A MODIFIED K-NEAREST NEIGHBOR CLASSIFIER TO DEAL WITH UNBALANCED CLASSES

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1 A MODIFIED K-NEAREST NEIGHBOR CLASSIFIER TO DEAL WITH UNBALANCED CLASSES Aram AlSuer, Ahmed Al-An and Amr Atya 2 Faculty of Engneerng and Informaton Technology, Unversty of Technology, Sydney, Australa alsuer@eng.uts.edu.au, ahmed@eng.uts.edu.au 2 Department of Computer Engneerng, Caro Unversty, Gza, Egypt amr@alumn.caltech.edu Keywords: Abstract: NN classfer, Unbalanced classes, Class-wse classfcaton accuracy. We present n ths paper a smple, yet valuable mprovement to the tradtonal -Nearest Neghbor (NN) classfer. It ams at addressng the ssue of unbalanced classes by maxmzng the class-wse classfcaton accuracy. The proposed classfer also gves the opton of favorng a partcular class through evaluatng a small set of fuzzy rules. When tested on a number of UCI datasets, the proposed algorthm managed to acheve a unformly good performance. INTRODUCTION -nearest neghbor (NN) s a well-nown and wdely used classfcaton algorthm. Ths s due to ts smple mplementaton and relatvely low computatonal cost compared to other classfcaton methods. To classfy an unseen pattern, the algorthm uses the labels of ts -nearest neghbors and apples a votng crteron (Duda and Hart, 973). It has been shown that when both the number of patterns, N, and the number of neghbors,, approach nfnty such that /N, the error rate of the NN classfer approaches the optmal Bayes error rate (Cover and Hart, 967). In addton, (Tan, 25, Paredes and Vdal, 26) have showed that the tradtonal NN classfer can provde good results when dealng wth large dataset wth evenly dstrbuted patterns among the dfferent classes. A number of modfcatons have been proposed to the tradtonal NN classfer to mprove ts performance (Zeng et al., Tan, 25, Paredes and Vdal, 26). Most of these modfcatons le nto two categores: modfyng the dstance measure (Paredes and Vdal, 26, Duda and Hart, 973), or usng a specfc weghtng mechansm, where weghts are assgned to the neghbors, classes, features or a combnaton of them. (Dudan, 976) has proposed to assgn a specfc weght to each neghbor, nstead of equally weghng the neghbors. Hence, a weghed - nearest neghbor (wnn) classfer was ntroduced. It s mportant to menton that mprovng the classfcaton performance by some of the above methods comes wth the cost of notceably ncreasng the computatonal tme, and hence, losng one of the man attractons of the NN classfer. An mportant aspect of many classfcaton problems s that patterns are often not equally dstrbuted among classes, and the dfferent classes may vary n ther degrees of mportance. For nstance, n medcal dagnoss we may only have a small number of patents nfected by a certan dsease compared to the total number of persons that are tested. If the classfcaton system merely attempts to maxmze the overall classfcaton accuracy wthout tang nto consderaton the mportance of each class, then such a system would not be very benefcal. Many pattern classfcaton methods, ncludng NN and most of ts varants, do not tae the class balance and class mportance ssues nto consderaton. Thus, n order to overcome ths drawbac, t s mportant to consder the senstvty and specfcty measures, whch are defned as follows: True Postve Senstvty True Postve + False neagatve True Negatve Specfcty= True Negatve + False Postve () In ths paper we propose a new weghtng mechansm n order to maxmze the class-wse classfcaton accuracy, and hence acheve the rght 48

2 A MODIFIED K-NEAREST NEIGHBOR CLASSIFIER TO DEAL WITH UNBALANCED CLASSES balance between senstvty and specfcty. Moreover, as the ssue of favorng a partcular class s applcaton dependent, an extenson to the algorthm s presented to accommodate ths through the development of a smple fuzzy nference system. The next secton descrbes the tradtonal NN algorthm and number of ts varants, followed by a descrpton of the proposed algorthm. The ssue of favorng a specfc class s presented n secton four. Secton fve presents the expermental results and a concluson s gven n secton sx. 2 K-NEAREST NEIGHBOR CLASSIFICATION OVERVIEW A -nearest neghbor (NN) classfer s mplemented by dentfyng the -nearest neghbor tranng patterns to each unnown test pattern. The nearest neghbor can be found usng a dstance measure. The most wdely used dstance measure s the Eucldean, whch s defned usng the followng formula: n 2 ( ) (2) d x y where x = {x, x n } and y = {y, y n } are two samples of the same dmenson. A more general dstance measure s the Mnows dstance, whch s defned as: n (3) d x y Note that the Eucldean dstance s a specal case of the Mnows dstance (wth = 2). Another specal case of the Mnows dstance s the Manhattan dstance, whch s obtaned by assgnng =. The votng of the -nearest neghbors can be ether unweghted or weghted. In unweghted votng the class labels are assgned accordng to the maorty vote, hence, all neghbors have the same weght. In weghted votng the weght assgned to each neghbor, w, s proportonal to ts dstance from the underlyng test pattern, x, as follow (Yong et al., 29): w d( x, x) d( x, x) f, d( x, x) d ( x, x) d( x, x) d( x, x) f, d( x, x) d ( x, x) (4) where x and x represent the nearest and farthest th neghbor to the test pattern x respectvely. It has been found that ths weghtng scheme can produce better results for most cases. As descrbed n (Tan, 25), the tradtonal NN classfer, ts weghted verson and many of ts varants fal to provde good results when dealng wth unbalanced data,.e. patterns are not evenly dstrbuted among classes. Tan proposed a classweghtng approach, whch assgns a lower weght to the class that has large number of patterns. For a bnary-class problem, the weght for each class, s obtaned by: w Num C Mn Num C l ( x x )/ { ( ),2} / l (5) where >, Num(C x ) represents the number of neghbors that belong to class when consderng the testng pattern x. Based on the recall and precson measures, ths method acheved better results than the tradtonal NN when appled to unbalanced text corpus. The next secton descrbes our proposed weghtng scheme. 3 THE PROPOSED ALGORITHM Let s consder the case of bnary classfcaton wth two unbalanced classes such that Pr(C ) < Pr(C 2 ),.e., the probabltes of the two classes are dfferent. Let L be the set of tranng patterns of sze M, L the set of tranng patterns that belong to Class C, and L 2 the set of tranng patterns that belong to Class C 2. If L and L 2 consst of M and M 2 patterns respectvely, then M < M 2, M + M 2 = M. It s obvous that patterns of L 2 wll on average contrbute more n dentfyng the -nearest neghbors of the avalable patterns than those of L. The basc dea of the proposed algorthm s to ncrease the nfluence of the underrepresented set, L, n the dentfcaton of the -nearest neghbors. Below are the mplementaton steps: Compute the mean dstance of the -nearest neghbors from each pattern, x m, n L to the tranng patterns of L (denote t by d, m, where ={,2}, m=:m). Compute a dstance weghtng functon for each class C. 49

3 IJCCI 29 - Internatonal Jont Conference on Computatonal Intellgence Q M d m, m 2 M d l m l, m (6) Gven pattern x m that needs to be classfed, the -nearest neghbors are computed by reweghtng the dstances accordng to the dstance weghtng functon. Specfcally we multply all dstances from x m to the tranng patterns of L by Q. The resultng dstances are then sorted and the smallest neghbors are selected to gve the new -nearest neghbors. Accordng to ths procedure, f class C s underrepresented, then t s expected to have Q < Q 2. Ths wll mae patterns of L more represented n the new -nearest neghbors. Please note that ths procedure can easly be extended to deal wth mult classes by smply mang ranges between and the number of classes, J (the summaton n the denomnator of (6) would be over all classes). Also, we found that t would be better to mpose a lower lmt to the value of Q n case that one of the classes s severely underrepresented. 4 FAVORING A SPECIFIC CLASS The dea of ncreasng/decreasng the nfluence of a certan set, L, s used here to favor/dsfavor class C. Ths s mplemented through developng a smple Fuzzy Inference System (FIS), whch has two nputs: Q and a favorng factor, Fav, and one output,. The membershp functons of these varables are shown n Fg.. Frstly, Q s calculated, as explaned n the prevous secton, whle Fav needs to be specfed by the user. Both Q and Fav wll be used to evaluate the fuzzy nference system. If the user would le to favor C, then Fav needs to be assgned a value greater than.5. A value that s less than.5 wll dsfavor C, whle a value of.5 (neutral) means that none of the two classes wll be favored. The output of the FIS,, wll be used to update the value for Q, as shown n Eq. 7. s allowed to range between mn and max, whch are calculated usng Eq. 8, where s a constant. Ths approach wll be useful n producng Recever Operatng Characterstcs curves (ROC), whch s a graphcal plot of the senstvty vs. - specfcty. The ROC curve gves a better ndcaton about the performance of dfferent classfers than merely relyng on the overall classfcaton accuracy. Good curves le closer to the top left corner and the worst case s a dagonal lne. Fgure 2 shows dfferent cases for ROC curves, case 4 represents the random guess, whle a classfer wth an ROC curve smlar to that of case s consdered optmal (Westn, 2). Q Q Q Q (7) max mn * Q *( Q ) (8) Low Med Hgh Q Decr Neutral Incr Fav Hgh - Low - Around Low + Hgh mn max.2 Fgure : Membershps of the nputs, Q and Fav, and output of the FIS. Fgure 2: ROC curves for dfferent classfers. ) If Fav s Neutral then s Around 2) If Q s Med and Fav s Decrement then s Hgh + 3) If Q s Med and Fav s Increment then s Hgh - 4) If Q s Low and Fav s Decrement then s Hgh + 5) If Q s Hgh and Fav s Increment then s Hgh - 6) If Q s Hgh and Fav s Decrement then s Low + 7) If Q s Low and Fav s Increment then s Low - Fgure 3: Rules of the fuzzy nference system. The fuzzy rules are shown n Fg. 3. The frst rule bascally mples that f Fav s neutral then Q should not be changed. Rules 2 and 3 ndcate that n 4

4 A MODIFIED K-NEAREST NEIGHBOR CLASSIFIER TO DEAL WITH UNBALANCED CLASSES order to dsfavor/favor C, gven that Q s medum, then Q needs to be moved towards mn / max (ncreased/decreased wth a value proportonal to Fav). The same concept s appled to rules 4 and 5. Rules 6 and 7 are ntroduced to control the amount of ncrement/decrement when Q s already hgh/low. 5 EXPERIMENTAL RESULTS We tested the algorthm on a number of real-world pattern classfcaton problems. In the frst experment, the followng NN varants are consdered: The tradtonal -nearest neghbor classfer (NN), the weghted -nearest neghbor classfer (wnn), the modfed NN accordng to the proposed dstance measure descrbed n secton 3 (mnn), mnn wth weghted neghbors (wmnn), the Fuzzy NN (FNN), adopted from (Keller et al.), the Evdental NN based on the Dempster-Shafer theory of evdence (Denoeux, 995) (DSNN), the neghborhood component analyss (NCA) (Goldberger et al.), and Tan s class weghtng NN (CWNN) descrbed above. Table : Dataset Descrpton. Name # Patterns # Attrbutes C/C2 rato Pma Hll Cmc Sonar Mamm Hearts Btrans Heart Bands Gcredt Teach Wdbc Acredt Haber Ion In our experments, we have used 6 datasets from the UCI repostory webste (Newman, 27), as shown n Table. 8% of the patterns were used for tranng and 2% for testng. For each method several values of have been used, = {3, 5,.., 5}, and the one that gave the best performance usng a cross-valdaton scheme was chosen. In order to evaluate the performance of each method, class-wse classfcaton accuracy was used (Ac s the accuracy of class C ). We also calculated the average classfcaton accuracy of the two classes Acv= (Ac +Ac 2 )/2. The Acv results of the eght NN varants are presented n Table 2. The table shows that when NN and wnn produce dfferent performance of the two classes, consderable mprovement can be acheved usng the proposed method (mnn and wmnn). The mean of Acv over all tested datasets show that both mnn and wmnn can notceably mprove the accuracy of the underrepresented class as well as Acv. It s worth mentonng that CWNN fals when appled to certan datasets as t does not taes nto account the dstances between neghbors of dfferent classes,.e. the weghts only depend on the number of patterns that belong to each class. Addtonally, ths method needs tunng of the exponent. On the other hand, CWNN performed slghtly better than the proposed method when appled to datasets that have relatvely small number of pattern, such as Heart and HeartS, where n such case dstances between neghbors of dfferent classes may not gve a good estmate of the weghts. In the second experment, the ssue of favorng a partcular class s consdered by applyng the FIS explaned n secton 4, we referred to t as FISNN, to selected datasets from Table. The value of Fav was vared between.9 and., and the obtaned results are shown n Table 3. We can see that n all of the examned datasets, FISNN managed to adust the value of Q, such that qute a hgh classfcaton accuracy of the desred class s acheved. Ths, of course, comes wth the expense of reducng the accuracy of the other class, where the hgher the accuracy of one class, the lower the accuracy of the other. As explaned earler, ths represents an addtonal opton gven to the user n case that he/she wants to gve more emphass to a partcular class. It s worth mentonng that the hghest value of the mean of Acv s acheved around Fav =.5, whch s bascally the wmnn descrbed n secton 3. Ths s also the value that produces the mnmum dfference between the mean of Ac and that of Ac 2,.e., the best compromse between senstvty and specfcty. Fg. 4 shows the ROC curves of the dfferent classfers for the Mamm, Bands and Pma datasets. As the tradtonal NN and ts varants do not gve the opton of favorng a partcular class, the curves are drawn usng three ponts only, {,}, {,} and the average class-wse accuracy of those classfers. The proposed method on the other hand has the ablty to construct the full curve and t clearly shows the behavor of the classfer. Those curves wll be qute benefcal f the user would le to now the tradeoff of favorng a partcular class. The graphs also show that the proposed algorthm 4

5 IJCCI 29 - Internatonal Jont Conference on Computatonal Intellgence acheved on average much better result than the rest of the classfers. REFERENCES Cover, T. & Hart, P. (967) Nearest neghbor pattern classfcaton. Informaton Theory, IEEE Transactons on, 3, Denoeux, T. (995) A -nearest neghbor classfcaton rule based on Dempster-Shafer theory. Systems, Man and Cybernetcs, IEEE Transactons on, 25, Duda, R. O. & Hart, P. E. (973) Pattern classfcaton and scene analyss, N.Y., Wley. Dudan, S. A. (976) The Dstance-Weghted -Nearest- Neghbor Rule. SMC, 6, Goldberger, J., Rowes, S., Hnton, G. & Salahutdnov, R. Neghborhood Component Analyss. NIPS. Keller, J. M., Gray, M. R. & Gvens, J. A. A fuzzy - nearest neghbor algorthm. Newman, A. A. A. D. J. (27) UCI Machne Learnng Repostory. Unversty of Calforna, Irvne, School of Informaton and Computer Scences. Paredes, R. & Vdal, E. (26) Learnng weghted metrcs to mnmze nearest-neghbor classfcaton error. Pattern Analyss and Machne Intellgence, IEEE Transactons on, 28, -. Tan, S. (25) Neghbor-weghted K-nearest neghbor for unbalanced text corpus. Expert Systems wth Applcatons, 28, Westn, L. K. (2) Recever operatng characterstc (ROC) analyss. Department of Computng Scence,Umeå Unversty,sweden. Yong, Z., Yupu, Y. & Lang, Z. (29) Pseudo nearest neghbor rule for pattern classfcaton. Expert Syst. Appl., 36, Zeng, Y., Yang, Y. & Zhao, L. Pseudo nearest neghbor rule for pattern classfcaton. Expert Systems wth Applcatons, In Press, Corrected Proof. Fgure 4: ROC of the dfferent methods. 6 CONCLUSIONS In ths paper we proposed a new modfcaton to the tradtonal NN classfer that s able to maxmze the class-wse classfcaton accuracy, and hence produce good compromse between senstvty and specfcty. In addton, a fuzzy nference system has been added to the classfer, whch enables the user to favor a partcular class. Results obtaned usng a number of UCI datasets demonstrate the ablty of the proposed method n achevng better performance than the tradtonal NN classfer and a number of ts varants. 42

6 A MODIFIED K-NEAREST NEIGHBOR CLASSIFIER TO DEAL WITH UNBALANCED CLASSES Table 2: Average class-wse accuracy for a selected UCI dataset usng dfferent classfcaton methods. Dataset NN wnn mnn wmnn FNN DSNN NCA CWKNN Acredt 84.5± ± ± ± ± ± ± ±.97 Bands 62.9± ± ± ± ± ± ± ±.76 Btrans 6.84± ± ± ± ± ± ± ±.6 Cancer 96.54± ± ± ± ± ± ± ±.6 Cmc 58.55± ± ± ± ± ± ± ±.27 Gcredt 59.7± ± ± ± ±.6 6.9± ± ±.7 Haber 56.4± ±2. 63.± ± ± ± ± ±2.26 Heart 75.97± ± ± ± ± ± ± ±.87 HeartS 78.39± ± ± ± ± ± ± ±2.3 Hll 55.48± ± ± ± ± ± ± ±.8 Ion 78.32± ± ± ± ± ± ± ±.42 Mamm 79.6± ± ± ± ± ± ± ±.92 Pma 7.44± ± ± ± ± ± ± ±.35 Sonar 8.89± ± ± ± ± ± ± ±2.6 Teach 63.5± ± ± ± ± ± ± ±4.6 Wdbc 95.92± ± ± ± ± ± ± ±.44 Mean 72.36± ± ± ± ± ± ± ±.54 Table 3: Class-wse classfcaton accuracy for dfferent Fav values. Fav % Pma Ac Ac mean Hll Ac Ac mean Cmc Ac Ac mean Sonar Ac Ac mean Mamm Ac Ac mean Hearts Ac Ac mean Btrans Ac Ac mean Heart Ac Ac mean Bands Ac Ac mean mean(ac) mean(ac2) mean(acv)

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