A Novel Feature Line Segment Approach for Pattern Classification

Transcription

1 12th International Conference on Information Fuion Seattle, WA, USA, July 6-9, 2009 A Novel Feature Line Segment Approach for Pattern Claification Yi Yang Intitute of Integrated Automation Xi an Jiaotong Univerity Xi an, Shaanxi, China P.R. jiafeiyy@163.com Chongzhao Han Intitute of Integrated Automation Xi an Jiaotong Univerity Xi an, Shaanxi, China P.R. czhan@mail.xjtu.edu.cn Deqiang Han Intitute of Integrated Automation Xi an Jiaotong Univerity Xi an, Shaanxi, China P.R. deqhan@mail.xjtu.edu.cn Abtract - In thi paper, a novel pattern claification approach i propoed called hortet feature line egment (SFLS). It retain the idea and advantage of nearet feature line (NFL) and it can uppre the drawback of NFL, i.e., the extrapolation inaccuracy, interpolation inaccuracy and high computational complexity. SFLS ue length of the feature line egment atifying given geometric relation contraint, intead of the perpendicular ditance from query point to feature line in NFL. SFLS ha clear geometric-theory foundation and it implementation i relatively imple. In experiment baed on artificial dataet and real-world dataet, comparion between SFLS and other claification method are provided, including nearet neighbor (NN), k-nn, NFL and ome refined NFL method. Experimental reult how that SFLS i a imple yet effective claification approach. Keyword: Claification, nearet neighbor (NN), nearet feature line (NFL), interpolation inaccuracy and extrapolation inaccuracy. 1 Introduction With the development of pattern claification[1], variou claification method have been propoed till now. The nearet neighbor (NN) rule i a non-parametric claification method which i imple, yet effective. It ha been proved that NN ha aymptotic error rate that i at mot twice the Baye error rate [2]. NN alo ha it own drawback, i.e., the repreentational capacity of the dataet and the error rate depend on how the training ample are choen to account for poible variation and alo how many training ample are available [3]. Lot of approache have been propoed which aim to counteract the drawback. k-nn, urrounding neighbor (SN) [4], graphic neighbor (GN) [5] and nearet feature line (NFL) [3,6] are all repreentative. NFL i a non-parametrilaifier which attempt to improve the repreentational capacity of a ample et with limited ize by uing the traight line which pa through each pair of the ample from the ame cla. A good argument for doing thi i that it add extra information to the ample et. NFL ha hown wonderful performance in many application, including face recognition [3], audio retrieval [7], peaker identification [8], image claification [6], etc. Though ucceful in improving the claification ability, there are alo drawback in NFL. Two kind of trepaing can caue two type of inaccuracie: extrapolation inaccuracy and interpolation inaccuracy [9,10]. In addition, NFL ha relatively high computational complexity. A mentioned by the author of [9]: extrainformation i a double-edged word. There are everal method propoed to rectify the NFL to counteract it drawback. In [11], center-baed nearet neighbor (CNN) method wa propoed to leen the computation burden of the original NFL method by defining another kind of line called center-baed line (CL). CL connect a ample point with known label and the center of the ample correponding cla, intead of two labeled ample point in the NFL method. In [10], the author pointed out one of the inaccuracie referred above extrapolation inaccuracy, and propoed a olution called nearet neighbor line (NNL). NNL alo can ignificantly reduce the computation time. In[12], a tunable nearet neighbor (TNN) method wa propoed to improve the performance of NFL. In [13], nearet feature midpoint (NFM) wa propoed to refine NFL by defining the ditance metric a the minimum Euclidean ditance between the query point and the feature midpoint. The computation complexity of NFM i ignificantly le than NFL. All thee refined method are effective and rational, but they till have their own drawback, i.e. they can not thoroughly reolve the problem of extrapolation inaccuracy and interpolation inaccuracy. Rectified nearet feature line egment (RNFLS) wa propoed in [9] to overcome both of the two inaccuracie mentioned above. Thu it can ignificantly improve the claification performance when compared to NFL. RNFLS alo ha it deficiencie which are dicued in next ection. In thi paper, we propoe a imple yet effective approach called hortet feature line egment (SFLS), which can uppre both the extrapolation inaccuracy and interpolation inaccuracy. Our approach ha clear geometrictheory foundation and retain the idea and advantage of original NFL, i.e., it ue a linear model of pair of ample point within the ame cla. It ue line feature intead of ISIF 490

2 point feature. Intead of calculating the ditance between the query point and the feature line, SFLS attempt to find the hortet feature line egment which atifie given geometric relation requirement. Experimental reult how the wonderful performance of SFLS in artificial dataet and real-world dataet. Comparion between SFLS and ome other claification method uch a NN, k-nn, NFL and refined NFL are alo provided to verify the rationality and efficacy of SFLS propoed. 2 Nearet feature line method The nearet feature line (NFL) method i deigned to utilize the information provided by each pair of point in the ame cla by contituting ome feature line (FL) pace. The NFL ditance i defined a the Euclidean ditance from the query point to the FL, i.e., the ditance between a query point and it projection onto the FL a illutrated in Figure 1. NFL i conitently uperior to the NN method baed on conventional ditance [12]. Figure 1. Nearet feature line 2.1 Brief review of NFL Let x, x j be two training ample from the ame cla, = 1,..., M, where M repreent the number of cla. Each training ample feature vector ha n dimenion. xi x j denote the traight line paing through x i and x j, which i called a feature line (FL) of the cla. Let x q be the query point and x ij be the projection point of x q on FL x i x j. x ij can be calculated i a follow ( x q, x i and where x j are all column vector): xij = (1 µ ) xi + µ x j (1) T ( xq xi ) ( xj xi ) µ = T ( x x ) ( x x ) j i j i (2) The ditance from the query point x q to the FL xi x j can be calculated a: d ( x, x x ) = x x (3) q i j q ij where denote ome norm. The claification deciion can be done a follow: = arg min d (4) 1 M where i the cla label aigned to query point. dnfl = d i called the NFL ditance. The repreentational capacity of training ample can be generalized by NFL, i.e., NFL can add extra information to training ample. For a given training ample et, quantity of line feature baed on NFL i larger than that of point feature. However the extra information might be a doubleedged word. Though ucceful in improving the claification performance, there are till ome drawback in NFL that will limit it further application in practice. Two main drawback can be ummarized a follow: (a) It will encounter large computation complexity problem when there are too many ample in each cla [11]. If there are N training ample belong to cla, there will be N( N 1)/2feature line. Suppoe that each ample feature vector i n dimenion, for traditional NFL, there will be (3 n+ 1) ( N ( N 1)/2) multiplication operation. In [8], a fat algorithm to calculating d( xq, xi x j) i propoed, computation time of which i only 1/3of traditional one. (b) It will encounter the problem of extrapolation inaccuracy and interpolation inaccuracy, which will be dicued further afterward. 2.2 Extrapolation inaccuracy and interpolation inaccuracy of NFL In NFL, only perpendicular ditance between query ample and feature line i ued a the deciion criterion. When a traight line of one cla trepae into of other clae area, it may lead to claification error. Two type of trepaing will be dicued in thi ection, which can caue two type of inaccuracie: extrapolation inaccuracy and interpolation inaccuracy [9] Extrapolation inaccuracy In Figure 2, query ample x q i urrounded by ample belong to cla Star, but it i claified to cla Circle with the criterion of NFL. Thi claification error i due to the extrapolating part of FL xx

3 cla Circle. Thi i due to the interpolating part of FL xx 1 2. To overcome the interpolation inaccuracy, author in [9] propoed a method for ample election. They contruct rectified nearet feature line egment (RNFLS) ubpace to repreent each cla by having thoe line egment trepaing the cla-area of other clae removed. Figure 2. Extrapolation inaccuracy. Figure 4. Interpolation inaccuracy. Figure 3. Sample point with the ame NFL ditance. We can ee that, all the point at two parallel dah line have the ame NFL ditance according to FL x 1 x 2. All the query point with the ame NFL ditance will be dealt without dicrimination. Thi i an important reaon for generation of extrapolation inaccuracy. It can be proved that extrapolation inaccuracy can be ignored if the feature pace dimenion i large enough [9]. But in a feature pace of low dimenion, it actually harm. To counteract the extrapolation inaccuracy, the author in [10] propoed nearet neighbor line (NNL). In NNL, only one feature line linking the nearet two ample point to the query i ued to repreent the cla. NNL can avoid the extrapolation problem, but it may alo loe ome claification ability [9]. In [13], nearet feature midpoint (NFM) wa propoed which attempt to overcome the extrapolation inaccuracy by defining the ditance metric a the minimum Euclidean ditance between the query point and the feature line midpoint. In other reearche uch a tunable nearet neighbor (TNN) [12], rectified nearet feature line egment (RNFLS) [9] and extended nearet feature line(enfl) [14], they modified the calculation method of the NFL ditance metric baed on the relative poition relationhip between the query point and the feature line. Thu the query point with the ame original NFL ditance can be dealt with dicrimination and extrapolation inaccuracy can be counteracted Interpolation inaccuracy An example of interpolation inaccuracy in NFL i illutrated in Figure 4. The query ample x q i urrounded by the ample belong to cla Star, but xq i claified to Figure 5. Sample ditribution cauing ignificant interpolation and extrapolation inaccuracy problem. Figure 5 illutrate a cae which can produce both the extrapolation inaccuracy and interpolation inaccuracy. The refined NFL method including NNL, NFM, TNN, can only uppre the extrapolation inaccuracy. The RNFLS method can overcome both the extrapolation and interpolation inaccuracy, o it i an effective modified NFL claification method. It can implement claification well in the environment illutrated in Figure 5. In the ame environment, the claification error will increae ignificantly for NFL. However, RNFLS alo ha it drawback. It implementation i relatively complex. It ha the procedure of ample ubpace election and calculation method of the ditance metric are variou according to the relative poition relationhip between the query point and the feature line egment. It computational complexity i relatively high. In thi paper, we propoe a novel feature line egment approach, which i imple yet effective and can uppre both the interpolation inaccuracy and extrapolation inaccuracy. 3 Shortet feature line egment claification approach In thi ection, a novel claification approach called hortet feature line egment (SFLS) i introduced. SFLS doe not calculate the ditance between the query point and the feature line. Intead, it attempt to find the hortet feature line egment atifying given geometric relation 492

4 contraint. Two ample in the ame cla contitute a feature line egment. If the query point i inide or on the hyperphere centered at the midpoint of the feature line egment and whoe diameter i length of the feature line egment, the correponding feature line egment will be tagged. Then find out the hortet feature line egment in all the tagged feature line egment and aign the hortet feature line cla label to the query point. The pecific claification procedure i a follow: Let x i, x j be two training ample from the ame cla, = 1,..., M. M repreent number of cla. If there are N ample belonging to cla in training et, there will be N( N 1)/2 feature line egment. For a query point x q, execute the tep a follow: 1) Calculate the angle (denoted byα ) between vector: x q x i and x q x j baed on formula (5): T 180 ( xq xi ) ( xq xj) α = arcco (5) π x x x x Where q i q j x q, xi and x j are all column vector and denote the Euclidean norm. The unit of α i deg. If α i an acute angle (i.e., 0 α < 90 ), leave feature line egment xi x j untagged. If α i a right angle or an obtue angle (i.e., 90 α 180 ), tag feature line egment x i x j. Such feature line egment are called tagged feature line. 2) Find out the hortet tagged feature line egment: i j x x, label cla to query point x q. If there i no tagged feature line, then the correponding query point i rejected to be claified. Baed on the baic geometric theorem, a illutrated in Fig. 6, if α i an acute angle, then x q i outide the hyperphere; if α i a right angle, then x q i on the hyperphere; if α i an obtue angle, then x q i inide the hyperphere. The hyperphere center i the midpoint of the feature line egment xi x j and it diameter i the length of the feature line egment xi x j. (a) (b) (c) Figure 6. Relative poition relation between query point and feature line egment. (a) Inide hyperphere; (b) On hyperphere; Outide hyperphere. The claification procedure can be briefly by an example illutrated in Figure 7. Figure 7. Shortet feature line egment claification. Query point x q i inide hyperphere x q centered at midpoint of xx 1 2 and whoe diameter i x 1 x 2. x q i alo inide hyperphere C entered at midpoint of xx 1 2 and whoe diameter i x1 x 2. Feature line egment 1 2 xx and xx 1 2 are both tagged. xx 1 2 i horter than xx 1 2, o query point xq i labeled cla Circle. 4 Analyi of SFLS SFLS propoed in thi paper ha powerful claification ability and it can uppre ome ignificant drawback of NFL. SFLS i alo relatively eay to be implemented and it ha relatively low computational complexity. 4.1 Claification ability analyi (1) SFLS retain the idea and advantage of NFL SFLS ue a linear model of each pair of ample point within the ame cla to generalize the repreentational capacity of ample et. It ue line feature intead of point feature. (2)SFLS ha the concentration property of feature line egment SFLS ue the feature line egment intead of feature line. The feature pace contituted by feature line egment ha a good property, i.e., the concentration[9]. It can be concluded that tronger concentration property can bring the improvement of claification accuracy[9]. (3)SFLS ha more reaonable imilarity definition NFL only ue the perpendicular ditance from the query point to the feature line a the criterion for claification deciion. The perpendicular ditance i in fact the imilarity between point feature and line feature. What we concern i the imilarity between the query ample and the given cla. Only uch a ditance i not ufficient to repreent the imilarity between query ample and the feature line correponding cla. Le perpendicular ditance doe not alway repreent more imilarity. Such a propoition can be upported by the two kind of inaccuracie aforementioned. In SLFS, a referred above, 493

5 both the ditance and the volume of the hyperphere contituted by feature line egment and query point are emphaized to define the imilarity. For a tagged feature line egment, the horter the feature line egment i, the le the volume of the hyperphere i, which cover the query point and the feature line egment. Le volume repreent more imilarity between query point and feature line egment. According to the imilarity definition in SFLS, there will be a good property: the query point i alway near to both the two point contituting the hortet feature line egment. It can better reflect the imilarity between the query point and the correponding cla. (4)SFLS can uppre inaccuracie caued by trepa Firt, SFLS ue line egment intead of traight line. Intuitively, a line egment ha finite length while a traight line extend to infinity. Thu the probability to produce trepa problem can be reduced when uing feature line egment. Second, SFLS ha no problem of extrapolation inaccuracy. The poition parameter µ can be calculated. In SFLS, according to Formula (2), 0 µ 1definitely come into exitence. It mean that the projection of x q onto the feature line i an interpolating point of the tagged feature line egment. Third, in ome cae, the interpolation inaccuracy can be uppreed in SFLS a illutrated in Figure 8. Figure 8. NFL and SFLS in interpolation inaccuracy problem. For query point x q, the hortet feature line egment in cla circle i i j xx and in cla Star i xx. The query point x q urrounded by ample of cla Star i claified to cla Circle by NFL due to the c interpolation inaccuracy ( d < d ). But it i correctly claified to cla Star by SFLS becaue xx i j > xx i j. But it hould be noted that SFLS can not thoroughly reolve the interpolation inaccuracy. i j 4.2 Computational complexity analyi If there are N ample belong to cla in training et, we will have N( N 1)/2 feature line. Suppoe that feature vector of each ample ha n dimenion, the computation in SFLS for each cla include (3 n) ( N ( N 1)/2) multiplication operation which i le than that of the original NFL approach: (3 n+ 1) ( N ( N 1)/2). And it hould be noted that there i no offline ample ubpace election or preparation or the claification baed on SFLS. In RNFLS, offline preparation i needed, which ha computation time 3 complexity of ON ( ), where N denote the number of training ample. In general, SFLS propoed in thi paper i a imple yet effective claification method. It ha clear geometric theory foundation. It retain the main idea and advantage of NFL and at ame time it can counteract extrapolation inaccuracy and interpolation inaccuracy. SFLS attempt to find the hortet feature line egment which i near to the query point. However, the near i not defined baed on the perpendicular ditance from the query point to the feature line. It ue the geometric relation to define the near, which repreent the imilarity between the query and the feature line egment. 5 Experiment The claification performance of SFLS i compared with claification approache including NN, k-nn, traditional NFL, NNL, TNN, RNFLS. The experiment are executed on two artificial dataet and ome real-world dataet. 5.1 Artificial dataet: Two-piral The two piral curve can be generated baed on formula (6) and formula (7) a follow [9,15-17]: x = k co( ) piral1: (6) y = k in( ) x = k co( + π) piral2: (7) y = k in( + π) Set parameter π /2 3π and uppoe that each cla probability denity i uniform along the correponding curve. The two-piral dataet for experiment i polluted by Gauian noie whoe mean i zero and variance iσ = 1.5. The two piral curve and the polluted two piral curve for experiment are illutrated in Figure 9. Totally 500 ample are generated. Cla 1 and cla 2 each have 250 ample. Randomly elect 125 training ample from each cla, the remainder are reerved for tet. Then we ue NN, k-nn, NFL, NNL, RNFLS and the SFLS propoed to perform the claification. The k parameter in k-nn i et to 3. The claification procedure referred above i executed for 10 time. In each time, ample for training and tet are re- 494

6 elected randomly. The experimental reult are lited in Table 1. where µ denote the ditance between the two Gauian center. Totally 500 ample are generated. Cla 1 and cla 2 each have 250 ample. Randomly elect 125 training ample from each cla, the remainder are reerved a tet ample. 10 different µ value are ued: ( µ = 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5 ). We ue NN, k-nn, NFL, NNL, RNFLS and the SFLS propoed to perform the claification. The k parameter in k-nn i et to 3. The experimental reult are hown in Figure 10 and Table 2. The Bayeian claifier ue the true ditribution (Guaian). Figure 9. Two-piral curve. (a) Two-piral; (b) Two-piral polluted with Gauian noie. Table 1 Two-Spiral Claification performance Claification Approach Correct Rate (Min value) Correct Rate (Max value) Correct Rate (Mean) NN 93.6% 96.8% 95.5% k-nn 94.8% 98.4% 96.6% NFL 52.0% 63.2% 56.0% NNL 85.2% 91.6% 89.2% RNFLS 94.0% 97.2% 96.0% SFLS 94.0% 97.2% 96.0% It can be een that NFL i not proper to be ued in two-piral claification problem. It i due to the interpolation inaccuracy and extrapolation inaccuracy referred above. RNFLS gain better performance due to the building of the RNFLS ubpace and the new definition of ditance metric baed on the relative poition relationhip between query point and feature line egment. SFLS propoed in thi paper i relatively imple but it alo achieve the ame claification accuracy a RNFLS. 5.2 Claification problem of two clae with Gauian ditribution To compare the claification performance of the SFLS and other claification method with that of Bayeian claifier, we generate the artificial two-cla dataet with known probability denity- Guaian Ditribution. Any ample in our dataet ha two dimenion ( x, y ). x i a Gauian ditributed variable and y i a uniform ditributed variable. Their joint-probability denity function i a follow [9]: 1 1 x 2 exp[ ( ) ], a y b, pcla1( x, y) = 2 πσ( b a) 2 σ 0, otherwie; 1 1 x µ 2 exp[ ( ) ], a y b, pcla2( x, y) = 2 πσ( b a) 2 σ 0, otherwie; (8) Figure 20. Claification performance of two-gauian ditribution ample. Table 2 Claification performance of two-gauian ditribution ample Claification approach Mean claification correct rate of experiment with variou ditance of Gauian center Bayeian 88.1% NN 83.1% 3-NN 85.0% NFL 69.7% NNL 80.8% RNFLS 85.7% SFLS 85.7% Baed on the reult, it can be concluded that the NFL i not fit for thi type of dataet. Thi i mainly due to the extrapolation inaccuracy. The NN, k-nn and the refined NFL approache can achieve relatively good claification performance. SFLS and RNFLS perform the bet except for Bayeian approach. However, SFLS i impler. 5.3 Claification problem baed on realworld dataet The artificial dataet ued above are two-cla. To verify the SFLS performance in multi-cla (cla number 3) claification problem, we ue three multiple-cla real- 495

7 world dataet from UCI [18] including Wine (3-cla), Iri (3-cla) and Gla (6-cla). For Wine and Gla dataet, all attribute value are normalized to lie between 0 and 1 baed on formula (9): vi min vi ai = (9) max vi min vi where v i denote the actual value of attribute i. Maximum and minimum operation are over all ample. Each dataet i randomly divided into 5 dijoint ubet of approximately equal ize. For each time, we elect three ubet to contitute the training et and treat the remainder a the teting et. Thu, there are totally C = different trial over each dataet. The average claification performance acro the 10 trial are computed. The k parameter in k-nn i et to 3. The experimental reult are illutrated in Figure 11. Baed on the experimental reult illutrated in Figure 11, we can conclude that the SFLS approach propoed in thi paper alway achieve better performance than the other over the 3 real-world dataet. Figure 11. Claification performance baed on three dataet from UCI. 5.4 Dicuion Claification baed on line feature are not definitely better than thoe baed on point feature. When the cale of ample et i limited and when the feature vector of ample point have relatively low dimenion, the advantage of line feature (or line egment feature) can be ditinctly hown, which can alo been upported by the provided experimental reult. There are everal NFL-refined claification approache ued in the experiment. SFLS and RNFLS alway perform better than original NFL and other NFL-refined method. When compared to other method baed on line feature, SFLS ha more rational and more comprehenive definition of imilarity, which i crucial for the claification problem. Compared to RNFLS, both SFLS and RNFLS ue feature line egment which ha the good property of concentration and they can uppre the inaccuracie caued by trepa, but SFLS doe not need the preparation procedure for ample ubpace. SFLS ue imply the length of feature line egment intead of variou ditance definition in RNFLS. SFLS i relatively imple yet effective which can do the claification at leat a well a RNFLS in average when the number of cla i relatively low and SFLS perform better than RNFLS when the number of cla i relatively high. Thee can be upported by the provided experimental reult. It hould be noted that for SFLS, in the wort cae where there i no tagged feature line for a tet ample, if rejection deciion i not permitted, jut ue the rule of nearet neighbor (NN) intead to make the claification deciion for the ample which are reject to be claified by SFLS. 6 Concluion In thi paper, a novel claification approach called SFLS i propoed. SFLS can be regarded a a refined NFL method which can ubtantially improve the claification performance of NFL. Theoretical jutification and analyi of SFLS rationality are provided. Experimental reult baed on artificial dataet and real-world dataet (including UCI, etc.) alo how that SFLS i an effective claification approach. In the future, we will do further reearch in SFLS. The computational cot of SFLS i till not a low a expected. Fater and more effective algorithm for SFLS are worth reearching. Some modification to SFLS can be made according to other effective claification approache. For example, baed on SFLS, k-sfls can be developed, jut like the idea of k-nn, better performance can be expected. In thi paper, we only dicu about the claification problem with Euclidean ditance. Other ditance definition can alo be ued to implement the SFLS approach. Thi can alo be our work in the future. Acknowledgement Thi work i upported in part by the National Natural Science Foundation of China, No and the Grant for State Key Program for Baic Reearch of China (973) No.2007CB and in part by National Natural Science Foundation of China (No ) and China Potdoctoral Science Foundation (No ). Reference [1] S. Theodoridi, and K. Koutroumba, Pattern Recognition, 2nd ed., Elevier, Singapore, [2] R. O. Duda, and P.E. Hart, D.G. Stork, Pattern Claification, 2nd ed., Wiley Inter-cience publication, New York, [3] S. Z. Li, and J. W. Lu, Face recognition uing the nearet feature line method, IEEE Tranaction on Neural Network, Vol. 10, No. 2, pp ,

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