Selection and Classification of Statistical Data Using Fuzzy Logic

Transcription

1 Selecton and Classfcaton of Statstcal Data Usng Fuzzy Logc Mroslav Hudec (1), Mrko Vujoševć (2) (1) INFOSTAT Insttute of Informatcs and Statstcs, Bratslava, Slovaka (2) Faculty of Organzatonal Scences, Jove Ilća 154, Beograd, Serba Abstract Lngustc expressons lke: hgh rate of unemployment, or hgh mgraton level etc., are very often used n lfe and n statstcs. The goal of ths research s to capture these expressons and make them sutable for database queres and classfcaton tasks. Ths paper shows dfferences between usual and fuzzy approaches n database queres and classfcaton and ponts out advantages of the fuzzy approach. The fuzzy logc deals wth reasonng that s approxmate rather than precse to solve problems n a way that more resembles human logc. For queryng process the generalsed logcal condton n the WHERE part of the SQL was developed. In order to classfy data, fuzzy queres are generated from fuzzy rules. The objectve of ntegraton s to use the same generalsed logcal condton n data selecton and data classfcaton. The proposed fuzzy approach provdes flexblty when users cannot unambguously set the boundares between data. The fuzzy approach also extracts addtonal valuable hdden nformaton n comparson wth the usual, crsp approaches. In ths way, queres based on lngustc expressons on clent s sde are supported and are accessng relatonal databases n the same way as the classcal SQL enables. Keywords: fuzzy SQL, fuzzy classfcaton, database. 1. Introducton Ths paper examnes two often used processes: data selecton (database queres) and data classfcaton. The am of ths paper s to emphasze stuatons when classcal {true, false} logc s not adequate n these two processes and offers fuzzy logc because the fuzzy logc s an approach to computng based on "degrees of truth" rather than the usual "true or false" logc. Fuzzy approach s sutable for statstcal databases. Lngustc expressons lke: hgh rate of unemployment or medum mgraton level etc., are very often used and t s useful to catch them and use n database queres and classfcaton. Statstcal ndcators are often collected wth some errors and vagueness and classcal technques may nvolve some nadequately selected, or classfed data. Ths paper after short ntroducton of fuzzy logc dea presents our research n fuzzy database queryng area, where we have created generalzed logcal condton (GLC). Ths fuzzy query dea s explaned n one case study. Durng research n database queres and classfcaton by fuzzy systems we got a new dea of data classfcaton technque. The new approach to treatng classfcaton s based on the same GLC. Ths

2 dea leads to ntegraton of data selecton and classfcaton nto one tool. In ths paper the characterstcs of a new approach are explaned on one case study. The dea how to mprove the whole soluton to satsfy all Boolean axoms s mentoned and fnally some conclusons are drawn. 2. Fuzzy dea The core of both classcal and fuzzy logc s the dea of a set. In classcal set theory an element belongs or does not belong to a set. For example consder a set called hgh unemployment (HU) defned as follows: HU={x unemployment(x)>=10%} where x s a regon. It means that regon wth 9.95% unemployment does not belong to the HU but regon wth 10% belongs. These constrants are drawback when the boundares between values of some attrbutes are contnuous. The fuzzy logc theory brngs a paradgm n work wth the graduaton, uncertanty and ambguty descrbed by lngustc expressons. The fuzzy set theory permts the gradaton of the membershp of the element n a set. Ths gradaton s descrbed by a membershp functon µ valued n the nterval [0, 1]. The HU example can be presented by fuzzy sets shown n fgure 1. User could defne that the unemployment equal and bgger than 10% (L p n fgure 1) s HU, the unemployment smaller than 8% (L d n fgure 1) defntely s not HU and unemployment between 8% and 10% partally belongs to the HU concept. The fuzzy approach uses knowledge that does not have clearly defned boundares. Many of the phenomena from real world fall nto ths class. Fgure 1: Fuzzy sets for bg unemployment concept 3. Database queres An answer for the queston why users search databases could be as follows: To retreve data needed to make a decson. The structured query language (SQL) s used to obtan data from relatonal databases. The SQL query s as follows: select attrbute_1,,attrbute_n from T (1) where attrbute_p > P and attrbute_r < R. The result of the query s shown n graphcal mode n fgure 2. Values P and R delmt the space of nterestng data. Small squares n the graph show database records. In the graph s obvously shown that two records are very close to satsfyng the query crteron. These two records could be two potental customers and drect marketng could attract them or two muncpaltes whch almost satsfy crteron for some fnancal support.

3 Fgure 2: The result of a SQL query SQL makes crsp selecton. It means that the record would not be selected even f t s extremely close to the ntent of a query. Ths s the penalty pad for usng the crsp logc n selecton crteron. In cases when the user can not unambguously separate nterested data from not nterested by sharp boundares or when the user wants to obtan data that are very close to meet the query crteron and to know the ndex of dstance to full query satsfacton, t s necessary to adapt the SQL to these requrements. 3.1 Fuzzy SQL queres A fuzzy query system s an nterface to users to get nformaton from database usng (quas) natural language sentences. The answer to a fuzzy query sentence s generally a lst of records, ranked by the degree of matchng (Branco et. al., 2005). The query compatblty ndex (QCI) s used to ndcate how the selected record satsfes a query. The QCI has values from the [0,1] nterval: 0 record does not satsfy a query, 1-record has full query satsfacton, nterval (0,1) record partally satsfes a query. The GLC formula for the WHERE part of the SQL based on lngustc expressons was bult n (Hudec, 2007). The mplementaton of GLC for statstcal databases s explaned n (Hudec, 2008). In ths paper the process of fuzzy selecton by GLC s mentoned and examned n a case study. The GLC has the followng structure: n WHERE ( a o L ) (2) = 1 x where n denotes number of fuzzy constrants n a WHERE clause of a query, and = or where and and or are fuzzy logcal operators, and a > Ld, a s Bg a o Lx = a < L, a s Small. g a > Ld and a < Lg, a s About where a s a database attrbute, L d s the lower bound and L g s upper bound of a lngustc expresson descrbed by fuzzy set shown n fgure 3.

4 The queryng process conssts of the two steps. In the frst step lower and/or upper bounds of lngustc expressons (fuzzy sets) are used as parameters for database queres. It means that all records that have QCI greater than zero are selected. In the second step the chosen analytcal form of the fuzzy set s used to calculate the membershp degree of each selected record to approprate fuzzy set. Fnally, approprate t-norms or t-conorms are used to calculate QCI values for all retreved records. a) Bg value b) Small value c) About value Fgure 3: Fuzzy sets In many-valued logc there exst many functons descrbng and operator (t-norms) and or operator (t-conorms) because each of condtons nsde the WHERE clause can be partally satsfed. The followng functons can be used for t-norms: mnmum: QCI = mn( µ (a )), =1,...,n (3) product: QCI = ( µ (a )), =1,...,n (4) bounded dfference (BD): QCI = max(0, µ ( a ) n + 1) (5) where µ ) denotes the membershp degree of the attrbute a to the -th fuzzy set. ( a Mn t-norm takes nto account the lowest value of membershp degrees to fuzzy sets. Product t-norm takes nto account all membershp degrees n the WHERE clause. A fuzzy query nterpreter whch transforms fuzzy queres to the SQL structure was developed. In ths way, queres based on lngustc expressons on the clent sde are supported and accessng relatonal databases n the same way as the SQL s enabled. 3.2 Case study Ths system s tested on data from the Urban and Muncpalty Statstcs database used n the Statstcal Offce of the Slovak Republc (Benčč and Hudec, 2002). In ths case study, dstrcts wth hgh length of road and small area sze are sought. The bg road nfrastructure densty s analyzed as an llustratve example. The query has the followng form: select dstrct, roads, area from T where roads s Bg and area s Small The length of road ndcator s represented by Bg value fuzzy set wth these parameters L d =200 km and L p =300 km and the shape as from Fgure 3a). The Small value fuzzy set wth parameters L p =450 km 2 and L g =650 km 2 and shape as from Fgure 3b) descrbes the area attrbute Result of fuzzy query s shown n Table 1. The value of mn t-norm (3) s used for dstrct rankng. The Table 1 shows sx dstrcts fully satsfyng the query; one dstrct s extremely close to satsfyng the query and another two dstrcts are close to the n = 1

5 query crteron. It means for example that even small changes n dstrcts attrbutes could nvolve that another records fully satsfy the query. If SQL was used, ths addtonal nformaton would reman hdden. Table 1: Result of fuzzy query. Dstrct Roads [km] Area [km 2 ] QCI Detva 567, Myjava 563, Žarnovca 366, Bratslava I 335, Púchov 320, Pešťany 305, Považská Bystrca 324, ,935 Kysucké N. M. 269, ,7 Senec 269, ,69 Žar nad Hronom 249, ,5 Nové Mesto n. V. 528, ,35 Krupna 334, , In cases when user uses SQL and wants to obtan smlar results lke result presented n Table 1 t s needed to make small changes n crteron parameters and to execute larger number of queres. The query from example (1) could be modfed as follows: (attrbute_p > P and attrbute_r < R) extract records that satsfy ntal condtons. (attrbute_p > P-p 1 and attrbute_p < P) and (attrbute_r < R+r 1 and attrbute_r < R) select records that meet query crtera wth value of 0.9 or almost meet the query crtera (where p 1 and r 1 are small real values greater than zero), (attrbute_p > P-p 2 and attrbute_p < P-p 1 ) and (attrbute_r < R+r 2 and attrbute_r < R+r 1 ) select records that meet query crtera wth value of 0.8 or records that are very close to query crtera (where p 1 <p 2, r 2 >r 1 ), etc. The followng concluson appears: for the very soft gradaton, the unlmted number of SQL queres has to be used. In case of fuzzy queres, only one query meets ths requrement. 4. Data classfcaton Users classfy data n order to fnd n whch class each of classfed record (terrtoral unt, customer) belongs. Expert systems are one of technques capable of data classfcaton. The usual classfcaton by expert system s llustrated on the example of estmatng and plannng of road mantenance needs n the wnter. Muncpaltes are classfed accordng to the length of roads n km and the number of days wth snowng. Classfcaton dagram s presented n fgure 4. The fgure 4 shows the classcal classfcaton. Muncpaltes are dvded nto four classes from class C1 (the smallest needs for the mantenance) to class C4 (the bggest needs for the mantenance). Ths method treats the top rated terrtoral unt T4 n the same way as T3. Unts T2 and T3 have smlar length of roads as well as smlar number of days wth snowng. However, T2 and T3 are treated n dfferent classes.

6 Fgure 4: Classcal classes 4.1 Fuzzy classfcaton Fuzzy classes reflect realty better and allow decson makers or analytcs to descrbe nput attrbutes and output classes more ntutvely usng lngustc varables, overlappng classes and approxmate reasonng. Objects that belong to more than one class are treated n all classes where they have partal membershp. In order to solve a fuzzy classfcaton problem wthn a knowledge-based fuzzy nference system (FS) t s necessary to fuzzfy attrbutes, determne all IF-THEN rules (rule base), process them and to provde result n a usable and understandable form. More about fuzzy classfcaton s n (Hudec and Vujoševć, 2005). The advantages of fuzzy systems are as follows. They enable the creaton of logcal nference system based on human mnd ncludng uncertantes n membershp degrees to the approprate fuzzy sets. support the nference process based on IF-THEN rules. enable accessble, understandable and easy to use and modfy knowledge base. There are many fuzzy system softwares capable to solve classfcaton tasks, for example MatLab or FLOPS. These softwares have been produced to solve a wde area of tasks but they are complcated for users. In order to solve a classfcaton task, the decson maker needs the assstance to prepare the nput data from database nto proper format for the FS and to present the results nto a useful and understandable form. Ths part could be programmed but t s not a trval task. The decson maker also needs assstance to set the most sutable mathematcal functons nsde the FS. 4.2 Integraton of data selecton and classfcaton If t s goal to create easy to use soft computng tool for classfcaton, fuzzy systems are not very sutable from users pont of vew. The new dea for classfcaton and how to ntegrate data selecton and data classfcaton has been found durng our work on fuzzy queres. IF part of a fuzzy rule corresponds to a WHERE clause of a query, and all rules related to the same output class (THEN part of a rule) are aggregated wth an OR operator nto the same query. A fuzzy query would return all records together wth ther membershp degrees to the query crtera satsfacton. Ths membershp degree s also membershp degree to the approprate output class. The classfcaton query language s desgned n the sprt of the above descrbed fuzzy SQL. The dfference s n the added clause classfy_nto. The classfy_nto clause specfes the name of the output class to whch selected records are classfed. The structure s as follows:

7 classfy_nto [class ] select [attrbute 1 ],.[attrbute n ] from [tables, relatons] K where ( a o L ) n k = 1 = 1 x where s AND operator, n s the number of attrbutes nsde the IF part of the rule, s OR operator whch connects those k antecedents n IF part that have common THEN part or the same output class. The results of all queres are objects selected nto overlappng classes. The fnal rank for each record can be calculated from the equaton: m R O = µ P (6) = 1 Oc where m s number of classes, and P s coeffcent descrbng class C. µ Oc s the membershp degree of object O to class C Advantages of ths approach are as follows: Queres select only records that wll be classfed. Records that do not belong to any class are not needlessly selected; Data preparaton to adequate nput vector or matrx s not needed; Presentaton of results n useful and understandable form for example n xls format or on thematc maps could be easy mplemented. 4.3 Case study The system s appled and tested for muncpalty classfcaton usng data of Banská Bystrca regon from the same Urban and Muncpalty Statstcs database. In ths case study muncpaltes are classfed accordng to the percentage of needs for the wnter road mantenance. In ths example two attrbutes are used and fuzzfed nto two sets: length of roads n klometers (Road) and number of days wth snow (Snow). These sets are shown n fgure 5. Ths example contans four fuzzy rules wth the followng structure: If Road s Small and Snow s Small Then Mantenance s Small. If Road s Small and Sow s Bg Then Mantenance s Medum. If Road s Bg and Snow s Small Then Mantenance s Medum. If Road s Bg and Snow s Bg Then Mantenance s Bg. µ µ Fgure 5: fuzzy sets Small (S) and Bg (B) for Roads and Snow ndcators. Three fuzzy queres are created from these four rules. The query for output class Medum s as follows:

8 Classfy_nto M select muncpalty, roads, snow from T where (roads s Small and snow s Bg) or (roads s Bg and snow s Small) The percentage of needs can be assocated wth each output class. For nstance, class S (Small) gets a percentage of needs of 10% or accordng to equaton (6): P s =0.1, class M (medum) gets 50% and muncpaltes from class B (Bg) get 90% from consdered needs. Ths tool classfed 126 muncpaltes. 10 of them fully belong to class S, 74 fully belong to class M, 14 fully belong to class B and other 28 partally belong to more than one class. If classcal classfcaton was used, ths addtonal valuable nformaton would be hdden. Table 2 shows rankng results for some muncpaltes. Table 2: Some of classfed muncpaltes. Muncpalty Coeffcent of Coeffcent of Muncpalty needs (P) needs (P) Radzovce 0,1 D. Harmanec 0,5 Dudnce 0,1 Kremnca 0,5 Lpovany 0,1268 Trnavá Hora 0,5 Tornaľa 0,2103 Poltár 0,5146 Kalnovo 0,2868 H. Tsovník 0,62 Hajnáčka 0,34 Donovaly 0,86 Vnca 0,34 B. Bystrca 0,9 Čebovce 0,3932 Ľubetová 0,9 From membershp degree to each class and equaton (6) the coeffcent of needs (R) s calculated. Two muncpaltes are taken as example: Čebovce belongs to class S wth degree of and M wth degree of P(Čebovce)=0.267* *0.5=0,3932. Banská Bystrca fully belongs to class B. P(Banská Bystrca)=1*0.9=0.9. In case of classcal classfcaton two muncpaltes wth very smlar ndcators values near the boundary value may be classfed nto dfferent classes and t wll cause dfference between obtaned needs. To avod ths dsadvantage the user has to create sgnfcantly greater number of output classes and rules f he wants to use crsp tools n comparson wth the fuzzy approach. Indeed for very soft rankng by crsp tools user needs nearly nfnty number of nput ranges, rules and output classes. 5. Dscusson about proposed model The usefulness of proposed fuzzy queryng approach depends also on the theoretcal and practcal development of fuzzy database systems. In fuzzy databases the fuzzy query s a part of the fuzzy database management system and a specal fuzzy query development s not necessary. Many statstcal databases are developed n relatonal database management systems. Ths trend domnates n development of new statstcal nformaton systems and databases so the fuzzy selecton presented n ths work have the sgnfcant perspectve for further usage. However, further development of

9 ntegrated selecton and classfcaton does not depend on fuzzy database development and ts practcal applcatons. The GLC s developed on axoms of fuzzy logc. Fuzzy logc s based on the same prncple as classcal logc, the prncple of truth-functonalty. Logc s truth functonal f the truth value of a compound sentence depends only on the truth values of the consttuent atomc sentences, not on ther meanng or structure (Radojevć, 2008). In case of all many-valued logcs, ncludng fuzzy logc, ths prncple s not sufcent and as a consequence these logcs are not n the frame of Boolean algebra (BA). Zadeh, the nnovator of fuzzy sets and fuzzy logc, descrbed fuzzy logc as precse many-valued logc where axoms of contradcton and excluded mddle are not satsfed. Frst way how to use gradaton n mathematcs s to leave these axoms and accept the prncple of truth functonalty wth all consequences. Radojevć n (Radojevć, 2008) got postve answer to the queston: Can the dea of fuzzness be realzed n a Boolean frame? New approach to treatng fuzznes or gradaton n logc s based on nterpolatve realzaton of Boolean algebra (IBA). IBA has symbolc level (fnte BA) and semantc or valued level. Above mentoned problem does not appear n fuzzy classfcaton. There are some stuatons when ths problem can arse n fuzzy queryng. It s possble to avod ths problem by selectng adequate t-norm or t-conorm functon for queres. Although ths knd of choosng adequate functons satsfes demands, t s very nterestng to mprove fuzzy queres wth IBA concept. The IBA could provde better flexblty and avod theoretcal possble stuatons when napproprate functons are chosen. The web applcaton wth a fuzzy module for data dssemnaton s another way of mprovement of ths fuzzy query approach. Statstcal nsttutons put vast amount of data nto ther webstes and provdng a selecton crtera by lngustc expresson gves natural way for data selecton. 6. Concluson It s proven n our research that the proposed fuzzy system for selecton and classfcaton may be successfully used. If crsp sets and sharp boundares n queres are used then the small error n data values or cases when user can not unambguously defne the crtera by crsps sets may nvolve some nadequate selected or classfed data. Fuzzy logc provdes answer how to avod these dsadvantages. User gets two advantages: the model s descrbed n natural language and the result gves addtonal valuable nformaton that mght reman hdden f usual methods are used. FS are powerful tools but t s not easy to work wth them. Both, advantages of FS and advantages of our queryng process are ncluded n our classfcaton concept. The knowledge base s understandable, easy to modfy and use. Our classfcaton approach exempts users from data preparaton to adequate nput matrx, preparaton of results nto adequate form and settng approprate functons nsde FS.

10 The fuzzy methods are n ths approach ndependent modules. The modularty of here mentoned modules allow ther modfcatons and mprovements ndependently. Implementaton of the IBA n queryng process s very nterestng topc for further research. The formulas for transformng fuzzy queres to the classcal ones and fuzzy rules to fuzzy queres were bult and a fuzzy query nterpreter based on these formulas s under developng. In ths way, queres based on lngustc expressons on clent sde are supported and accessng relatonal databases n the same way as the classcal SQL s enabled. The nterpreter also creates fuzzy queres from fuzzy rules and converts them nto classcal SQL. At the end of classfcaton process nterpreter puts records nto approprate classes. No modfcaton of databases has to be undertaken. Ths approach could be reused for another databases or purposes. The core of ths tool (GLC) remans the same, only an nput and output parts have to be adapted to acheve new needs. If users are prepared to accept a less accurate system that contans approxmate reasonng, they look for one whch s fully comprehensble to them. Our approach could meet ther needs. References Benčč A., Hudec M. (2002) MOŠ/MIS Urban and muncpal statstcs project and nformaton system of the Slovak Republc, Symposum on operatonal research, Vuletć Prnt, XXI-32--XXI-35 Bosc P., Pvert O. (2000) SQLf Query Functonalty on Top of a Regular Relatonal Database Management System, n: Knowledge Management n Fuzzy Databases, Pons M. et al. (Eds.), Physca Publsher Branco A., Evsukoff A., Ebecken N. (2005) Generatng Fuzzy Queres from Weghted Fuzzy Classfer Rules, ICDM workshop on Computatonal Intellgence n Data Mnng, IOS Press, Cox E. (2005) Fuzzy modelng and genetc algorthms for data mnng and exploraton, Morgan Kaufmann Publshers. San Francsco. Hudec M., Vujoševć M., Fuzzy systems and neuro-fuzzy systems for the muncpaltes classfcaton. In EUROFUSE, 2005, Eurofuse annversary workshop on Fuzzy for Better. M. Pupn Insttute, Hudec M. (2007) Fuzzy mprovement of the SQL. Balkan-Conference-on- Operatonal-Research, Hudec M. (2008) Fuzzy SQL for statstcal databases, UNECE-EUROSTAT-OECD Meetng on Management of Statstcal Informaton Systems, Radojevć D. (2008) Interpolatve Realzaton of Boolean Algebra as a Consstent Frame for Gradaton and/or Fuzzness, n: Forgng New Fronters: Fuzzy Poneers II Studes n Fuzzness and Soft Computng, Nkravesh M. et al. (Eds.), Sprnger, Sler W., Buckley J. (2005) Fuzzy expert sytems and fuzzy reasonng. John Wley & Sons, Inc. New Jersey. Werro N. et al. (2008) Concept and Implementaton of a Fuzzy Classfcaton Query Language, Internatonal Conference on Data Mnng, CSREA Press,