developing the criteria for the global minimum of the sum of squares in nonlinear regression. In particular, the concept of the local unimodality is introduced; a function is called locally unimodal on a level set if it has a unique local minimum in each component of that level set. We show that the level of the local unimodality of the sum of squares is equal to the minimum squared radius of the intrinsic curvature of the nonlinear regression function. A new class of unidirected nonlinear regression models is introduced with an interpretation in terms of differential geometry. The criteria are illustrated by several popular nonlinear regression models. [Copyright &y& Elsevier]

Least absolute deviations (LAD) regression, i.e. L1 regression, is more resistant to the outliers in the response variable than the least-squares regression, but is relatively sensitive to outlying observations in explanatory variables. However, some but few attentions have been contributed to the influence assessment for LAD regression, especially for LAD nonlinear regression. In this paper, we propose several diagnostic measures, which can be used for LAD regression models. The quasi-likelihood distance based on the L1 objective function, Cook distance based on the elliptical norm and some other diagnostic measures are introduced for LAD regression, and two examples are given to illustrate the use of these measures. The diagnostic models for LAD regression are also investigated. It is proved that the estimators of the case deletion model (CDM) and the mean shift outlier model (MSOM) are equal in linear and nonlinear LAD regression models. [Copyright &y& Elsevier]

In the paper, the large deviation inequalities of the LS estimator for the nonlinear regression model with martingale differences errors are established. The assumptions for the errors are (conditional) exponential integrability which weaken the bounded condition in Hu (1993). As an application, we give the large deviation inequalities of LS estimator for the simple Michaelis–Menten model. [ABSTRACT FROM AUTHOR]

• Modeling of adulthood obesity in Spain using surveys from 1987 to 2017. • Compartmental model considering social contacts based on differential equations. • Nonlinear regression is conducted when white noise error is included. • Bayesian inference is performed using the Metropolis algorithm. • Sensitivity analyses highlight the importance of prevention strategies. The excess weight population is growing riskily in Spain. This is confirmed by the Spanish National Health Survey 2017, which gathers the percentage of overweight and obese adults in Spain from 1987 to 2017. In this paper we propose a mathematical model based on differential equations to calibrate the incidence of excess weight in the Spanish adulthood population. The main principle is that fatness is an epidemic, where there are three stages, namely normal weight, overweight and obese, and healthy individuals may become "infected" by social contact. We start with a well-posed deterministic formulation of the model, where the parameters are fitted by minimizing the mean square error objective function. The long-term behavior of the system shows that 37% and 24% of Spanish adults will be overweight and obese in the long run, respectively. Due to the incomplete knowledge of the underlying phenomenon and error measurements, randomness must be incorporated into the model formulation. A white noise error term is thus added to conduct nonlinear regression. The response becomes a stochastic process, whose prediction interval captures the variability of the overweight and obese populations correctly. We go a step beyond by treating the model parameters as random variables. The Bayesian inference method allows for quantifying the propagation of uncertainty in this setting, by running the Metropolis algorithm (both brute-force and Adaptive). The numerical simulations show that it provides similar results to the frequentist analysis. The sensitivity analyses from the different methods agree and suggest that prevention strategies are more important than treatment strategies to control adulthood obesity. Specially, the treatment of the transition from the obese to the overweight stages is the least recommendable for reducing the obesity epidemic. [ABSTRACT FROM AUTHOR]