《模式识别》课程教学资源（书籍文献）Neural Networks for Classification - A Survey

EEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS-PART C:APPLICATIONS AND REVIEWS,VOL.30,NO.4,NOVEMBER 2000 451 Neural Networks for Classification:A Survey Guoqiang Peter Zhang Abstract-Classification is one of the most active research and [79.Since any classification procedure seeks a functional application areas of neural networks.The literature is vast and relationship between the group membership and the attributes growing.This paper summarizes the some of the most important developments in neural network classification research.Specifi- of the object,accurate identification of this underlying function cally,the issues of posterior probability estimation,the link be- is doubtlessly important.Third,neural networks are nonlinear tween neural and conventional classifiers,learning and general- models,which makes them flexible in modeling real world ization tradeoff in classification,the feature variable selection,as complex relationships.Finally,neural networks are able to well as the effect of misclassification costs are examined.Our pur- estimate the posterior probabilities,which provides the basis pose is to provide a synthesis of the published research in this area for establishing classification rule and performing statistical and stimulate further research interests and efforts in the identi- fied topics. analysis [138]. Index Terms-Bayesian classifier,classification,ensemble On the other hand.the effectiveness of neural network clas- methods,feature variable selection,learning and generalization, sification has been tested empirically.Neural networks have misclassification costs,neural networks. been successfully applied to a variety of real world classification tasks in industry,business and science [186.Applications in- I.INTRODUCTION clude bankruptcy prediction [2],[96],[101],[167],[187],[195], handwriting recognition [61],[92],[98],[100],[113],speech YLASSIFICATION is one of the most frequently en- recognition[25l,[106],product inspection[9刀，[130]，fault de- countered decision making tasks of human activity.A tection [11],[80],medical diagnosis [19],[20],[30],[31],and classification problem occurs when an object needs to be bond rating [44],[163],[174].A number of performance com- assigned into a predefined group or class based on a number parisons between neural and conventional classifiers have been of observed attributes related to that object.Many problems in made by many studies [36],[82],[115].In addition,several business,science,industry,and medicine can be treated as clas- computer experimental evaluations of neural networks for clas- sification problems.Examples include bankruptcy prediction, sification problems have been conducted under a variety of con- credit scoring,medical diagnosis,quality control,handwritten ditions[127刀，[161]. character recognition,and speech recognition. Traditional statistical classification procedures such as dis- Although significant progress has been made in classification criminant analysis are built on the Bayesian decision theory related areas of neural networks,a number of issues in applying neural networks still remain and have not been solved success- [42].In these procedures,an underlying probability model must be assumed in order to calculate the posterior probability upon fully or completely.In this paper,some theoretical as well as empirical issues of neural networks are reviewed and discussed. which the classification decision is made.One major limitation of the statistical models is that they work well only when the The vast research topics and extensive literature makes it impos- underlying assumptions are satisfied.The effectiveness of these sible for one review to cover all of the work in the filed.This re- view aims to provide a summary of the most important advances methods depends to a large extent on the various assumptions or in neural network classification.The current research status and conditions under which the models are developed.Users must have a good knowledge of both data properties and model capa- issues as well as the future research opportunities are also dis- cussed.Although many types of neural networks can be used bilities before the models can be successfully applied. Neural networks have emerged as an important tool for for classification purposes [105],our focus nonetheless is on the feedforward multilayer networks or multilayer perceptrons classification.The recent vast research activities in neural (MLPs)which are the most widely studied and used neural net- classification have established that neural networks are a promising alternative to various conventional classification work classifiers.Most of the issues discussed in the paper can methods.The advantage of neural networks lies in the fol- also apply to other neural network models. lowing theoretical aspects.First,neural networks are data The overall organization of the paper is as follows.After the driven self-adaptive methods in that they can adjust themselves introduction,we present fundamental issues of neural classifica- tion in Section II,including the Bayesian classification theory, to the data without any explicit specification of functional or the role of posterior probability in classification,posterior prob- distributional form for the underlying model.Second,they are ability estimation via neural networks,and the relationships be- universal functional approximators in that neural networks can tween neural networks and the conventional classifiers.Sec- approximate any function with arbitrary accuracy [37],[78], tion III examines theoretical issues of learning and generaliza- tion in classification as well as various practical approaches to Manuscript received July 28,1999:revised July 6,2000. improving neural classifier performance in learning and gener- G.P.Zhang is with the J.Mack Robinson College of Business,Georgia State University,Atlanta,GA 30303 USA(e-mail:gpzhang@gsu.edu). alization.Feature variable selection and the effect of misclassi- Publisher Item Identifier S 1094-6977(00)11206-4. fication costs-two important problems unique to classification 1094-6977/00S10.00©20001EEE

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 4, NOVEMBER 2000 451 Neural Networks for Classification: A Survey Guoqiang Peter Zhang Abstract—Classification is one of the most active research and application areas of neural networks. The literature is vast and growing. This paper summarizes the some of the most important developments in neural network classification research. Specifi￾cally, the issues of posterior probability estimation, the link be￾tween neural and conventional classifiers, learning and general￾ization tradeoff in classification, the feature variable selection, as well as the effect of misclassification costs are examined. Our pur￾pose is to provide a synthesis of the published research in this area and stimulate further research interests and efforts in the identi￾fied topics. Index Terms—Bayesian classifier, classification, ensemble methods, feature variable selection, learning and generalization, misclassification costs, neural networks. I. INTRODUCTION CLASSIFICATION is one of the most frequently en￾countered decision making tasks of human activity. A classification problem occurs when an object needs to be assigned into a predefined group or class based on a number of observed attributes related to that object. Many problems in business, science, industry, and medicine can be treated as clas￾sification problems. Examples include bankruptcy prediction, credit scoring, medical diagnosis, quality control, handwritten character recognition, and speech recognition. Traditional statistical classification procedures such as dis￾criminant analysis are built on the Bayesian decision theory [42]. In these procedures, an underlying probability model must be assumed in order to calculate the posterior probability upon which the classification decision is made. One major limitation of the statistical models is that they work well only when the underlying assumptions are satisfied. The effectiveness of these methods depends to a large extent on the various assumptions or conditions under which the models are developed. Users must have a good knowledge of both data properties and model capa￾bilities before the models can be successfully applied. Neural networks have emerged as an important tool for classification. The recent vast research activities in neural classification have established that neural networks are a promising alternative to various conventional classification methods. The advantage of neural networks lies in the fol￾lowing theoretical aspects. First, neural networks are data driven self-adaptive methods in that they can adjust themselves to the data without any explicit specification of functional or distributional form for the underlying model. Second, they are universal functional approximators in that neural networks can approximate any function with arbitrary accuracy [37], [78], Manuscript received July 28, 1999; revised July 6, 2000. G. P. Zhang is with the J. Mack Robinson College of Business, Georgia State University, Atlanta, GA 30303 USA (e-mail: gpzhang@gsu.edu). Publisher Item Identifier S 1094-6977(00)11206-4. [79]. Since any classification procedure seeks a functional relationship between the group membership and the attributes of the object, accurate identification of this underlying function is doubtlessly important. Third, neural networks are nonlinear models, which makes them flexible in modeling real world complex relationships. Finally, neural networks are able to estimate the posterior probabilities, which provides the basis for establishing classification rule and performing statistical analysis [138]. On the other hand, the effectiveness of neural network clas￾sification has been tested empirically. Neural networks have been successfully applied to a variety of real world classification tasks in industry, business and science [186]. Applications in￾clude bankruptcy prediction [2], [96], [101], [167], [187], [195], handwriting recognition [61], [92], [98], [100], [113], speech recognition [25], [106], product inspection [97], [130], fault de￾tection [11], [80], medical diagnosis [19], [20], [30], [31], and bond rating [44], [163], [174]. A number of performance com￾parisons between neural and conventional classifiers have been made by many studies [36], [82], [115]. In addition, several computer experimental evaluations of neural networks for clas￾sification problems have been conducted under a variety of con￾ditions [127], [161]. Although significant progress has been made in classification related areas of neural networks, a number of issues in applying neural networks still remain and have not been solved success￾fully or completely. In this paper, some theoretical as well as empirical issues of neural networks are reviewed and discussed. The vast research topics and extensive literature makes it impos￾sible for one review to cover all of the work in the filed. This re￾view aims to provide a summary of the most important advances in neural network classification. The current research status and issues as well as the future research opportunities are also dis￾cussed. Although many types of neural networks can be used for classification purposes [105], our focus nonetheless is on the feedforward multilayer networks or multilayer perceptrons (MLPs) which are the most widely studied and used neural net￾work classifiers. Most of the issues discussed in the paper can also apply to other neural network models. The overall organization of the paper is as follows. After the introduction, we present fundamental issues of neural classifica￾tion in Section II, including the Bayesian classification theory, the role of posterior probability in classification, posterior prob￾ability estimation via neural networks, and the relationships be￾tween neural networks and the conventional classifiers. Sec￾tion III examines theoretical issues of learning and generaliza￾tion in classification as well as various practical approaches to improving neural classifier performance in learning and gener￾alization. Feature variable selection and the effect of misclassi￾fication costs—two important problems unique to classification 1094–6977/00$10.00 © 2000 IEEE

452 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS-PART C:APPLICATIONS AND REVIEWS,VOL.30,NO.4,NOVEMBER 2000 problems-are discussed in Sections IV and V,respectively.Fi-C(x)is also known as the conditional risk function.The op- nally,Section VI concludes the paper. timal Bayesian decision rule that minimizes the overall expected cost is II.NEURAL NETWORKS AND TRADITIONAL CLASSIFIERS A.Bayesian Classification Theory Decide for x if). (5) Bayesian decision theory is the basis of statistical classifi- cation methods [42].It provides the fundamental probability When the misclassification costs are equal (0-1 cost function), model for well-known classification procedures such as the sta- then we have the special case (2)of the Bayesian classification tistical discriminant analysis. rule.Note the role of posterior probabilities in the decision rules Consider a general M-group classification problem in which (2)and(5). each object has an associated attribute vector x ofd dimensions. From(1)and (4)and note that the denominator is common to Let w denote the membership variable that takes a value of wi all classes,Bayesian decision rule (5)is equivalent to:Decidew M if an object is belong to group j.Define Pwi)as the prior for x iff()is the minimum.Consider probability of group j and f(xwias the probability density the special two-group case with two classes of wl and w2.We function.According to the Bayes rule should assign x to class 1 if P(wjlx)=f)P() (1) c21(xX)P(w2)f(x|2)<C2(X)P(w1)f(x|w) f(x or where P(wix)is the posterior probability of group j and f(x)is the probability density function:f(x)= fo u nOP) f(xwj)P(wj). (6) f(x w2)c12()P(w1) Now suppose that an object with a particular feature vector x is observed and a decision is to be made about its group mem- Expression(6)shows the interaction of prior probabilities and bership.The probability of classification error is misclassification cost in defining the classification rule,which can be exploited in building practical classification models to P(Error|x)=∑P(,lx) alleviate the difficulty in estimation of misclassification costs. 千) =1-P(wj x)if we decide wj. B.Posterior Probability Estimation via Neural Networks Hence if the purpose is to minimize the probability of total clas- In classification problems,neural networks provide direct es- sification error(misclassification rate),then we have the fol- timation of the posterior probabilities [58],[138],[156],[178]. lowing widely used Bayesian classification rule The importance of this capability is summarized by Richard and Lippmann [138]: Decide%forxiP代a冈==lMyP氏alX. (2) "Interpretation of network outputs as Bayesian probabilities allows outputs from multiple networks to be combined for This simple rule is the basis for other statistical classifiers. higher level decision making,simplifies creation of rejection For example,linear and quadratic discriminant functions can be thresholds,makes it possible to compensate for difference derived with the assumption of the multivariate normal distribu- between pattern class probabilities in training and test data, tion for the conditional density f(x of attribute vector x. allows output to be used to minimize alternative risk functions. There are two problems in applying the simple Bayes decision and suggests alternative measures of network performance." rule(2).First,in most practical situations,the density functions A neural network for a classification problem can be viewed are not known or can not be assumed to be normal and there- as a mapping function,:Rd-RM,where d-dimensional fore the posterior probabilities can not be determined directly. input x is submitted to the network and an M-vectored network Second,by using(2),the decision goal is simply to minimize output y is obtained to make the classification decision.The net- the probability of misclassifying a new object.In this way,we work is typically built such that an overall error measure such as are indifferent with regard to the consequences of misclassifica- the mean squared errors(MSE)is minimized.From the famous tion errors.In other words,we assume that the misclassification least squares estimation theory in statistics(see [126]),the map- costs for different groups are equal.This may not be the case for ping function x-y which minimizes the expected squared many real world applications where the cost of a wrong assign- error ment is quite different for different groups. If we can assign a cost to a misclassification error,we may Ely-F(x)2 (7) use that information to improve our decision.Let ci(x)be the cost of misclassifying x to group i when it actually belongs to is the conditional expectation of y given x group j.The expected cost associated with assigning x to group i IS (x)=y|. (8) M Ci()=cj(x)P(),i=1,2....,M.(4)In the classification problem,the desired output y is i=1 a vector of binary values and is the jth basis vector

452 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 4, NOVEMBER 2000 problems—are discussed in Sections IV and V, respectively. Fi￾nally, Section VI concludes the paper. II. NEURAL NETWORKS AND TRADITIONAL CLASSIFIERS A. Bayesian Classification Theory Bayesian decision theory is the basis of statistical classifi￾cation methods [42]. It provides the fundamental probability model for well-known classification procedures such as the sta￾tistical discriminant analysis. Consider a general -group classification problem in which each object has an associated attribute vector of dimensions. Let denote the membership variable that takes a value of if an object is belong to group . Define as the prior probability of group and as the probability density function. According to the Bayes rule (1) where is the posterior probability of group and is the probability density function: . Now suppose that an object with a particular feature vector is observed and a decision is to be made about its group mem￾bership. The probability of classification error is Error if we decide Hence if the purpose is to minimize the probability of total clas￾sification error (misclassification rate), then we have the fol￾lowing widely used Bayesian classification rule Decide for if (2) This simple rule is the basis for other statistical classifiers. For example, linear and quadratic discriminant functions can be derived with the assumption of the multivariate normal distribu￾tion for the conditional density of attribute vector . There are two problems in applying the simple Bayes decision rule (2). First, in most practical situations, the density functions are not known or can not be assumed to be normal and there￾fore the posterior probabilities can not be determined directly. Second, by using (2), the decision goal is simply to minimize the probability of misclassifying a new object. In this way, we are indifferent with regard to the consequences of misclassifica￾tion errors. In other words, we assume that the misclassification costs for different groups are equal. This may not be the case for many real world applications where the cost of a wrong assign￾ment is quite different for different groups. If we can assign a cost to a misclassification error, we may use that information to improve our decision. Let be the cost of misclassifying to group when it actually belongs to group . The expected cost associated with assigning to group is (4) is also known as the conditional risk function. The op￾timal Bayesian decision rule that minimizes the overall expected cost is Decide for if (5) When the misclassification costs are equal (0–1 cost function), then we have the special case (2) of the Bayesian classification rule. Note the role of posterior probabilities in the decision rules (2) and (5). From (1) and (4) and note that the denominator is common to all classes, Bayesian decision rule (5) is equivalent to: Decide for if is the minimum. Consider the special two-group case with two classes of and . We should assign to class 1 if or (6) Expression (6) shows the interaction of prior probabilities and misclassification cost in defining the classification rule, which can be exploited in building practical classification models to alleviate the difficulty in estimation of misclassification costs. B. Posterior Probability Estimation via Neural Networks In classification problems, neural networks provide direct es￾timation of the posterior probabilities [58], [138], [156], [178]. The importance of this capability is summarized by Richard and Lippmann [138]: “Interpretation of network outputs as Bayesian probabilities allows outputs from multiple networks to be combined for higher level decision making, simplifies creation of rejection thresholds, makes it possible to compensate for difference between pattern class probabilities in training and test data, allows output to be used to minimize alternative risk functions, and suggests alternative measures of network performance.” A neural network for a classification problem can be viewed as a mapping function, , where -dimensional input is submitted to the network and an -vectored network output is obtained to make the classification decision. The net￾work is typically built such that an overall error measure such as the mean squared errors (MSE) is minimized. From the famous least squares estimation theory in statistics (see [126]), the map￾ping function which minimizes the expected squared error (7) is the conditional expectation of given (8) In the classification problem, the desired output is a vector of binary values and is the th basis vector

ZHANG:NEURAL NETWORKS FOR CLASSIFICATION 453 ej=(0,...,0,1,0,...,0t if x group j.Hence the networks and statistical classifiers.The direct comparison be- ith element of F(x)is given by tween them may not be possible since neural networks are non- linear model-free method while statistical methods are basically F(x)=Elvilx linear and model based. =1·P(=1|x)+0·P(物=0|x) By appropriate coding of the desired output membership =P(=1|x) values,we may let neural networks directly model some dis- criminant functions.For example,in a two-group classification =P(wjx). (9) problem,if the desired output is coded as 1 if the object is from class I and-1 if it is from class 2.Then,from(9)the neural That is,the least squares estimate for the mapping function in a network estimates the following discriminant function: classification problem is exactly the posterior probability. Neural networks are universal approximators [37]and in theory can approximate any function arbitrarily closely. g(x)=P(w1 x)-P(w2 x). (11) However,the mapping function represented by a network is not perfect due to the local minima problem,suboptimal network The discriminating rule is simply:assign x to w ifg(x)>0 or architecture and the finite sample data in neural network w2 if g(x)<0.Any monotone increasing function of the poste- training.Therefore,it is clear that neural networks actually rior probability can be used to replace the posterior probability in(1 1)to form a different discriminant function but essentially provide estimates of the posterior probabilities. The mean squared error function(7)can be derived [143], the same classification rule [83]as As the statistical counterpart of neural networks,discriminant analysis is a well-known supervised classifier.Gallinari et al. [54]describe a general framework to establish the link between MSE [F(x)-P((x)dx discriminant analysis and neural network models.They find that in quite general conditions the hidden layers of an MLP project the input data onto different clusters in a way that these clus- P(wil x(1-P(wj x))f(x)dx.(10) ters can be further aggregated into different classes.For linear MLPs,the projection performed by the hidden layer is shown theoretically equivalent to the linear discriminant analysis.The The second term of the right-hand side is called the approxima- nonlinear MLPs,on the other hand,have been demonstrated tion error [14]and is independent of neural networks.It reflects through experiments the capability in performing more pow- the inherent irreducible error due to randomness of the data.The erful nonlinear discriminant analysis.Their work helps under- first term termed as the estimation error is affected by the effec- stand the underlying function and behavior of the hidden layer tiveness of neural network mapping.Theoretically speaking,it for classification problems and also explains why the neural net- may need a large network as well as large sample data in order works in principle can provide superior performance over linear to get satisfactory approximation.For example,Funahashi [53] discriminant analysis.The discriminant feature extraction by shows that for the two-group d-dimensional Gaussian classifi- the network with nonlinear hidden nodes has also been demon- cation problem,neural networks with at least 2d hidden nodes strated in Asoh and Otsu [6]and Webb and Lowe [181].Lim. have the capability to approximate the posterior probability with Alder and Hadingham [103]show that neural networks can per- arbitrary accuracy when infinite data is available and the training form quadratic discriminant analysis. proceeds ideally.Empirically,it is found that sample size is crit- Raudys [134],[135]presents a detailed analysis of nonlinear ical in learning but the number of hidden nodes may not be so single layer perceptron(SLP).He shows that during the adap- important [83],[138]. tive training process of SLP,by purposefully controlling the That the outputs of neural networks are least square estimates SLP classifier complexity through adjusting the target values, of the Bayesian a posteriori probabilities is also valid for other learning-steps,number of iterations and using regularization types of cost or error function such as the cross entropy function terms,the decision boundaries of SLP classifiers are equivalent [63],[138].The cross entropy function can be a more appro- or close to those of seven statistical classifiers.These statistical priate criterion than the squared error cost function in training classifiers include the Enclidean distance classifier,the Fisher neural networks for classification problems because of their bi- linear discriminant function,the Fisher linear discriminant nary output characteristic [144].Improved performance and re- function with pseudo-inversion of the covariance matrix,the duced training time have been reported with the cross entropy generalized Fisher linear discriminant function,the regularized function [75],[77].Miyake and Kanaya [116]show that neural linear discriminant analysis,the minimum empirical error networks trained with a generalized mean-squared error objec- classifier,and the maximum margin classifier [134].Kanaya tive function can yield the optimal Bayes rule. and Miyake [88]and Miyake and Kanaya [116]also illustrate theoretically and empirically the link between neural networks C.Neural Networks and Conventional Classifiers and the optimal Bayes rule in statistical decision problems. Statistical pattern classifiers are based on the Bayes decision Logistic regression is another important classification tool. theory in which posterior probabilities play a central role.The In fact,it is a standard statistical approach used in medical fact that neural networks can in fact provide estimates of pos- diagnosis and epidemiologic studies [91].Logistic regression terior probability implicitly establishes the link between neural is often preferred over discriminant analysis in practice [65]

ZHANG: NEURAL NETWORKS FOR CLASSIFICATION 453 if group . Hence the th element of is given by (9) That is, the least squares estimate for the mapping function in a classification problem is exactly the posterior probability. Neural networks are universal approximators [37] and in theory can approximate any function arbitrarily closely. However, the mapping function represented by a network is not perfect due to the local minima problem, suboptimal network architecture and the finite sample data in neural network training. Therefore, it is clear that neural networks actually provide estimates of the posterior probabilities. The mean squared error function (7) can be derived [143], [83] as (10) The second term of the right-hand side is called the approxima￾tion error [14] and is independent of neural networks. It reflects the inherent irreducible error due to randomness of the data. The first term termed as the estimation error is affected by the effec￾tiveness of neural network mapping. Theoretically speaking, it may need a large network as well as large sample data in order to get satisfactory approximation. For example, Funahashi [53] shows that for the two-group -dimensional Gaussian classifi￾cation problem, neural networks with at least hidden nodes have the capability to approximate the posterior probability with arbitrary accuracy when infinite data is available and the training proceeds ideally. Empirically, it is found that sample size is crit￾ical in learning but the number of hidden nodes may not be so important [83], [138]. That the outputs of neural networks are least square estimates of the Bayesian a posteriori probabilities is also valid for other types of cost or error function such as the cross entropy function [63], [138]. The cross entropy function can be a more appro￾priate criterion than the squared error cost function in training neural networks for classification problems because of their bi￾nary output characteristic [144]. Improved performance and re￾duced training time have been reported with the cross entropy function [75], [77]. Miyake and Kanaya [116] show that neural networks trained with a generalized mean-squared error objec￾tive function can yield the optimal Bayes rule. C. Neural Networks and Conventional Classifiers Statistical pattern classifiers are based on the Bayes decision theory in which posterior probabilities play a central role. The fact that neural networks can in fact provide estimates of pos￾terior probability implicitly establishes the link between neural networks and statistical classifiers. The direct comparison be￾tween them may not be possible since neural networks are non￾linear model-free method while statistical methods are basically linear and model based. By appropriate coding of the desired output membership values, we may let neural networks directly model some dis￾criminant functions. For example, in a two-group classification problem, if the desired output is coded as 1 if the object is from class 1 and if it is from class 2. Then, from (9) the neural network estimates the following discriminant function: (11) The discriminating rule is simply: assign to if or if . Any monotone increasing function of the poste￾rior probability can be used to replace the posterior probability in (11) to form a different discriminant function but essentially the same classification rule. As the statistical counterpart of neural networks, discriminant analysis is a well-known supervised classifier. Gallinari et al. [54] describe a general framework to establish the link between discriminant analysis and neural network models. They find that in quite general conditions the hidden layers of an MLP project the input data onto different clusters in a way that these clus￾ters can be further aggregated into different classes. For linear MLPs, the projection performed by the hidden layer is shown theoretically equivalent to the linear discriminant analysis. The nonlinear MLPs, on the other hand, have been demonstrated through experiments the capability in performing more pow￾erful nonlinear discriminant analysis. Their work helps under￾stand the underlying function and behavior of the hidden layer for classification problems and also explains why the neural net￾works in principle can provide superior performance over linear discriminant analysis. The discriminant feature extraction by the network with nonlinear hidden nodes has also been demon￾strated in Asoh and Otsu [6] and Webb and Lowe [181]. Lim, Alder and Hadingham [103] show that neural networks can per￾form quadratic discriminant analysis. Raudys [134], [135] presents a detailed analysis of nonlinear single layer perceptron (SLP). He shows that during the adap￾tive training process of SLP, by purposefully controlling the SLP classifier complexity through adjusting the target values, learning-steps, number of iterations and using regularization terms, the decision boundaries of SLP classifiers are equivalent or close to those of seven statistical classifiers. These statistical classifiers include the Enclidean distance classifier, the Fisher linear discriminant function, the Fisher linear discriminant function with pseudo-inversion of the covariance matrix, the generalized Fisher linear discriminant function, the regularized linear discriminant analysis, the minimum empirical error classifier, and the maximum margin classifier [134]. Kanaya and Miyake [88] and Miyake and Kanaya [116] also illustrate theoretically and empirically the link between neural networks and the optimal Bayes rule in statistical decision problems. Logistic regression is another important classification tool. In fact, it is a standard statistical approach used in medical diagnosis and epidemiologic studies [91]. Logistic regression is often preferred over discriminant analysis in practice [65]

454 IEEE TRANSACTIONS ON SYSTEMS,MAN,AND CYBERNETICS-PART C:APPLICATIONS AND REVIEWS,VOL.30,NO.4,NOVEMBER 2000 [132].In addition,the model can be interpreted as posterior that both underfitting and overfitting will affect generalization probability or odds ratio.It is a simple fact that when the capability ofa model.Therefore a model should be built in such logistic transfer function is used for the output nodes,simple a way that only the underlying systematic pattern of the popu- neural networks without hidden layers are identical to logistic lation is learned and represented by the model. regression models.Another connection is that the maximum The underfitting and overfitting phenomena in many data likelihood function of logistic regression is essentially the modeling procedures can be well analyzed through the cross-entropy cost function which is often used in training well-known bias-plus-variance decomposition of the prediction neural network classifiers.Schumacher et al.[149]make a error.In this section,the basic concepts of bias and variance detailed comparison between neural networks and logistic as well as their connection to neural network classifiers are regression.They find that the added modeling flexibility of discussed.Then the methods to improve learning and gener- neural networks due to hidden layers does not automatically alization ability through bias and/or variance reductions are guarantee their superiority over logistic regression because of reviewed. the possible overfitting and other inherent problems with neural networks [176]. A.Bias and Variance Composition of the Prediction Error Links between neural and other conventional classifiers have Geman et al.[57]give a thorough analysis of the relationship been illustrated by[32],[33],[741,[1391,[1401,[151],[175]. between learning and generalization in neural networks based Ripley [139],[140]empirically compares neural networks with on the concepts of model bias and model variance.A prespec- various classifiers such as classification tree,projection pursuit ified model which is less dependent on the data may misrepre- regression,linear vector quantization,multivariate adaptive re- sent the true functional relationship and have a large bias.On gression splines and nearest neighbor methods. the other hand,a model-free or data-driven model may be too A large number of studies have been devoted to empirical dependent on the specific data and have a large variance.Bias comparisons between neural and conventional classifiers.The and variance are often incompatible.With a fixed data set,the most comprehensive one can be found in Michie et al.[115] effort of reducing one will inevitably cause the other increasing. which reports a large-scale comparative study-the StatLog A good tradeoff between model bias and model variance is nec- project.In this project,three general classification approaches essary and desired in building a useful neural network classifier. of neural networks,statistical classifiers and machine learning Without loss of generality,consider a two-group classifica- with 23 methods are compared using more than 20 different real tion problem in which the binary output variable yE t0,1}is data sets.Their general conclusion is that no single classifier related to a set of input variables(feature vector)x by is the best for all data sets although the feedforward neural networks do have good performance over a wide range of prob- y=F(x)+E lems.Neural networks have also been compared with decision trees [28],[36],[66],[104],[155],discriminant analysis [36], where x)is the target or underlying function and e is assumed [127],[146],[161],[193],CART [7],[40],k-nearest-neighbor to be a zero-mean random variable.From(8)and(9),the target [82],[127],and linear programming method [127]. function is the conditional expectation ofy givenx,that is FX=E(y x=Pw1 x. (12) III.LEARNING AND GENERALIZATION Given a particular training data set Dy of size N,the goal of Learning and generalization is perhaps the most important modeling is to find an estimate,f(x;D),of Fx)such that an topic in neural network research [3],[18],[157],[185].Learning overall estimation error can be minimized.The most commonly is the ability to approximate the underlying behavior adaptively used performance measure is the mean squared error from the training data while generalization is the ability to pre- dict well beyond the training data.Powerful data fitting or func- MSE =E[-f(x;DN))2] tion approximation capability of neural networks also makes =E[(u-F(x)]+(fx;Dx)-Fx).(13) them susceptible to the overfitting problem.The symptom of It is important to notice that the MSE depends on the particular an overfitting model is that it fits the training sample very well data set Dy.A change of the data set and/or sample size may but has poor generalization capability when used for prediction result in a change in the estimation function and hence the esti- purposes.Generalization is a more desirable and critical feature mation error.In most applications,the training data set Dy rep- because the most common use of a classifier is to make good resents a random sample from the population of all possible data prediction on new or unknown objects.A number of practical sets of size N.Considering the random nature of the training network design issues related to learning and generalization in- data,the overall prediction error of the model can be written as clude network size,sample size,model selection,and feature se- lection.Wolpert [188]addresses most of these issues oflearning EDE[(y-f(x;DN) and generalization within a general Bayesian framework. =E[(-F())]+ED[(f(x;DN)-F(x))(14) In general,a simple or inflexible model such as a linear clas- sifier may not have the power to learn enough the underlying re-where Ep denotes the expectation over all possible random lationship and hence underfit the data.On the other hand,com-samples of sample size N.In the following,D will be used plex flexible models such as neural networks tend to overfit the to represent the data set with the fixed sample size N for con- data and cause the model unstable when extrapolating.It is clear venience.Since the first term on the right hand side,E[(y-

454 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 4, NOVEMBER 2000 [132]. In addition, the model can be interpreted as posterior probability or odds ratio. It is a simple fact that when the logistic transfer function is used for the output nodes, simple neural networks without hidden layers are identical to logistic regression models. Another connection is that the maximum likelihood function of logistic regression is essentially the cross-entropy cost function which is often used in training neural network classifiers. Schumacher et al. [149] make a detailed comparison between neural networks and logistic regression. They find that the added modeling flexibility of neural networks due to hidden layers does not automatically guarantee their superiority over logistic regression because of the possible overfitting and other inherent problems with neural networks [176]. Links between neural and other conventional classifiers have been illustrated by [32], [33], [74], [139], [140], [151], [175]. Ripley [139], [140] empirically compares neural networks with various classifiers such as classification tree, projection pursuit regression, linear vector quantization, multivariate adaptive re￾gression splines and nearest neighbor methods. A large number of studies have been devoted to empirical comparisons between neural and conventional classifiers. The most comprehensive one can be found in Michie et al. [115] which reports a large-scale comparative study—the StatLog project. In this project, three general classification approaches of neural networks, statistical classifiers and machine learning with 23 methods are compared using more than 20 different real data sets. Their general conclusion is that no single classifier is the best for all data sets although the feedforward neural networks do have good performance over a wide range of prob￾lems. Neural networks have also been compared with decision trees [28], [36], [66], [104], [155], discriminant analysis [36], [127], [146], [161], [193], CART [7], [40], -nearest-neighbor [82], [127], and linear programming method [127]. III. LEARNING AND GENERALIZATION Learning and generalization is perhaps the most important topic in neural network research [3], [18], [157], [185]. Learning is the ability to approximate the underlying behavior adaptively from the training data while generalization is the ability to pre￾dict well beyond the training data. Powerful data fitting or func￾tion approximation capability of neural networks also makes them susceptible to the overfitting problem. The symptom of an overfitting model is that it fits the training sample very well but has poor generalization capability when used for prediction purposes. Generalization is a more desirable and critical feature because the most common use of a classifier is to make good prediction on new or unknown objects. A number of practical network design issues related to learning and generalization in￾clude network size, sample size, model selection, and feature se￾lection. Wolpert [188] addresses most of these issues of learning and generalization within a general Bayesian framework. In general, a simple or inflexible model such as a linear clas￾sifier may not have the power to learn enough the underlying re￾lationship and hence underfit the data. On the other hand, com￾plex flexible models such as neural networks tend to overfit the data and cause the model unstable when extrapolating. It is clear that both underfitting and overfitting will affect generalization capability of a model. Therefore a model should be built in such a way that only the underlying systematic pattern of the popu￾lation is learned and represented by the model. The underfitting and overfitting phenomena in many data modeling procedures can be well analyzed through the well-known bias-plus-variance decomposition of the prediction error. In this section, the basic concepts of bias and variance as well as their connection to neural network classifiers are discussed. Then the methods to improve learning and gener￾alization ability through bias and/or variance reductions are reviewed. A. Bias and Variance Composition of the Prediction Error Geman et al. [57] give a thorough analysis of the relationship between learning and generalization in neural networks based on the concepts of model bias and model variance. A prespec￾ified model which is less dependent on the data may misrepre￾sent the true functional relationship and have a large bias. On the other hand, a model-free or data-driven model may be too dependent on the specific data and have a large variance. Bias and variance are often incompatible. With a fixed data set, the effort of reducing one will inevitably cause the other increasing. A good tradeoff between model bias and model variance is nec￾essary and desired in building a useful neural network classifier. Without loss of generality, consider a two-group classifica￾tion problem in which the binary output variable is related to a set of input variables (feature vector) by where is the target or underlying function and is assumed to be a zero-mean random variable. From (8) and (9), the target function is the conditional expectation of given , that is (12) Given a particular training data set of size , the goal of modeling is to find an estimate, , of such that an overall estimation error can be minimized. The most commonly used performance measure is the mean squared error (13) It is important to notice that the MSE depends on the particular data set . A change of the data set and/or sample size may result in a change in the estimation function and hence the esti￾mation error. In most applications, the training data set rep￾resents a random sample from the population of all possible data sets of size . Considering the random nature of the training data, the overall prediction error of the model can be written as (14) where denotes the expectation over all possible random samples of sample size . In the following, will be used to represent the data set with the fixed sample size for con￾venience. Since the first term on the right hand side

ZHANG:NEURAL NETWORKS FOR CLASSIFICATION 455 F(x))2]=Ele2],is independent of both the training sample to produce accurate classification.That simple classifiers often and the underlying function,it reflects the irreducible estima- perform well in practice [76]seems to support Friedman's find- tion error because of the intrinsic noise of the data.The second ings term on the right hand side of(14),therefore,is a nature measure of the effectiveness of f(x;D)as a predictor of y.This term can B.Methods for Reducing Prediction Error be further decomposed as [57] As a flexible"model-free"approach to classification,neural ED[(f(x:D)-E(x2 networks often tend to fit the training data very well and thus have low bias.But the potential risk is the overfitting that causes ={ED[fxD】-E(y|x)}2 high variance in generalization.Dietterich and Kong [41]point +ED{(f(x:D)-ED[f(x;D)]2). (15) out in the machine learning context that the variance is a more important factor than the learning bias in poor prediction perfor- The first term on the right hand side is the square of the bias mance.Breiman [26]finds that neural network classifiers be- and is for simplicity called model bias while the second one is long to unstable prediction methods in that small changes in termed as model variance.This is the famous bias plus variance the training sample could cause large variations in the test re- decomposition of the prediction error. sults.Much attention has been paid to this problem ofoverfitting Ideally,the optimal model that minimizes the overall MSE or high variance in the literature.A majority of research effort in (14)is given by f(;D)=E(yx),which leaves the min- has been devoted to developing methods to reduce the overfit- imum MSE to be the intrinsic error Ele2].In reality,however, ting effect.Such methods include cross validation [118,[184]. because of the randomness of the limited data set D.the esti- training with penalty terms [182],and weight decay and node mate f(x;D)is also a random variable which will hardly be the pruning [137],[148].Weigend [183]analyzes overfitting phe- best possible function E(y x)for a given data set.The bias and nomena by introducing the concept of the effective number of variance terms in(15)hence provide useful information on how hidden nodes.An interesting observation by Dietterich [39]is the estimation differs from the desired function.The model bias that improving the optimization algorithms in training does not measures the extent to which the average of the estimation func- have positive effect on the testing performance and hence the tion over all possible data sets with the same size differs from the overfitting effect may be reduced by "undercomputing." desired function.The model variance,on the other hand,mea- Wang [179]points out the unpredictability of neural networks sures the sensitivity of the estimation function to the training in classification applications in the context of learning and gen- data set.Although it is desirable to have both low bias and low eralization.He proposes a global smoothing training strategy variance,we can not reduce both at the same time for a given by imposing monotonic constraints on network training,which data set because these goals are conflicting.A model that is less seems effective in solving classification problems [5] dependent on the data tends to have low variance but high bias Ensemble method or combining multiple classifiers [21],[8], if the model is incorrect.On the other hand,a model that fits the [64],[67],[87刀，[128]，[129]，[192]is another active research data well tends to have low bias but high variance when applied area to reduce generalization error [153].By averaging or voting to different data sets.Hence a good model should balance well the prediction results from multiple networks,the model vari- between model bias and model variance ance can be significantly reduced.The motivation of combining The work by Geman et al.[57]on bias and variance tradeoff several neural networks is to improve the out-of-sample clas- under the quadratic objective function has stimulated a lot of sification performance over individual classifiers or to guard research interest and activities in the neural network,machine against the failure of individual component networks.It has been learning,and statistical communities.Wolpert [190]extends the shown theoretically that the performance of the ensemble can bias-plus-variance dilemma to a more general bias-variance-co- not be worse than any single model used separately if the pre- variance tradeoff in the Bayesian context.Jacobs [85]studies dictions of individual classifier are unbiased and uncorrelated various properties of bias and variance components for mix- [129].Tumer and Ghosh [172]provide an analytical frame- tures-of-experts architectures.Dietterich and Kong [41],Kong work to understand the reasons why linearly combined neural and Dietterich [94],Breiman [26],Kohavi and Wolpert [93], classifiers work and how to quantify the improvement achieved Tibshirani [168],James and Hastie [86],and Heskes [71]have by combining.Kittler et al.[90]present a general theoretical developed different versions of bias-variance decomposition for framework for classifier ensembles.They review and compare zero-one loss functions of classification problems.These alter- many existing classifier combination schemes and show that native decompositions provide insights into the nature of gen- many different ensemble methods can be treated as special cases eralization error from different perspectives.Each decomposi- of compound classification where all the pattern representations tion formula has its own merits as well as demerits.Noticing are used jointly to make decisions. that all formulations of the bias and variance decomposition in An ensemble can be formed by multiple network architec- classification are in additive forms,Friedman [48]points out tures,same architecture trained with different algorithms,dif- that the bias and variance components are not necessarily addi- ferent initial random weights,or even different classifiers.The tive and instead they can be "interactive in a multiplicative and component networks can also be developed by training with dif- highly nonlinear way."He finds that this interaction may be ex- ferent data such as the resampling data.The mixed combination ploited to reduce classification errors because bias terms may of neural networks with traditional statistical classifiers has also be cancelled by low-variance but potentially high-bias methods been suggested [35],[112]

ZHANG: NEURAL NETWORKS FOR CLASSIFICATION 455 , is independent of both the training sample and the underlying function, it reflects the irreducible estima￾tion error because of the intrinsic noise of the data. The second term on the right hand side of (14), therefore, is a nature measure of the effectiveness of as a predictor of . This term can be further decomposed as [57] (15) The first term on the right hand side is the square of the bias and is for simplicity called model bias while the second one is termed as model variance. This is the famous bias plus variance decomposition of the prediction error. Ideally, the optimal model that minimizes the overall MSE in (14) is given by , which leaves the min￾imum MSE to be the intrinsic error . In reality, however, because of the randomness of the limited data set , the esti￾mate is also a random variable which will hardly be the best possible function for a given data set. The bias and variance terms in (15) hence provide useful information on how the estimation differs from the desired function. The model bias measures the extent to which the average of the estimation func￾tion over all possible data sets with the same size differs from the desired function. The model variance, on the other hand, mea￾sures the sensitivity of the estimation function to the training data set. Although it is desirable to have both low bias and low variance, we can not reduce both at the same time for a given data set because these goals are conflicting. A model that is less dependent on the data tends to have low variance but high bias if the model is incorrect. On the other hand, a model that fits the data well tends to have low bias but high variance when applied to different data sets. Hence a good model should balance well between model bias and model variance. The work by Geman et al. [57] on bias and variance tradeoff under the quadratic objective function has stimulated a lot of research interest and activities in the neural network, machine learning, and statistical communities. Wolpert [190] extends the bias-plus-variance dilemma to a more general bias-variance-co￾variance tradeoff in the Bayesian context. Jacobs [85] studies various properties of bias and variance components for mix￾tures-of-experts architectures. Dietterich and Kong [41], Kong and Dietterich [94], Breiman [26], Kohavi and Wolpert [93], Tibshirani [168], James and Hastie [86], and Heskes [71] have developed different versions of bias-variance decomposition for zero-one loss functions of classification problems. These alter￾native decompositions provide insights into the nature of gen￾eralization error from different perspectives. Each decomposi￾tion formula has its own merits as well as demerits. Noticing that all formulations of the bias and variance decomposition in classification are in additive forms, Friedman [48] points out that the bias and variance components are not necessarily addi￾tive and instead they can be “interactive in a multiplicative and highly nonlinear way.” He finds that this interaction may be ex￾ploited to reduce classification errors because bias terms may be cancelled by low-variance but potentially high-bias methods to produce accurate classification. That simple classifiers often perform well in practice [76] seems to support Friedman’s find￾ings. B. Methods for Reducing Prediction Error As a flexible “model-free” approach to classification, neural networks often tend to fit the training data very well and thus have low bias. But the potential risk is the overfitting that causes high variance in generalization. Dietterich and Kong [41] point out in the machine learning context that the variance is a more important factor than the learning bias in poor prediction perfor￾mance. Breiman [26] finds that neural network classifiers be￾long to unstable prediction methods in that small changes in the training sample could cause large variations in the test re￾sults. Much attention has been paid to this problem of overfitting or high variance in the literature. A majority of research effort has been devoted to developing methods to reduce the overfit￾ting effect. Such methods include cross validation [118], [184], training with penalty terms [182], and weight decay and node pruning [137], [148]. Weigend [183] analyzes overfitting phe￾nomena by introducing the concept of the effective number of hidden nodes. An interesting observation by Dietterich [39] is that improving the optimization algorithms in training does not have positive effect on the testing performance and hence the overfitting effect may be reduced by “undercomputing.” Wang [179] points out the unpredictability of neural networks in classification applications in the context of learning and gen￾eralization. He proposes a global smoothing training strategy by imposing monotonic constraints on network training, which seems effective in solving classification problems [5]. Ensemble method or combining multiple classifiers [21], [8], [64], [67], [87], [128], [129], [192] is another active research area to reduce generalization error [153]. By averaging or voting the prediction results from multiple networks, the model vari￾ance can be significantly reduced. The motivation of combining several neural networks is to improve the out-of-sample clas￾sification performance over individual classifiers or to guard against the failure of individual component networks. It has been shown theoretically that the performance of the ensemble can not be worse than any single model used separately if the pre￾dictions of individual classifier are unbiased and uncorrelated [129]. Tumer and Ghosh [172] provide an analytical frame￾work to understand the reasons why linearly combined neural classifiers work and how to quantify the improvement achieved by combining. Kittler et al. [90] present a general theoretical framework for classifier ensembles. They review and compare many existing classifier combination schemes and show that many different ensemble methods can be treated as special cases of compound classification where all the pattern representations are used jointly to make decisions. An ensemble can be formed by multiple network architec￾tures, same architecture trained with different algorithms, dif￾ferent initial random weights, or even different classifiers. The component networks can also be developed by training with dif￾ferent data such as the resampling data. The mixed combination of neural networks with traditional statistical classifiers has also been suggested [35], [112]