Analysis of nonlinear regression models with cubic spline functions: a systematic approach

The paper outlines a method of analysis of intricically non‐linear regression functions. The nonlinear model is first approximated by a cubic spline function. Thereafter, the standard methods of analysis in linear models are applied to obtain estimates and test statistics.

Tridiagonal matrix algorithm is known to be more efficient in the computational process than the Gaussian elimination method in solving linear system problems involving tridiagonal matrices. In this paper, the tridiagonal matrix algorithm is applied to the cubic spline interpolation problem with natural boundary conditions. In this case, the tridiagonal matrix algorithm plays a role in finding the second derivative of each cubic spline sub-function so that it is more efficient in obtaining the coefficients of the third order polynomials that form the cubic spline function

Wiley-Interscience Paperback Series Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists. The authors have put together an extraordinary presentation of concepts and methods concerning the use and analysis of nonlinear regression models ...highly recommend[ed] ...for anyone needing to use and/or understand issues concerning the analysis of nonlinear regression -Technometrics [This book] provides a good balance of relevant theory and application with many examples ...[and it] provides the most balanced approach to theory and application appropriate for a first course in nonlinear regression modeling for graduate statistics students. -Mathematical Reviews [This book] joins a distinguished list of publications with a reputation for balancing technical rigor with readability, and theory with application. [It] upholds tradition ...[and is] a worthwhile reference for the marketing researcher with a serious interest in linear models. -Journal of Marketing Research This book offers a balanced presentation of the theoretical, practical, and computational aspects of nonlinear regression and provides background material on linear regression, including the geometrical development for linear and nonlinear least squares. authors employ real data sets throughout, and their extensive use of geometric constructs and continuing examples makes the progression of ideas appear very natural. book also includes pseudocode for computing algorithms.

Random norming aids analysis of non-linear regression models with sequential informative dose selection

The temperature induced response of long-span cable-stayed bridge in cantilever state is significant, which is of great interest to study the temperature characteristics during construction period. A method of analyzing the eigenvalue and its extremum of daily temperature based on cubic spline function (CSF) is proposed. By setting the fixed time interval reasonably, introducing variable time interval and extracting nodes at the MinMax of daily temperature, the obtained CSF can approach the measured temperature curve with high accuracy. Based on CSF, the temperature characteristics at three levels of measuring point, section and component are analyzed in turn. The temperature monitoring data of a cable-stayed bridge with main span of 938 m and side span of steel-concrete composited box girder (CBG) during construction are analyzed. The results show that the temperature variation of steel box girder is remarkable; the steel beam of CBG is similar to steel box girder before composited, and it turns stable after composited; the influence of PE color on cable temperature is notable than that of the cable specification; as blue PE cable, the temperature difference of cable vs pylon and cable vs CBG exceed 17°C and 13°C.

Research on the non-smooth problems in the nonlinear support vector regression. A nonlinear smooth support vector regression model is proposed. Using a generalized cubic spline function approach the non-smooth part in the support vector regression model. The model of the nonlinear smooth support vector regression is solved by BFGS-Armijo. Then, the approximation accuracy and the astringency of the generalized cubic spline function to the insensitive loss function were analyzed. As a result, we found the four-order and six times spline function's approximation effect is better than other smooth functions, and the nonlinear smooth support vector regression model, which be proposed in this paper is convergent.

Objective To investigate the dose-response relationship between body mass index (BMI) and hyperuricemia (HUA) among aged 30-60 adults in Beijing. Methods A questionnaire survey was conducted on 8 866 adults in Beijing Electric Power Hospital in 2012.Multivariate logistic regression and restricted cubic spline methods were used to explore the relationship between BMI and hyperuricemia management and dose-response analysis. Results Among 8 866 participants, there were 5 623 males (63.4%) and 3 243 females (36.6%). The prevalence rate of HUA was 18.4%, and male was higher than female (24.9% vs. 7.1%). Age, sex, smoking, BMI, blood pressure (BP), glucose and serum lipid level were significantly different (all P<0.05). After adjusting for factors such as age, smoking, BP, glucose, total cholesterol, triglyceride, HDL-C and LDL-C, results showed that overweight men (OR=1.68, 95%CI: 1.42-1.99), obesity men (OR=2.16, 95%CI: 1.76-2.65), overweight women (OR=1.59, 95%CI: 1.15-2.20) and obesity women (OR=2.77, 95%CI: 1.79-4.27) had increased the risk of HUA (all P<0.0001). Restricted cubic spline analysis showed a non-linear dose-response relationship between the continuous change in BMI and hyperuricemia in the male population (χ2=15.49, P=0.0004) and a linear relationship in female (χ2=0.57, P=0.7521). Conclusions There is a certain correlation between continuous changes of BMI and HUA. A non-linear relationship is found in male and a linear relationship was found in female. Key words: Hyperuricemia; Overweight; Obesity; Restricted cubic spline

این مطالعه به منظور بررسی پیش بینی شاخص های عملکرد تولیدی در مرغان تخم گذار با استفاده از شبکه های عصبی مصنوعی و رگرسیون غیرخطی چندگانه انجام شد. بررسی بر روی اطلاعات چهار دوره متوالی پرورش در یک واحد پرورش مرغ تخم گذار صورت گرفت. روش های داده-کاوی شامل رگرسیون خطی و غیر خطی، شبکه عصبی پرسپترون سه لایه، شبکه عصبی پرسپترون چهار لایه و شبکه عصبی با تابع پایه ای شعاعی بود. در این مدل ها از متغیرهای سن گله، میزان خوراک مصرفی و فصل تولید به عنوان متغیر پیشگو و شاخص های عملکرد تولیدی شامل درصد تخم-گذاری، وزن توده ای تخم مرغ تولیدی و ضریب تبدیل غذایی به عنوان متغیر پاسخ استفاده شد. نتایج نهایی رگرسیون های خطی نشان داد که برای تمامی متغیرهای وابسته مورد مطالعه متغیر مستقل سن گله معنی دار می باشد. بنابراین رگرسیون غیر خطی شاخص های عملکرد تولیدی در مقابل سن برای مقایسه با شبکه های عصبی مختلف مورد بررسی قرار گرفت و برای مقایسه کلیه مدل ها از ضریب تعیین (R2) و میانگین قدر مطلق خطا (MAE) استفاده شد. نتایج نشان داد بین شبکه های عصبی مصنوعی مختلف مورد مطالعه، شبکه با تابع پایه ای شعاعی بهتر از سایر مدل های در پیش‌بینی شاخص های عملکرد تولیدی مرغان تخم گذار عمل می کند.

In most articles, the leverage effect is often analyzed by ARCH type models from the angle of volatility, and VaR is assumed to have linear relationship with volatility. In this paper, a new nonlinear quantile regression model is proposed, and we argue that the constant and slope varies with the realized volatility and yield. An empirical analysis of stock market of China based on above model is presented at last section. In addition, the CVaR is estimated conditioned on realized volatility, which can also verify leverage effect in stock market.

BackgroundThe most frequently used method to quantitatively describe the response to ionizing irradiation in terms of clonogenic survival is the linear-quadratic (LQ) model. In the LQ model, the logarithm of the surviving fraction is regressed linearly on the radiation dose by means of a second-degree polynomial. The ratio of the estimated parameters for the linear and quadratic term, respectively, represents the dose at which both terms have the same weight in the abrogation of clonogenic survival. This ratio is known as the α/β ratio. However, there are plausible scenarios in which the α/β ratio fails to sufficiently reflect differences between dose-response curves, for example when curves with similar α/β ratio but different overall steepness are being compared. In such situations, the interpretation of the LQ model is severely limited.MethodsColony formation assays were performed in order to measure the clonogenic survival of nine human pancreatic cancer cell lines and immortalized human pancreatic ductal epithelial cells upon irradiation at 0-10 Gy. The resulting dataset was subjected to LQ regression and non-linear log-logistic regression. Dimensionality reduction of the data was performed by cluster analysis and principal component analysis.ResultsBoth the LQ model and the non-linear log-logistic regression model resulted in accurate approximations of the observed dose-response relationships in the dataset of clonogenic survival. However, in contrast to the LQ model the non-linear regression model allowed the discrimination of curves with different overall steepness but similar α/β ratio and revealed an improved goodness-of-fit. Additionally, the estimated parameters in the non-linear model exhibit a more direct interpretation than the α/β ratio. Dimensionality reduction of clonogenic survival data by means of cluster analysis was shown to be a useful tool for classifying radioresistant and sensitive cell lines. More quantitatively, principal component analysis allowed the extraction of scores of radioresistance, which displayed significant correlations with the estimated parameters of the regression models.ConclusionsUndoubtedly, LQ regression is a robust method for the analysis of clonogenic survival data. Nevertheless, alternative approaches including non-linear regression and multivariate techniques such as cluster analysis and principal component analysis represent versatile tools for the extraction of parameters and/or scores of the cellular response towards ionizing irradiation with a more intuitive biological interpretation. The latter are highly informative for correlation analyses with other types of data, including functional genomics data that are increasingly beinggenerated.

Pharmacokinetic-Pharmacodynamic Modeling and Simulation

ABSTRACTSinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.

We study the parameter estimation in a nonlinear regression model with a general error's structure, strong consistency and strong consistency rate of the least squares estimator are obtained.

In this paper, heteroscedastic nonlinear regression (HNLR) models under the flexible class of two–piece distributions based on the scale mixtures of normal (TP–SMN) family were examined. This novel class of nonlinear regression (NLR) models is a generalization of the well-known heteroscedastic symmetrical nonlinear regression models. The TP–SMN is a rich class of distributions that covers symmetric and asymmetric as well as heavy-tailed distributions. Using the suitable hierarchical representation of the family, the researchers first derived an EM–type algorithm for iteratively computing maximum likelihood (ML) estimates of the parameters. Then, in order to examine the performance of the proposed models and methods, some simulation studies were presented to show the robust aspect of this flexible class against outlying and also atypical data. As the last step, a natural real dataset was fitted under the proposed HNLR models.

A fundamental mathematical field with many applications in science and engineering is linear algebra. This paper investigates the various applications of linear algebra in the fields of traffic flow analysis, electric circuits, cubic spline interpolation, and genetics. This research delves into individual applications while emphasizing cross-disciplinary insights, fostering innovative solutions through the convergence of genetics, cubic spline interpolation, circuits, and traffic flow analysis. The research employs specific methodologies in each application area to demonstrate the practical integration of linear algebra in genetics, cubic spline interpolation, electric circuits, and traffic flow analysis. In genetics, linear algebra techniques are utilized to represent genetic data using matrices, analyze genotype distributions across generations, and identify genotype-phenotype associations. For cubic spline interpolation, linear algebra is employed to construct smooth interpolating curves, involving the derivation of equations for spline functions and the determination of coefficients using boundary conditions and continuity requirements. In electric circuit analysis, linear algebra is crucial for modeling circuit elements, formulating systems of linear equations based on Kirchhoff's laws, and solving for voltage and current distributions in circuits. In traffic flow analysis, linear algebra techniques are used to represent traffic movement in networks, formulate systems of linear equations representing traffic flow dynamics, and solve for traffic flow solutions to optimize transportation networks. By addressing contemporary challenges, emerging research frontiers, and future trajectories at the intersection of linear algebra and diverse domains, this study underscores the profound impact of mathematical tools in advancing understanding and resolving complex real-world problems across multiple fields.