Compared to other methods for the estimation in additive models the new approach neither requires observations on a regular grid nor the estimation. Gams were originally developed by trevor hastie and robert tibshirani to blend properties of generalized linear models with additive models. Linear smoothers and additive models stanford university. Although, the asymptotic properties of the integration estimator, and to some extent the backfitting, method too, are well understood, its small sample properties are not well investigated. In this paper, a new likelihood approach for fitting generalized additive models is proposed. Backfitting and the ace algorithm additive regression models and backfitting the backtting algorithm is used to estimate smooth functions in. Park2 university of mannheim and seoul national university in this paper a new smooth back.
In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. Unlike linear regression models, which are fitted by using weighted least squares and have an exact solution, the estimation procedure for a gam or a glm. Gaussseidel algorithm, generalized additive models, numerical analysis, projection, backfitting. Chapter 10 additive models, gam, and neural networks rafalab. At each iteration, an adjusted dependent variable is formed and an additive regression model is fit using the backfitting algorithm. General additive models and their application in modelling zooplankton lifecycle dynamics. The estimate has the simple structure of nadarayawatson smooth backfitting but at the same time achieves the oracle property of local linear smooth backfitting. In this paper a new smooth backfitting estimate is proposed for additive regression models. Backfitting algorithm estimates the approximating regression surface, working around the curse of dimentionality. Friedman and werner stuetzle 1981 and is an essential part of the ace algorithm.
It is rateoptimal and its implementation based on local linear estimation achieves the same bias. Continuously additive models for nonlinear functional regression. Backfitting and smooth backfitting for additive quantile. However, the inferences of the models have not been very well developed, due partially to the complexity of the backfitting estimators. Pdf smooth backfitting in generalized additive models. We show that backfitting is the gaussseidel iterative method for solving a set of normal equations associated with the additive model. Backfitting and smooth backfitting for additive quantile models.
Use the logit transform to translate the probability estimation problem into a regression problem, as we did in section 4. Efficient estimation of generalized additive nonparametric. Pdf the backfitting algorithm is an iterative procedure for fitting additive. Asymptotic proper ties of the estimator and convergence of the algorithm are discussed. A smooth backfitting procedure is developed and asymptotic normality of the resulting estimator is established. Structured sparse additive models junmingjunming yin and eric yin and eric xingxing lecture 27, april 24, 20. Smooth backfitting in additive inverse regression springerlink. See class website 1 outline znonparametric regression and kernel smoothing zadditive models zsparse additive models spam zstructured sparse additive models. Bandwidth selection for smooth backfitting in additive models arxiv. The smooth backfitting method does not have these drawbacks. In the first part of this paper we examine certain aspects. Chapter 10 additive models, gam, and neural networks. Nonparametric lag selection for additive models based on the.
Compared to other methods for the estimation in additive models the new approach neither requires observations on a regular grid nor the. Despite this on going research, there is still no answer, applicable to general linear smoothers, to the questions of convergence of the backfitting algorithm and uniqueness of the estimators for additive models of dimension greater than 2. The backfitting algorithm is a general algorithm that can fit an additive model with any regression type fitting mechanisms. In statistics, an additive model am is a nonparametric regression method. Convergence of the algorithm has been studied by buja, hastie, and tibshirani 1989.
Pdf convergence of the backfitting algorithm for additive. It is shown that our proposal based on local linear fit. On the backfitting algorithm for additive regression. Probabilistic graphical modelsprobabilistic graphical models structured sparse additive models junmingjunming yin and eric yin and eric xingxing lecture 27, april 24, 20. Smooth backfitting for errorsinvariables varying coefficient regression models. We propose an automatic structure recovery method for additive models, based on a backfitting algorithm coupled with local polynomial smoothing, in conjunction with a new kernelbased variable selection strategy. Pdf convergence of the backfitting algorithm for additive models.
An iterative algorithm based on smooth backfitting is developed from the newtonkantorovich theorem. At each stage, add the model that maximizes the probability of the data given the ensemble classifier. Integration and backfitting methods in additive modelsfinite sample properties and comparison, test. Inla software for bayesian inference with gams and more. Integration and backfitting methods in additive modelsfinite. We analyse additive regression model fitting via the backfitting algorithm. Many ways are available to approach the formulation and estimation of additive models.
We consider the problem of estimating an additive regression function in an inverse regression model with a convolution type operator. Smooth backfitting in generalized additive models by kyusang yu,1 byeong u. The backfitting algorithm is an iterative procedure for fitting additive models in which, at each step, one component is estimated keeping the other components fixed, the algorithm proceeding. The class of additive models is a useful compromise. However, the theoretical properties of the local scoring estimator are poorly understood. Park2 and enno mammen1 university of mannheim, seoul national university and university of mannheim generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with. This approach provides a class of exible functional nonlinear regression models, where random predictor curves are coupled with scalar responses. Backfitting algorithms for totalvariation and empiricalnorm. Series a 57 1994, 316329 convergence of the backfitting algorithm for additive models craig f. And the class of all possible smooths is too large the cod makes it hard to smooth in high dimensions. We show that in the case of a large class of curve estimators, which includes regressograms, simple step. Because of this, it is less affected by the curse of dimensionality than e.
Additive models are a class of nonparametric regression models of the form. The local scoring algorithm is analogous to the iterative reweighted least squares algorithm for solving likelihood and nonlinear regression equations. We show that these backfitting quantile estimators are asymptotically equivalent to the corresponding backfitting estimators of the additive components in a speciallydesigned additive mean regression model. There are few tools available to answer some important and frequently asked questions, such as whether a specific additive component is significant or admits. On the use of generalized additive models in timeseries. It can be used with different smoothers such as smoothing splines and local regression smoothers.
Although, the asymptotic properties of the integration estimator, and to some extent the backfitting method too, are well understood, its small sample properties are not well investigated. The backfitting algorithm is an iterative procedure for fitting additive models in which, at each step, one component is estimated keeping the other components fixed, the algorithm proceeding component by component and iterating until convergence. Ansley and robert kohn received 14 june 1991 revise. The estimation of binary nonparametric regression model. Recently, there has been considerable research on sparse additive models in highdimensional settings, where p is close to or greater than n, but the number of nonzero functions f j. Time series analysis with generalized additive models. Probabilistic graphical modelsprobabilistic graphical models structured sparse additive models junmingjunming yin and eric yin and eric xingxing lecture 27, april 24, 20 reading. The backfitting algorithm is suitable for fitting any additive model, and in gam it is used within the local scoring iteration when several smooth functions are included in the model. The additive functions are estimated by solving a system of nonlinear integral equations. We examine and compare the finite sample performance of the competing backfitting and integration methods for estimating additive nonparametric regression using simulated data.
Whenever you spot a trend plotted against time, you would be looking at a time series. View backfit from sta 457 at university of toronto. Consistency of our procedure is established under very general conditions, including heteroskedasticity. Pdf optimal estimation in additive regression models. Opsomer, 2000 show that if covariates are independent, then the asymptotic bias of the linear smoother obtained through a backfitting algorithm becomes very small.
Nonparametric lag selection for additive models based on. We introduce continuously additive models, which can be motivated as extensions of additive regression models with vector predictors to the case of in nitedimensional predictors. A simple smooth backfitting method for additive models by enno mammen1 and byeong u. Pdf this paper is concerned with optimal estimation of the additive. In statistics, a generalized additive model gam is a generalized linear model in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. It was introduced in 1985 by leo breiman and jerome friedman along with generalized additive models.
We provide conditions for consistency and nondegeneracy and prove. Additive models with backfitting algorithms are popular multivariate nonparametric fitting techniques. Continuously additive models for nonlinear functional. Additive regression an overview sciencedirect topics. Concepts of degrees of freedom and corresponding akaike or bayesian information criteria, particularly useful for regularization and variable selection in highdimensional covariate spaces, are discussed as well. The backfitting algorithm is a general algorithm that can fit an additive model with any regressiontype fitting mechanisms. Backfitting algorithms for totalvariation and empirical. Before illustrating how additive models work in practice, lets talk about why wed want to use them. By deriving explicit expressions for the estimators of. Despite this ongoing research, there is still no answer, applicable to general linear smoothers, to the questions of convergence of the backfitting algorithm and uniqueness of the estimators for additive models of dimension greater than 2. Chapter 10 additive models, gam, and neural networks in observational studies we usually have observed predictors or covariates x 1,x 2. Mar 26, 2015 we consider the problem of estimating an additive regression function in an inverse regression model with a convolution type operator. Convergence of the backfitting algorithm for additive models. The backfitting algorithm is used to fit additive models.
Pdf a simple smooth backfitting method for additive. Each component is estimated with the same asymptotic accuracy as if the other components were known. The existence of a unique solution and the convergence of the associated backfitting algorithm are established theoretically. Thus, the backfitting algorithm that performs best in an additive model can simplify the estimation of the parameters of the response surface model. On the backfitting algorithm for additive regression models. In this paper, we study the ordinary backfitting and smooth backfitting as methods of fitting additive quantile models. This function is not intended to be called directly. In most cases, the backfitting algorithm is equivalent to the gaussseidel method algorithm for solving a certain linear system of equations. Nonparametric lag selection for additive models based on the smooth backfitting estimator author. This procedure can be expected to perform well because of its wellknown finite sample performance of the smooth backfitting estimator. Aug 01, 2002 the backfitting algorithm is suitable for fitting any additive model, and in gam it is used within the local scoring iteration when several smooth functions are included in the model. Integration and backfitting methods in additive models.
May 02, 2019 backfitting algorithm estimates the approximating regression surface, working around the curse of dimentionality. An official journal of the spanish society of statistics and operations research, springer. Smooth backfitting in generalized additive models arxiv. Generalized additive models are a popular class of multivariate nonparametric regression models, due in large part to the ease of use of the local scoring estimation algorithm. It has been proved successful in the application of the sbf technique to additive models and partially additive models with measurement errors in the predictors, see han and park 2018 and lee et al. This algorithm consists of two loops that are scoring step outer loop is iterated until the.
1367 1445 1595 776 1411 1461 1426 637 315 31 948 1029 424 177 1579 1261 483 835 831 1597 1325 1025 290 562 8 538 769 832 646 913 677 776