I Soft Computing Approach to Pattern Classification In conventional approach, position of each patten(say, Pi) on the finite range of the pattern space is represented by a pattern vector/feature vector Fc and we always try to discriminate among patterns by classifying the pattern vector/feature vector Fci Therefore, from the given data set (i.e. the training data set), we always know where the patterns are located and then try to separate them by some appropriate decision function. Subsequently, we use the said decision function to classify the test pattern vector/feature vector if we go by mimic ognitive process of and Simon 1972; Zadeh et al. 1975)for pattern classification(object recognition) then the problem is, from a given set of imprecise observations stated in terms of fuzzy If-Then rules, how to represent the patterns on the pattern space for developing suitable inferencing technique for classification. As the observations (knowledge)are given in terms of fuzzy If-Then rules, it is clear that at every observation(rule), features are given in the form of fuzzy set defined over the feature universe. Therefore, in such a case we cannot represent/locate individual pattern by a vector. Instead, from a given observation (rule), we can represent/ locate a population of patterns in an area(in case of R; but region in general) the pattern space by fuzzy pattern vector/feature vector(see Fig. A I of Appendix A)which is a very natural representation of the antecedent parts of a multidi- mensional implication(i.e. R3)represented by a fuzzy set, i.e. a multidimensional fuzzy implication(MFD)(Sugeno and Takagi 1983; Tsukamoto 1979). Now, we stipulate the following definition Definition 1.1. Let F, be a fuzzy vector having"n"components, each of which is a fuzzy set d' defined over the universe Ui of the feature axis Fi. The fuz zev vector Fr is a fuzzy set in the quantized product space U, U2x.Un.Each element of the fuzzy set is a vector having same initial point but different terminal poin hich are the elements of the fuzzy point which is a fuzzy set. Each terminal point of each vector in the set carries one membership value indicating its(vectors) s to the set F. A fuzzy vector F is represented as 可={g(可)可 VIEUx U2x whep:1xb2x…xUn→0. is the membership function of F, and "E(V)is the grade of membership of Vin F The process of defuzzification of the fuzzy vector Fr is performed based on selecting the elements of the fuzzy vector F/(which is a fuzzy set) having highest membership values. The defuzzified version of the fuzzy vector F is a fuzzy vector which is a fuzzy set. In case the defuzzified version of the fuzzy vector F/
In conventional approach, position of each pattern (say, Pi) on the finite range of the pattern space is represented by a pattern vector/feature vector ~FCi ~ and we always try to discriminate among patterns by classifying the pattern vector/feature vector ~FCi ~ . Therefore, from the given data set (i.e. the training data set), we always know where the patterns are located and then try to separate them by some appropriate decision function. Subsequently, we use the said decision function to classify the test pattern vector/feature vector. But, if we go by mimicking the cognitive process of human reasoning (Newell and Simon 1972; Zadeh et al. 1975) for pattern classification (object recognition) then the problem is, from a given set of imprecise observations stated in terms of fuzzy If–Then rules, how to represent the patterns on the pattern space for developing suitable inferencing technique for classification. As the observations (knowledge) are given in terms of fuzzy If–Then rules, it is clear that at every observation (rule), features are given in the form of fuzzy set defined over the feature universe. Therefore, in such a case we cannot represent/locate individual pattern by a vector. Instead, from a given observation (rule), we can represent/ locate a population of patterns in an area (in case of R2 ; but region in general) on the pattern space by fuzzy pattern vector/feature vector (see Fig. A.1 of AppendixA) which is a very natural representation of the antecedent parts of a multidimensional implication (i.e. R3) represented by a fuzzy set, i.e. a multidimensional fuzzy implication (MFI) (Sugeno and Takagi 1983; Tsukamoto 1979). Now, we stipulate the following definition; Definition 1.1. Let F !f be a fuzzy vector having ‘‘n’’ components, each of which is a fuzzy set Di defined over the universe Ui of the feature axis Fi. The fuzzy vector F !f is a fuzzy set in the quantized product space U1 U2 ...Un. Each element of the fuzzy set is a vector having same initial point but different terminal points which are the elements of the fuzzy point which is a fuzzy set. Each terminal point of each vector in the set carries one membership value indicating its (vector’s) degree of belonginness to the set F !f . A fuzzy vector F !f is represented as F !f ¼ l F !f V ! : V ! 8 V !e U1 U2 ... Un where l F !f : U1 U2 ... Un ! ½0; 1 is the membership function of F !f and l F !f ðV !Þ is the grade of membership of V !in F !f . The process of defuzzification of the fuzzy vector F !f is performed based on selecting the elements of the fuzzy vector F !f (which is a fuzzy set) having highest membership values. The defuzzified version of the fuzzy vector F !f is a fuzzy vector which is a fuzzy set. In case the defuzzified version of the fuzzy vector F !f 6 1 Soft Computing Approach to Pattern Classification
1.2 Passage Between Conventional Approach to Pattern Classification represents a fuzzy set which is a fuzzy singleton then the defuzzified version of F becomes the crisp version(see Example 1.1). The fuzzy set D as mentioned in the Definition 1.1 is an element of the term set as discussed in fig. 1.1. The univer of the components of F, ie the fuzzy set D, may be continuous/discrete and the universe of F may be continuous/discrete. In case universes are discrete we should follow the numerical definition of membership functions; otherwise, we should follow the functional definition. In case the defuzzified version of F reduces to the crisp vector as stated earlier the membership value at the terminal point of the vector Vi of F can alternatively be interpreted as the highest possibility of Vi to hold the property of the fuzzy vector F. By the term property we want to mean a particular combination of the elements of different term sets For instance, with respect to Fig(A1)of Appendix-A the property assumed by fuzzy vector F is(medium, medium). Like this we can have property(medium, small),(medium, big)etc. (also see Fig. 1.5). The crisp vector V15.13 of Fig(A 1) of Appendix-a obtained by the process of defuzzification of the fuzzy vector F, has the highest possibility to hold the above said property(medium= MI medium= M2). In case we obtain an induced fuzzy vector F which is basically a cylindrical extension of a primary fuzzy set Dover U, all the elements of the defuzzified version of F, have equal possibility values which are the highest possibility to hold the property of the element D of the term set over U;. For nstance,in Fig. A 2 of Appendix-A, the vectors from V15.1 to V15. 25 have equal possibility values which are the highest possibility to hold the property of the fuzzy set M, which is an element of the term set S, Mi, Bi defined over the universe of U1. The relation of the generic element represented by the line segment (9< uio<10)defined over the universe of U2 with all the elements of the fuzzy MI is written by 001/V101o,0.2/V V13020.8/V140,1/V1510,0./V610,0.6/V10,04/Vs0,0.2/V190o,001 /V20. 10. The array itself is a fuzzy set where the vectors VI 10,10,V11,10 V1210,V1310V V1s10.V1610.V17 10. VI810. V V20.10 are the elements of the said fuzzy set(see Fig. A2 of Appendix-A).The possibility of the vector V,ojo to hold the property of the vector VIs.10 is 0.01 The fuzzy set M,, which is linguistically stated as fuzzy set medium over the universe U, of FI, is basically obtained by a process of fuzzification of the discrete element represented by the line segment 14<uis <15. Thus a point on real line or a discrete line segment(on real line) which is the quantized version of the point is represented by a fuzzy set on real line. Intuitively speaking, the quantize point 15 represented by the line segment 14<uis <15 on the range of feature axis Fi,i.e 0<UI<28(see Fig. A. I and A 2 of Appendix-A)is approximately the 'mid int'. To represent the 'midpoint by linguistically stated fuzzy set Mi defined
represents a fuzzy set which is a fuzzy singleton then the defuzzified version of F !f becomes the crisp version (see Example 1.1). The fuzzy set Di as mentioned in the Definition 1.1 is an element of the term set as discussed in Fig. 1.1. The universe of the components of F !f , i.e. the fuzzy set Di , may be continuous/discrete and the universe of F !f may be continuous/discrete. In case universes are discrete we should follow the numerical definition of membership functions; otherwise, we should follow the functional definition. In case the defuzzified version of F !f reduces to the crisp vector as stated earlier the membership value at the terminal point of the vector V !ij of F !f can alternatively be interpreted as the highest possibility of V !ij to hold the property of the fuzzy vector F !f . By the term property we want to mean a particular combination of the elements of different term sets. For instance, with respect to Fig. (A.1) of Appendix-A the property assumed by fuzzy vector F !f is (medium, medium). Like this we can have property (medium, small), (medium, big) etc. (also see Fig. 1.5). The crisp vector V !15;13 of Fig. (A.1) of Appendix-A obtained by the process of defuzzification of the fuzzy vector F !f has the highest possibility to hold the above said property (medium = M1, medium = M2). In case we obtain an induced fuzzy vector F !f which is basically a cylindrical extension of a primary fuzzy set Di over Ui, all the elements of the defuzzified version of F !f have equal possibility values which are the highest possibility to hold the property of the element Di of the term set over Ui. For instance, in Fig. A.2 of Appendix-A, the vectors from V !15;1 to V !15;25 have equal possibility values which are the highest possibility to hold the property of the fuzzy set M1 which is an element of the term set {S1, M1, B1} defined over the universe of U1. The relation of the generic element represented by the line segment 9 u2 10\10 defined over the universe of U2 with all the elements of the fuzzy set M1 is written by an array f0:01=V !10;10; 0:2=V !11;10; 0:4=V !12;10; 0:6= V !13;10; 0:8=V !14;10; 1=V !15;10; 0:8=V !16;10; 0:6=V !17;10; 0:4=V !18;10; 0:2=V !19;10; 0:01 =V !20;10g. The array itself is a fuzzy set where the vectors V !10;10; V !11;10; V !12;10; V !13;10 V !14;10; V !15;10; V !16;10; V !17;10; V !18;10; V !19;10; V !20;10 are the elements of the said fuzzy set (see Fig. A.2 of Appendix-A). The possibility of the vector V !10;10 to hold the property of the vector V !15;10 is 0.01. The fuzzy set M1, which is linguistically stated as fuzzy set medium over the universe Ui of F1, is basically obtained by a process of fuzzification of the discrete element represented by the line segment 14 u1 15\15. Thus a point on real line or a discrete line segment (on real line) which is the quantized version of the point is represented by a fuzzy set on real line. Intuitively speaking, the quantize point 15 represented by the line segment 14 u1 15\15 on the range of feature axis F1, i.e. 0 B U1\28 (see Fig. A.1 and A.2 of Appendix-A) is approximately the ‘midpoint’. To represent the ‘midpoint’ by linguistically stated fuzzy set M1 defined 1.2 Passage Between Conventional Approach to Pattern Classification 7
I Soft Computing Approach to Pattern Classification over the discrete universe of UI we consider some spread on both sides of the quantize point 15 which is the quantification of our perception about the said midpoint'15 which is fuzzified over the discrete universe U1. This quantized representation of the fuzzy set medium(M1) is not unique and may vary dependin upon the perception of an individual which is not unique. But whatever may be the variation of human perception, it always follows a particular trend. Thus the integrity of perception (i.e. the unique trend in perception) is reflected on the diversity of representation where lies the real flavour of fuzzine Remark l1 I. A fuzzy point on R is a straight line. For example, consider the fuzzy set Mion Fig. A I and A 2 of Appendix-A 2. A fuzzy point on R- is an area. For example, consider the ABCD of Fig. A I of Appendix-A and area EFGH of Fig. A2 of Appendix-A 3. A fuzzy point on Ris a volume 4. A fuzzy point on R is a region. Now we introduce the notion of fuzzy pattern vector/feature vector i.e. F which is an analogous representation of Fa on R2(see Fig. 1.3). Tip of the fuzzy pattern vector/feature vector no longer represents a single pattern on R; rather it represents a population of patterns(set of patterns Example 1. 1. Let us consider the following fuzzy feature vector/pattern vector (see Fig A l of Appendix-A). The primary fuzzy set Mi and M2 are defined over the discrete universe of U1 and U2 respectively. Each discrete element of the universe Ui Vi is represented by small line segment over Ui Vi F={(F(可,可)} r(可) F/(V) 001/V107)+…+(001/V0.n7)+ +…+(08/u)+…+(7s)+…+(00/7)} Where and 2 are in the set-theoretic sense and M, is a fuzzy set {001/9≤u<10),02/(10≤h1<11,0.4/(11≤n2<12),06/(12≤ <13),0.8/(13≤uh4<14),1/(14≤u<15),0.8/(15≤l6<16),0.6/(16≤l17 <17),04/(17≤as<18),0.2/(18≤9<19),001/(19≤2≤20)}onte features axis u1, M2isafuzzyset{0.01/(6≤<7),0.1/7≤<8),02/(8≤ <9),0.4/(9≤20<10),0.6/(10≤21<11,0.8/(11≤2<12),1/(12≤n23<
over the discrete universe of U1 we consider some spread on both sides of the quantize point 15 which is the quantification of our perception about the said ‘midpoint’ 15 which is fuzzified over the discrete universe U1. This quantized representation of the fuzzy set medium (M1) is not unique and may vary depending upon the perception of an individual which is not unique. But whatever may be the variation of human perception, it always follows a particular trend. Thus the integrity of perception (i.e. the unique trend in perception) is reflected on the diversity of representation where lies the real flavour of fuzziness. Remark 1.1. 1. A fuzzy point on R1 is a straight line. For example, consider the fuzzy set M1 on Fig. A.1 and A.2 of Appendix-A. 2. A fuzzy point on R2 is an area. For example, consider the ABCD of Fig. A.1 of Appendix-A and area EFGH of Fig. A.2 of Appendix-A. 3. A fuzzy point on R3 is a volume. 4. A fuzzy point on Rn is a region. Now we introduce the notion of fuzzy pattern vector/feature vector i.e. F !f which is an analogous representation of F !~ci on R2 (see Fig. 1.3). Tip of the fuzzy pattern vector/feature vector no longer represents a single pattern on R2 ; rather it represents a population of patterns (set of patterns). Example 1.1. Let us consider the following fuzzy feature vector/pattern vector (see Fig. A.1 of Appendix-A). The primary fuzzy set M1 and M2 are defined over the discrete universe of U1 and U2 respectively. Each discrete element of the universe Ui Vi is represented by small line segment over Ui Vi. F1 is M1 F2 is M2 ¼ F !f ¼ lF !f V ! ; V n o ! ¼ lF !f V ! =V n o ! ¼ X20 i¼1 X19 j¼1 lF !f V !ij =V !ij ¼ 0:01= V !10;7 þþ 0:01=V !10;17 þþ 0:01=V !10;19 n þþ 0:8=V !14;12 þþ 1=V !15;13 þþ 0:01=V !20;19 o Where þ and P are in the set theoretic sense and M1 is a fuzzy set 0:01=ð9 u1 10\10Þ; 0:2=ð10 u1 11\11Þ; 0:4=ð11 u1 12\12Þ; 0:6=ð12 u1 13 \13Þ; 0:8=ð13 u1 14\14Þ; 1=ð14 u1 15\15Þ; 0:8=ð15 u1 16\16Þ; 0:6=ð16 u1 17 \17Þ; 0:4=ð17 u1 18\18Þ; 0:2=ð18 u1 19\19Þ; 0:01=ð19 u1 20 20Þ on the features axis U1; M2isafuzzyset 0:01=ð6 u2 7\7Þ; 0:1=ð7 u2 8\8Þ; 0:2=ð8 u2 9 \9Þ:; 0:4=ð9 u2 10\10Þ; 0:6=ð10 u2 11\11Þ; 0:8=ð11 u2 12\12Þ; 1=ð12 u2 13\ 8 1 Soft Computing Approach to Pattern Classification
1.2 Passage Between Conventional Approach to Pattern Classification 13),0.8/(13≤424<14),0.6/(14≤25<15),0.4/(5≤l26<16),0./(16≤y <17),0.1/(17<u18<18),0.01(18<uig <19) on the feature axis F2 and each vector Vi which is an element of the fuzzy set F, represents one small quantized zone on the pattern space. Thus, instead of a point pattern a population of patterns (a set of patterns)is represented by F. Pattern pr of Fig. A 1 of Appendix-A is represented by an element V1s. 13 of the fuzzy set represented by F and pattern P4 is represented by V14.18 of F. Here the fuzzy set F, is defined over the universe UI X U2, i.e. the pattern space. In this case, also note that the defuzzified version of the fuzzy vector Fr is a fuzzy singleton represented by vector Definition 1.2. Length of a fuzzy vector F is represented as F ∑F(/d {nF(n)/++(m)/=m L(oonIv ionl) +(00/ 0/o Definition 1.3. Slope of a fuzzy vector Ff; represented as, F=∑∑F(V) F(V1a7)/∠v07+ o01/∠v)+ 01/∠v ,19 mple 1.2. Va2+b2 and V1513=tan"Ig where a, b are shown Depending upon the area of each class occupied by the tip of the F/(see Fig. A1 Appendix-A)we determine the possibility of occurrence of different classes patterns under that F. Such possibility of occurrence is represented by a fuzzy set which is the consequent part of a MFl and which is defined in the pattern space which
13Þ; 0:8=ð13 u2 14\14Þ; 0:6=ð14 u2 15\15Þ; 0:4=ð15 u2 16\16Þ; 0:2=ð16 u2 17 \17Þ; 0:1=ð17 u2 18\18Þ; 0:01=ð18 u2 19\19Þ on the feature axis F2 and each vector V !ij which is an element of the fuzzy set F !f represents one small quantized zone on the pattern space. Thus, instead of a point pattern a population of patterns (a set of patterns) is represented by F !f . Pattern p7 of Fig. A.1 of Appendix-A is represented by an element V !15;13 of the fuzzy set represented by F !f and pattern p4 is represented by V !14;18 of F !f . Here the fuzzy set F !f is defined over the universe U1 9 U2, i.e. the pattern space. In this case, also note that the defuzzified version of the fuzzy vector F !f is a fuzzy singleton represented by vector V !15;13. Definition 1.2. Length of a fuzzy vector F !f is represented as, j F !f j ¼ X 20 i¼1 X 19 j¼1 lF !fðV !ijÞ , jV !ijj ¼ l F !f V !10;7 .jV !10;7j þ . . .. . . þ l F !f V !20;19 .jV !20;19j ¼ 0:01. jV !10;7j þ . . .. . . þ 0:01. jV !10;7j þþ 1jV !15;13j þþ 0:01. jV !20;19j Definition 1.3. Slope of a fuzzy vector F !f ; represented as, \ F !f ¼ X 20 i¼1 X 19 j¼1 l F !fðV !ijÞ , \V !ij ¼ l F !f V !10;7 .\V !10;7 þ . . .. . . þ l F !f V !20;19 .\V !20;19 ¼ 0:01. \V !10;7 þ . . .. . . þ 0:01. \V !10;7 þ þ 1 \V !15;13 . þþ 0:01. \V !20;19 Example 1.2. V !15;13 ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ b2 p and \V !15;13 ¼ tan1 a b where a, b are shown in Fig. 1.4. Depending upon the area of each class occupied by the tip of the F !f (see Fig. A.1 of Appendix-A) we determine the possibility of occurrence of different classes of patterns under that F !f . Such possibility of occurrence is represented by a fuzzy set which is the consequent part of a MFI and which is defined in the pattern space which 1.2 Passage Between Conventional Approach to Pattern Classification 9
I Soft Computing Approach to Pattern Classification gment 13 15.131 15.13 segment 15 Fig. 1. 4 A portion of Fig. AI of Appendix-A is the universe of all classes of patterns. Thus, in the same pattern space, i.e. in the Cartesian product U1 X U2 (if n= 2)we define two types of fuzzy sets; one fuzzy set is represented by F, and the other fuzzy set is the consequent part of a MFI.The consequent part of a MFl, which is a fuzzy set, simply indicates the relative position of a fuzzy feature vector/pattern vector with respect to the different classes of patterns in the pattern space. Once the antecedent part and the consequent part of a MFI are represented by two types of fuzzy sets as stated above our next job is attach a meaningful interpretation to the said representations. One possible inter- pretation is given by Eq. AI of Appendix-A which is similar to R4 represented by fuzzy sets. The other interpretation is given by Eq. A4 of Appendix-A. A particular interpretation simply provides the information how the antecedent part of a MFI is related to its consequent part. Depending upon different types of interpretations, w are having different types of model for fuzzy reasoning Incidentally, note that by fuzzy feature vector/pattern vector we basically describe the entire pattern space by few quantized zones where neighboring zones overlap each other(see Fig. 1.5). The overlapped zones are called fuzzy zones of our description of the pattern space So far we have discussed fuzzy If?Then rules wher measurable by some sensors, e.g. standard of living, behavior, appearance, etc. Such features are fuzzified(see Fig. 1. 1)for the purpose of representing our perception of objects(patterns). But, apart from these, we can have features which can be sensed by standard transducers or which are measurable, e.g. formant frequency of a vowel, curvature and internal angle at a dominant point of a 2-D closed curve, etc (Koch and Kashyap 1987). Under such circumstances, given
is the universe of all classes of patterns. Thus, in the same pattern space, i.e. in the Cartesian product U1 9 U2 (if n = 2) we define two types of fuzzy sets; one fuzzy set is represented by F !f and the other fuzzy set is the consequent part of a MFI. The consequent part of a MFI, which is a fuzzy set, simply indicates the relative position of a fuzzy feature vector/pattern vector with respect to the different classes of patterns in the pattern space. Once the antecedent part and the consequent part of a MFI are represented by two types of fuzzy sets as stated above our next job is to attach a meaningful interpretation to the said representations. One possible interpretation is given by Eq. A.1 of Appendix-A which is similar to R4 represented by fuzzy sets. The other interpretation is given by Eq. A4 of Appendix-A. A particular interpretation simply provides the information how the antecedent part of a MFI is related to its consequent part. Depending upon different types of interpretations, we are having different types of model for fuzzy reasoning. Incidentally, note that by fuzzy feature vector/pattern vector we basically describe the entire pattern space by few quantized zones where neighboring zones overlap each other (see Fig. 1.5). The overlapped zones are called fuzzy zones of our description of the pattern space. So far we have discussed fuzzy If?Then rules where features are not directly measurable by some sensors, e.g. standard of living, behavior, appearance, etc. Such features are fuzzified (see Fig. 1.1) for the purpose of representing our perception of objects (patterns). But, apart from these, we can have features which can be sensed by standard transducers or which are measurable, e.g. formant frequency of a vowel, curvature and internal angle at a dominant point of a 2-D closed curve, etc (Koch and Kashyap 1987). Under such circumstances, given Fig. 1.4 A portion of Fig. A.1 of Appendix-A 10 1 Soft Computing Approach to Pattern Classification