Using an ensemble of classifiers, instead of a single classifier, can lead to improved generalization. The gains obtained by combining however, are often affected more by the selection of what is presented to the combiner, than by the actual combining method that is chosen. In this paper we focus on data selection and classifier training methods, in order to "prepare" classifiers for combining. We review a combining framework for classification problems that quantifies the need for reducing the correlation among individual classifiers. Then, we discuss several methods that make the classifiers in an ensemble more complementary. Experimental results are provided to illustrate the benefits and pitfalls of reducing the correlation among classifiers, especially when the training data is in limited supply. 2 1 Introduction A classifier's ability to meaningfully respond to novel patterns, or generalize, is perhaps its most important property (Levin et al., 1990; Wolpert, 1990). In...

This paper reviews research on combining artificial neural nets, and provides an overview of, and an introduction to, the papers contained this Special Issue, and its companion (Connection Science, 9, 1). Two main approaches, ensemble-based, and modular, are identified and considered. An ensemble, or committee, is made up of a set of nets, each of which is a general function approximator. The members of the ensemble are combined in order to obtain better generalisation performance than would be achieved by any of the individual nets. The main issues considered here under the heading of ensemble-based approaches, are (a) how to combine the outputs of the ensemble members (b) how to create candidate ensemble members and (c) which methods lead to the most effective ensembles? Under the heading of modular approaches we begin by considering a divide-and-conquer approach by which a function is automatically decomposed into a number of subfunctions which are treated by specialis...

Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers. This chapter provides an analytical framework to quantify the improvements in classification results due to combining. The results apply to both linear combiners and order statistics combiners. We first show that to a first order approximation, the error rate obtained over and above the Bayes error rate, is directly proportional to the variance of the actual decision boundaries around the Bayes optimum boundary. Combining classifiers in output space reduces this variance, and hence reduces the "added" error. If N unbiased classifiers are combined by simple averaging, the added error rate can be reduced by a factor of N if the individual errors in approximating the decision boundaries are uncorrelated. Expressions are then derived for linear combiners which are biased or correlated, and the effect of output correlations on ensemble performance is quantified. For order statistics based non-linear combiners, we derive expressions that indicate how much the median, the maximum and in general the ith order statistic can improve classifier performance. The analysis presented here facilitates the understanding of the relationships among error rates, classifier boundary distributions, and combining in output space. Experimental results on several public domain data sets are provided to illustrate the benefits of combining and to support the analytical results.

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: Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers. This paper provides an analytical framework to quantify the improvements in classification results due to combining. The results apply to both linear combiners and the order statistics combiners introduced in this paper. We show that combining networks in output space reduces the variance of the actual decision region boundaries around the optimum boundary. For linear combiners, we show that in the absence of classifier bias, the added classification error is proportional to the boundary variance. For non-linear combiners, we show analytically that the selection of the median, the maximum and in general the ith order statistic improves classifier performance. The analysis presented here facilitates the understanding of the relationships among error rates, classifier boundary distributions...

Several researchers have experimentally shown that substantial improvements can be obtained in difficult pattern recognition problems by combining or integrating the outputs of multiple classifiers. This paper summarizes our recent theoretical results that quantify the improvements due to multiple classifier combining. Furthermore, we present an extension of this theory that leads to an estimate of the Bayes error rate. Practical aspects such as expressing the confidences in decisions and determining the best data partition/classifier selection are also discussed. Keywords: Linear combining, order statistics combining, Bayes error, error correlation, error reduction, ensemble networks, performance limits. Introduction Given infinite training data, consistent classifiers approximate the Bayesian decision boundaries to arbitrary precision, therefore providing similar generalizations (Geman, Bienenstock, & Doursat 1992). However, often only a limited portion of the pattern space is avai...

Integrating the outputs of multiple classifiers via combiners or meta-learners has led to substantial improvements in several difficult pattern recognition problems. In this article we investigate a family of combiners based on order statistics, for robust handling of situations where there are large discrepancies in performance of individual classifiers. Based on a mathematical modeling of how the decision boundaries are affected by order statistic combiners, we derive expressions for the reductions in error expected when simple output combination methods based on the the median, the maximum and in general, the i th order statistic, are used. Furthermore, we analyze the trim and spread combiners, both based on linear combinations of the ordered classifier outputs, and show that in the presence of uneven classifier performance, they often provide substantial gains over both linear and simple order statistics combiners. Experimental results on both real world data and standard public domain data sets corroborate these findings.

Broad classes of statistical classification algorithms have been developed and applied successfully to a wide range of real world domains. In general, ensuring that the particular classification algorithm matches the properties of the data is crucial in providing results that meet the needs of the particular application domain. One way in which the impact of this algorithm/application match can be alleviated is by using ensembles of classifiers, where a variety of classifiers (either different types of classifiers or different instantiations of the same classifier) are pooled before a final classification decision is made. Intuitively, classifier ensembles allow the different needs of a difficult problem to be handled by classifiers suited to those particular needs. Mathematically, classifier ensembles provide an extra degree of freedom in the classical bias/variance tradeoff, allowing solutions that would be difficult (if not impossible) to reach with only a single classifier. Because of these advantages, classifier ensembles have been applied to many difficult real world problems. In this paper, we survey select applications of ensemble methods to problems that have historically been most representative of the difficulties in classification. In particular, we survey applications of ensemble methods to remote sensing, person recognition, one vs. all recognition, and medicine.

Combining the outputs of multiple neural networks has led to substantial improvements in several difficult pattern recognition problems. In this article, we introduce and investigate robust combiners, a family of classifiers based on order statistics. We focus our study to the analysis of the decision boundaries, and how these boundaries are affected by order statistics combiners. In particular, we show that using the ith order statistic, or a linear combination of the ordered classifier outputs is quite beneficial in the presence of outliers or uneven classifier performance. Experimental results on several public domain data sets corroborate these findings. I. Introduction In recent years, a great deal of attention has been focused on pooling as a means to improve the generalization ability of neural networks [22]. Approaches to pooling classifiers can be separated into two main categories: simple combiners, e.g., voting [4], [5] or averaging [16], and computationally expensive combi...

Integrating the outputs of multiple classifiers via combiners or meta-learners has led to substantial improvements in several difficult pattern recognition problems. In this article we investigate a family of combiners based on order statistics, for robust handling of situations where there are large discrepancies in performance of individual classifiers. Based on a mathematical modeling of how the decision boundaries are affected by order statistic combiners, we derive expressions for the reductions in error expected when simple output combination methods based on the the median, the maximum and in general, the ith order statistic, are used. Furthermore, we analyze the trim and spread combiners, both based on linear combinations of the ordered classifier outputs, and show that in the presence of uneven classifier performance, they often provide substantial gains over both linear and simple order statistics combiners. Experimental results on both real world data and standard public domain data sets corroborate these findings.