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238
Bagging Predictors
 Machine Learning
, 1996
"... Bagging predictors is a method for generating multiple versions of a predictor and using these to get an aggregated predictor. The aggregation averages over the versions when predicting a numerical outcome and does a plurality vote when predicting a class. The multiple versions are formed by making ..."
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Cited by 2558 (1 self)
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Bagging predictors is a method for generating multiple versions of a predictor and using these to get an aggregated predictor. The aggregation averages over the versions when predicting a numerical outcome and does a plurality vote when predicting a class. The multiple versions are formed by making bootstrap replicates of the learning set and using these as new learning sets. Tests on real and simulated data sets using classification and regression trees and subset selection in linear regression show that bagging can give substantial gains in accuracy. The vital element is the instability of the prediction method. If perturbing the learning set can cause significant changes in the predictor constructed, then bagging can improve accuracy. 1. Introduction A learning set of L consists of data f(y n ; x n ), n = 1; : : : ; Ng where the y's are either class labels or a numerical response. We have a procedure for using this learning set to form a predictor '(x; L)  if the input is x we ...
Regression Shrinkage and Selection Via the Lasso
 Journal of the Royal Statistical Society, Series B
, 1994
"... We propose a new method for estimation in linear models. The "lasso" minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactl ..."
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Cited by 1905 (37 self)
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We propose a new method for estimation in linear models. The "lasso" minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly zero and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and treebased models are briefly described. Keywords: regression, subset selection, shrinkage, quadratic programming. 1 Introduction Consider the usual regression situation: we h...
Generalized Additive Models
, 1990
"... Liklihood based regression models, such as the normal linear regression model and the linear logistic model, assume a linear (or some other parametric) form for the covariate effects. We introduce the Local Scotinq procedure which replaces the liner form C Xjpj by a sum of smooth functions C Sj(Xj)a ..."
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Cited by 1368 (34 self)
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Liklihood based regression models, such as the normal linear regression model and the linear logistic model, assume a linear (or some other parametric) form for the covariate effects. We introduce the Local Scotinq procedure which replaces the liner form C Xjpj by a sum of smooth functions C Sj(Xj)a The Sj(.) ‘s are unspecified functions that are estimated using scatterplot smoothers. The technique is applicable to any likelihoodbased regression model: the class of Generalized Linear Models contains many of these. In this class, the Locul Scoring procedure replaces the linear predictor VI = C Xj@j by the additive predictor C ai ( hence, the name Generalized Additive Modeb. Local Scoring can also be applied to nonstandard models like Cox’s proportional hazards model for survival data. In a number of real data examples, the Local Scoring procedure proves to be useful in uncovering nonlinear covariate effects. It has the advantage of being completely automatic, i.e. no “detective work ” is needed on the part of the statistician. In a further generalization, the technique is modified to estimate the form of the link function for generalized linear models. The Local Scoring procedure is shown to be asymptotically equivalent to Local Likelihood estimation, another technique for estimating smooth covariate functions. They are seen to produce very similar results with real data, with Local Scoring being considerably faster. As a theoretical underpinning, we view Local Scoring and Local Likelihood as empirical maximizers of the ezpected loglikelihood, and this makes clear their connection to standard maximum likelihood estimation. A method for estimating the “degrees of freedom ” of the procedures is also given.
Additive Logistic Regression: a Statistical View of Boosting
 Annals of Statistics
, 1998
"... Boosting (Freund & Schapire 1996, Schapire & Singer 1998) is one of the most important recent developments in classification methodology. The performance of many classification algorithms can often be dramatically improved by sequentially applying them to reweighted versions of the input dat ..."
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Cited by 1250 (21 self)
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Boosting (Freund & Schapire 1996, Schapire & Singer 1998) is one of the most important recent developments in classification methodology. The performance of many classification algorithms can often be dramatically improved by sequentially applying them to reweighted versions of the input data, and taking a weighted majority vote of the sequence of classifiers thereby produced. We show that this seemingly mysterious phenomenon can be understood in terms of well known statistical principles, namely additive modeling and maximum likelihood. For the twoclass problem, boosting can be viewed as an approximation to additive modeling on the logistic scale using maximum Bernoulli likelihood as a criterion. We develop more direct approximations and show that they exhibit nearly identical results to boosting. Direct multiclass generalizations based on multinomial likelihood are derived that exhibit performance comparable to other recently proposed multiclass generalizations of boosting in most...
Greedy Function Approximation: A Gradient Boosting Machine
 Annals of Statistics
, 2000
"... Function approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest{descent minimization. A general gradient{descent \boosting" paradigm is developed for additi ..."
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Cited by 587 (12 self)
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Function approximation is viewed from the perspective of numerical optimization in function space, rather than parameter space. A connection is made between stagewise additive expansions and steepest{descent minimization. A general gradient{descent \boosting" paradigm is developed for additive expansions based on any tting criterion. Specic algorithms are presented for least{squares, least{absolute{deviation, and Huber{M loss functions for regression, and multi{class logistic likelihood for classication. Special enhancements are derived for the particular case where the individual additive components are regression trees, and tools for interpreting such \TreeBoost" models are presented. Gradient boosting of regression trees produces competitive, highly robust, interpretable procedures for both regression and classication, especially appropriate for mining less than clean data. Connections between this approach and the boosting methods of Freund and Shapire 1996, and Frie...
Classification by pairwise coupling
, 1998
"... We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the BradleyTerry method for paired comparisons. We study the nature of the class probability estim ..."
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Cited by 279 (0 self)
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We discuss a strategy for polychotomous classification that involves estimating class probabilities for each pair of classes, and then coupling the estimates together. The coupling model is similar to the BradleyTerry method for paired comparisons. We study the nature of the class probability estimates that arise, and examine the performance of the procedure in real and simulated datasets. Classifiers used include linear discriminants, nearest neighbors, and the support vector machine.
A Comparison of Prediction Accuracy, Complexity, and Training Time of Thirtythree Old and New Classification Algorithms
, 2000
"... . Twentytwo decision tree, nine statistical, and two neural network algorithms are compared on thirtytwo datasets in terms of classication accuracy, training time, and (in the case of trees) number of leaves. Classication accuracy is measured by mean error rate and mean rank of error rate. Both cr ..."
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Cited by 172 (7 self)
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. Twentytwo decision tree, nine statistical, and two neural network algorithms are compared on thirtytwo datasets in terms of classication accuracy, training time, and (in the case of trees) number of leaves. Classication accuracy is measured by mean error rate and mean rank of error rate. Both criteria place a statistical, splinebased, algorithm called Polyclass at the top, although it is not statistically signicantly dierent from twenty other algorithms. Another statistical algorithm, logistic regression, is second with respect to the two accuracy criteria. The most accurate decision tree algorithm is Quest with linear splits, which ranks fourth and fth, respectively. Although splinebased statistical algorithms tend to have good accuracy, they also require relatively long training times. Polyclass, for example, is third last in terms of median training time. It often requires hours of training compared to seconds for other algorithms. The Quest and logistic regression algor...
Discriminant Analysis by Gaussian Mixtures
 Journal of the Royal Statistical Society, Series B
, 1996
"... FisherRao linear discriminant analysis (LDA) is a valuable tool for multigroup classification. LDA is equivalent to maximum likelihood classification assuming Gaussian distributions for each class. In this paper, we fit Gaussian mixtures to each class to facilitate effective classification in nonn ..."
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Cited by 152 (10 self)
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FisherRao linear discriminant analysis (LDA) is a valuable tool for multigroup classification. LDA is equivalent to maximum likelihood classification assuming Gaussian distributions for each class. In this paper, we fit Gaussian mixtures to each class to facilitate effective classification in nonnormal settings, especially when the classes are clustered. Low dimensional views are an important byproduct of LDAour new techniques inherit this feature. We are able to control the withinclass spread of the subclass centers relative to the betweenclass spread. Our technique for fitting these models permits a natural blend with nonparametric versions of LDA. Keywords: Classification, Pattern Recognition, Clustering, Nonparametric, Penalized. 1 Introduction In the generic classification or discrimination problem, the outcome of interest G falls into J unordered classes, which for convenience we denote by the set J = f1; 2; 3; \Delta \Delta \Delta Jg. We wish to build a rule for pred...
Polynomial Splines and Their Tensor Products in Extended Linear Modeling
 Ann. Statist
, 1997
"... ANOVA type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function, or spectral density function. Polynomial splines are used to m ..."
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Cited by 142 (14 self)
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ANOVA type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function, or spectral density function. Polynomial splines are used to model the main effects, and their tensor products are used to model any interaction components that are included. In the special context of survival analysis, the baseline hazard function is modeled and nonproportionality is allowed. In general, the theory involves the L 2 rate of convergence for the fitted model and its components. The methodology involves least squares and maximum likelihood estimation, stepwise addition of basis functions using Rao statistics, stepwise deletion using Wald statistics, and model selection using BIC, crossvalidation or an independent test set. Publically available software, written in C and interfaced to S/SPLUS, is used to apply this methodology to...
Flexible Metric Nearest Neighbor Classification
, 1994
"... The Knearestneighbor decision rule assigns an object of unknown class to the plurality class among the K labeled "training" objects that are closest to it. Closeness is usually defined in terms of a metric distance on the Euclidean space with the input measurement variables as axes. The ..."
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Cited by 123 (2 self)
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The Knearestneighbor decision rule assigns an object of unknown class to the plurality class among the K labeled "training" objects that are closest to it. Closeness is usually defined in terms of a metric distance on the Euclidean space with the input measurement variables as axes. The metric chosen to define this distance can strongly effect performance. An optimal choice depends on the problem at hand as characterized by the respective class distributions on the input measurement space, and within a given problem, on the location of the unknown object in that space. In this paper new types of Knearestneighbor procedures are described that estimate the local relevance of each input variable, or their linear combinations, for each individual point to be classified. This information is then used to separately customize the metric used to define distance from that object in finding its nearest neighbors. These procedures are a hybrid between regular Knearestneighbor methods and treestructured recursive partitioning techniques popular in statistics and machine learning.