Li Zheng2 School of Management Fujian University of Technology Fujian Province, , China

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1 Iteratioal Coferece of Orgaizatioal Iovatio (ICOI 07) Usig Fuzzy FP-Growth for Miig Associatio Rules Chie-Hua Wag* * Correspodig author: wagthuck@gmail.com Li Zheg zzheglimail@gmail.com Xuelia Yu Yuxuelia_00@6.com XiDua Zheg @qq.com Abstract This paper aims to use fuzzy set theory ad FPgrowth derived from fuzzy associatio rules. At first, we apply fuzzy partitio method ad decide a membership fuctio of quatitative value for each trasactio item. Next, we implemet FP-growth to deal with the process of data miig. I additio, i order to uderstad the impact of fuzzy FP-growth algorithm ad other fuzzy data miig algorithms o the executio time ad the umbers of geerated associatio rule, the experimet will be performed by usig differet thresholds. Lastly, the experimet results show fuzzy FP-growth algorithm is more efficiet tha other existig methods. Keywords data miig; fuzzy associatio rule; FP-growth I. INTRODUCTION Data miig is a methodology for the extractio of ew kowledge from data. This kowledge may relate to problems that we wat to solve []. Thus, data miig ca ease the kowledge acquisitio bottleeck i buildig prototype systems []. If data miig extractig ca effectively be applied o all varieties of aalysis, the process of decisiomakig i busiess will be fairly smooth. I commo trasactios, the associatio rule (X Y) is the most popular mea. The purpose is to search for the relatio that exists amog items of database. The relatio reflects that exists (X) appear, other items(y) are likely to appear as well []. For istace, whe a customer purchases bread, oe might also get milk alog with it. Thus associatio rules ca assist decisio makers to scope out the possible items that are likely to be purchased by cosumers. Meawhile, it facilitates plaig marketig strategies []. This paper maily focuses o the associatio rule techique. I the covetioal associatio rule algorithm, scaig database takes eormous time particularly whe oe uses Apriori algorithm, which ofte affects the efficiecy i data miig. To solve the drawback aforemetioed, Ha et al. [] proposed a miig method, called Frequet-Patter growth (FP-growth), which does ot eed to geerate cadidate itemsets ad is cosidered more efficiet. FPgrowth is costructed by readig the data set oe trasactio at a time ad mappig each trasactio oto a path i a Frequet Patter-tree (FP-tree). Sice differet trasactios ca have several items i commo at the same time, their paths may overlap. The more paths overlap with oe aother, the more compressio we ca achieve by usig the FP-tree structure. If the size of the FP-tree is small eough to fit ito the mai memory, we ca extract frequet itemsets directly from the structure i memory istead of makig repeated passes over the data stored o disks []. Therefore, without geeratig cadidate itemsets, oe oly eeds to sca the database twice. Additioally, i regard to the matter of decisio makig, oe has to take users' perceptio ad cogitive ucertaity of subjective decisios ito cosideratio. Zadeh proposed the Fuzzy Set Theory [] i 96 to deal with cogitive ucertaity of vagueess ad ambiguity. Sice liguistic variables ad liguistic values [6-8] ca be described with fuzzy cocepts to subjectively correspod with the possible cogitio of a decisio maker, they are hady i carryig out aalysis of decisios-makig. Fuzzy data miig has the recetly become a importat research matter. Therefore, this paper proposes a fuzzy data miig method - Fuzzy Frequet Patter Growth, which treats each item from a trasactio database as a liguistic variable, ad each liguistic variable is partitioed based o its liguistic value. I so doig, the atural laguage ca be utilized to fully explai fuzzy associatio rules. There are two phases i the proposed method. Oe is to fid frequet -itemset by scaig the database oce, ad the other is to establish a membership fuctio FP-tree by scaig the database twice. The, oe coditioal patter base ad oe coditioal membership fuctio FP-tree will be extracted from each ode i a membership fuctio FP-tree to geerate the fuzzy associatio rules. Copyright 07, the Authors. Published by Atlatis Press. This is a ope access article uder the CC BY-NC licese ( 8

2 II. FUZZY PARTITION METHOD The cocept of liguistic variables plays a fudametal role i describig fuzzy logic ad approximate reasoig. To deal with liguistic variables, the fuzzy partitio method [9, ] ca divide quatitative variables ito fuzzy sets with membership fuctios. For example, a liguistic age ca be partitioed ito youg, middle ad old. Ad their membership fuctios i [0, 60] are (0, 0, 0), (0, 0, 60), (0, 60, 60), respectively. Commo membership fuctios are triagular or trapezoid membership fuctios. I this paper, symmetric triagleshaped liguistic variables are used for simplicity. We assume that a quatitative variable a is partitioed ito K fuzzy sets { Ak,i, Ak,i,K, Ak,ik }, Ak,ik is the ith fuzzy grid ad is defied by the triagular membership fuctio [7-0]: µ k,i k ( x) = max{ x aikk i / b K i,0}, () where aik i = mi + (ma mi )(ik ) /( K k ), b K i = (ma mi k ) /( K k ), ad ma ad mi are the maximum ad miimum values of the domai iterval of Ak,i k. aiki is the top where k the membership degree is equal to ad b K i is the spread of the membership fuctio of Ak,i k. I the data set we use fuzzy partitio method to trasform quatitative attributes ito fuzzy grids. A quatitative attribute xk is represeted as: ( µ k,i Ak,i + µ k, i Ak,i +L+ µ k, ik Ak,ik ) Ak,ik : ikth liguistic value of Kk liguistic values defied i liguistic variable xk, where i K; µ k,i k : membership fuctio of Ak,ik ; tp: cout k,ik : cout kmax : Gkmax : pth trasactio data; the summatio of µ k,ik value for to ; the maximum cout value amog cout k,ik value, ik = to K; the fuzzy grid xi with cout kmax ; α: the user-specified miimum fuzzy support value; β: the user-specified miimum fuzzy cofidece value; SPr: the set of frequet legths with r attributes. B. The proposed algorithm INPUT: A body of trasactio data; each liguistic variable with K liguistic values; a user-specified miimum fuzzy support (mi FS) α; ad a user-specified miimum fuzzy cofidece (mi FC) β. OUTPUT: A set of fuzzy associatio rules. () usig the triagular membership fuctio defied by above formula, where Ak,i k is the ith fuzzy grid of K liguistic terms defied i attribute xk of pth trasactio data. Fig.. The membership fuctios are used i the example. III. NOTATIONS AND ALGORITHM The proposed costructio algorithm for geeratig fuzzy associatio rules from the trasactio database is described i this sectio. The otatios used i the proposed algorithm are firstly stated below. A example, e.g. TABLE I is give to illustrate the proposed algorithm behid steps. The assumptios of the membership fuctios for the item quatities are show i Fig.. Additioally, assume that the predefied miimum fuzzy support value ad miimum fuzzy cofidece value are 0. ad 0., respectively. A. Notatios : rhe umber of trasactio database; d: mumber of attributes used to describe each sample data, where m; xi: ith attribute; Ki: mumber of liguistic values i the ith attribute where i d; TABLE I. THE DATA SETS ARE USED IN THIS EXAMPLE ITMS (A, ), (B, 8), (D, ), (E, ) (B, 7), (C, ), (D, ) (A, ), (D, 7), (E, 6) (A, ), (B, ), (E, ) (B, ), (C, ), (D, 8) STEP : Trasform the quatitative attribute Ak,ik for each item xk i each trasactio datum tp (p = to ) ito a fuzzy set µk,ik. For example, the results are show i TABLE II. STEP : Calculate the cout of each fuzzy grid (liguistic term) µ k,ik i the trasactio data, ad the results are show i TABLE III. m cout k,ik = µ p = k,i k () 9

3 TABLE II. THE FUZZY SETS TRANSFORMED FROM THE DATA SET IN TABLE. ITMS (0.667/A.Low + 0./A.Middle), (0.8/B.Middle /B.Large) + (0./D.Low + 0./D.Middle) + (0./E.Middle /E.High) (.0/B.Middle), (0.8/C.Low /C.Middle), (0.667/D.Low + 0./D.Middle) (0.8/A.Low /A.Middle), (.0/D.Middle), (0.667/E.Low + 0./E.Middle) (0.667/A.Low + 0./A.Middle), (0.8/B.Low /B.Middle), (0./E.Low /E.Middle) (0./B.Low /B.Middle), (0.667/C.Low + 0./C.Middle), (0.8/D.Middle /D.Large) TABLE III. Item A.Low A.Middle A.Large B.Low B.Middle B.Large C.Low C.Middle THE COUNT OF THE FUZZY GRIDS. Cout Item C.Large D.Low D.Middle D.Large E.Low E.Middle E.Large oe aother based o the same trasactios. I the example, the fuzzy FP-tree is show i Fig.. Cout TABLE V. STEP : Fid coutkmax. Let Gkmax be the grid with THE NEW FUZZY SETS FROM TABLE. Fuzzy sets (0.8/B.Middle), (0./D.Low), (0.667/A.Low), (0./E.Middle) (.0/B.Middle), (0./D.Low), (0.8/A.Low) (.0/D.Middle), (0.8/A.Low), (0./E.Middle) (0.67/B.Middle), (0.667/A.Low), (0.667/E.Middle) (0.667/B.Middle), (0.8/D. Middle), (0./C.Low) coutkmax for item xi. G kmax will be used to represet this attribute i later FP-Growth miig processig. This step is repeated for the other items. Thus, Low is chose for A, Middle is chose for B, Low is chose for C, Middle is chose for D ad Middle is chose for E. STEP : Collect each attribute grid to form the -dim patter (SP). STEP : Check whether each coutkmax of each Gkmax is larger tha or equal to the predefied miimum support value α. If Gkmax satisfies the above coditio, put it i the set of frequet legths. That is: () SP = {Gkmax coutkmax α, i K } Here, miimum fuzzy support is 0., the its cout is.. So the couts values of A.Low, B.Middle, C.Low, D.Middle ad E.Middle are all larger tha., these -dim patters are put i SP. STEP 6: While frequet fuzzy grids (SP) are ot ull, do the followig steps. STEP 7: This step begis to proceed with FP-Growth. Establish a descedig data table called Header Table, show as TABLE IV. TABLE IV. HEADER TABLE. -dim patter B.Middle D.Middle A.Low E.Middle C.Low Cout STEP 8: Sca the fuzzy set table at the first time to rebuild a ew fuzzy set table, e.g. TABLE V, that the fuzzy sets ate sorted by the fuzzy grids from the Header Table. STEP 9: Set a ode {ROOT} of a fuzzy FP-tree. The the secod time to sca the ew fuzzy set table could geerate the odes of the fuzzy grids i the tree the lik the odes with Fig.. The membership fuctios for FP-tree. STEP 0: Mie the possible frequet patters through fuzzy FP-tree. Scaig from the bottom of Header Table derives coditioal patter bases of every item from the paths i the fuzzy FP-tree. Next, the coditioal membership fuctio FP-trees of every item could be costructed based o the coditioal patter bases. The, the whole set of possible patters of each item would be geerated from the correspodig coditioal fuzzy FP-tree. STEP : The followig correspodig frequet patters. substeps are doe for (a) Calculate the fuzzy value of each trasactio data tp i s as µ si = µ si Λµ si K Λµ si r +, where µ si is the membership fuctio of tp i grids sk. ad the miimum operator is used for the itersectio, the r + µ si = mi µ si k k = () (b) If fuzzy support for each frequet patter is calculated as 0

4 couts = µ si () Also, we simulate eve oe hudred thousad to compare these three algorithms which set miimum fuzzy cofidece at 0. The results are show i TABLE VI. (c) If fuzzy support value is larger or equal to the userspecified miimum support value α, put it is SPr. I the example, the possible patters of (B.Middle, C.Low), (A.Low, E.Middle), (D.Middle, A.Low), (B.Middle, D.Middle) are kept i SP. Therefore, they are frequet patters. STEP : Costruct effective associatio rules for each frequet patter ( s, s,..., sq, q ) usig the followig. (a) List all possible frequet patters. sλs K Λsq sl, l = to q (b) Calculate the cofidece values of all associatio rules usig µ si ( µ si Λµ si Λ L Λµ si q ) () STEP : Output the associatio rules with cofidece values larger tha or equal to the user-specified cofidece threshold β. Here, the mi fuzzy cofidece is 0.. Sice (A.Low, E.Middle) is larger tha 0., therefore this frequet patter is a effective rule. I fact, the followig four frequet patters form associatio rules ad are a output to users.. (C.Low, B.Middle) with fuzzy cofidece =.0: If a low umber of C is bought the a middle of B is bought with a cofidece of.0.. (A.Low, E.Middle) with fuzzy cofidece = 0.86: If a low umber of A is bought the a middle of E is bought with a cofidece of (E.Middle, A.Low) with fuzzy cofidece =.0: If a middle umber of E is bought the a low of A is bought with a cofidece of.0.. (A.Low, D.Middle) with fuzzy cofidece = 0.6: If a low umber of A is bought the a middle of D is bought with a cofidece of 0.6. The four rules above are the output as meta-kowledge cocerig the give trasactio. IV. EXPERIMENTAL RESULTS The sectio reports o experimets made to show the performace of the proposed approach. They were implemeted i VB o a Petium-IV.7 Persoal Computer. Trasactio database from a supermarket of a retail busiess i Kime, Taiwa, were used to show the feasibility. A total of 86 trasactios were icluded i the database. Each trasactio records the purchasig iformatio for a customer ad uses membership fuctio as Fig.. The experimetal results were made to compare the accuracy of the proposed fuzzy data miig algorithm. Hog et al.'s ad Hu's proposed algorithm ([, 6]) i which the miimum fuzzy support values set at 0. is show i Figure. Fig.. The compariso of the accuracy of three fuzzy data miig algorithms. TABLE VI. Mi FS THE EXPERIMENT RESULTS FOR THREE DATA MININGS ALGORITHMS. The proposed method Hu s mehod Hog et al s method From Fig., it is easily see that the accuracy of the proposed method was higher tha that of Hog et al ad Hu's proposed method for various miimum fuzzy cofidece values. I additio, from TABLE VI, it is easily see that miimum fuzzy support is smaller, which geerates more associatio rules. I execute time, the proposed method is the best tha the other methods, the miimum fuzzy support is smaller, especially. But smaller miimum fuzzy support does ot ifluece the whole execute time. With the miimum fuzzy support icreases, the execute time will be more obvious. V. CONCLUSIONS I this paper, we have proposed a fuzzy data miig algorithm, which combies fuzzy set theory ad FP-Growth to deal with quatitative values ad fid iterestig patters amog them. The rules mied represet quatitative regularity for large trasactio database ad ca be adopted to provide some suggestios to appropriate supervisors. The proposed algorithm ca solve disadvatages foud i Apriori ad ehace the whole efficiecy. The experimetal results with the data i a supermarket of retail busiess show the feasibility of the proposed miig algorithm. Whe compared with Hog et al. ad Hu's fuzzy miig method ([, 6]), our approach ca get better miig results due to the characteristic of FPgrowth. It ows feature of without cadidate geeratio, ad scas two for all process. The experimetal results proouced the proposed method is more excellet tha Apriori miig method.

5 I additio, as for defiitios of liguistic values, maagers ca select his ow preferece, ad refer to past experieces ad relate cogitive ability to desig a umber of liguistic values ad shapes, such as Gaussia distributio ad trapezoid membership fuctio. So, it cofirms a maager's cogitive i subject. However, it is ot easy for users to specify these parameters, the GA may be cosidered as a appropriate tool for automatically determiig the appropriate parameter specificatios. As regard to mi FS, mi FC ad partitio umbers i each quatitative attribute, these parameters are easily user-specified for idividual problems. Ad GA ows to automatically search for characteristic. I the future, we will combie GA ad the proposed method to solve various problem domais. [6] [7] [8] [9] [0] [] [] Refereces [] [] [] [] [] Berry, M., Lioff, G., Data Miig Techiques: for Marketig, Sales, ad Customer Support, Joh Wiley & Ss, Now York, 997. Ha, J. W., Kamber, M., Data Miig: Cocepts ad Techiques, Morga Kaufma, Sa Fracisco, 00. Ha, J., Pei, J., Yi, Y., Miig frequet patters without cadidate geeratio, I proceedig SIGMOD 00 Proceedig of the 000 ACM SIGMOD iteratioal coferece o Maagemet of data, pp. -, 000. Hog, T. P., Che, J. B., Fid relevat attributes ad membership fuctios, Fuzzy Sets ad Systems, vol. 0, pp.89-0, 999. Hog, T. P., Kuo, C. S., Chi, S. C., Trade-off betwee computatio time ad umber of rules for fuzzy miig from quatitative data, Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems, 9, pp , 00. [] [] [] [6] [7] [8] Hu, Y. C., Miig associatio rules at a cocept hierarchy usig fuzzy partitio, Joural of Iformatio Maagemet, vol., p. 6-80, 006. Hu, Y. C., Che, R. S., ad Tzeg, G. H., Fidig fuzzy classificatio rules usig data miig techiques, Patter Recogitio Letters, vol., pp. 09-9, 00. Hu, Y. C., Fidig useful fuzzy cocepts for patter classificatio usig geetic algorithm, Iformatio Scieces, vol. 7, pp. -9, 00. Ishibuchi, H., Nozaki, K., Yamaoto, N., Taaks, Selectig fuzzy if-the rules for classificatio problem usig geetic algorithm, IEEE Trasactios o Fuzzy Systems, vol., pp , 99. Ishibchi, H., Nakashima, T., Murata, T., Performace evaluatio of fuzzy classifier systems for multidimesioal patter classificatio problems, IEEE Trasactio o Systems, Ma, ad Cyberetics, vol. 9, pp , 999. Jag, J.S.R., ANFIS: adaptive-etwork-based fuzzy iferece systems, IEEE Trasactios o Systems, Ma, ad Cyberetics,, pp , 99. Myra, S., Web usage miig for web site evaluatio, Commuicatio of the ACM,, pp. 7-, ACM, New York, 000. Ta, P. N., Steibach, Michael, Kumar Vipi, Itroductio to Data Miig, Pearso Addiso Wesley, Bosto, 00. Wag, L. X., Medel, J. M., Geeratig fuzzy rules by learig from examples. IEEE Trasactios o Systems, Ma, ad Cyberetics, vol., pp. -7, 99. Zadeh, L. A., Fuzzy Sets, Iformatio Cotrol, vol. 8, pp. 8-, 96. Zadeh, L. A., The cocept of a liguistic variable ad its applicatio t approximate reasoig I, Iformatio Sciece, vol. 8, pp. 99-9, 97a. Zadeh, L. A., The cocept of a liguistic variable ad its applicatio t approximate reasoig II. Iformatio Sciece, vol. 8, pp. 0-7, 97b. Zadeh, L. A., The cocept of a liguistic variable ad its applicatio t approximate reasoig III, Iformatio Sciece, vol. 9, pp. -80, 976.