Clustering Algorithm of Similarity Segmentation based on Point Sorting

Transcription

1 Internatonal onference on Logstcs Engneerng, Management and omputer Scence (LEMS 2015) lusterng Algorthm of Smlarty Segmentaton based on Pont Sortng Hanbng L, Yan Wang*, Lan Huang, Mngda L, Yng Sun, Hanyuan Zhang ollege of omputer Scence and Technology, Jln Unversty hangchun, Jln, , hna orrespondng author: wy m Abstract We propose a clusterng algorthm of smlarty segmentaton based on pont sortng to mprove the clusterng performance. Takng full advantage of segmentaton sortng of the clusterng algorthm based on mnmum spannng tree, the algorthm use a varety of methods for dfferent stuatons to sort these cluster elements wth ther smlarty and segment them where there are large changes n ther smlarty to obtan cluster results. In order to compare the performance of the method, we select some tradtonal cluster analyss methods lke k-means, herarchcal clusterng and densty clusterng wth nose data, etc. In the expermental testng, we select three sets of two-dmensonal artfcal data sets and four sets of real data sets as test data. And three evaluaton ndexes are appled to measure the qualty of clusterng. The smulaton results n test data show that ths algorthm can mprove the accuracy of the algorthm effectvely and acheved good clusterng performance. Keywords-smlarty; pont sortng; segmentaton clusterng; Wavelet flter I. INTRODUTION luster analyss [1, 2] method s a procedure that all the elements n a set wll be dvded nto multple sets consderng some crtera, and the elements are dvded nto the same set have more sgnfcant smlarty than those n dfferent sets. Namely, the elements concentrated n the same set, whch s usually called a cluster, are more smlar wth ther smlarty or dstance as the dvson standard. ommonly used cluster analyss methods can be roughly dvded nto fve categores that are classfed clusterng, herarchcal clusterng [3, 4, 5], densty clusterng, grd clusterng [6], clusterng model. These categores are represented as one or more of specfc analyss methods, such as k- means clusterng [7, 8, 9] s a representatve of classfed clusterng, and DBSAN [10, 11] s a representatve of densty clusterng, etc. These clusterng methods are wdely used n the prevous studes, and are appled to dfferent stuatons consderng ther own specally advantages and dsadvantages. In the artcle, by our understandng of the nature of cluster analyss, the clusterng problem s converted to arrange the data elements n a certan order to a onedmensonal array based on the smlarty and segment them to cluster groups accordng to certan rules. II. SIMILARITY SEGMENTATION BASED ON POINT SORTING A. Algorthmc Thnkng As mentoned n the ntroducton, the cluster analyss s a procedure that all the elements wll be dvded nto multple sets, and the data n the same set after the dvson has more sgnfcant smlarty, whle the data n dfferent sets has a lower degree of smlarty. The mathematcal descrpton of the concept s as follows: Defnton 1. Set U as a lmted data set, the set contans n elements 1, 2, 3,..., n, each element contans m propertes. Namely can be expressed as the vector form 1, 2, 3,..., m, d 1, 2 s the dstance between, j. lusterng s the process of dvdng set nto K non-empty set, 1, 2, 3,..., k U 1, 2, 3,..., n,,,..., k by a certan rule, and the set, the followng condtons: 1) 1,2,3,..., k 2) 1,2,3,..., k, j 1,2,3,..., k, j 3) k j U 1, need to satsfy Accordng to the defnton 1 and the data n the same set after the dvson has more sgnfcant smlarty, whle the data n dfferent sets has low smlarty. lusterng should be subject to the followng constrants n the deal condtons:,,,, d, d, m p n q j m n m p 1,2,3,..., k, j 1,2,3,..., k, j, m, p 1,2,3,..., card, n, q 1,2,3,..., card. j The dstance of any two elements n the same class should be shorter than these n dfferent classes. Assumng that one-dmensonal array orderly formed wth cluster labels by sortng the clustered elements under deal condtons s as follows: k Now suppose the cluster has m elements, 1 namely 1, 1 2,..., 1 m 1 has n elements, namely 1, 1,..., 1 n 1 2 2, 2,the sortng order of the elements n The authors - Publshed by Atlants Press 475

2 adjacent clusters and 1 2 n the above array s as follows: 1 m 1 1 m After the contnuous calculaton of the dstance between the adjacent element nodes n the array, t s obvous to conclude that d m 1 d j, j 1,2,3,..., card 2, j. 1, 2 2, 2, In other words, calculatng all the dstances between every adjacent elements and segmentng the largest one wll lead to get n+1 clusters. Of course, the above stuaton s under deal condtons, but the actual stuaton whch s often encountered wth s that the dstance between the elements cannot work well to explan the smlarty between each par elements. For example, the elements n the two-dmensonal plane n the cluster, the most ntutve and commonly used Eucldean dstance s not very good to reflect element densty dstrbuton, whle the densty dstrbuton of the elements s a key reference for a lot of clusterng data. And when usng twodmensonal data s processed by the Eucldean dstance, t often happens that some elements are lkely to have the same dstance to two or more classes, namely d m, n d m, p, t s not n conformty wth the assumpton of deal condtons. Even so, we usually thnk that t stll meets the condton that s d 1 m, 21 d 1 m, 1 t, t 1,2,3,..., card 1 and the average dstance between the element and all the other elements n same the cluster should be shorter than the average dstance between the elements and all elements n a dfferent cluster, whch s concluded as follows: card 1 card 2 d 1, 1 d 1, 1 1 m t 1 m r (1) card 1 card 2 t 1,2,3..., card 1, r 1,2,3..., card 2 In the above condtons, clusterng can be performed accordng to ths method that the nodes n a array are sorted by the dstance, and then the dstance between the adjacent nodes s calculated, the segmentaton s appled n the maxmum dstance. Ths method can be realzed by a two-category algorthm as a result of the assumpton that the dstance between elements n the same cluster s shorter or equal to the dstance between the elements n dfferent clusters. Frstly, the dstance of each par elements and the sums of all dstances from every element to others are calculated successvely, and then the maxmum value of the sums of the dstance from an element to all the others s added n the head of an one dmensonal array, the element s noted as E1, then another element, noted as En whch has the maxmum dstance to E1 has been found out and put nto the tal of the one-dmensonal array. Under these condtons, E1 and En, should not be dvded n the same class n any case. In the next step, an element whch s the nearest dstance from array head lke E1(or an array of tal) s found out and nserted nto the poston where s adjacent to array head (or the tal of the array), noted as E2. An element whch has the nearest average dstance from all elements of the array head (or the tal of the array) s found out and nserted nto the poston where s adjacent to array head (or the tal of the array), untl all the elements n the array are arranged nto the approprate poston n ths order. alculate the dstance of the adjacent nodes n the sorted array, segmentaton n the maxmum dstance, and two clusters are formed at last. If there s a need to get more clusters, run the algorthm agan n one of the exsted clusters, then there wll be one more cluster obtaned, and so on. The above algorthm s based on the deal condtons that the dstance between elements n the same cluster s shorter or equal to the dstance between the dfferent clusters. However, n many non-convex data, especally n the uneven densty dstrbuton of the data set, the constrants cannot be effectvely guaranteed. Therefore, n the actual stuaton, accordng to two strateges, whch are that the dstance between the adjacent the clusterng center s mnmum and the average densty of the adjacent elements s maxmum, to sort all elements n dfferent clusters to the array by one of the strateges, and the clusterng segmentaton pont s stll n the longest dstance between the adjacent nodes n the array. All the cluster centers can be regarded as a complete graph, the recprocal of the dstance between centers s used as the weght to get mnmum spannng tree [12], use the weght to traverse the mnmum spannng tree n depth frst or breadth frst, the traversal results wll be put n a array. The procedure mentoned above s to complete the process of pont sortng. The dstance between the adjacent elements n a sorted array s calculated, segment n the maxmum dstance. Then the clusterng results are obtaned. B. Algorthmc Descrpton Smlarty algorthm based on ponts sortng has two steps ncludng ponts sortng and, dfferent strateges can be adopted n each step for dfferent stuatons. 1) Pont Sortng: The multdmensonal dstance between the elements can be mapped nto a onedmensonal dstance n an array by ponts sortng. Dfferent sortng methods can be adopted for dfferent data sets, the sortng result from the exstng cluster analyss method can also be used. For example, herarchcal clusterng produces nested and dsjont tree (Fg. 1) whle t s also sortng the element nodes n the data set. Although sometmes ths knd of sortng s arbtrary, t can stll reflect the degree of smlarty between nodes to a large extent. The sortng result s to put nto a one-dmensonal array and the adjacent elements nodes n the array usually have a hgh smlarty. The applcat on based on ths method wll be further descrbed later. Pont sortng can also be sorted agan by the sortng results of the later cluster analyss method. The ultmate goal of ponts sortng s to make two nodes wth a more closely dstance between adjacent nodes. At ths pont, some of the adjacent nodes are close enough to each other can be thought n the same cluster. The maxmum dstance area between adjacent nodes s called clusterng segmentaton. 476

3 Herarchcal luterng Tree Fgure 1. The order of 7, 12, 8, 13, 1, 2...whch s below the herarchcal clusterng can be nput one dmensonal array to form a sort of element nodes. 2) Segmentaton lusterng: For the segmentaton clusterng of pont sorted array, calculate the dstance between each adjacent nodes n the sorted array, the dstance between adjacent nodes become drawn curve, as shown n Fg. 2. The dstance between the adjacent nodes wth strong ampltude curve means ths dstance between several nodes on both sdes of the change s longer enough to regard ths poston, as a clusterng segmentaton pont. Ths correspondng relatonshp between the dstance curve and the smlarty matrx of adjacent nodes are shown n Fg. 3. Dstance Dstance Element Node Fgure 4. Dstance curve between the adjacent nodes fltered by usng db2 wavelet. After wavelet de-nosng, the curve has become relatvely smooth, so that the dstance between the adjacent nodes changes obvously. You can gve a certan threshold and clear the part of the curve whch s less than the threshold. That means that the several elements whose dstance between the adjacent nodes s less than a certan threshold are n the same cluster and they should not break up nto others. The threshold value s typcally the average of the ampltude of the curve. Ths curve has been dvded nto a number of segments whch are dscrete, as shown n Fg. 5. Dstance Element Node Fgure 2. Dstance curve between the adjacent nodes. The dstance between the adjacent nodes curve and the correspondng relaton of smlarty matrx Element Node Fgure 5. After fltered, dstance curve between the adjacent nodes s dvded nto several dsconnected segments. Fgure 3. The maxmum dstance between adjacent nodes whch s also regarded as the maxmum ampltude s located n smlarty matrx for the most sutable for clusterng segmentaton. In ths case, to fnd the clusterng segmentaton ponts s to search the maxmum dstance between the adjacent nodes. All segmentaton ponts are sorted and they wll be the preferred pont of dvson when the dstance s larger. In other words, clusterng for dchotomous classes s to select the frst sorted ponts for segmentaton and for three classes s to select the top two sorted ponts for segmentaton and so on, untl the requred number of clusters are obtaned. 477

4 III. SIMULATION EPERIMENT A. Experment Test Data Set For expermental nput data format, the rows represent dfferent elements and the columns represent dfferent attrbutes of the element. The expermental output data are one-dmensonal array by consstng of cluster labels wth correspondng to the row elements of the nput data. For non-standard partton data and twodmensonal data, the result s estmated by subjectve judgment of the observer by a graphcal representaton of the results. In the experment, n order to compare the feasblty and performance of algorthm based on pont sortng, specally selected three groups of two-dmensonal artfcal data sets, whch are three rectangular data wth vsble boundary (Fg. 6), a data set based on Gaussan dstrbuton (Fg. 7), and a clusterng data sets based on the dstncton of densty (Fg. 7), and the other four groups are real data sets commonly used n studes of cluster analyss. The four groups of data are rs, alcohol, breast cancer, heart dsease data sets whch are from the Unversty of alforna campuses ear Bay (UI) machne learnng [13, 14] database. For the data defned by exsted crteron, the qualty of clusterng results was measured by Rand Index [15] and Adjusted Rand ndex [16]. In experments made by smple pont sortng, we choose the data set more easly dvded to verfy the feasblty of pont sortng algorthm, and then use the algorthm to process the real data sets wth the comparson result to other methods. The next experment manly tests the effectveness of clusterng algorthm of smlarty segmentaton based on adjacent nodes, usng three (sngle connecton, average and, complete connecton) herarchcal clusterng method, and usng two knds of dstance (Eucldean dstance [17], the Standard Eucldean dstance [17] respectvely for the three methods. For the twodmensonal data use Eucldean dstance and for hghdmensonal data use Eucldean dstance and the Standard Eucldean dstance to deal wth. To compare the nfluence on the result, we use the clusterng method of segmentaton based on adjacent nodes smlarty, not only when comparng wth classfcaton number for the standard partton, but also n the number of nonstandard classfcaton dvson. Fnally, clusterng algorthm of smlarty segmentaton based on pont sortng wll be verfed by the experment. Fgure 6. Three rectangular data set. Fgure 7. The data set based on the Gaussan Dstrbuton and the dstncton of densty. B. Smple Pont Sortng Method As mentoned earler, the appled pont sortng method s vared for dfferent data and the smlarty. In ths secton, ths artcle wll use a relatvely smple method of pont sortng for cluster analyss to prove the feasblty of the pont sortng method tself. Ths smple sortng method s frstly to fnd the farthest node from all of the nodes n the data set and put the node nto the frst poston of sortng array. Then put the node whch s the farthest away from the frst sorted node nto the last poston of sortng array. Next nsert the node whch s the closest to the frst or the last node n the sortng array nto the adjacent sde of the frst or the last node n the sortng array, n order to form an orderly array and complete the pont sortng process. Fnally fnd the maxmum dstance between adjacent nodes to segment n sorted array to form two clusters. As a result, the clusters have been formed through the teratve process for mult-classfcaton. The specfc process s as follows: 1) Generate an empty array whose sze s the number of all elements n the data set. 2) Fnd the farthest node from the sum of all the elements of a data set and put t n the frst poston of the array to store. 3) Fnd the farthest node from the frst sorted node and put t n the last poston of the array to store. 4) Judge whether the array s full. If t s full, go to step 5, otherwse go to step 6. 5) alculate the dstance between adjacent nodes n the array and segment the maxmum dstance, then judge whether the number of clusters has been reached to the requrements, f meet the requrements, then the algorthm ends, otherwse go to step 7. 6) Fnd out the node whch s the closest to the frst or the last node n the sortng array and nsert nto the adjacent sde of the frst or the last node n the sortng array, repeat step 4. 7) ompare the dstance between adjacent element nodes on both sdes of the clusterng segmentaton ponts. Fnd out the maxmum dstance as the next teraton of the data set, return to step 1. Usng the data sets of three rectangular and data sets based on Gaussan dstrbuton whch are more easly to dstngush n the experment, manually set the clusterng results for 3 clusters and the smlarty measure usng the Eucldean dstance, the clusterng results are shown n Fg

5 obtaned by usng adjacent nodes based on smlarty method and orgnal herarchcal clusterng method, the results are shown n Fg. 9. Eucldean dstance Standard Eucldean dstance Irs Average connecton herarchcal clusterng Average connecton smlarty Average connecton herarchcal clusterng Average connecton smlarty Fgure 8. Treatment of the three rectangular data set and the data set based on the Gaussan Dstrbuton usng smple ponts sortng algorthm.. Herarchcal lusterng ombned wth Segmentaton Herarchcal clusterng method tself has a functon of ponts sortng, although ths knd of ponts sortng may be arbtrary to some extent. On the whole, t stll meets the requrements that the dstance between the adjacent nodes should be close n ponts sortng. In ths secton, n order to valdate the feasblty and actual effect on smlarty between adjacent nodes usng the exstng pont sortng method, we get the sortng result from sngle connecton herarchcal clusterng method and average herarchcal clusterng method, then splt sequencng nodes array to use of smlarty method and obtan the clusterng results fnally. Its algorthm process s lsted as follows: 1) Frstly form herarchcal cluster tree usng herarchcal clusterng method, get a group result of ponts sortng after preorder traversal for the leaf node of the tree. 2) Put the sortng result nto an array. 3) Draw dstance curve of adjacent nodes. 4) Wavelet flterng and de-nose processng of the dstance curve. 5) Return to zero for the parts below the global average and get the dstance curve of sectonal adjacent nodes. 6) Fnd out the poston of maxmum dstance between the adjacent nodes n every curve and dvde t, namely clusterng segmentaton pont. 7) Order all clusterng segmentaton ponts accordng to the dstance, larger dstance ponts segmented frstly. 8) Select N segmentaton ponts of the largest dstance and go on segmentaton to form N+1 clusters. In ths experment, we use three sets of real data sets, namely rs, alcohol and breast cancer data. For the dstance between nodes, we use the Eucldean dstance and the standard Eucldean dstance to perform the experments n each set of data. The expermental evaluaton ndex s measured accordng to accuracy. The number of the clusters we have taken s 2 to 5, then calculate the accuracy for every cluster. ompare the algorthm performance even f the number of clusters does not equal to the standard dvson (whch s often encountered n practce, because the number of standard dvson clusters cannot always be predcted n advance). Makng a comparson between the clusterng results Wne Breast cancer Average connecton herarchcal clusterng Average connecton smlarty Average connecton herarchcal clusterng Average connecton smlarty Average connecton herarchcal clusterng Average connecton smlarty Average connecton herarchcal clusterng Average connecton smlarty Fgure 9. omparson of accuracy wth Herarchcal clusterng and smlarty between the adjacent nodes. From the above expermental results, wthout changng the pont sortng, the result of usng of the smlarty between adjacent nodes s sgnfcantly better than the dvson result of herarchcal clusterng method n most cases. That s manly because the use of wavelet flterng for smoothng treatment of dstance curve between adjacent nodes can weaken the nose and the mpact of abnormal nodes on clusterng results. We do not make a separate dvson of abnormal nodes for the small sze. We thnk t can have a better grasp of the whole for such data sets, but t wll also weaken the potental of algorthms n anomaly detecton and other aspects. D. K-means algorthm combned wth pont sortng and segmentaton lusterng Pont sortng clusterng methods not only can be used as a sngle cluster analyss method, but also can be combned wth other clusterng methods to complement and optmze the exstng methods. In ths secton, combned the pont sortng clusterng method wth the K-means algorthm, the experment s carred on to compare wth the results of K-means algorthm. The specfc process combned wth the K-means for pont sortng and clusterng segmentaton s as follows: 1) Use k-means algorthm to get N clusterng groups. 2) alculate the center of these N clusters. 3) hoose two closest centers of these N clusters, marked as a andb. The two clusters are marked as A and B. 4) Judge f A s from other clusters mergng, f t s, then go to step 6, else go to step 5. 5) alculate and merge A and B, sort the nodes nsde the clusters to form a new cluster by the formula A, B, then go to step

6 6) alculate the dstance of each par of b, a ' and b ' ( a ' and b ' are formed from last teraton), merge cluster A and B, f a' b bb' and a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula B, A', B', ; when a' b bb' and a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula A', B', B, when a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula A', B, B'. 7) heck f there are other clusters needed to be merged, f yes, return to step 1; else, ordered array has been formed, then go to step 8. 8) Go on for the ordered array and get the clusterng result. The expermental data s frstly used by ntutve two-dmensonal data based on the Gaussan dstrbuton and the densty-based dvson. In the experments, the data s frst to cluster nto 10 clusters usng K-means algorthm, and then the data based on the Gaussan dstrbuton s clustered nto 3 clusters, whle the data base on the densty nto 2 clusters, the expermental results are shown n Fg. 10 and Fg. 11: Fgure 10. The clusterng result of the data set based on the Gaussan Dstrbuton and the data set based on dstncton of densty obtaned by usng K-means algorthm. Fgure 11. The clusterng result of the data set based on the Gaussan Dstrbuton and the data set based on dstncton of densty obtaned by usng K-means algorthm combned wth ponts sortng. In Fg. 10 and Fg. 11, we clearly see that k-means algorthm, whch was unable to orgnally handle the non-convex data, have been able to work well wth densty-based dvson of data sets after combnng of pont sortng, and the performance n the data set based on the Gaussan dstrbuton of s also greatly mproved. Fgure 12. The clusterng result of the data set based on dstncton of densty obtaned by usng DBSAN algorthm and K-means algorthm combned wth ponts sortng. Also t needs to pay attenton that the k-means method tself s unstable, you cannot always get the best clusterng results. But even so, the results from k-means algorthm combned wth pont sortng segmentaton clusterng are more satsfactory than the orgnal method. It can be found from the comparson between Fg. 11 (rght) and Fg. 12 (rght) (n the fgure usng a denstybased clusterng algorthm for comparson). At the end of ths secton, 4 groups of real data used n ths paper usng the K-means clusterng segmentaton method combned wth pont sortng. Usually the performance on heart dsease data s worst wth the K- means clusterng algorthm. In the experment, frstly, usng the K-means method to cluster the heart dsease data nto 8 clusters, and then nto 2 clusters wth the based on pont sortng. The accuracy and Rand ndex reached respectvely 71.62% and 63.77%. It has been great ncreased, compared wth the 52.57% and 49.91% for heart dsease data analyzed by the orgnal K-means algorthm. The expermental results prove that performance and adaptablty of the K- means algorthm combned wth of pont sortng have greatly mproved. E. Herarchcal luster combned wth the Methods of Ponts Sortng and Segmentaton lusterng Because the result of ponts sortng obtaned from the herarchcal cluster tree s usually arbtrary, t can only show the successvely order of merged clusterng and can t change the order of nodes merged. In other words, t can t change the order of the nodes whch are merged earler and can t nsert the new mergng pont of the cluster nto the cluster havng been merged. Then we can consder the changes of clusterng center merged every tme when they were merged. Thus before mergng operaton, we can frstly calculate the current clusterng center, and compare the dstance between the clusterng center. When the current clusterng center was needed to be merged at now locates between the two clusterng centers be merged last tme, we nsert the current clusterng nodes nto them for nodes sortng. Ths s equvalent to backtrack the process of herarchcal clusterng to a certan extent. In ths experment, we use average connecton herarchcal clusterng based on clusterng order to modfy sortng method. entrod s used nstead of clusterng center for smplcty. entrod of the clusterng s as follows: 1 n x 1 (2) n 480

7 Its algorthm process s lsted as follows: 1) Fnd out two clusters needed to be merged usng average connecton herarchcal clusterng method and marked as A and B, then calculate the clusterng center of the both clusters, marked as a and b. 2) Judge f A s from other clusters mergng, f t s, then go to step 4, else go to step 3. 3) alculate and merge A and B, sort the nodes nsde the clusters to form a new cluster by the formula A, B, then go to step 5. 4) alculate the dstance of each par of b, a ' and b ' ( a ' and b ' are formed from last teraton), merge cluster A and B, f a' b bb' and a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula B, A', B', ; when a' b bb' and a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula A', B', B, when a' b a' b', sort the nodes nsde the clusters to form a new cluster by the formula A', B, B'. 5) heck f there are other clusters needed to be merged, f yes, return to step 1; else, ordered array has been formed, then go to step 6. 6) Go on for the ordered array and get the clusterng result. In order to test the algorthm presented n ths secton has a better grasp of clusterng and the ablty to resst nose, the data set based on densty dstncton s frstly used. Then we apply three herarchcal clusterng methods, dvde the data set nto two to fve clusters and dsplay the result of classfcaton. ompare t wth the result obtaned by usng smlarty segmentaton algorthm based on pont sortng presented n ths secton. The expermental result s showed n Fg. 13: The expermental results n above show that the three herarchcal clusterng result s satsfactory when dvdng the data set nto fve clusters usng sngle connecton herarchcal clusterng. However, no matter how many clusters are dvded usng the other two herarchcal clusterng, the results cannot meet the requrements. By compared to sngle connecton herarchcal clusterng, and dvdng the data set nto two clusters usng smlarty based on ponts sortng, we stll can get a satsfyng result. We can get a better result than the result of herarchcal cluster even when choosng wrong clusterng number. After choosng proper ponts sortng algorthm, average herarchcal clusterng method that can t be used to deal wth non-convex data has a good ablty of dealng wth the expermental data. In addton, t can have a better grasp of clusterng than sngle connecton herarchcal clusterng and are much less exposed to the nose nterference. Then we make a full test of performance of the algorthm proposed n ths secton usng three groups of real data sets. We perform smlarty measure for every data set employng Eucldean dstance and standard Eucldean dstance and calculate the accuracy respectvely when dvded nto two to fve clusters usng all algorthms. Then we choose the one havng the best performance and consder the results from the other several mportant clusterng methods whch are Sngle onnecton Herarchcal lusterng(sh), Average onnecton Herarchcal lusterng(ah), omplete onnecton Herarchcal lusterng(h), K-means lusterng(kms), Densty-based Spatal lusterng of Applcatons wth Nose(DBSAN), Markov lusterng(m), Affnty Propagaton lusterng(ap), Herarchcal lusterng combned wth Pont Sortng(HPS), Smple Pont Sortng (SPS) as a comparson. At last, we draw the graph and the result s showed n Fgure Segmentaton lusterng of smlarty based on pont sortng Average connecton herarchcal clusterng Sngle connecton herarchcal clusterng omplete connecton herarchcal clusterng Fgure 14. omparson of the accuracy of smlarty calculated by usng Eucldean dstance and Standard Eucldean dstance. 2 class 3 class 4 class 5 class Fgure 13. The data set based on dstncton of densty s dvded nto two to fve clusters by usng three herarchcal clusterng methods and smlarty based on ponts sortng. Fgure 15. omparson of the Rand ndex of smlarty calculated by usng Eucldean dstance and Standard Eucldean dstance. 481

8 Fgure 16. omparson of the Adjust Rand ndex of smlarty calculated by usng Eucldean dstance and Standard Eucldean dstance. As you can see from the above experment results, the accuracy, Rand ndex and Adjusted Rand ndex n the treatment of the data sets of rs, alcohol, breast cancer wth the algorthm of based on pont sortng s much better than the rest of the clusterng algorthms. It well llustrates that the performance of the clusterng algorthm based on pont sortng s very good and can be used n the applcaton and research. Especally n the structure of the data set wth pror knowledge, t can use the correspondng pont sortng method to get best functon. IV. ONLUSIONS The cluster algorthm of smlarty segmentaton based on pont sortng was proposed n ths paper, ts key step s usng certan rules for the dataset element nodes. The elements are mapped to a set of onedmensonal array n orderly arrangement and calculated the dstance between neghborng nodes n the ordered set, and these dstance are used to segment clusterng n large varatons of the smlarty. Namely the two core steps that are pont sortng and can be used alone to mprove the exstng methods, and also can be used n combnaton wth other cluster method nto a new cluster analyss method. Experments demonstrated the effectveness of each of these two steps, respectvely. But one of the major problems s the selecton of the smlarty n cluster analyss and data structures [18] matchng. The obtaned results may vary greatly for the same data set wth the same algorthm processng, when usng the dfferent smlarty measures. How to use the most approprate pont sortng for the data sets and smlarty of the known characterstcs to acheve a mnmum measurement error s stll pendng further study. AKNOWLEDGMENT Ths work was supported by the Natural Scence Foundaton of hna (Grant No ) and Development Project of Jln Provnce of hna (Grant No J). REFERENES [1] Tryon R. luster analyss: correlaton profle and orthometrc (factor) analyss for the solaton of untes n mnd and personalty. Edwards brother, Incorporated, lthoprnters and publshers, [2] R. B, The descrpton of personalty: basc trats resolved nto clusters, J. Abnorm. Soc. Psychol., vol. 38, no. 4, pp , [3] R. Sbson, SLINK: an optmally effcent algorthm for the sngle-lnk cluster method, omput. J., vol. 16, no. 1, pp , [4] D. Defays, An effcent algorthm for a complete lnk method, omput. J., vol. 20, no. 4, pp , Jan [5] J. H. W. Jr, Herarchcal Groupng to Optmze an Objectve Functon, J. Am. Stat. Assoc., vol. 58, no. 301, pp , [6] M.-. Su and.-h. hou, A modfed verson of the K-means algorthm wth a dstance based on cluster symmetry, IEEE Trans. Pattern Anal. Mach. Intell., no. 6, pp , [7] J. MacQueen, Some methods for classfcaton and analyss of multvarate observatons, n Proceedngs of the ffth Berkeley symposum on mathematcal statstcs and probablty, 1967, vol. 1, pp [8] S. Lloyd, Least squares quantzaton n PM, IEEE Trans. Inf. Theory, vol. 28, no. 2, pp , [9] E. W. FORGY, luster analyss of multvarate data : effcency versus nterpretablty of classfcatons, Bometrcs, vol. 21, pp , [10] M. Ester, H.-P. Kregel, J. Sander, and. u, A densty-based algorthm for dscoverng clusters n large spatal databases wth nose., n Kdd, 1996, vol. 96, pp [11] M. Ankerst, M. M. Breung, H.-P. Kregel, and J. Sander, OPTIS: Orderng Ponts to Identfy the lusterng Structure, n Proceedngs of the 1999 AM SIGMOD Internatonal onference on Management of Data, New York, NY, USA, 1999, pp [12] Y. u, V. Olman, and D. u, lusterng gene expresson data usng a graph-theoretc approach: an applcaton of mnmum spannng trees, Bonformatcs, vol. 18, no. 4, pp , Apr [13] T. Kanungo, D. M. Mount, N. S. Netanyahu,. D. Patko, R. Slverman, and A. Y. Wu, An effcent k-means clusterng algorthm: analyss and mplementaton, IEEE Trans. Pattern Anal. Mach. Intell., vol. 24, no. 7, pp , [14] M. Melă and D. Heckerman, An Expermental omparson of Model-Based lusterng Methods, Mach. Learn., vol. 42, no. 1 2, pp. 9 29, Jan [15] E. B. Fowlkes and. L. Mallows, A Method for omparng Two Herarchcal lusterngs, J. Am. Stat. Assoc., vol. 78, no. 383, pp , [16] L. Hubert and P. Arabe, omparng parttons, J. lassf., vol. 2, no. 1, pp , Dec [17] Deza M M, Deza E. Encyclopeda of dstances. Sprnger Berln Hedelberg, [18] Jan A K, Dubes R. Algorthms for clusterng data. Englewood lffs: Prentce hall,