SCIENTIFIC PROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Information Technology and Management Science 2002

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1 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence METHODS OF FUZZY ATTERN RECOGNITION R Grekovs Keywords: pattern recognton, fuzzy sets, coposton of fuzzy relatons, fuzzy rules, fuzzy clusterng Introducton Technque of learnng fuzzy rules The frst approaches to learnng fuzzy rules fro data were based on neural network concepts (perceptron-based ethods [] The underlyng prncpal of all these odels s that they need an evaluaton of the output that the fuzzy rules represented by the neural network produce for a gven nput Ths can be an error easure or sply a bnary evaluaton Then the odels a s at provng the output by changng the fuzzy sets appearng n the rules n a sutable way Ths change s usually derved by soe back propagaton technque or sply by heurstc algorth They are usually desgned for the adaptaton of the fuzzy sets, but soetes also nvolve a sple strategy for choosng an ntal set of rules, addng or deletng rules whle learnng They strongly rely on the sple archtecture that allows dentfy how changes n the fuzzy sets effect the output Therefore, they are qute suted for tunng the fuzzy sets n ths type of rules, but are not desgned for learnng rules Another popular approach to learnng fuzzy rules s based on evolutonary algorths Evolutonary algorths are a qute general technque for paraeter optsaton that s otvated by concepts of bologcal evoluton It starts wth an ntal 'populaton' of possble solutons of the optsaton proble that s usually randoly generated Then changes soe of the values of the populaton ('utaton' and apply 'bologcal' operators lke crossover that exchanges values between dfferent solutons Then the best solutons are selected to for the next generaton whch s treated n the sae way untl a axu nuber of generatons s coputed or a soluton wth a desred qualty appears Evolutonary algorths requre a paraetersed forulaton of the proble wth a qute lted nuber of paraeters Therefore, they can be appled to fuzzy systes where the nuber of possble rules s strctly lted An alternatve approach to learnng fuzzy rules s fuzzy clusterng Each fuzzy rule specfes a vague pont of the graph of the descrbed functon Many fuzzy clusterng algorths are pursung the followng strategy: A fuzzy cluster s represented by a typcal eleent usually the cluster centre and the ebershp degree of a datu to the cluster s decreasng wth ncreasng, soetes transfored dstance to the cluster centre Many fuzzy clusterng approaches characterze each cluster by a set of paraeters - prototype Gven a set X {x,, x N } IR p of saple data, the a s to deterne prototypes nzng the obectve functon: c N 2 J ( X, U, v ( u d ( v, x, k where u k - the ebershp degree of datu x k to cluster ; d(v,x k - the dstance of datu x k to the cluster, represented by the prototype v ; с - the nuber of clusters To nze data ponts x k wth a sall dstance d(v,x k, to the cluster should be assgned a hgh ebershp degree whereas data wth larger dstances should have low ebershp degrees Clusterng algorth begns wth a rando ntalsaton and updates u k and v n an teratve procedure k k

2 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence It s obvous that fuzzy clusterng reles on the nterpretaton of the rules and s therefore well suted for learnng such rules However, fuzzy clusterng does not yeld optal fuzzy sets due to the loss of nforaton caused by the proecton clusters 2 Feature analyss n fuzzy pattern classfcaton To select ost effectve eleents of the space a concept of coposton of shadows of fuzzy sets nto 2-densonal subspaces s ntroduced n [2] As a crteron of evaluaton a power of reducton, whch s receved durng the transforaton of an ntal feature set, s used Orthogonal shadows of these sets on co-ordnate planes wll be as follows: sup y,, ( X Y, S A XY S B A sup y,, ( X Z y Y B A B If bnary shadows S and S ZY are used as arguents of coposton Q, then reducton factors δ wll serve as separaton power of paraeter Z when classfyng patterns A and B Thus, two copostons could be constructed by one and the sae paraeter and per each coposton two reducton factors are obtaned The total reducton power could be A B B A defned as δ Z ( δ δ ZY ( δ δ ZY, where operatons and are used as expressons x y x + y - xy; x y x y araeter Z s not able to separate classes A and B, f the degree of an obect s ebershp to classes does not depend on that paraeter: A A ( x, ( X Z S S X L z y y z y Z Y B (, B (, (, SZY SY A A B B After operatons of coposton are perfored S S, S S Fro ths t A B A B follows that δ δ ZY 0 and δ ZY δ 0 Thus, δ Z 0 araeter Z has axu separaton power n coparson wth paraeters X and Y, f t enables correct obect classfcaton whereas the other paraeters are not able to perfor t The paper [3] ntroduces the dea of a graph of coposton of fuzzy relatons As an exaple, two fuzzy bnary relatons are used: and Q are defned as follows: X Z, Q Z Y Relatons and Q are set through the ebershp functons: : X Z [0, ], Q : Z Y [0, ] ZY ZY x x n xn X x,, x } { n X Z G z z k z k Z z,, z } { k Q Z Y y y y Y y Fg Graph of coposton º Q {,, y }

3 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence The coposton of relatons º Q, wll be a bnary relaton R º Q X Y wth the ebershp functon as follows: sup{ ( z, }, ( X Y R Q Coposton graph G, s obtaned through the aggregaton of fuzzy graphs of relatons and Q over ther coon set Z (Fg For each par ( X Y there s soe non-epty subset Z Z, n whch for each of ts eleent z the followng holds: { ( z, } z Q ( z, R sup Q More forally, t could be wrtten as follows: ( ( X Y ( Z Z( z ( z Q ( z, R For any coposton a appng of X Y nto Z could be constructed that possesses property (: Γ : X Y Z (2 Γ( deternes soe subset Z wth the followng propertes: z I( z ( z, Z Q R For all eleents z of set Z \ I( (те z Z z I( the followng holds: z I( z ( z, < Q R roectons of appng (2 nto sets X and Y that correspond to I X and I Y could be constructed: I ( x I(, x X ; I X Y y Y ( I(, y Y Next, reduced arguents of coposton - ˆ and Qˆ are deterned wth the graph G ˆ It can be seen that reducton leads to the decrease n cardnalty of coposton arguents and Q The cardnalty of fuzzy relaton could be found by sung up the ebershp degrees for all the eleents of relaton : Next, a factor that evaluates the decrease n sets and Q s ntroduced Value ˆ wll vary wthn lts ˆ and ˆ It s obvous that ˆ Let s fnd a possble value for ˆ n : n ax ˆ + ˆ ˆ ˆ z ΓX ( x \ ΓX ( x As 0 for each eleents z Z \ Γ ( x, that ˆ X z Γ ax ˆ z X ˆ ( x Value ˆ n, f Γ X (x n In the case of the strongest reducton Γ X (x n the reducton ndex δ [0,] 0, but when there s no reducton (Fg2 A B C D E F Fg2 Mappng of a set of values ˆ nto a set of reducton degrees δ

4 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence The lower horzontal axs represents an axs of reducton degree δ The upper axs s an axs of power ont D corresponds to the axu value of reducton ˆ ax, but pont F nu value of reducton ˆ n In pont Е a value of reducton s equal to ˆ If А 0, В δ, С, that the proportonalty expresson wll appear: BC EF ˆ ˆ δ n ˆ ax ˆ or or δ AB DE δ ˆ ˆ ˆ ˆ Thus, X Z X Z ax ˆ X Z δ n The reducton ndex δ Q for relaton Q, could be calculated by analogy 3 Usng a fuzzy logc n qualtatve odellng The ethod of qualtatve odelng s subdvded nto two parts [4]: fuzzy odelng and lngustc approxaton Qualtatve odelng s not very popular ethod, but ts concepton s defned by fuzzy odelng or lngustc (qualtatve odelng Before enterng the an subect, let s overvew a fuzzy odelng n fuzzy theory and qualtatve reasonng n AI There are any nterpretatons of fuzzy odelng For nstance, we can consder a fuzzy set as a fuzzy odel, whch uses a descrpton language based on fuzzy logc wth fuzzy predcates In a broader sense we can nterpret the fuzzy odelng as a qualtatve odelng schee by whch we qualtatvely descrbe syste behavor usng a natural language Qualtatve odel s a generalzed fuzzy odel consstng of lngustc explanatons about syste behavor Thus, to descrbe control rules, lngustc were used There are sall dstnctons and bg slartes between fuzzy odelng and qualtatve reasonng One dstncton s that fuzzy odelng starts fro the fact that a precse atheatcal odel of a coplex syste cannot be obtaned, whereas qualtatve reasonng starts fro the fact that, although a coplete ay be avalable, t cannot provde nsght nto the syste; a descrpton based on deep knowledge s needed Slartes are found n the confdence of the advantage of qualtatve expressons, n the goals, and n soe parts of descrpton languages for odelng Qualtatve reasonng akes use of a quantty space on whch landarks are defned Usually one ark 0 s set, and then three values slar to fuzzy {+, 0, -} are used Qualtatve odelng s a lngustc odel Lngustc odel s a odel that s descrbed usng lngustc ters nstead of atheatcal equatons wth nuercal values or conventonal logcal forula wth logcal sybols Identfcaton of fuzzy odels can be dvded nto two types structure dentfcaton and paraeter dentfcaton Structure dentfcaton has to solve two probles: fnd nput varables and fnd nput-output relatons It s necessary to fnd nput canddates (often heurstcall and nput varables, whch affect the output Also, t s necessary to fnd the nuber of rules n a fuzzy odel and the dvson of an nput space araeters are coeffcents n a odel of functonal syste (for the fuzzy odel paraeters of ebershp functon A ethod of qualtatve odelng conssts of two steps: revew of a fuzzy odel n ters of fuzzy nubers and lngustc nterpretaton of fuzzy odel Frst, defne nput varables Fro the set of possble canddates nput varables are selected heurstcally and ˆ ax n

5 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence those affected the output are fnally selected If take a odel wth four possble nput varables, then search algorth wll be as follows: on the bass of a fuzzy odel wth one nput four other odels are created everyone for the defned nput Then regulaton crteron s calculated (RC for each odel and one odel s selected wth one nput to nze RC Next we fx the one nput selected above and add another nput to our fuzzy odel fro aong the reanng three canddates Fuzzy odel has two nputs at ths stage We select the second nput as at the frst step, accordng to the value of RC Ths process s repeated untl RC s ncreasng The search of a dvson of the space goes wth FCM algorth (the fuzzy c-eans ethod As a result each output y s assocated wth a ebershp degree to the fuzzy cluster В Fuzzy cluster А n the nput set can be nduced by proectons of cluster А nto co-ordnate axs x and x 2 (Fg 3 A2 x 2 B A Output space y Input space A Fg 3 roecton of fuzzy cluster Thus, the followng rule can be derved: f x s А and x 2 s А 2, then y s В In ths case fuzzy dvson of nput space s the drect result of fuzzy clusterng Fndng the nuber of clusters s the ost portant aspect of clusterng The followng crteron s used to solve ths proble: n c S( c ( ( x v v x, k where n: nuber of data to be clustered; c: nuber of clusters, с 2; x k : kth data, usually vector; x : average of data: x, x 2,, x n ; v : vector expressng the center of th cluster; : nor; k : grade of kth data belongng to th cluster; : adustable weght (usually 5 3 k To get qualtatve odel on a bass of a fuzzy odel the ethod of lngustc approxaton of fuzzy sets s used It s necessary to fnd correspondng word or phrase fro ntal set of words for lngustc approxaton After ths procedure we wll get qualtatve odel wth lngustc rules 4 Fuzzy clusterng and fuzzy rules Clusterng algorth begns wth rando ntalzaton and nteractvely changes ebershp degrees u k and prototypes v The ost sple algorth s the fuzzy c-eans algorth whose prototypes are sply the cluster centers n the for of vectors v IR p, and the dstance d(v, x k s the Eucldean dstance between x k and cluster center Gustafson and k 2 x 2

6 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence Kessel enrched each prototype wth syetrc, postve defnte atrx С and copute the dstance as: 2 / p Τ d ( v, xk (det C ( xk v C ( xk v The ethod desgned by Gath and Geva ntroduces an addtonal for each prototype that allows the algorth to adopt to clusters of dfferent szes The prncpal dea to apply fuzzy clusterng n order to derve fuzzy rule s that each cluster nduces a rule by proectng the cluster to the correspondng coordnate spaces The proecton of a cluster to the th doan s obtaned by takng the th coordnate of each data pont and assgnng to t the ebershp degree of the orgnal data pont to the cluster Thus, cluster nduces the rule If ξ s and and ξ р- s р-, then ξ р s р, where denotes th proecton of the cluster and ξ,, ξ р- nput varables and ξ р output varable Therefore, fuzzy sets are defned by the proectons of the cluster, whereas the concluson ust be lnear functon n the nput varables However, fuzzy clusterng s not very precse dervng the rules Fro one hand the shape of a ebershp functon s qute unusual, fro the other fuzzy clusterng s developed to dvde product space, but not a sple doan One approach s based on developng of Takag-Sugeno type controllers fro the data wth constant functons n the conclusons of the rules Thus the paraeters that have to be deterned are the constant output values n the conclusons of the rules and the tps of the trangular ebershp functons n each nput doan The good results of ths approach otvated to develop a fuzzy clusterng algorth that as at fndng fuzzy parttons for the sngle doans on the bass of ultdensonal data To fnd fuzzy parttons t s necessary to deterne a sutable grd n the ult-densonal space The ebershp degree of a pont to the cluster represented by a grd pont as the nu of the ebershp degrees of the trangular ebershp functons whose tps are the proectons of the grd pont Fuzzy clusterng starts wth equdstant trangular ebershp functons on the doans We copute the proectons of the data and the ebershp degrees of these proectons to the trangular ebershp functons Then we update the trangular ebershp functons by coputng new tps as the cluster centers: n k ( x x ( v ( v t k, v k, v old new n ( v ( x k t k, v t ( v ( v where t and t actualzed, respectvely old tp and xk, v denotes vth proecton of datu x k, ( v told new old ( v s the trangular fuzzy set wth ts tp at t old An advantage of ths ethod s that clusters do not have an nfnte range, thus data ponts that are covered by other clusters far away fro one cluster do not have any nfluence on ths cluster old, 5 Fuzzy KORA-Ω algorth Fuzzy KORA-Ω algorth s a fuzzy varant of КОRА-3 algorth, ntroduced by ММBongard [6] n 966 КОRА-3 s developed to solve classfcaton proble n geology area Algorth works wth two dsont classes wth obects descrbed n ters of Boolean varables and has three stages learnng, relearnng and classfcaton Fuzzy KORA-Ω works n an analogous way to the KORA-3, and the essental dfferences are based on the followng fundaentals:

7 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence Let R{x,, x n } the feature set used to descrbe the obects O,, O the obects of the tranng saple МА K - the obects of the class K n МА,,, r C :M M G a coparson crteron of values for feature x, where M s the value set assocated to x β:(m M s 2 G partal slarty easure for any {x,, x s } R ΩO a partal descrpton of an obect О usng only the features of Ω {x,, x s } R (O obect's ebershp degree to K A value cobnaton a (a,, a p for the correspondng features Ω {x,, x p } for a fuzzy δ -coplex feature (а, Ω for K, wth degree ((а,,,, r, f: O K x ( O a ^^ xp ( O a 2 p β ( ΩO, ( O δ β ( ΩO, ( O ' 3 β ( ΩO, ( ( O δ 4 (( a,, ( O where δ > 0 and δ ' > 0 are thresholds In the hard case (( a, In case when β s an equalty functon, 3 Ω and ( { 0,} O КОРА-3 Let RC( K ' the set of δ -coplex feature for that, ths defnton concdes wth one for the ( a, Ω RC( K β ( ΩO, < η s called fuzzy η reans and s denoted by R( K K The set of all obectso K ' s A value cobnaton a (a,, a p for the correspondng features Ω {x,, x p } for a fuzzy δ -coplex feature (а, Ω for K, wth degree ((а,,,, r, f: O K x ( O a ^^ x ( O a ' p p 2 β ( ΩO, ( O δ O r ( K β ( ΩO, ( O ' O r ( K 3 β ( ΩO, ( ( O δ 4 (( a,, ( O where δ > 0 and δ ' > 0 are thresholds, teraton nuber In the hard case (( a, In case when β s equalty, 3 ( 0,, ths defnton concdes Ω and { } wth one for the КОРА-3 Let (а, Ω be a fuzzy δ -coplex feature Its nforatonal weght s calculated as: (( a, ( x β ( ΩO, ( O, xk Ω k O ' K where (x k and (O are the nforatonal weghts of feature x k and O obect n the class K, respectvely O

8 SCIENTIFIC ROCEEDINGS OF RIGA TECHNICAL UNIVERSITY Coputer Scence Inforaton Technology and Manageent Scence When a new obect О s gong to be classfed, t s copared wth all fuzzy δ - coplex features of each class (copleentary or not The ebershp degree of О for the class K s gven by: ( a, Ω RC( K β ( ΩO, (( a, (( a, ( O ax, where ax ax ( ( β ( ΩO, (( a, ' O O K ( a, Ω RC( K The proposed Fuzzy KORA-Ω algorth can be used to solve supervsed classfcaton probles wth any classes (fuzzy and hard classes dsount or not, wth partcpatons of obects descrbed wth any knd and nuber of features, any knd of coparson crteron for each feature, any knds of slarty easure and any set of feature subsets to fnd coplex features References Klawonn F, Keller A Learnng Fuzzy Rules fro Data In: 2 Ozols Y, Borsov A A Coparatve Analyss of the Features n the Fuzzy atterns Classfcaton // In: roc Fourth European Congress on Intellgent Technques and Soft Coputng, EUFIT 96 - Aachen, Gerany, Septeber 2-5, 996 p Ozols Y, Borsov A attern Classfcaton and Feature Extracton on the Bass of Coposton of Fuzzy Relatons // In: roc Seventh Internatonal Fuzzy Systes Assocaton World Congress, IFSA 97 - rague, Czech Republc, June, 25-29, 997 p Sugeno M, Yasukawa T A Fuzzy-Logc-Based Approach to Qualtatve Modellng // In: IEEE Transactons on Fuzzy Systes, Vol, No, February Klawonn F, Keller A Fuzzy Clusterng and Fuzzy Rules // In: roc Seventh Internatonal Fuzzy Systes Assocaton World Congress, IFSA 97 - rague, Czech Republc, June, 25-29, Комплексная интерпретация геологических и геофизических данных на вычислительных машинах Под ред ШАГубермана - Москва, De-la-Vega-Dora LA, Carrasco-Ochoa JA, Ruz-Shulcloper J Fuzzy KORA-Ω Algorth // In: EUFIT 98, Aachen, Gerany, Septeber 7-0, 998 Roans Grekovs, hd student n the Insttute of Inforaton Technology at the Techncal Unversty of Rga Address: Decson Support Systes Group, Insttute of Inforaton Technology, Techncal Unversty of Rga, Kalkyu Street, Rga LV-658, Latva E-al: roansgrekovs@navgatorlv Grekovs R Izplūdušo tēlu atpazīšanas etodes Šs raksts zskata dažādus darbus zplūdušo tēlu atpazīšanas oā Atpazīšanas etodes tek klasfcēt Arī aprakstīt nēperceptronskas tēlu atpazīšanas etodes kā arī etodes, kur zanto zplūdušas proekcas tēlu atpazīšanā un klasteru analīzē Grekovs R Methods of Fuzzy attern Recognton Ths paper s dedcated to the revew of soe papers on the thee of fuzzy pattern recognton There s soe classfcaton of recognton ethods Descrbed non-perceptron based ethods of pattern recognton, ncludng those, whch use fuzzy proectons n pattern recognton and cluster analyss Греков Р Методы нечеткого распознавания образов Данная статья посвящена обзору работ по теме распознавания нечетких образов Приведена классификация методов распознавания Рассмотрены неперцептронные методы распознавания образов, включая методы, использующие нечеткие проекции в распознавании образов и анализе кластеров