A mathematical introduction to least angle regression r. Least angle regression aka lars is a model selection method for linear regression when youre worried about overfitting or want your model to be easily interpretable. Increasing upper bound b, coefficients gradually turn on few critical values of b where support changes nonzero coefficients increase or decrease linearly between critical points can solve for critical values analytically. In statistics, least angle regression lars is an algorithm for fitting linear regression models to highdimensional data, developed by bradley efron, trevor hastie, iain johnstone and robert tibshirani. Uraibia,b, habshah midib,c, sohel ranab,d a department of statistics, college of administration and economics, university of alqadisiyah, 50082, iraq b institute for mathematical research, university putra malaysia, 43400 upm, serdang, malaysia. Least angle regression, forward stagewise and the lasso brad efron, trevor hastie, iain johnstone and robert tibshirani. Least angle regression lars proposed by efron et al. Use iterative weighted least squares iwls goodness of.
A sound understanding of the multiple regression model will help you to understand these other applications. The procedure most commonly used is the least squares criterion, and the regression line that results from this is called the least squares regression line. Computation of the lasso solutions the computation of the lasso solutions is a quadratic programming problem, and can be tackled by standard numerical analysis algorithms. Least angle regression and its lasso extension involve varying sets of predictors, and we also make use of updating techniques for the qr factorization to accomodate subsets of predictors in linear regression. What is least angle regression and when should it be used. A mathematical introduction to least angle regression. Least angle regression lars, a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. It is motivated by a geometric argument and tracks a path along which the predictors enter successively and the active predictors always maintain the same absolute correlation angle with the residual vector. Least angle regression lar provides answers to these questions, and an e. Least angle regression with discussions article pdf available in the annals of statistics 322 january 2004 with 1,993 reads how we measure reads. Forward selection starts with no variables in the model, and at each step it adds to the model the variable. Least angle regression is a promising technique for variable selection applications, o.
It provides an explanation for the similar behavior of lasso l 1penalized regression and forward stagewise. But the least angle regression procedure is a better approach. I move in least squares direction until another variable is as correlated tim hesterberg, insightful corp. If b is the current stagewise estimate, let cb be the vector of current correlations 1. Package lars february 20, 2015 type package version 1. The result of a regression analysis is an equation that can be used to predict a response from the value of a given predictor. Methodlar specifies least angle regression lar, which is supported in the hpreg procedure. Computation of least angle regression coefficient profiles. Note that the variable most correlated with the residual is equivalently the one that makes the least angle with the residual, whence the name. Jul 22, 2014 least angle regression is a modelbuilding algorithm that considers parsimony as well as prediction accuracy. Third, multiple regression offers our first glimpse into statistical models that use more than two quantitative. It is also an algorithm for efficiently finding all knots in the solution path for the aforementioned this regression procedure, as well as for lasso l1regularized linear regression.
Least angle regression efron, bradley, hastie, trevor, johnstone, iain, and tibshirani, robert, the annals of statistics, 2004. Least angle regression, forward stagewise and the lasso. Least angle regression 5 function in successive small steps. B rst step for least angle regression e point on stagewise path tim hesterberg, insightful corp. Significance testing in nonparametric regression based on the bootstrap delgado, miguel a. Abstract least angle regression is a promising technique for variable selection applications, offering a nice alternative to stepwise regression. One of the most common statistical modeling tools used, regression is a technique that treats one variable as a function of another. While not all steps in the derivation of this line are shown here, the following explanation should provide an intuitive idea of the rationale for the derivation. Second, multiple regression is an extraordinarily versatile calculation, underlying many widely used statistics methods. Least angle regression efron, bradley, hastie, trevor, johnstone, iain, and tibshirani, robert, the annals of statistics, 2004 significance testing in nonparametric regression based on the bootstrap delgado, miguel a.
B rst step for leastangle regression e point on stagewise path tim hesterberg, insightful corp. Linear least squares lls is the least squares approximation of linear functions to data. Least angle regression start with empty set select xj that is most correlated with residuals y. Forward stagewise regression takes a di erent approach among those. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Not only does this algorithm provide a selection method in its own right, but with one additional modification it can be used to efficiently produce lasso solutions.
Weve spent a lot of time discussing simple linear regression, but simple linear regression is, well, simple in the sense that there is usually more than one variable that helps explain the variation in the response variable. Splus and r package for least angle regression tim hesterberg, chris fraley insightful corp. This method is covered in detail by the paper efron, hastie, johnstone and tibshirani 2004, published in the annals of statistics. Robust multivariate least angle regression hassan s. Linear regression here is a version of least squares boosting for multiple linear. Not only does this algorithm provide a selection method in its own right, but with one additional modification, it can be used to efficiently produce lasso solutions. Find the variable x most correlated with the residual. Suppose we expect a response variable to be determined by a linear combination of a subset of potential covariates. Proceed in the direction of xj until another variable xk is equally correlated with residuals choose equiangular direction between xj and xk proceed until third variable enters the active set, etc step is always shorter than in ols p. Least angle regression is a variable selectionshrinkage procedure for highdimensional data. Least angle regression lar least angle regression was introduced by efron et al. Least angle regression is interesting in its own right, its simple structure lending itself to inferential analysis. It provides an explanation for the similar behavior of lasso. Sections 5 and 6 verify the connections stated in section 3.
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