Classification Data Mining with Hybrid Fuzzy Logic Aggregation

Transcription

1 Classificatio Data Miig with Hybrid Fuzzy Logic ggregatio JOHN F. SNFORD Iformatio Systems Philadelphia Uiversity Philadelphia, P US [email protected] LES M. SZTNDER Computer Iformatio Systems Philadelphia Uiversity Philadelphia, P US [email protected] bstract - Fuzzy logic is applied to the category discrimiatio problem related to idetificatio of mammary lesios as beig or maligat. Results of other similar studies are reviewed. The curret aalysis expads the fuzzy logic approach by usig the ormal distributio fuctio as set membership fuctios ad usig a geetic algorithm to optimize performace with the traiig partitio. The approach is applicable to problems havig arbitrarily large umber of parameters. Two differet data sets are examied. Data is portioed ito a traiig set ad validatio set ad each set is segregated ito beig ad maligat records. Values of mea ad stadard deviatio are iitially computed from the associated attributes ad are differet for the beig ad maligat records. I oe traiig method the stadard deviatios are adjusted to miimize overall error. I a secod method a bias adjusts the importace of each membership fuctio. Defuzzificatio is accomplished i three ways: modified averagig ad OR process; compariso of multiplied fuzzy set values; ad compariso of the multiplied squared set values. Results are compared with results obtaied through statistical logistic regressio. Key Words: fuzzy data-aalysis discrimiatio statistical aalysis screeig 1. Itroductio 1.1 Geeral Sice its itroductio 1965 by Lotfi Zadeh [1] fuzzy logic has foud extesive applicatio i cotrol systems from small appliaces to cameras ad eve heavy equipmet. lthough cotrol systems are its forte, ot surprisigly it has foud its way ito most if ot all areas of data miig as a seriously competig aalytic tool. To this repertoire we would add umerous iformatio search ad utilizatio techiques that have spawed eve a fuzzy database query laguage developed by Takihashi i 1995 [2]. Fuzzy logic shows itself to great advatage whe it ca simplify a otherwise extremely difficult algebraic formulatio for a cotrol system without apparet loss of effectiveess. I fact the fuzzy approach may be more effective. For example i the automatic cotrol of a camera the iput parameters are light itesity, distace to object, motio of the object relative to the camera, ad capability of the recordig media. Of these the oly costat parameter is the capability of recordig media. Other parameters may exhibit variatios over a wide rage. Formulatio of a crisp (exact) cotrol system is quite difficult. The fuzzy approach classifies the iput parameters ito fuzzy sets. For example, these sets might be dim light, medium light, bright light, earby, some distace, ad remote distace. The task the is to determie the percet of membership that the actual parameter has i each associated set. For example light may have 80% membership i medium, 10 % membership i bright ad zero membership i dim. The settig of camera speed ad aperture the ISSN: Issue 11, Volume 10, November 2011

2 is based o liguistic rules. Oe such rule might be, If bright light ad remote distace ad zero relative motio the aperture settig is 16 ad speed is 1/100. other might be if medium light ad remote distace ad zero relative motio the aperture settig is 11ad speed is 1/100 Liguistic rules are logic statemets. They iclude itersectio ad uio. Typically these are resolved through a set of rules as follows. ND B where ad B are limited to the rage (0, 1), becomes equivalet to mi (, B) ad OR B, where ad B are limited to the rage (0, 1), becomes equivalet to max (, B). The capability of classifyig patters for subsequet decisio makig processes is oe of the most fudametal characteristics of busiess itelligece ad data miig. s a field of study, classificatio has bee evolvig sice the early 1950s, closely followig the emergece ad evolutio of computer techology ad classificatio techiques. s previously metioed fuzzy logic has bee applied to this type of problem. 1.2 Other papers i the area. There is a perfusio of articles o the topic of fuzzy data discrimiatio. We metio a few that deal with medical diagosis which is also the directio of the curret work. Cios J K, et al. [3] reported o Fuzzy logic used for classificatio of coroary steosis from plaar thallium-201 scitigraphs. The object was diagosis of steosis which might be occurrig i ay of the three major arteries, left aterior descedig (LD), right coroary artery (RC), the circumflex artery (CCX). The scitigraphy cosisted of three views, left vetricle-aterior (NT), left lateral (LT), ad aterior oblique (LO). These data are themselves fuzzy with cosiderable overlap i perfusio patters that might be created by steosis. Some other approaches to solvig this problem iclude: fuzzy sets with probability assiged membership fuctios described by Cios al i 1991[4], machie learig algorithms described by Cios al i 1994 [5] ad Cios et al i 1994 [6], ad a expert system approach described by Cios et al i 1990 [7]. Cios et al i 1994 [3] used a data set cotaiig 64 patiets 46 with steosis ad 18 without. They used trapezoidal membership fuctios for each of the three scitigraphy views. They evaluated three kow learig algorithms, LFS, CLILP2, ad EXP. They formulated 13 differet liguistic statemets to provide crisp results. The otatio used i formulatig the if-the statemets relates to the medical parameters described i the paper but oe statemet is preseted here as a demostratio of defuzzificatio approach. IF LT4 has perfusio defect value i the rage of [0,4], [15,34],[45,54],[85,1001] with (0.2) ND LT8 has perfusio defect value i the rage of [5,24],[35,44],[75,100] with (0.4) ND LT5 has perfusio defect value i the rage of [0,4] with (0.4) THEN STENOSIS is i CCX. They reported o the results of the three differet algorithms for defuzzificatio. ccuracy for classifyig ormal arteries raged from 96% to 100%. However accuracy for correctly classifyig steosis raged from 85% to 93% for two algorithms. third had lower accuracy. Malek J, et al [7] discussed a automated breast cacer diagosis system utilizig fuzzy aggregatio. They proposed a fuzzy istrumet for hardware implemetatio of the automated system usig a CMOS itegrated circuit. They reported o active research coducted with 200 patiets to prove the cocept ad fially proposed a cocept desig for the CMOS itegrated circuit. For the active research, cells were extracted from the patiets lesios ad fixed o microscope slides. For each patiet 19 attributes are aalyzed. The GVF- Sake model was used to idetify the cotour of a typical ucleus withi the object. Texture-aalysisbased feature extractio ivolved wavelet trasforms. Statistical attributes icluded mea, variace, etropy, ad ormalized Shao etropy. The process is mathematically itesive. The fuzzy clusterig C- meas algorithm as is available i MTLB was use to assig fuzzy set memberships. Each ew patter was assiged a degree of membership of maligat class or beig class. The sum of all membership fuctios at ay poit equals 100% They reported 97% correct classificatio of beig cases ad 93% correct classificatio of maligat cases. ISSN: Issue 11, Volume 10, November 2011

3 Bagher-Ebadia H, et al [8] i 2004 used a automated eural etwork ad fuzzy clusterig for diagosis of coroary artery disease. They report two studies, oe with 58 subjects ad oe with 115. Data was obtaied from plaar images i three projectios, aterior, left aterior oblique, ad left lateral usig a collimator. Images were take at rest ad agai uder stress. Backgroud oise was suppressed by segmetig myocardium from the backgroud usig the fuzzy clusterig Picard iteratio algorithm. Euclidea distace was used as the distace metric. Each myocardium was partitioed ito four sectios ad the ceter of activity foud for each sectio. Samplig radii were take every 2 degrees to project the sectors o a x-y plae. There were three trasformed projectios divided ito 4 regios for at rest ad the same for stress. Thus there were two vectors of dimesio12. The mea ad variace row vectors were defied as feature vectors. The umber of features was reduced through a selectio process based o maximum separatio i multidimesioal feature space The artificial eural etwork was traied ad the evaluated. The true values, ormal or abormal, were obtaied from coroary agiographies take withi three moths of the imagig. ccuracy i predictio positive results varied from 77% to 86% ad predictio of egative results from 63% to 66%. I 2010 Licata [10] preseted a most iterestig demostratio of fuzzy logic as applied to the more geeral diagostic process. This is differet from the previous citatios i that it does ot deal with data aalysis to idetify oe of two possible outcomes as a test for a specific disease. Rather, he suggested that the applicatio of fuzzy set theory i place of the physicia s usual reliace solely o probabilistic logic ca improve the geeral diagosis process which attempts to idetify the cause of observed symptoms. He developed the fuzzy set theory process through a example applicatio to oe case study. Specific symptoms i this case are: dyspea at rest (a), oedema (b), tachyarrhythmias (c), epatomegally (d), ascites (e), pleural effusio (f). Possible diagoses would be hepatic cirrhosis (t), ephrosic sydrome (u), peumoia (w), myocardial ischemia (x) cogestive heart failure (y), worseig of the supravetricular arrhythmias (z). typical If-the rule would have the form, If a ad b ad c ad d ad e ad f the t or w or x or y or Severity of these symptoms a, b, c, etc. are typically liguistic ad fuzzy such as tachyarrhythmias is slow, moderate, or fast. Numerous laboratory tests provide more data for assigmet of fuzzy membership fuctios. Liguistic observatios ad laboratory results are assiged set memberships betwee 0 ad 1 through the use of charts that have bee prepared. Defuzzificatio ivolves a cosiderable umber of predicate logic statemets. These are evaluated usig geeralized modus poes ad sometimes more directly. This likely ivolves a good deal more skill i logic tha most physicias would poses. Licata s work poits out that the fuzzy logic approach is i lieu of the probabilistic approach geerally employed. Licata suggests that the fuzzy set theory solutio is more exactig ad probably superior to the usual process where idividual decisios i the chai leadig to a coclusio are made o the basis of probabilistic estimates. This paper is of course ot able to preset accuracy figures sice there is oly oe case cosidered. It is recouted here as a idicatio of the pervasive ifluece of fuzzy logic o the medical disciplie. 1.3 Differet approach The work we preset i the curret paper offers aother variatio. It employs the ormal distributio as set membership fuctios. These fuctios are idividually adjusted by usig weight ad varyig each stadard deviatio. Optimizatio is obtaied through applicatio of a. Liguistic rules are ot employed ad hece the fudametal process is applicable to data sets cotaiig may parameters ad to classificatio ito more tha two categories. The process is applied to several differet data sets ad results are compared with the more commo statistical regressio aalysis. 2. alytic approach Step 1 of the process ivolves assigig data to memberships i the fuzzy sets. ISSN: Issue 11, Volume 10, November 2011

4 2.1 Triagular membership fuctio Numerous membership fuctios are recogized for creatig the fuzzy sets [11] These iclude triagle, rectagle, parabola, trapezoid, ad eve projectios of oe operatig characteristic oto aother plae. If, for example, a isosceles triagle is used the the height would geerally be cosidered equal to 1 ad the apex is located at the parameter s average value over the totality of past data. particular data record will geerally have a value for this parameter that is ot at the average. Membership will be determied by the height of the lie C i Figure 1. This height has a simple relatioship show i equatios 1 ad 2 B B OB O O (1) C = h = 1 = = 1 OB OB OB OB Membership = 1 = 0 O OB For O OB Otherwise It should be oted that the value of O eed ot be a exact match to the average value of the parameter for all existig data records. I some cases, particularly i cases ivolvig dyamic cotrol systems, the value of O may be desigated i accordace with egieerig aalysis or other cosideratios. compariso of performace with triagular membership fuctio ad the expoetial fuctio was reported i a earlier paper [14]. I this case some adjustmet was made to triagular base ad expoetial width i a effort to fid the better fuctio. The expoetial fuctio performed better i that istace but triagular, expoetial (or sie wave), ad trapezoidal are all commo. 2.2 Trapezoidal membership fuctios Suppose that there are geeralized fuzzy umbers 1, 2,...,, with trapezoidal membership fuctios i = (c i, a i, b i, d i, w i ). The trapezoidal membership fuctio of a geeralized fuzzy umber, i, is the give by: (3) i (x)= a i x b i (x-ai)/(bi-ai) b i x c i 1 c i x d i (d i -x)/(d i -c i ) 2.3 Expoetial membership fuctio Expoetial membership fuctios ad sig-wave fuctios have also bee employed. I view of the fact that all parameters i the data sets uder study here deal with physical attributes that are most likely ormally distributed, the ormal distributio suggested itself as a membership fuctio for the curret aalysis. This distributio is expoetial ad probability is measured as betwee zero ad oe. The membership fuctio has the happy property that o parameter will actually have zero membership but will have a dimiishigly small membership as its locatio diverges far from the mea. The probability of beig at the mea is ½ ad ot 1 so for o other reaso tha esthetics we divided our membership percet by ½ i order to produce membership of 1 for parameters actually at the mea. The mea for each membership fuctio is the mea of that parameter over existig data. Spread of the membership fuctio is achieved by adjustig the stadard deviatio. Before traiig this value is set at the stadard deviatio of that parameter over existig data. d membership percetage is determied by the probability of the tail. Equatio 3 describes this. for X (4) Membership = P( X ) /.5 = (1 P( X )) /.5for X > Thus for a give parameter such as thickess Stadard deviatio = σ Mea = Measured value = x If x > the membership = (1- P(x))/0.5 If x < the membership = P(x)/ Geeral ggregatio Operatios ggregatio operatios are operatios by which several fuzzy sets are combied ito a sigle set. I geeral, ay aggregatio operatio is defied by a mappig [14] ISSN: Issue 11, Volume 10, November 2011

5 (5) h:[0,1] [0,1] for >=2. Whe applied to fuzzy sets 1, 2,..., defied o X, h produces a aggregate fuzzy set by operatig o the membership grades, (x) of each x ε X i the aggregated sets. Thus, we have: (6) (x)=h ( 1 (x), 2 (x),. (x)) for each x εx: The followig requiremets express the essece of the otio of aggregatio, after Klir ad Yua (Klir ad Yua, 1995): xiom 1. h (0,0,...,0) = 0 ad h (1,... 1) = 1. xiom 2. For ay pair of umbers a i, b i, where a i ε [0,1], b i ε [0, 1], if a i b i for all i the h(a i ) h(b i ), that is, h is mootoic odecreasig i all its argumets. itersectio, respectively. Geeralized Dombi's operatios [13] possess properties of fuzzy uios ad itersectios, which will be utilized i this work. They are defied as follows: Dombi's Fuzzy Uio: where λ is a parameter by which differet uios are distiguished, ad λ ε (0, ) Dombi's Fuzzy Itersectio Two additioal axioms are also employed to characterize aggregatio operatios. xiom 3. h is a cotiuous fuctio, that is it guaratees that a ifiitesimal variatio i ay argumet of h does ot produce a oticeable chage i the aggregate. xiom 4. h is symmetric fuctio i all its argumets, that is h(a i ) = h(a p ) for ay permutatio, p, of the argumets, that is the aggregated sets are equally importat. Fuzzy uios ad itersectios ca be qualified as aggregatio operatios o oly two argumets, their property of associativity, provides a mechaism for extedig their defiitio to ay umber of argumets. lso, i the case of bi-symmetry ad strict mootoicity, the extesios are straightforward. Thus, geeralized fuzzy uios ad itersectios ca be viewed as special aggregatio operatios that are symmetric, usually cotiuous, ad are required to satisfy cojuctive ad disjuctive attitudes. s a result, geeralized fuzzy uios ad itersectios ca produce oly aggregates that are subject to the followig restrictios [12] (7) max (a, b) (a, b) max (a, b) where λ is a parameter by which differet itersectios are distiguished, λ ε (0, ). From the above iequalities, oe ca see that geeralized fuzzy uios ad itersectios do ot produce ay aggregates that geerate values betwee mi ad max. Hece the geeral rules previously stated. ND B where ad B are limited to the rage (0, 1), becomes equivalet to mi (, B) ad OR B, where ad B are limited to the rage (0, 1), becomes equivalet to max (, B). ggregates that are ot restricted i this way, however, are allowed by xioms 1 through 4. Operatios that produce them are called averagig operatios. There are several classes of averagig operatios are a compromise betwee mm ad mm Oe such class, which covers the etire iterval betwee the mi ad max operatios cosists of geeralized meas. This class of operatios was used i fuzzy predictio, defied after (Klir ad Yua, 1995) as: (8) ad i mi (a, b) i(a, b) mi (a, b) where u(a,b) ad i(a, b) is the fuzzy uio, ad fuzzy ISSN: Issue 11, Volume 10, November 2011

6 α α a1 + a a (11) ha ( a1, a2,... a) = where α is a parameter by which differet meas are distiguished, ad α ε R (α is ot equal to 0). I our applicatio, we opted for α = 4, based o our experiece. Fuctio ha clearly satisfies all axioms of aggregatio operatios ad, cosequetly, it represets a parameterized class of cotiuous ad symmetric aggregatio operatios. variat of this approach is used i the followig aalysis. The approach uses weightig values for each alpha ad these weightig value may be adjusted for optimum performace. further variat that is ivestigated ivolves multiplicatio rather tha averagig. I that case the idividual weightig factors are of o cosequece. Liguistic statemets are usually associated with defuzzificatio. However i a classificatio problem, whe there are may parameters of equal importace, it is difficult to formulate such liguistic statemets. 2.5 Fuzzy Sets ad Data Miig The typical aalytic approach to data miig employs radom selectio to partitio the data ito two groups (sometimes 3). The first group is used to trai the data miig algorithm. For example if regressio is employed, the traiig partitio is used to set the parameters used i the regressio formula. The usual measures for performace are used for this purpose, residual error, p-values of coefficiets, ad so o. Whe two partitios are use they are usually based o a 60/40 or 70/30 ratio. These are geerally accepted ratios but it ot really possible to demostrate a theoretical optimum ad if the data set is small more may be required for traiig. Oce the data miig model has bee selected it is evaluated agaist the secod partitio, usually called the validatio partitio, to see how it performs. If the data has bee divided ito three partitios the differet models may be evaluated agaist the secod data set ad small further adjustmets may be made to the most promisig model before it is evaluated agaist the third partitio of data. Oly two partitios were employed for the aalysis reported here. Data i the first, traiig partitio are separated ito α 1 α two groups. Oe cotaiig date of kow beig istaces ad the other cotaiig data of the kow maligat istaces. For each set the mea ad stadard deviatio is computed for each of the attributes. These values are the use to set the membership fuctio for each attribute as described i equatio (4) above. Whe the model is ru the data from each record is evaluated for membership i each of the two groups of fuzzy sets. The results are aggregated ad compared with the decisio for beig or maligat beig made o the basis of which aggregatio has the larger value. Data from both the first, traiig, partitio ad the secod, validatio, partitio are evaluated by the model. Usually the traiig data performs better as might be expected. 2.6 Methods employed here Several aggregatio methods have bee compared. There is ot clear aalytical presidet for some of the selectios. Establishig a data miig model is i some ways a empirical process. That is why the traiig partitio ad the validatio partitio methodology is employed. The averagig method would be defied as follows. I this case the idividual membership fuctios are ot weighted but the aggregate is. (12) x ) = ( x ) additive ( k= 1 k k ( k= 1 k k k (13) x ) = b ( x ) additive with idividual b k weight adjustmet. is the membership fuctio for class. 1 is the membership fuctio for class, parameter umber 1 2 is the membership fuctio for class, parameter umber 2. x1is the value of parameter # 1 associated with object x. x2 is the value of parameter # 2 associated with object x. Equatios (12) ad (13) would be repeated for B where is associated maligacy ad B with beig. The multiplicative aggregates are defied as: ISSN: Issue 11, Volume 10, November 2011

7 (14) ( x) = k = k ( x 1 k ) multiplicative 2 (15) ( x) = k = ( k ( x 1 k )) multiply squares I each istace the idividual k ( x k ) membership fuctios may be adjusted durig the traiig phase to achieve miimum classificatio error. 2.7 Traiig Traiig is the distictive feature of the somewhat heuristic model preseted here. Durig the traiig stage the stadard deviatio used i each membership fuctio ca be adjusted to improve overall performace of the traiig partitio data. Icreasig idividual stadard deviatio has the effect of icreasig the width of that membership expoetial fuctio. The fuctio ow departs from the true statistical probability fuctio, but the choice of the ormal probability fuctio for the expoetial was arbitrary ad ot usually used i fuzzy logic ayway. The data represets physical attributes ad such attributes might be assumed to have ormal distributios. However the data was obtaied from huma observatios that are quatized ad hece ot ormal. The uaided model performed quite well. However performace was improved by adjustig the value used for stadard deviatio of each idividual parameter associated with each class. The additive method of combiig idividual parameter memberships ad the product method performed comparably i practice. It would seem that usig the squares of membership values would emphasize parameters where there is good membership. However empirical work suggests that there was o sigificat differece. adjustig coefficiet may be used whe addig membership values. This allows a small amout of improvemet i some istaces. With some data miig models over traiig is possible. I that case the error percetage is very good for the traiig partitio but markedly degraded for the validatio set. This situatio did ot preset itself. 2.8 Traiig Optimizatio There are may parameters ad each oe has a adjustable stadard deviatio. djustig by eye is time cosumig. was used i a attempt to optimize this process. The geetic algorithm used was a Microsoft Excel add-i called xl bit ( [15] 3. Results with Data Set #1 2.1 Statistical Method Classificatio data miig is ofte accomplished o a statistical basis with a logistic regressio model. I order to compare performace of the fuzzy model logistic regressio was applied to the same data. The vehicle used was a Microsoft Excel add-i called XLMier [16]. We report results obtaied with several data sets here. Data set umber 1 is Mammographic Mass Data made public by Schulz-Wedtlad [17]. The mammographic Mass database cosisted of the followig parameters. 1. BI-RDS assessmet: 1 to 5 (ordial) 2. ge: patiet's age i years (iteger) 3. Shape: mass shape: roud=1 oval=2 lobular=3 irregular=4 (omial) 4. Margi: mass margi: circumscribed=1 microlobulated=2 obscured=3 ill-defied=4 spiculated=5 (omial) 5. Desity: mass desity high=1 iso=2 low=3 fatcotaiig=4 (ordial) 6. Severity: beig=0 or maligat=1 (biomial) Missig ttribute Values: - BI-RDS assessmet: 2 - ge: 5 - Shape: 31 - Margi: 48 - Desity: 76 - Severity: 0 The predicted classificatio is either beig or maligat. Cases with missig variables were removed. Summary of results are i the table. For these calculatios the gai of each idividual fuzzy set was left costat durig the aggregatio (averagig) process but idividual stadard deviatios were adjusted so as to arrow or icrease the width of the membership fuctio. Percetages show are error ISSN: Issue 11, Volume 10, November 2011

8 TBLE logistic regressio logistic regressio Fuzzy Logic Model sum of membership values Fuzzy Logic sum of membership values Fuzzy Logic Model usig membership product Fuzzy Logic Model usig membership product Membership product & geetic algorithm to adjust Sigma Membership product & geetic algorithm to adjust sigma usig sum of to adjust Sigma usig sum of to adjust sigma Beig 15.95% 41/ % 30/ % 51/ % 27/ / / % 46/ % 35/ % 39/ % 26/167 Cacer 15.61% 37 / % 26/ % 40/ % 29/ / / % 27/ % 20/ % 37/ % 27/163 Overall Error 15.79% 78/ % 56/ % 91/ % 56/ / / % 73/ % 55/ % 76/ /330 percetages. The ratios, such as 7/270, show error cout ad total records i that classificatio. Validatio scores are the importat oes from the poit of view of data aalysis. lso there is ot a great deal of variatio amog results. This seems to match results from some other researchers i this area. Traiig ad validatio scores are show. s aticipated, traiig scores are slightly better. Usig the to optimize the membership fuctios cotributed a slight improvemet whe the decisio rule was based o the sum of membership values ad made o improvemet i overall accuracy whe the product of membership values was used. I this latter istace it altered the probabilities for failures i detectig maligacy by icreasig them with a cosequet decrease i failures to classify as beig. The, like other traiig adjustmets, worked to miimize the overall error i the traiig process. The hope is that this will project ito improved performace with the validatio partitio data. It is true that some adjustmet of the sigma values i the idividual membership fuctios ca be used to shift errors from oe type to aother while still maitaiig overall error rate ear costat. I lieu of adjustig the stadard deviatios a weightig factor for each attribute ca be adjusted. Of course, this is oly possible whe the aggregatio fuctio is a summatio. Thus: (16) x ) = b ( x ) ( k= 1 k k k The coefficiet b k applies to the attribute k. Results i this simple case are show i Table B. d they are ot markedly differet from results previously obtaied ISSN: Issue 11, Volume 10, November 2011

9 TBLE B sum of to adjust weight sum of to adjust weights Beig 13.23% 34/ % 20/167 Cacer 20.25% 48/ % 33/163 Overall Error 16.6% 82/ /330 t least i some istaces weightig the membership fuctios is as useful as adjustig the idividual shapes of the membership fuctios. It is ot obvious that this would be the case. It does say that simple dyamic weightig adjustmet durig the traiig phase may be as useful as aythig else. Results also suggest that the statistical logistic regressio approach is about as valuable as the fuzzy logic approach. This is ot the case i cotrol systems where fuzzy logic performs exceedigly well. This paper reports o a empirical ivestigatio with a limited amout of data. It offers o proof that statistical methods will always perform as well or less well as fuzzy methods. Not all researchers have reported a compariso so oe hesitates to make geeralized observatios. 4. Results Data Set #2 Data set #2 was Breast Cacer data made public by Magasaria & Wolberg [18] [19]. This data set was also briefly reported i our previous paper. The methodology has ow bee greatly expaded with adjustmet of idividual membership fuctios ad use of the. Data set parameters are as follows: 1. Sample code umber id umber 2. Clump Thickess Uiformity of Cell Size Uiformity of Cell Shape Margial dhesio Sigle Epithelial Cell Size Bare Nuclei Blad Chromati Normal Nucleoli Mitoses Class 2= beig; 4 = maligat TBLE C logistic regressio logistic regressio usig sum of to adjust Sigma usig sum of to adjust sigma Beig 1.11% 3/ % 7/ % 11/ % 2/174 Cacer / % 3/ % 19/ % 10/98 Overall Error 1.71% 7/ % 10/ % 30/ % 12/272 The predicted classificatio is either beig or maligat. Stadard deviatio of membership fuctios were adjusted usig the. Sets were aggregated as a averagig process similar to equatio (11) ad selectio made o the basis of which was the larger. Summary of results are i the table C. gai results from the fuzzy approach are very good but ot better that statistical logistical regressio. They are i some respect comparable. Results for data set #2 were cosiderably better tha those for data set #1 possibly because data set #2 had more attributes from which to establish set memberships. ll results were really comparable with results of other researchers as reported earlier i this paper. The Fitess chart developed by the software for data set #2 is preseted as Figure 2. It shows that the has approached asymptotically a horizotal lie ad further iteratios may ot produce improvemet uless there are local miimums. Cosiderig the ature of the problem, local miimums are ulikely. ISSN: Issue 11, Volume 10, November 2011

10 the result. It ca lead to icorrect aswers whe usig regressio to fit higher order polyomials. But this is ot a problem here. I the heuristic approach the model is based o actual performace. No oe attempts so say it would be valid for classes differet from the oes with which the model is validated. 5. Solutio Efficacy There is aturally a questio as to why these aalytic attempts are ot more successful i discrimiatig amog the two classes. I most medical ad probably most biological etities various attributes will be highly correlated. Figure 3 preset the correlatio matrix for data set #2. Here it ca be see that a high degree of correlatio exists. I geeral this makes statistical regressio less accurate or questioable at least. This limitatio will certaily be preset i all attempts at discrimiatio of biological etities util or uless orthogoal characteristics ca be foud. 6. Observatios alysis shows that the parameters of data sets are strogly correlated. This is ot surprisig sice they usually represet physical characteristics of some class of objects. The multiple liear regressio model assumes idepedet variables. Colliearity may make it impossible to see the cotributios of each variable to The fuzzy logic approach preseted here is ot limited to selectio amog two categories. Clearly the resolutio through a logical OR operatio ca be applied to three or eve more categories ad presumably eve more. [14] There is o practical limit provided there is sufficiet data. Three classes requires that traiig data be divided ito three sets; for classes would require 4 sets, etc. Statistical evaluatio of results would require some reasoable umber of records for each class. 7. Coclusios It is clear that fuzzy logic employig a expoetial membership fuctio, weighted averagig, ad logical OR operatio is a effective tool i data aalysis ivolvig discrimiatio where may parameters are ivolved. It performs as well as results reported by other researchers lookig at similar problems with differet fuzzy methods ad hybrid fuzzy methods. Because membership fuctios may be adjusted idividually the usual method of dividig data ito a traiig partitio ad a validatio partitio is appropriate. Rather extesive adjustmet ( learig ) durig the traiig stage improves performace. This may iclude use of a. Results observed here must be classified as aecdotal because oly two data sets were used. However, from these observatios it appears that fuzzy logic is comparable but does ot outperform usual data aalysis methods such as logistic regressio. I istaces where image aalysis is a pricipal feature (as reported i other papers oted earlier) the situatio does ot led itself to stadard statistical regressio methods. Refereces [1] Zadeh, L.. "Fuzzy sets", Iformatio ad Cotrol 8 (3): [2] Takahashi Y. (1995) fuzzy Query Laguage for Relatioal Databases, i Fuzziess i Database Maagemet Systems, Physica Publisher, Heidelberg, ISSN: Issue 11, Volume 10, November 2011

11 [3] Cios K J, Goodeday L S, Sztadera L M Hybrid itelligece system for diagosig coroary steosis. Combiig fuzzy geeralized operators with decisio rules geerated by machie learig algorithms, Egieerig i Medicie ad Biology Magazie, IEEE, Vol. 13 Issue 5, 1994 pages [4] Cios K J, Shi I, ad Goodeday LS Usig Fuzzy Sets to Diagose Coroary rtery Steosis, IEEE Computer Magazie Pages 57-63, 1991 [5] Cios K J, Liu N (1994) machie Learig algoriothm Usig Iterger Liear Programmig KYBERNETES, MCB Uiversity Press,U.K [6] Cios K J, Noraes I 91991) iductive Learig lgorithm KYBERNETES, MCB Uiversity Press, U.K. 920:3), pages [7] Cios K J, Freasier R E, Goodeday L S, drews L T expert system for diagosis of coroary artery steosis based o 201TI scitigrams usig the Dempster-Shafer theory of evidece, Oxford Uiversity Press, U.K. (6) No. 4 pages [8] Malek J, Sebri, Mabrouk S, Torki K, Tourki R Joural of Sigal Processig Systems Volume 55 Issue 1-3, pages 49-66, pril 2009 Kluwer cademic Publishers [9] Bagher-Ebadia H, Soltaia-Zadeh H, Setayeshi S, Smith S T Neural etwork ad fuzzy clusterig approach for automatic diagosis of coroary artery disease i uclear medicie IEEE Trasactios o Nuclear Sciece Vol51 Issue 1 pages (Feb 2004) [10] Licata G "Employig fuzzy logic i the diagosis of a cliical case" Health., Vol.2, No.3, pages March 2010 [14] Saford J, Sztadera L (2010) Classificatio Data Miig Usig Fuzzy Logic, Iteratioal cademy of Busiess ad Public dmiistratio Disciplies New Orleas Coferece Proceedigs October 2010, pages [15] XLBIT Geetic algorithm for Microsoft Excel from based i Kuala Lumpur, Malaysia. [16] XL-Mier is a Excel dd-i developed by Cytel Software Corporatio ad distributed by Resamplig Stats Ic. [17] Schulz-Wedtlad 2007 Mammographic Mass Data. Origial ower: Prof. Dr. Rüdiger Schulz- Wedtlad, Istitute of Radiology, Gyaecological Radiology, Uiversity Erlage-Nuremberg, Uiversitätsstraße 21-23, Erlage, Germay. Relevat Paper: M. Elter, R. Schulz-Wedtlad ad T. Witteberg (2007), The predictio of breast cacer biopsy outcomes usig two CD approaches that both emphasize a itelligible decisio process. Medical Physics 34(11), pp [17] Magasaria O. & Wolberg W. (1990) "Cacer diagosis via liear programmig", SIM News, 23(5), [18] Magasaria O & Wolberg W, 1990, "Cacer diagosis via liear programmig", SIM News, Volume 23, Number 5, September 1990, pp 1 & 18. [19] Wolberg W. & Magasaria O. (1990) "Multisurface method of patter separatio for medical diagosis applied to breast cytology", Proceedigs of the Natioal cademy of Scieces, U.S.., Volume 87, pp [11] Pedrycz W. Fuzzy Cotrol ad Fuzzy Systems Joh Wiley & Sos, ISBN [12] Klir G. J. ad Yua B. (1995), Fuzzy Sets ad Fuzzy Logic Theory ad pplicatios, Pretice Hall, Eglewood Cliffs. [13] Sztadera L. M., Goodeday, L. S., ad Cios K. J. (1996), Neuro-Fuzzy lgorithm for Diagosis of Coroary rtery Steosis, Computers i Biology ad Medicie Joural, 26(2), ISSN: Issue 11, Volume 10, November 2011