Estimation of nonlinear models finds applications in every field of engineering and the sciences. Much work has been done to build solid statistical theories for its use and interpretation. However, there has been little analysis of the tolerance of nonlinear model estimators to deviations from assumptions and normality. We focus on analyzing the robustness properties of M-estimators of nonlinear models by studying the effects of deviations from assumptions and normality on these estimators. We discuss St. Laurent and Cook's Jacobian Leverage and identify the relationship of the technique to the robustness concept of influence. We derive influence functions for M-estimators of nonlinear models and show that influence of position becomes, more generally, influence of model. The result shows that, for M-estimators, we must bound not only influence of residual but also influence of model. Several examples highlight the unique problems of nonlinear model estimation and demonstrate the utility of the influence function. / Master of Science

Estimation procedures and optimal designs for estimation of the individual parameters and of the global parameters are discussed under various conditions of prior knowledge. The extension to nonlinear parametrization of the response function ís based on the asymptotical validity of the results for the linear parametrization. For the case where the error variance and the dispersion matrix are unknown, an iterative estimation procedure is suggested. An example based on dental plaque pH profiles demonstrates the improvement that is achieved (a) through using the optimal design or a design that ís close to the optimal, and (b) through taking into account prior information. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

Varying diameter ratios associated with smooth concentric tube-in-tube heat exchangers are known to have an effect on its convective heat transfer capabilities. Much literature exists for predicting the inner tube’s heat transfer coefficients, however, limited research has been conducted for the annulus and some of the existing correlations are known to have large errors. Linear and nonlinear regression models exist for determining the heat transfer coefficients, however, these are complex and time consuming methods and require much experimental data in order to obtain accurate solutions. A direct solution to obtain the heat transfer coefficients in the annulus is sought after. In this study a large dataset of experimental measurements on heat exchangers with annular diameter ratios of 0.483, 0.579, 0.593 and 0.712 was gathered. The annular diameter ratio is defined as the ratio of the outer diameter of the inner tube to the inner diameter of the outer tube. Using various methods, the data was processed to determine local and average Nusselt numbers in the turbulent flow regime. These methods included the modified Wilson plot technique, a nonlinear regression scheme, as well as the log mean temperature difference method. The inner tube Reynolds number exponent was assumed to be a constant 0.8 for both the modified Wilson plot and nonlinear regression methods. The logarithmic mean temperature difference method was used for both a mean analysis on the full length of the heat exchanger, and a local analysis on finite control volumes. Friction factors were calculated directly from measured pressure drops across the annuli. The heat exchangers were tested for both a heated and cooled annulus, and arranged in a horizontal counter-flow configuration with water as the working medium. Data was gathered for Reynolds numbers (based on the hydraulic diameter) varying from 10 000 to 28 000 for a heated annulus and 10 000 to 45 000 for a cooled annulus. Local inner wall temperatures which are generally difficult to determine, were measured with thermocouples embedded within the wall. Flow obstructions within the annuli were minimized, with only the support structures maintaining concentricity of the inner and outer tubes impeding flow. / Dissertation (MEng)--University of Pretoria, 2013. / gm2014 / Mechanical and Aeronautical Engineering / unrestricted

The field of nonlinear regression is a long way from reaching a consensus. Once a method decides to explore nonlinear combinations of predictors, a number of questions are raised, such as what nonlinear combinations to permit and how best to search the resulting model space. Genetic Association Studies comprise an area that stands to gain greatly from the development of more sophisticated regression methods. While these studies' ability to interrogate the genome has advanced rapidly over recent years, it is thought that a lack of suitable regression tools prevents them from achieving their full potential. I have tried to investigate the area of regression in a methodical manner. In Chapter 1, I explain the regression problem and outline existing methods. I observe that both linear and nonlinear methods can be categorised according to the restrictions enforced by their underlying model assumptions and speculate that a method with as few restrictions as possible might prove more powerful. In order to design such a method, I begin by assuming each predictor is tertiary (takes no more than three distinct values). In Chapters 2 and 3, I propose the method Sparse Partitioning. Its name derives from the way it searches for high scoring partitions of the predictor set, where each partition defines groups of predictors that jointly contribute towards the response. A sparsity assumption supposes most predictors belong in the 'null group' indicating they have no effect on the outcome. In Chapter 4, I compare the performance of Sparse Partitioning to existing methods using simulated and real data. The results highlight how greatly a method's power depends on the validity of its model assumptions. For this reason, Sparse Partitioning appears to offer a robust alternative to current methods, as its lack of restrictions allows it to maintain power in scenarios where other methods will fail. Sparse Partitioning relies on Markov chain Monte Carlo estimation, which limits the size of problem on which it can be used. Therefore, in Chapter 5, I propose a deterministic version ofthe method which, although less powerful, is not affected by convergence issues. In Chapter 6, I describe Bayesian Projection Pursuit, which adds spline fitting into the method to cope withnon-tertiary predictors.

Application of composite materials to structures has presented the need for engineering analysis and modeling to understand the failure mechanisms. Unfortunately, composite materials, especially in a tubular geometry, present a situation where it is difficult to generate simple stress states that allow for the characterization of the ply-level properties. The present work focuses on calculating the mechanical characteristics, both on a global and local level, for composite laminate tubes. Global responses to axisymmetric test conditions (axial tension, torsion, and internal pressure) are measured on sections of the material. New analysis techniques are developed to use the global responses to calculate the ply level properties for tubular composite structures. Error analyses are performed to illustrate the sensitivity of the nonlinear regression methods to error in the experimental data. Ideal test matrices are proposed to provide the best data sets for improved accuracy of the property estimates. With these values, the stress and strain states can be calculated through the thickness of the material, enabling the application of failure criteria, and the calculation of failure envelopes. / Ph. D.

Modelos não lineares como o Logístico e o Gompertz são amplamente usados para descrever vários processos biológicos por meio da curva de crescimento dada pela equação do modelo. O objetivo deste trabalho foi ajustar o modelo Chanter, assim como o Logístico e o Gompertz, utilizando um conjunto de dados do fruto do cacaueiro. O modelo Chanter é um híbrido entre o modelo Logístico e o modelo Gompertz cujos parâmetros podem ser interpretados similarmente. A comparação sobre a qualidade do ajuste entre os modelos foi feita utilizando as seguintes medidas estatísticas: o critério de informação de Akaike (AIC), o critério Peso de Akaike, o critério de informação de Bayes (BIC), o desvio padrão residual (DPR) e as medidas de não linearidade vício de Box e curvatura de Bates e Watts além de um estudo de simulação. Verificou-se que o modelo Chanter dentre os modelos estudados neste trabalho é o mais adequado para o ajuste dos dados do fruto do cacaueiro. / Nonlinear models such as Logistic and Gompertz are widely used to describe several biological processes using a growth curve given by the equation of the model. The objective of this work was to adjust the Chanter model, as well as the Logistic and the Gompertz, using a data set of cocoa fruit. The Chanter model is a hybrid between the Logistic model and the Gompertz model whose parameters can be interpreted similarly. A comparison of the quality of fit between the models was made using the following statistical measures: the Akaike information criterion (AIC), the Akaike weight criterion, Bayes information criterion (BIC), residual standard deviation (RSD), and measures of non-linearity Box addiction and Bates and Watts curvature as well as a simulation study. It was verified that the Chanter model is the most suitable one among the studied models for modeling the cocoa data.

In many scientific studies, the response variable bears a generalized nonlinear regression relationship with a certain covariate of interest, which may, however, be confounded by other covariates with unknown functional form. We propose a new class of models, the partly parametric generalized additive model (PPGAM) for doing generalized nonlinear regression with the confounding covariate effects adjusted nonparametrically. To avoid the curse of dimensionality, the PPGAM specifies that, conditional on the covariates, the response distribution belongs to the exponential family with the mean linked to an additive predictor comprising a nonlinear parametric function that is of main interest, plus additive, smooth functions of other covariates. The PPGAM extends both the generalized additive model (GAM) and the generalized nonlinear regression model. We propose to estimate a PPGAM by the method of penalized likelihood. We derive some asymptotic properties of the penalized likelihood estimator, including consistency and asymptotic normality of the parametric estimator of the nonlinear regression component. We propose a model selection criterion for the PPGAM, which resembles the BIC. We illustrate the new methodologies by simulations and real applications. We have developed an R package PPGAM that implements the methodologies expounded herein.

The symbolic regression problem is to find a function, in symbolic form, that fits a given data set. Symbolic regression provides a means for function identification. This research describes an adaptive hybrid system for symbolic function identification of thermo-physical model that combines the genetic programming and a modified Marquardt nonlinear regression algorithm. Genetic Programming (GP) system can extract knowledge from the data in the form of symbolic expressions, i.e. a tree structure, that are used to model and derive equation of state, mixing rules and phase behavior from the experimental data (properties estimation). During the automatic evolution process of GP, the function structure of generated individual module could be highly complicated. To ensure the convergence of the regression, a modified Marquardt regression algorithm is used. Two stop criteria are attached to the traditional Marquardt algorithm to enforce the algorithm repeat the regression process before it stops. Statistic analysis is applied to the fitted model. Residual plot is used to test the goodness of fit. The χ2-test is used to test the model's adequacy. Ten experiments are run with different form of input variables, number of data points, standard errors added to data set, and fitness functions. The results show that the system is able to find models and optimize for its parameters successfully.

Estimation of the ground motion parameters is extremely important for engineers to make the structures safer and more economical, so it is one of the main issues of Earthquake Engineering. Peak values of the ground motions obtained either from existing records or with the help of attenuation relationships, have been used as a useful parameter to estimate the effect of an earthquake on a specific location. Peak Ground Velocities (PGV) of a ground motion is used extensively in the recent years as a measure of intensity and as the primary source of energy-related analysis of structures. Consequently, PGV values are used to construct emergency response systems like Shake Maps or to determine the deformation demands of structures. Despite the importance of the earthquakes for Turkey, there is a lack of suitable attenuation relationships for velocity developed specifically for the country. The aim of this study is to address this deficiency by developing an attenuation relationship for the Peak Ground Velocities of the chosen database based on the strong ground motion records of Turkey. A database is processed with the established techniques and corrected database for the chosen ground motions is formed. Five different forms of equations that were used in the previous studies are selected to be used as models and by using nonlinear regression analysis, best fitted mathematical relation for attenuation is obtained. The result of this study can be used as an effective tool for seismic hazard assessment studies for Turkey. Besides, being a by-product of this study, a corrected database of strong ground motion recordings of Turkey may prone to be a valuable source for the future researchers.

This work studied the productivity evaluation and well test analysis of horizontal wells. The major components of this work consist of a 3D coupled reservoir/wellbore model, a productivity evaluation, a deconvolution technique, and a nonlinear regression technique improving horizontal well test interpretation. A 3D coupled reservoir/wellbore model was developed using the boundary element method for realistic description of the performance behavior of horizontal wells. The model is able to flexibly handle multiple types of inner and outer boundary conditions, and can accurately simulate transient tests and long-term production of horizontal wells. Thus, it can serve as a powerful tool in productivity evaluation and analysis of well tests for horizontal wells. Uncertainty of productivity prediction was preliminarily explored. It was demonstrated that the productivity estimates can be distributed in a broad range because of the uncertainties of reservoir/well parameters. A new deconvolution method based on a fast-Fourier-transform algorithm is presented. This new technique can denoise "noisy" pressure and rate data, and can deconvolve pressure drawdown and buildup test data distorted by wellbore storage. For cases with no rate measurements, a "blind" deconvolution method was developed to restore the pressure response free of wellbore storage distortion, and to detect the afterflow/unloading rate function using Fourier analysis of the observed pressure data. This new deconvolution method can unveil the early time behavior of a reservoir system masked by variable-wellbore-storage distortion, and thus provides a powerful tool to improve pressure transient test interpretation. The applicability of the method is demonstrated with a variety of synthetic and actual field cases for both oil and gas wells. A practical nonlinear regression technique for analysis of horizontal well testing is presented. This technique can provide accurate and reliable estimation of well-reservoir parameters if the downhole flow rate data are available. In the situation without flow rate measurement, reasonably reliable parameter estimation can be achieved by using the detected flow rate from blind deconvolution. It has the advantages of eliminating the need for estimation of the wellbore storage coefficient and providing reasonable estimates of effective wellbore length. This technique provides a practical tool for enhancement of horizontal well test interpretation, and its practical significance is illustrated by synthetic and actual field cases.