Appendix D: Nonlinear Least Squares for Data Fitting
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Least-Squares Fitting of Circles and Ellipses∗
Least-Squares Fitting of Circles and Ellipses∗ Walter Gander Gene H. Golub Rolf Strebel Dedicated to Ake˚ Bj¨orck on the occasion of his 60th birthday. Abstract Fitting circles and ellipses to given points in the plane is a problem that arises in many application areas, e.g. computer graphics [1], coordinate metrol- ogy [2], petroleum engineering [11], statistics [7]. In the past, algorithms have been given which fit circles and ellipses in some least squares sense without minimizing the geometric distance to the given points [1], [6]. In this paper we present several algorithms which compute the ellipse for which the sum of the squares of the distances to the given points is minimal. These algorithms are compared with classical simple and iterative methods. Circles and ellipses may be represented algebraically i.e. by an equation of the form F(x) = 0. If a point is on the curve then its coordinates x are a zero of the function F. Alternatively, curves may be represented in parametric form, which is well suited for minimizing the sum of the squares of the distances. 1 Preliminaries and Introduction Ellipses, for which the sum of the squares of the distances to the given points is minimal will be referred to as \best fit” or \geometric fit", and the algorithms will be called \geometric". Determining the parameters of the algebraic equation F (x)=0intheleast squares sense will be denoted by \algebraic fit" and the algorithms will be called \algebraic". We will use the well known Gauss-Newton method to solve the nonlinear least T squares problem (cf. -
Lesson 4 – Instrumental Variables
Ph.D. IN ENGINEERING AND Advanced methods for system APPLIED SCIENCES identification Ph.D. course Lesson 4: Instrumental variables TEACHER Mirko Mazzoleni PLACE University of Bergamo Outline 1. Introduction to error-in-variables problems 2. Least Squares revised 3. The instrumental variable method 4. Estimate an ARX model from ARMAX generated data 5. Application study: the VRFT approach for direct data driven control 6. Conclusions 2 /44 Outline 1. Introduction to error-in-variables problems 2. Least Squares revised 3. The instrumental variable method 4. Estimate an ARX model from ARMAX generated data 5. Application study: the VRFT approach for direct data driven control 6. Conclusions 3 /44 Introduction Many different solutions have been presented for system identification of linear dynamic systems from noise-corrupted output measurements On the other hand, estimation of the parameters for linear dynamic systems when also the input is affected by noise is recognized as a more difficult problem Representations where errors or measurement noises are present on both inputs and outputs are usually called Errors-in-Variables (EIV) models In case of static systems, errors-in-variables representations are closely related to other well-known topics such as latent variables models and factor models 4 /44 Introduction Errors-in-variables models can be motivated in several situations: • modeling of the dynamics between the noise-free input and noise-free output. The reason can be to have a better understanding of the underlying relations (e.g. in econometrics), rather than to make a good prediction from noisy data • when the user lacks enough information to classify the available signals into inputs and outputs and prefer to use a “symmetric” system model. -
A Toolbox for Nonlinear Regression in R: the Package Nlstools
JSS Journal of Statistical Software August 2015, Volume 66, Issue 5. http://www.jstatsoft.org/ A Toolbox for Nonlinear Regression in R: The Package nlstools Florent Baty Christian Ritz Sandrine Charles Cantonal Hospital St. Gallen University of Copenhagen University of Lyon Martin Brutsche Jean-Pierre Flandrois Cantonal Hospital St. Gallen University of Lyon Marie-Laure Delignette-Muller University of Lyon Abstract Nonlinear regression models are applied in a broad variety of scientific fields. Various R functions are already dedicated to fitting such models, among which the function nls() has a prominent position. Unlike linear regression fitting of nonlinear models relies on non-trivial assumptions and therefore users are required to carefully ensure and validate the entire modeling. Parameter estimation is carried out using some variant of the least- squares criterion involving an iterative process that ideally leads to the determination of the optimal parameter estimates. Therefore, users need to have a clear understanding of the model and its parameterization in the context of the application and data consid- ered, an a priori idea about plausible values for parameter estimates, knowledge of model diagnostics procedures available for checking crucial assumptions, and, finally, an under- standing of the limitations in the validity of the underlying hypotheses of the fitted model and its implication for the precision of parameter estimates. Current nonlinear regression modules lack dedicated diagnostic functionality. So there is a need to provide users with an extended toolbox of functions enabling a careful evaluation of nonlinear regression fits. To this end, we introduce a unified diagnostic framework with the R package nlstools. -
An Efficient Nonlinear Regression Approach for Genome-Wide
An Efficient Nonlinear Regression Approach for Genome-wide Detection of Marginal and Interacting Genetic Variations Seunghak Lee1, Aur´elieLozano2, Prabhanjan Kambadur3, and Eric P. Xing1;? 1School of Computer Science, Carnegie Mellon University, USA 2IBM T. J. Watson Research Center, USA 3Bloomberg L.P., USA [email protected] Abstract. Genome-wide association studies have revealed individual genetic variants associated with phenotypic traits such as disease risk and gene expressions. However, detecting pairwise in- teraction effects of genetic variants on traits still remains a challenge due to a large number of combinations of variants (∼ 1011 SNP pairs in the human genome), and relatively small sample sizes (typically < 104). Despite recent breakthroughs in detecting interaction effects, there are still several open problems, including: (1) how to quickly process a large number of SNP pairs, (2) how to distinguish between true signals and SNPs/SNP pairs merely correlated with true sig- nals, (3) how to detect non-linear associations between SNP pairs and traits given small sam- ple sizes, and (4) how to control false positives? In this paper, we present a unified framework, called SPHINX, which addresses the aforementioned challenges. We first propose a piecewise linear model for interaction detection because it is simple enough to estimate model parameters given small sample sizes but complex enough to capture non-linear interaction effects. Then, based on the piecewise linear model, we introduce randomized group lasso under stability selection, and a screening algorithm to address the statistical and computational challenges mentioned above. In our experiments, we first demonstrate that SPHINX achieves better power than existing methods for interaction detection under false positive control. -
Overview of Total Least Squares Methods
Overview of total least squares methods ,1 2 Ivan Markovsky∗ and Sabine Van Huffel 1 — School of Electronics and Computer Science, University of Southampton, SO17 1BJ, UK, [email protected] 2 — Katholieke Universiteit Leuven, ESAT-SCD/SISTA, Kasteelpark Arenberg 10 B–3001 Leuven, Belgium, [email protected] Abstract We review the development and extensions of the classical total least squares method and describe algorithms for its generalization to weighted and structured approximation problems. In the generic case, the classical total least squares problem has a unique solution, which is given in analytic form in terms of the singular value de- composition of the data matrix. The weighted and structured total least squares problems have no such analytic solution and are currently solved numerically by local optimization methods. We explain how special structure of the weight matrix and the data matrix can be exploited for efficient cost function and first derivative computation. This allows to obtain computationally efficient solution methods. The total least squares family of methods has a wide range of applications in system theory, signal processing, and computer algebra. We describe the applications for deconvolution, linear prediction, and errors-in-variables system identification. Keywords: Total least squares; Orthogonal regression; Errors-in-variables model; Deconvolution; Linear pre- diction; System identification. 1 Introduction The total least squares method was introduced by Golub and Van Loan [25, 27] as a solution technique for an overde- termined system of equations AX B, where A Rm n and B Rm d are the given data and X Rn d is unknown. -
Uncorrected Proof
Annals of Operations Research 124, 69–79, 2003 2003 Kluwer Academic Publishers. Manufactured in The Netherlands. 1 1 2 2 3 3 4 Multiple Neutral Data Fitting 4 5 5 6 6 CHRIS TOFALLIS [email protected] 7 7 Deptament of Stats, Econ, Accounting and Management Systems, Business School, 8 University of Hertfordshire, UK 8 9 9 10 10 11 Abstract. A method is proposed for estimating the relationship between a number of variables; this differs 11 12 from regression where the emphasis is on predicting one of the variables. Regression assumes that only 12 13 one of the variables has error or natural variability, whereas our technique does not make this assumption; 13 instead, it treats all variables in the same way and produces models which are units invariant – this is 14 important for ensuring physically meaningful relationships. It is thus superior to orthogonal regression in 14 15 that it does not suffer from being scale-dependent. We show that the solution to the estimation problem is 15 16 a unique and global optimum. For two variables the method has appeared under different names in various 16 17 disciplines, with two Nobel laureates having published work on it. 17 18 18 Keywords: functional relation, data fitting, regression, model fitting 19 19 20 20 21 Introduction 21 22 22 23 23 24 The method of regression is undoubtedly one of the most widely used quantitative meth- 24 25 ods in both the natural and social sciences. For most users of the technique the term is 25 26 taken to refer to the fitting of a function to data by means of the least squares criterion. -
ROBUST SOLUTIONS to LEAST-SQUARES PROBLEMS with UNCERTAIN DATA ∗ ´ LAURENT EL GHAOUI† and HERVE LEBRET† Abstract
SIAM J. MATRIX ANAL. APPL. c 1997 Society for Industrial and Applied Mathematics Vol. 18, No. 4, pp. 1035–1064, October 1997 015 ROBUST SOLUTIONS TO LEAST-SQUARES PROBLEMS WITH UNCERTAIN DATA ∗ ´ LAURENT EL GHAOUI† AND HERVE LEBRET† Abstract. We consider least-squares problems where the coefficient matrices A, b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone program- ming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it pro- vides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A, b are rational functions of an unknown-but-bounded perturbation vector. We show how to mini- mize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation. Key words. least-squares problems, uncertainty, robustness, second-order cone programming, semidefinite programming, ill-conditioned problem, regularization, robust identification, robust in- terpolation AMS subject classifications. 15A06, 65F10, 65F35, 65K10, 65Y20 PII. S0895479896298130 Notation. For a matrix X, X denotes the largest singular value and X F k k k k the Frobenius norm. If x is a vector, maxi xi is denoted by x . For a matrix A, | | k k∞ A† denotes the Moore–Penrose pseudoinverse of A. -
Model Selection, Transformations and Variance Estimation in Nonlinear Regression
Model Selection, Transformations and Variance Estimation in Nonlinear Regression Olaf Bunke1, Bernd Droge1 and J¨org Polzehl2 1 Institut f¨ur Mathematik, Humboldt-Universit¨at zu Berlin PSF 1297, D-10099 Berlin, Germany 2 Konrad-Zuse-Zentrum f¨ur Informationstechnik Heilbronner Str. 10, D-10711 Berlin, Germany Abstract The results of analyzing experimental data using a parametric model may heavily depend on the chosen model. In this paper we propose procedures for the ade- quate selection of nonlinear regression models if the intended use of the model is among the following: 1. prediction of future values of the response variable, 2. estimation of the un- known regression function, 3. calibration or 4. estimation of some parameter with a certain meaning in the corresponding field of application. Moreover, we propose procedures for variance modelling and for selecting an appropriate nonlinear trans- formation of the observations which may lead to an improved accuracy. We show how to assess the accuracy of the parameter estimators by a ”moment oriented bootstrap procedure”. This procedure may also be used for the construction of confidence, prediction and calibration intervals. Programs written in Splus which realize our strategy for nonlinear regression modelling and parameter estimation are described as well. The performance of the selected model is discussed, and the behaviour of the procedures is illustrated by examples. Key words: Nonlinear regression, model selection, bootstrap, cross-validation, variable transformation, variance modelling, calibration, mean squared error for prediction, computing in nonlinear regression. AMS 1991 subject classifications: 62J99, 62J02, 62P10. 1 1 Selection of regression models 1.1 Preliminary discussion In many papers and books it is discussed how to analyse experimental data estimating the parameters in a linear or nonlinear regression model, see e.g. -
Nonlinear Regression, Nonlinear Least Squares, and Nonlinear Mixed Models in R
Nonlinear Regression, Nonlinear Least Squares, and Nonlinear Mixed Models in R An Appendix to An R Companion to Applied Regression, third edition John Fox & Sanford Weisberg last revision: 2018-06-02 Abstract The nonlinear regression model generalizes the linear regression model by allowing for mean functions like E(yjx) = θ1= f1 + exp[−(θ2 + θ3x)]g, in which the parameters, the θs in this model, enter the mean function nonlinearly. If we assume additive errors, then the parameters in models like this one are often estimated via least squares. In this appendix to Fox and Weisberg (2019) we describe how the nls() function in R can be used to obtain estimates, and briefly discuss some of the major issues with nonlinear least squares estimation. We also describe how to use the nlme() function in the nlme package to fit nonlinear mixed-effects models. Functions in the car package than can be helpful with nonlinear regression are also illustrated. The nonlinear regression model is a generalization of the linear regression model in which the conditional mean of the response variable is not a linear function of the parameters. As a simple example, the data frame USPop in the carData package, which we load along with the car package, has decennial U. S. Census population for the United States (in millions), from 1790 through 2000. The data are shown in Figure 1 (a):1 library("car") Loading required package: carData brief(USPop) 22 x 2 data.frame (17 rows omitted) year population [i] [n] 1 1790 3.9292 2 1800 5.3085 3 1810 7.2399 .. -
Thesis Would Not Be Possible
SYMMETRIC AND TRIMMED SOLUTIONS OF SIMPLE LINEAR REGRESSION by Chao Cheng A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (STATISTICS) December 2006 Copyright 2006 Chao Cheng Dedication to my grandmother to my parents ii Acknowledgments First and foremost, I would like to thank my advisor, Dr. Lei Li, for his patience and guidance during my graduate study. Without his constant support and encouragement, the completion of the thesis would not be possible. I feel privileged to have had the opportunity to work closely and learn from him. I am very grateful to Dr. Larry Goldstein and Dr. Jianfeng Zhang, in the Depart- ment of Mathematics, for their most valuable help in writing this thesis. Most of my knowledge about linear regression is learned from Dr. Larry Goldstein’s classes. I would also like to thank Dr. Xiaotu Ma, Dr. Rui Jiang, Dr. Ming Li, Huanying Ge, and all the past and current members in our research group for their suggestions and recommendations. I greatly benefit from the discussion and collaboration with them. Finally, I would like to thank my little sister and my parents for their love, support and encouragement. I would also like to thank my grandmother, who passed away in 2002, soon after I began my study in USC. I feel sorry that I did not accompany her in her last minute. This thesis is dedicated to her. iii Table of Contents Dedication ii Acknowledgments iii List of Tables vi List of Figures vii Abstract 1 Chapter 1: INTRODUCTION 2 1.1 Simple linear regression model . -
Total Least Squares Regression in Input Sparsity Time
Total Least Squares Regression in Input Sparsity Time Huaian Diao Zhao Song Northeast Normal University & KLAS of MOE University of Washington [email protected] [email protected] David P. Woodruff Xin Yang Carnegie Mellon University University of Washington [email protected] [email protected] Abstract In the total least squares problem, one is given an m n matrix A, and an m d × × matrix B, and one seeks to “correct” both A and B, obtaining matrices A and B, so that there exists an X satisfying the equation AX = B. Typically the problem is overconstrained, meaning that m max(n, d). The cost of the solutionb A,bB 2 2 b b is given by A A F + B B F . We give an algorithm for finding a solution k − k k − k 2 b2 b X to the linear system AX = B for which the cost A A F + B B F is at most a multiplicativeb (1 + )bfactor times the optimalk − cost,k up tok an− additivek error η that may be an arbitrarilyb b small function of n. Importantly,b our runningb time is O(nnz(A) + nnz(B)) + poly(n/) d, where for a matrix C, nnz(C) denotes its number of non-zero entries. Importantly,· our running time does not directlye depend on the large parameter m. As total least squares regression is known to be solvable via low rank approximation, a natural approach is to invoke fast algorithms for approximate low rank approximation, obtaining matrices A and B from this low rank approximation, and then solving for X so that AX = B. -
A Bayesian Approach to Nonlinear Parameter Identification for Rigid
A Bayesian Approach to Nonlinear Parameter Identification for Rigid Body Dynamics Jo-Anne Ting∗, Michael Mistry∗, Jan Peters∗, Stefan Schaal∗† and Jun Nakanishi†‡ ∗Computer Science, University of Southern California, Los Angeles, CA, 90089-2520 †ATR Computational Neuroscience Labs, Kyoto 619-0288, Japan ‡ICORP, Japan Science and Technology Agency Kyoto 619-0288, Japan Email: {joanneti, mmistry, jrpeters, sschaal}@usc.edu, [email protected] Abstract— For robots of increasing complexity such as hu- robots such as humanoid robots have significant additional manoid robots, conventional identification of rigid body dynamics nonlinear dynamics beyond the rigid body dynamics model, models based on CAD data and actuator models becomes due to actuator dynamics, routing of cables, use of protective difficult and inaccurate due to the large number of additional nonlinear effects in these systems, e.g., stemming from stiff shells and other sources. In such cases, instead of trying to wires, hydraulic hoses, protective shells, skin, etc. Data driven explicitly model all possible nonlinear effects in the robot, parameter estimation offers an alternative model identification empirical system identification methods appear to be more method, but it is often burdened by various other problems, useful. Under the assumption that a rigid body dynamics such as significant noise in all measured or inferred variables (RBD) model is sufficient to capture the entire robot dynamics, of the robot. The danger of physically inconsistent results also exists due to unmodeled nonlinearities or insufficiently rich data. this problem is theoretically straightforward as all unknown In this paper, we address all these problems by developing a parameters of the robot such as mass, center of mass and Bayesian parameter identification method that can automatically inertial parameters appear linearly in the rigid body dynamics detect noise in both input and output data for the regression equations [3].