## The Annals of Statistics

- Ann. Statist.
- Volume 28, Number 2 (2000), 337-407.

### Additive logistic regression: a statistical view of boosting (With discussion and a rejoinder by the authors)

Jerome Friedman, Trevor Hastie, and Robert Tibshirani

#### Abstract

Boosting is one of the most important recent developments in classification methodology. Boosting works by sequentially applying a classification algorithm to reweighted versions of the training data and then taking a weighted majority vote of the sequence of classifiers thus produced. For many classification algorithms, this simple strategy results in dramatic improvements in performance. 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 two-class 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 situations, and far superior in some. We suggest a minor modification to boosting that can reduce computation, often by factors of 10 to 50. Finally, we apply these insights to produce an alternative formulation of boosting decision trees. This approach, based on best-first truncated tree induction, often leads to better performance, and can provide interpretable descriptions of the aggregate decision rule. It is also much faster computationally, making it more suitable to large-scale data mining applications.

#### Article information

**Source**

Ann. Statist. Volume 28, Number 2 (2000), 337-407.

**Dates**

First available in Project Euclid: 15 March 2002

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1016218223

**Mathematical Reviews number (MathSciNet)**

MR1790002

**Digital Object Identifier**

doi:10.1214/aos/1016218223

**Zentralblatt MATH identifier**

01828945

**Subjects**

Primary: 62G05: Estimation 62G07: Density estimation 68T10: Pattern recognition, speech recognition {For cluster analysis, see 62H30} 68T05: Learning and adaptive systems [See also 68Q32, 91E40]

**Keywords**

classification tree nonparametric estimation stagewise fitting machine learning

#### Citation

Friedman, Jerome; Hastie, Trevor; Tibshirani, Robert. Additive logistic regression: a statistical view of boosting (With discussion and a rejoinder by the authors). The Annals of Statistics 28 (2000), no. 2, 337--407. doi:10.1214/aos/1016218223. http://projecteuclid.org/euclid.aos/1016218223.