DOCSLIB.ORG

On Efficient Designing of Nonlinear Experiments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Efficient Designing of Nonlinear Experiments Statistica Sinica 5(1995), 421-440 ON EFFICIENT DESIGNING OF NONLINEAR EXPERIMENTS Probal Chaudhuri and Per A. Mykland Indian Statistical Institute and University of Chicago Abstract: Adaptive designs that optimize the Fisher information asso ciated with a nonlinear exp erimentare considered. Asymptotic prop erties of the maximum like- liho o d estimate and related statistical inference based on dep endent data generated by sequentially designed adaptive nonlinear exp eriments are explored. Conditions on the exp erimental designs that ensure rst order eciency of the maximum likeliho o d estimate when the parametrization of the nonlinear mo del is suciently smo oth and regular are derived. A few interesting op en questions that arise naturally in course of the investigation are mentioned and brie y discussed. Key words and phrases: Adaptive sequential designs, rst order eciency, Fisher information, lo cal -optimality, martingales, maximum likeliho o d, nonlinear mo dels. 1. Intro duction: Adaptive Sequential Design of Nonlinear Exp eri- ments In a typical nonlinear set up involving a resp onse Y and a regressor X , the de- p endence of the resp onse on the regressor is mo deled using a family of probability distributions, whichinvolve an unknown Euclidean parameter (to be estimated from the data) in such a way that the parametrization is smo oth and regular, and the asso ciated Fisher information happ ens to b e a nonlinear function of that parameter of interest. Standard examples of such nonlinear mo dels include the usual nonlinear regression mo dels (see e.g. Gallant (1987), Bates and Watts (1988), Seb er and Wild (1989)), various generalized linear mo dels (see e.g. Mc- Cullagh and Nelder (1989)), and many p opular heteroscedastic regression mo dels (see e.g. Box and Hill (1974), Bickel (1978), Jobson and Fuller (1980), Carroll and Rupp ert (1988)). An awkward situation arises when one needs to design an exp eriment in order to ensure maximum eciency of parameter estimates in such a nonlinear problem. Since the Fisher information asso ciated with any such problem dep ends on the unknown parameter in a nonlinear way, an ecient designing of the exp eriment to guarantee optimal p erformance of the nonlinear least squares or the maximum likeliho o d estimate will require knowledge of that unknown parameter! (See Co chran (1973, pp: 771-772), Bates and Watts (1988, 422 PROBAL CHAUDHURI AND PER A. MYKLAND p: 129) for various interesting remarks in connection with this). An extensive discussion on p ossible resolutions of this problem together with several examples of nonlinear exp eriments can be found in Ford, Titterington and Kitsos (1989), Myers, Khuri and Carter (1989) and Chaudhuri and Mykland (1993). A few more interesting examples are discussed b elow. Example 1.1. Treloar (1974) investigated an enzymatic reaction in whichthe number of counts per minute of a radioactive pro duct from the reaction was measured, and from these counts the initial velo city (Y ) of the reaction was calculated. This velo city is related to the substrate concentration (X )chosen by the exp erimenter through the nonlinear regression equation (Michaelis-Menten 1 mo del) Y = X ( + X ) + e. Here = ( ; ) is the unknown parameter 0 1 0 1 of interest, and e is the random noise with zero mean as usual. The exp eriment was conducted once with the enzyme (Galactosyltransferase of Golgi Membranes) treated with Puromycin and once with the enzyme untreated. The main ob jective of the study was to investigate the e ect of the intro duction of Puromycin on the ultimate velo city parameter and the half velo city parameter . 0 1 Example 1.2. Cox and Snell (1989, pp: 10-11) while discussing binary resp onse regression mo dels mentioned ab out an industrial exp eriment, where the number of ingots that were not ready for rolling were counted out of a xed number of ingots tested. The number counted dep ends on twocovariates, which are heating time and soaking time determined by the exp erimenter at the time of pro cessing. This leads to a binomial regression mo del and the main interest is in getting insights into how variations in heating and soaking times in uence the number of ingots that turn out to b e not ready for rolling. Example 1.3. Powsner (1935) collected data from an exp eriment that was conducted to determine the e ect of exp erimental temp erature on the develop- mental stages (i.e. embryonic, egg-larval, larval and pupal stages) of the fruit y Drosophila Melanogaster. McCullagh and Nelder (1989) discussed the data and demonstrated how it can be analyzed using appropriate gamma regression mo dels with the exp erimental temp erature as the covariate. Chaudhuri and Mykland (1993) investigated an adaptive sequential scheme for designing nonlinear exp eriments and established asymptotic D-optimalityof 1=2 the design as well as n -consistency, asymptotic normalityand rst order e- ciency of the maximum likeliho o d estimate based on observations generated by their prop osed scheme in a nonlinear set up satisfying suitable regularity condi- tions. Adaptive sequential and batch sequential designs for certain very sp eci c typ es of nonlinear exp eriments related to some sp ecial mo dels were explored ear- DESIGNING NONLINEAR EXPERIMENTS 423 lier byBox and Hunter (1965), Ford and Silvey (1980), Ab delbasit and Plackett (1983), Ford, Titterington and Wu (1985), Wu (1985), Minkin (1987) and Mc- Cormick, Mallik and Reeves (1988) among others. Some of these authors made attempts to study the asymptotic behavior of their resp ective design schemes and resulting parameter estimates. The key idea lying at the ro ot of such design strategies is to divide available resources into several groups and to split the entire exp eriment into a number of steps. At each step, the exp eriment is conducted using only a single p ortion of divided resources. At the end of each step, an analysis is carried out and parameter estimates are up dated using the available data. The results emerging from this data analysis are used in ecient design- ing of the exp eriment at subsequent steps. As pointed out by several authors (see e.g. Cherno (1975), Fedorov (1972), Silvey (1980)), the most attractive feature of adaptive sequential exp eriments is their ability to optimally utilize the dynamics of the learning pro cess asso ciated with exp erimentation, data analysis and inference. At the b eginning, the scientist do es not have much information, and hence an initial exp eriment is b ound to be somewhat tentative in nature. As the exp erimentcontinues and more and more observations are obtained, the scientist is able to form a more precise impression of the underlying theory,and this more de nite p erception is used to design a more informative exp erimentin the future that is exp ected to give rise to more relevant and useful data. The main purp ose of this article is to explore some fundamental theoretical issues that are of critical imp ortance in studying the asymptotics of sequen- tially designed adaptive nonlinear exp eriments. In Section 2, we will explore the asymptotic p erformance of maximum likeliho o d estimates based on dep endent observations generated by adaptive and sequentially designed exp eriments in a very general nonlinear set up, which amply covers situations o ccurring in prac- tice and mo dels studied in the literature. Unlike Chaudhuri and Mykland (1993), who considered a very sp eci c adaptive sequential scheme, our fo cus here will b e on general adaptive pro cedures for sequentially designing exp eriments and some related asymptotics. In Section 3, we will discuss the limiting b ehavior of some adaptive design pro cedures and work out some useful conditions that guarantee their large sample optimality with resp ect to a broad class of optimal design criteria. Once again, unlikeChaudhuri and Mykland (1993), who concentrated on D-optimal designs only, we will consider a general family of optimal design criteria that includes D-optimality and many other optimality criteria as sp ecial cases. In the course of the development of the principal theoretical results in Sections 2 and 3, we observe certain intriguing facts, and some interesting ques- tions, which arise naturally, remain unanswered at this moment. All technical pro ofs are presented in App endix. 424 PROBAL CHAUDHURI AND PER A. MYKLAND 2. Maximum Likeliho o d and Related Statistical Inference Supp ose now that the conditional distribution of the resp onse Y given X = x has a probability density function or probability mass function f (y j; x). Here the form of the function f is assumed to b e known, and is an unknown d-dimensional Euclidean parameter of interest that takes its values in (the parameter space), d which will be assumed to be a convex and op en subset of R . The value of the regressor X is determined by the exp erimenter, who cho oses it from a set (the exp eriment space) before each trial of the exp eriment. In applications, can b e a nite set (e.g. factorial exp eriments with each factor having a nite numb er of levels) or some appropriate compact (i.e. closed and b ounded) interval or region in an Euclidean space (e.g. when the regressor X is continuous in nature) or a mixture of the two (e.g. analysis of covariance typ e problems). As pointed out in Chaudhuri and Mykland (1993), the conditional distribution of Y given X = x mayinvolve an unknown nuisance parameter (e.g. the unknown error variance in a homoscedastic normal regression mo del) in addition to the parameter of principal interest .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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 ..
    [Show full text]
  • Fitting Models to Biological Data Using Linear and Nonlinear Regression
    Fitting Models to Biological Data using Linear and Nonlinear Regression A practical guide to curve fitting Harvey Motulsky & Arthur Christopoulos Copyright 2003 GraphPad Software, Inc. All rights reserved. GraphPad Prism and Prism are registered trademarks of GraphPad Software, Inc. GraphPad is a trademark of GraphPad Software, Inc. Citation: H.J. Motulsky and A Christopoulos, Fitting models to biological data using linear and nonlinear regression. A practical guide to curve fitting. 2003, GraphPad Software Inc., San Diego CA, www.graphpad.com. To contact GraphPad Software, email [email protected] or [email protected]. Contents at a Glance A. Fitting data with nonlinear regression.................................... 13 B. Fitting data with linear regression..........................................47 C. Models ....................................................................................58 D. How nonlinear regression works........................................... 80 E. Confidence intervals of the parameters ..................................97 F. Comparing models................................................................ 134 G. How does a treatment change the curve?..............................160 H. Fitting radioligand and enzyme kinetics data ....................... 187 I. Fitting dose-response curves .................................................256 J. Fitting curves with GraphPad Prism......................................296 3 Contents Preface ........................................................................................................12
    [Show full text]
  • Nonlinear Regression and Nonlinear Least Squares
    Nonlinear Regression and Nonlinear Least Squares Appendix to An R and S-PLUS Companion to Applied Regression John Fox January 2002 1 Nonlinear Regression The normal linear regression model may be written yi = xiβ + εi where xi is a (row) vector of predictors for the ith of n observations, usually with a 1 in the first position representing the regression constant; β is the vector of regression parameters to be estimated; and εi is a random error, assumed to be normally distributed, independently of the errors for other observations, with 2 expectation 0 and constant variance: εi ∼ NID(0,σ ). In the more general normal nonlinear regression model, the function f(·) relating the response to the predictors is not necessarily linear: yi = f(β, xi)+εi As in the linear model, β is a vector of parameters and xi is a vector of predictors (but in the nonlinear 2 regression model, these vectors are not generally of the same dimension), and εi ∼ NID(0,σ ). The likelihood for the nonlinear regression model is n y − f β, x 2 L β,σ2 1 − i=1 i i ( )= n/2 exp 2 (2πσ2) 2σ This likelihood is maximized when the sum of squared residuals n 2 S(β)= yi − f β, xi i=1 is minimized. Differentiating S(β), ∂S β ∂f β, x ( ) − y − f β, x i ∂β = 2 i i ∂β Setting the partial derivatives to 0 produces estimating equations for the regression coefficients. Because these equations are in general nonlinear, they require solution by numerical optimization. As in a linear model, it is usual to estimate the error variance by dividing the residual sum of squares for the model by the number of observations less the number of parameters (in preference to the ML estimator, which divides by n).
    [Show full text]
  • Part III Nonlinear Regression
    Part III Nonlinear Regression 255 Chapter 13 Introduction to Nonlinear Regression We look at nonlinear regression models. 13.1 Linear and Nonlinear Regression Models We compare linear regression models, Yi = f(Xi; ¯) + "i 0 = Xi¯ + "i = ¯ + ¯ Xi + + ¯p¡ Xi;p¡ + "i 0 1 1 ¢ ¢ ¢ 1 1 with nonlinear regression models, Yi = f(Xi; γ) + "i where f is a nonlinear function of the parameters γ and Xi1 γ0 . Xi = 2 . 3 γ = 2 . 3 q£1 p£1 6 Xiq 7 6 γp¡1 7 4 5 4 5 In both the linear and nonlinear cases, the error terms "i are often (but not always) independent normal random variables with constant variance. The expected value in the linear case is 0 E Y = f(Xi; ¯) = X ¯ f g i and in the nonlinear case, E Y = f(Xi; γ) f g 257 258 Chapter 13. Introduction to Nonlinear Regression (ATTENDANCE 12) Exercise 13.1(Linear and Nonlinear Regression Models) Identify whether the following regression models are linear, intrinsically linear (nonlinear, but transformed easily into linear) or nonlinear. 1. Yi = ¯0 + ¯1Xi1 + "i This regression model is linear / intrinsically linear / nonlinear 2. Yi = ¯0 + ¯1pXi1 + "i This regression model is linear / intrinsically linear / nonlinear because Yi = ¯0 + ¯1 Xi1 + "i p0 = ¯0 + ¯1Xi1 + "i 3. ln Yi = ¯0 + ¯1Xi1 + "i This regression model is linear / intrinsically linear / nonlinear because ln Yi = ¯0 + ¯1 Xi1 + "i 0 p0 Yi = ¯0 + ¯1Xi1 + "i 0 where Yi = ln Yi 4. Yi = γ0 + γ1Xi1 + γ2Xi2 + "i This regression model1 is linear / intrinsically linear / nonlinear 5.
    [Show full text]
  • Nonlinear Regression for Split Plot Experiments
    Kansas State University Libraries New Prairie Press Conference on Applied Statistics in Agriculture 1990 - 2nd Annual Conference Proceedings NONLINEAR REGRESSION FOR SPLIT PLOT EXPERIMENTS Marcia L. Gumpertz John O. Rawlings Follow this and additional works at: https://newprairiepress.org/agstatconference Part of the Agriculture Commons, and the Applied Statistics Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Gumpertz, Marcia L. and Rawlings, John O. (1990). "NONLINEAR REGRESSION FOR SPLIT PLOT EXPERIMENTS," Conference on Applied Statistics in Agriculture. https://doi.org/10.4148/2475-7772.1441 This is brought to you for free and open access by the Conferences at New Prairie Press. It has been accepted for inclusion in Conference on Applied Statistics in Agriculture by an authorized administrator of New Prairie Press. For more information, please contact [email protected]. 159 Conference on Applied Statistics in Agriculture Kansas State University NONLINEAR REGRESSION FOR SPLIT PLOT EXPERIMENTS Marcia L. Gumpertz and John O. Rawlings North Carolina State University ABSTRACT Split plot experimental designs are common in studies of the effects of air pollutants on crop yields. Nonlinear functions such the Weibull function have been used extensively to model the effect of ozone exposure on yield of several crop species. The usual nonlinear regression model, which assumes independent errors, is not appropriate for data from nested or split plot designs in which there is more than one source of random variation. The nonlinear model with variance components combines a nonlinear model for the mean with additive random effects to describe the covariance structure.
    [Show full text]
  • Nonlinear Regression
    Nonlinear Regression © 2017 Griffin Chure. This work is licensed under a Creative Commons Attribution License CC-BY 4.0. All code contained herein is licensed under an MIT license In this tutorial... In this tutorial, we will learn how to perform nonlinear regression using the statistic by estimating the DNA binding energy of the lacI repressor to the O2 operator DNA sequence. What is probability? One of the most powerful skills a scientist can possess is a knowledge of probability and statistics. I can nearly guarantee that regardless of discipline, you will have to perform a nonlinear regression and report the value of a parameter. Before we get into the details, let's briefly touch on what we mean we report the "value of a parameter". What we seek to identify through parameter estimation is the value of a given parameter which best describes the data under a given model. Estimation of a parameter is intimately tied to our interpretation of probability of which there are two major schools of thought (Frequentism and Bayesian). A detailed discussion of these two interpretations is outside the scope of this tutorial, but see this wonderful 25 minute video on Bayesian vs Frequentist probability by Jake VanderPlas. Also see BE/Bi 103 - Data Analysis in the Biological Sciences by Justin Bois, which is one of the best courses I have ever taken. How do we estimate a parameter? Above, we defined a parameter value as that which best describes our data given a specific model, but what do we mean by "best describes"? Imagine we have some set of data ( and , which are linearly correlated, as is shown in the figure below.
    [Show full text]
  • Nonlinear Regression (Part 1)
    Nonlinear Regression (Part 1) Christof Seiler Stanford University, Spring 2016, STATS 205 Overview I Smoothing or estimating curves I Density estimation I Nonlinear regression I Rank-based linear regression Curve Estimation I A curve of interest can be a probability density function f I In density estimation, we observe X1,..., Xn from some unknown cdf F with density f X1,..., Xn ∼ f I The goal is to estimate density f Density Estimation 0.4 0.3 0.2 density 0.1 0.0 −2 0 2 4 rating Density Estimation 0.4 0.3 0.2 density 0.1 0.0 −2 0 2 4 rating Nonlinear Regression I A curve of interest can be a regression function r I In regression, we observe pairs (x1, Y1),..., (xn, Yn) that are related as Yi = r(xi ) + i with E(i ) = 0 I The goal is to estimate the regression function r Nonlinear Regression Bone Mineral Density Data 0.2 0.1 gender female male Change in BMD 0.0 10 15 20 25 Age Nonlinear Regression Bone Mineral Density Data 0.2 0.1 gender female male Change in BMD 0.0 −0.1 10 15 20 25 Age Nonlinear Regression Bone Mineral Density Data 0.2 0.1 gender female male Change in BMD 0.0 10 15 20 25 Age Nonlinear Regression Bone Mineral Density Data 0.2 0.1 gender female male Change in BMD 0.0 10 15 20 25 Age The Bias–Variance Tradeoff I Let fbn(x) be an estimate of a function f (x) I Define the squared error loss function as 2 Loss = L(f (x), fbn(x)) = (f (x) − fbn(x)) I Define average of this loss as risk or Mean Squared Error (MSE) MSE = R(f (x), fbn(x)) = E(Loss) I The expectation is taken with respect to fbn which is random I The MSE can be
    [Show full text]
  • Sketching Structured Matrices for Faster Nonlinear Regression
    Sketching Structured Matrices for Faster Nonlinear Regression Haim Avron Vikas Sindhwani David P. Woodruff IBM T.J. Watson Research Center IBM Almaden Research Center Yorktown Heights, NY 10598 San Jose, CA 95120 fhaimav,[email protected] [email protected] Abstract Motivated by the desire to extend fast randomized techniques to nonlinear lp re- gression, we consider a class of structured regression problems. These problems involve Vandermonde matrices which arise naturally in various statistical model- ing settings, including classical polynomial fitting problems, additive models and approximations to recently developed randomized techniques for scalable kernel methods. We show that this structure can be exploited to further accelerate the solution of the regression problem, achieving running times that are faster than “input sparsity”. We present empirical results confirming both the practical value of our modeling framework, as well as speedup benefits of randomized regression. 1 Introduction Recent literature has advocated the use of randomization as a key algorithmic device with which to dramatically accelerate statistical learning with lp regression or low-rank matrix approximation techniques [12, 6, 8, 10]. Consider the following class of regression problems, arg min kZx − bkp; where p = 1; 2 (1) x2C where C is a convex constraint set, Z 2 Rn×k is a sample-by-feature design matrix, and b 2 Rn is the target vector. We assume henceforth that the number of samples is large relative to data dimensionality (n k). The setting p = 2 corresponds to classical least squares regression, while p = 1 leads to least absolute deviations fit, which is of significant interest due to its robustness properties.
    [Show full text]
  • Likelihood Ratio Tests for Goodness-Of-Fit of a Nonlinear Regression Model
    Likelihood Ratio Tests for Goodness-of-Fit of a Nonlinear Regression Model Ciprian M. Crainiceanu¤ David Rupperty April 2, 2004 Abstract We propose likelihood and restricted likelihood ratio tests for goodness-of-fit of nonlinear regression. The first order Taylor approximation around the MLE of the regression parameters is used to approximate the null hypothesis and the alternative is modeled nonparametrically using penalized splines. The exact finite sample distribution of the test statistics is obtained for the linear model approximation and can be easily simulated. We recommend using the restricted likelihood instead of the likelihood ratio test because restricted maximum likelihood estimates are not as severely biased as the maximum likelihood estimates in the penalized splines framework. Short title: LRTs for nonlinear regression Keywords: Fan-Huang goodness-of-fit test, mixed models, Nelson-Siegel model for yield curves, penalized splines, REML. 1 Introduction Nonlinear regression models used in applications arise either from underlying theoretical principles or from the trained subjective choice of the statistician. From the perspective of goodness-of-fit testing these are null models and, once data become available, statistical testing may be used to validate or invalidate the original assumptions. There are three main approaches to goodness-of-fit testing of a null regression model. The standard approach is to nest the null parametric model into a parametric supermodel (sometimes called the “full model”) that is assumed to be true and use likelihood ratio tests (LRTs). This ¤Department of Biostatistics, Johns Hopkins University, 615 N. Wolfe Street, Baltimore, MD, 21205 USA. Email: [email protected] ySchool of Operational Research and Industrial Engineering, Cornell University, Rhodes Hall, NY 14853, USA.
    [Show full text]