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The Influence of Class Imbalance on Cost-Sensitive Learning

The Influence of Class Imbalance on Cost-Sensitive Learning
The Influence of Class Imbalance on Cost-Sensitive Learning

The In?uence of Class Imbalance on Cost-Sensitive Learning:

An Empirical Study

Xu-Ying Liu Zhi-Hua Zhou

National Laboratory for Novel Software Technology

Nanjing University,Nanjing210093,China

{liuxy,zhouzh}@https://www.wendangku.net/doc/a85530467.html,

Abstract

In real-world applications the number of examples in one class may overwhelm the other class,but the pri-mary interest is usually on the minor class.Cost-sensitive learning has been deeded as a good solution to these class-imbalanced tasks,yet it is not clear how does the class-imbalance affect cost-sensitive classi?ers.This paper presents an empirical study using38data sets,which dis-closes that class-imbalance often affects the performance of cost-sensitive classi?ers:When the misclassi?cation costs are not seriously unequal,cost-sensitive classi?ers gener-ally favor natural class distribution although it might be im-balanced;while when misclassi?cation costs are seriously unequal,a balanced class distribution is more favorable.

1.Introduction

In real-world applications,data sets are often imbal-anced,that is,the number of examples in one class may overwhelm the other class.This problem is prevalent in many applications,such as fraud/intrusion detection,medi-cal diagnosis/monitoring,etc.Learning algorithms that do not consider class-imbalance tend to be overwhelmed by the major class and ignore the minor one[3].However,in class-imbalance learning,usually the primary interest is on the minor class.That is,the cost of misclassifying a minor class example is usually more expensive than that of mis-classifying a major one[9,11].

Various cost-sensitive learning methods[5,7,10,13,15] have been developed to deal with unequal misclassi?ca-tion costs.Cost-sensitive learning has also been deemed as a good solution to class-imbalance learning[12].Mal-oof[8]indicated that,learning from imbalanced data sets and learning with unequal costs can be handled in a similar manner.On one hand,cost-sensitive learning is a good so-lution to class-imbalance problem[3];on the other hand,methods designed for class-imbalance problem can also help in cost-sensitive learning[15].In fact,cost-sensitive learning methods deal with class-imbalance by incurring different costs for different classes.Thus,it is feasible to handle unequal misclassi?cation costs and class-imbalance in a uni?ed framework[14].

However,previous research mainly focus on pure class-imbalance learning or pure cost-sensitive learning,largely ignoring the fact that class-imbalance and unequal misclas-si?cation costs usually occur simultaneously.Although it has been observed that class-imbalance has some in?uence on cost-sensitive classi?ers[2,10,14,15],up to now there is no thorough investigation about the in?uence of class-imbalance on cost-sensitive learning methods.

This paper presents an empirical study using38data sets on how class-imbalance would affect cost-sensitive learning methods.The results reveal that class-imbalance does often affect the performance of cost-sensitive classi?ers.When costs do not differ seriously,cost-sensitive classi?ers gen-erally favor natural class distribution;while when costs dif-fer seriously,cost-sensitive classi?ers favor a balanced class distribution instead.This suggests that in the future when we are dealing with a not-seriously imbalanced data set, we can simply apply cost-sensitive learning methods;while when we are facing with a seriously imbalanced data set, before applying cost-sensitive learning methods,we shall try to balance the class distribution.

2.Balancing Class Distribution

We focus on2-class tasks,where the positive class is of the primary interest and with higher misclassi?cation cost C+,while the other class is the negative class with lower cost C?.There is no cost for correct prediction.Assuming the positive class has N+examples and negative class has N?examples.Since costs can be normalized with the opti-mal decision unchanged[7],C?can always be set to1,and therefore C+is always bigger than1.

A popular approach to cost-sensitive learning is to rescale(or rebalance)the classes such that the in?uences of different classes on the learning process are in proportion to their costs[14].A typical process is to assign the train-ing examples of different classes with different weights, where the weights are in proportion to their corresponding misclassi?cation costs,and then pass the weighted exam-ples to a classical learning algorithm such as C4.5decision tree[7,10].Besides weighting the examples,the Rescaling approach can also be realized in many other ways,such as sampling the training examples[6–8]or moving the deci-sion thresholds[5,7,15]corresponding to their correspond-ing misclassi?cation costs.

In Rescaling,the optimal rescale ratio(called cost-sensitive rescale ratio)of positive class to negative class is rc+,?,as shown in Eq.1.

rc+,?=C+

C?

(1)

In order to deal with class-imbalance using Rescaling, different costs are to be incurred for different classes.So, the optimal rescale ratio(called imbalance rescale ratio)of positive class to negative class is ri+,?,as shown in Eq.2, where minor class will have bigger imbalance rescale ratio.

ri+,?=N?

N+

(2)

In order to handle unequal misclassi?cation costs and class-imbalance simultaneously,both the cost-sensitive rescale ratio rc and the imbalance rescale ratio ri should be considered.A natural way is to merge them into a single rescale ratio r,as shown in Eq.3.

r+,?=C+

C?

×

N?

N+

(3)

When minor class has higher misclassi?cation cost and less examples,in order to eliminate the in?uence of class-imbalance,the rescale ratio becomes larger than that of the case when only misclassi?cation costs are considered. From the view of cost-sensitive learning,the cost of the pos-itive class should be bigger than its given misclassi?cation cost,since the positive class has less examples.That is to say,misclassi?cation cost of a class should be normalized by the number of examples in this class,as shown in Eq.4.

C+ =C+

N+

,C? =

C?

N?

(4)

Note that Ciraco et al.[4]claimed that in general,chang-ing the class distribution is not equivalent to altering the cost ratio.However,in Rescaling they are equivalent,which can be found by comparing Eqs.3and4.Thus,Rescaling is a good basis for studying the in?uence of class-imbalance on cost-sensitive classi?ers.

3.Empirical Study

3.1.Settings

In the empirical study,the instance-weighting-based cost-sensitive C4.5decision tree(denoted by C45CS)[10] is used.In order to handle class-imbalance in cost-sensitive classi?ers,the strategy described in Section2is employed, which is denoted by B-C45CS.

Thirty-eight UCI data sets[1]are https://www.wendangku.net/doc/a85530467.html,rmation on these data sets is shown in Table1.The?rst part of Ta-ble1shows eleven imbalanced2-class data sets;the sec-ond part shows twenty-seven2-class data sets derived from multi-class data sets or relatively balanced2-class data sets according to one of the following strategies:1)treating one class of a multi-class data set as the positive class while the union of all other classes as the negative class(e.g.ann0), or2)selecting two classes of a multi-class data set,where one class is regarded as the positive class while the other is regarded as the negative class(e.g.balance1),or3)sam-pling a small subset from one class of a relatively balanced 2-class data set,which is regarded as the positive class while the other class is regarded as the negative class(e.g.adult and spamebase).The imbalance level(the ratio of major class size to minor class size)in these data sets varies from

1.9to66.

The misclassi?cation cost C?is always set to1,and C+is set to an integer varying from1to max(30,3×imbalance level).For each[C+,C?]combination,?ve times10-fold strati?ed cross validation are performed,and the average results are recorded.

3.2Results

In order to investigate the in?uence of class-imbalance on cost-sensitive classi?ers,the difference between the classi?ers’performance without/with considering class-imbalance is studied.

3.2.1Performance Tendency

Figure1depicts the performance tendency of C45CS and B-C45CS with the increasing cost ratio(ratio of C+to C?)on 8representative data sets,where the y-axes is the ratio of the classi?er’s total cost to its corresponding baseline method (standard C4.5).The horizontal dashed line at y=1indi-cates the performance of the baseline method.The vertical dashed line points out the imbalance level.Figures1(a)and 1(b)present typical performance tendency on data sets with slight class-imbalance.The imbalance level of german and abalone4is2.3and7.5,respectively.Twelve more data sets share such performance tendency,including breast-cancer, credit-g,haberman,spect,abalone2,3,5balance1,2,adult1, 2

Table1.Experimental Data Sets Dataset Size#Att#Class Target#Maj/#Min allbp-op2800292-20.1 breast-cancer28692- 2.4 breast-w69992- 1.9 credit-g1000202- 2.3 euthyroid3163252-9.8 german1000242- 2.3 haberman30632- 2.8 hepatitis155192- 3.8 hypothyroid3163252-20.0 sick3772292-15.3 spect267222- 3.9 abalone041778-Age≤4

55.4 abalone141778-Age≤521.1 abalone241778-Age≤68.3 abalone341778-Age>12 5.0 abalone441778-Age>137.5 abalone541778-Age>1410.5 abalone641778-Age>

1515.0 abalone741778-Age>1620.5 abalone841778-Age>1729.7 abalone941778-Age>1843.4 abalone1041778-Age>1966.4 ann07200213Class=1

42.4 ann17200213Class=218.6 ann27200213Class=312.5 balance062543B11.8 balance133743L/B 5.9 balance233743B/R 5.9 page-blocks05473105215.6 page-blocks1502810

51/542.7 nursery0459484very/prio13.0 adult012000142≤50K:1045

>50K:2000

adult111000142≤50K:10410

>50

K:1000

adult210500142≤50K:10420

>50K:500

adult310200142≤50K:10450

>50K:200 spambase032885720:2788,1:500 5.6 spambase12988

5720:2788,1:20013.9 spambase228885720:2788,1:10027.9

ann1and spambase0.The imbalance level of these data sets are mostly less than10.Figures1(c),1(d)and1(e) present typical performance tendency on data sets with se-rious class-imbalance.The imbalance level of hypothyroid, page-blocks1and spambase2is20.0,42.7and27.9,re-spectively.Thirteen more data sets share such performance tendency,including allbp-op,abalone0,1,6-10,

ann2,page-blocks0,adult2,3and spambase2.The imbalance level of these data sets are all greater than10.

On data sets described above,C45CS and B-C45CS have

Cost Ratio

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(g)nursery0(h)ann0

Figure1.Performance Tendency only one cross point.Before the point C45CS is superior to B-C45CS,while after the point C45CS is inferior to B-C45CS.There are some exceptions,as shown in Figure1(f) where there are more than one cross points.Data sets shar-ing such performance tendency include breast-w,euthyroid, hepatitis and sick.Note that the imbalance level of these data sets is less than10,and before the?rst cross point C45CS is superior to B-C45CS.In some cases C45CS al-ways dominates B-C45CS,as shown in Figure1(g).Data sets sharing such performance tendency include balance0, nrusery0and adult0.There is a very exceptional case(on the data set ann0),as shown in Figure1(h),where there is only one cross point but before the point C45CS is inferior to B-C45CS while after the point C45CS is superior to B-C45CS.

Among the38experimental data sets,C45CS and B-C45CS have cross points on35data sets.This means that when the minor class has higher misclassi?cation cost,

3

class-imbalance often affects the cost-sensitive classi?er C45CS.But class-imbalance will only take effect when the cost ratio in the concerned task is large enough.When the cost ratio is small,C45CS is generally superior to B-C45CS.Thus,it is not needed to consider class-imbalance in such cases.This is mainly because when the cost ratio is small,considering class-imbalance in cost-sensitive classi-?ers will make the rescale ratio overly large.3.2.2Quantitative Results

To quantitatively study the in?uence of class-imbalance on cost-sensitive classi?ers,min is observed.Figure 2illus-trates this criterion.Since performance tendency shows the performance of a cost-sensitive classi?er’s against the baseline method,the area under the performance tendency curve (denoted by AP T )can be regarded as a performance measure.Smaller AP T indicates better performance.In Figure 2,AP T (C 45CS )=area (A )+area (B ),and AP T (B -C 45CS )=area (A )+area (C ).Thus,when the performance tendency of two methods have cross points,area (A )is always smaller than the AP T values of both C45CS and B-C45CS.Otherwise,area (A )equals to the AP T value of the dominative classi?er.Thus,area (A )is helpful to evaluate the in?uence of class-imbalance,and its value is de?ned as min .The value of AP T (C 45CS )?min indicates how much C45CS favors balanced class distribu-tion:The larger the value,the more the balanced class dis-tribution is favored;while the value of AP T (B -C 45CS )?min indicates how much C45CS favors natural class dis-tribution:The larger the value,the more the natural class distribution is favored.Note that all these values should be normalized by the number of cost ratios,since the imbal-ance level of different data sets are not identical.

Cost Ratio

T o t a l C o s t

Figure 2.An Illustration of min

The total costs of C45CS and B-C45CS are tabulated in Table 2.Here min is also reported.The table entries present the real results of C4.5,and the ratio of the other methods against that of C4.5.Note that,the results for C45CS and B-C45CS are equivalent to the (normalized)AP T value.The bolded min values indicates that on the corresponding data set the min value is smaller than the

Table 2.Quantitative Results

Data Set C4.5C45CS B-C45CS min allbp-op

158.310.481.433.419breast-cancer 99.865.337.343.325breast-w 27.330.492.482.455credit-g 290.445.361.344.337euthyroid 59.805.670.719.663german 255.160.457.394.386haberman 103.835.345.346.325hepatitis 30.235.500.494.466hypothyroid 43.540.937.861.781sick 71.520.474.491.460spect 47.465.548.523.494abalone0248.340.538.221.217abalone1274.180.498.331.317abalone2310.950.469.475.447abalone3671.020.562.502.484abalone4573.690.609.528.503abalone5448.330.682.613.579abalone6515.400.663.515.489abalone7572.900.679.500.473abalone8619.000.731.519.494abalone9611.080.799.529.497abalone10616.940.740.476.455ann025.310.156.382.119ann115.200.442.590.404ann222.900.635.490.488balance090.650.675 1.062.675balance176.390.491.407.383balance276.390.502.405.380page-blocks090.850.718.716.645page-blocks1262.440.706.572.558nursery0169.020.283.593.283adult01695.300.441.501.441adult11050.520.591.644.556adult21144.470.634.671.571adult31252.430.735.746.632spambase0172.010.621.603.589spambase1154.410.814.663.649spambase2182.110.939.689.663avg.

345.519

.578

.536

.476

AP T value of C45CS,which implies that class-imbalance affects C45CS on that data set.

Table 2shows that,the difference between C45CS’AP T and min can be as much as 28.5%on data set abalone10which suffers from serious class-imbalance (the imbalance level is 66.4).Even on data sets with very slight class-imbalance,min can be smaller than C45CS’s AP T .For example,on breast-w ,min is 3.7%smaller than C45CS’s AP T .The average of min on all data sets is 0.476,which is 10.2%less than the average AP T of C45CS.

Generally speaking,class-imbalance does affect the cost-sensitive classi?er C45CS.Concretely,C45CS favors natural class distribution in case of small cost ratio,while

4

favors a balanced class distribution in case of big cost ratio. Moreover,the more serious the class-imbalance,the more signi?cant the in?uence.Unfortunately,the position of the critical point of C45CS and B-C45CS is hard to be decided since it is task-dependent.

Besides C45CS,we have also empirically investigated the in?uence of class-imbalance on several other cost-sensitive learning methods,such as instance-weighting-based cost-sensitive Na¨?ve Bayes classi?er[13],threshold-moving-based cost-sensitive neural networks[15],and MetaCost[5].The?ndings are similar to that have been reported in this paper.Due to the page limit,these results will be presented in a longer version.

4.Concluding Remarks

As far as we know,there is only one work concerned the in?uence of class-imbalance on cost-sensitive learning[2]. Based on results of three data sets,it was concluded that when misclassi?cation costs are unequal,a training set us-ing the natural class distribution is the best.In this paper, we report on an empirical study involving thirty-eight data sets.Our results disclose that,cost-sensitive classi?ers gen-erally favor natural class distribution when costs differ less, while a balanced class distribution is more favorable when costs differ seriously.This only partially agrees with the conclusion of the previous work[2].

It was indicated in[11]that traditional classi?ers(i.e. cost-insensitive classi?ers)favor a certain class distribu-tion for a certain evaluation metric.Our results disclose that even for the same evaluation metric,a cost-sensitive classi?er may favor different class distributions on different data sets.These may suggest that the in?uence of class-imbalance on traditional classi?ers and cost-sensitive clas-si?ers are somewhat different.

In this paper we employ Eq.3to merge the cost-sensitive weights as well as class-imbalance weights into a uni?ed framework.It is possible that our results have been biased because of the adoption of this merging scheme.Empiri-cal study with other kinds of merging schemes has been left for future work.Moreover,in this paper we only consid-ers natural class distribution and balanced class distribution. Whether it is better to take distributions other than natural or balanced distributions is an interesting issue to be explored in the future.Furthermore,studying the in?uence of class-imbalance on multi-class cost-sensitive learning methods is also an interesting future issue. Acknowledgments:Supported by the National Sci-ence Fund for Distinguished Young Scholars of China (60325207),the Jiangsu Science Foundation(BK2004001), and the973Fundamental Research Program of China (2002CB312002).

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一般过去式 时间状语:yesterday just now (刚刚) the day before three days ag0 a week ago in 1880 last month last year 1. I was in the classroom yesterday. I was not in the classroom yesterday. Were you in the classroom yesterday. 2. They went to see the film the day before. Did they go to see the film the day before. They did go to see the film the day before. 3. The man beat his wife yesterday. The man didn’t beat his wife yesterday. 4. I was a high student three years ago. 5. She became a teacher in 2009. 6. They began to study english a week ago 7. My mother brought a book from Canada last year. 8.My parents build a house to me four years ago . 9.He was husband ago. She was a cooker last mouth. My father was in the Xinjiang half a year ago. 10.My grandfather was a famer six years ago. 11.He burned in 1991

学生造句--Unit 1

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