Journal of Artfcal Intellgence Practce (206) : 8-3 Clausus Scentfc Press, Canada New Dstance Measures on Dual Hestant Fuzzy Sets and Ther Applcaton n Pattern Recognton L Xn a, Zhang Xaohong* b College of Arts and Scences, Shangha Martme Unversty, Shangha 20306, Chna a 4749275@qq.com, b zxhonghz@263.net, Keywords: Dual hestant fuzzy sets, Dstance Measure, Hestance degree, Pattern recognton. Abstract: The concept of dual hestant fuzzy sets (DHFSs), whch was frst ntroduced as a new extenson of fuzzy sets and hestant fuzzy sets n 202, s a useful tool to deal wth the vagueness and ambguty n many practcal problems under hestant fuzzy envronment. Normally, we use the defnton of dstance to descrbe the relatonshp of two DHFSs. However, consderng that the exstng dstance measures of DHFSs stll have some major shortcomngs, so n ths paper, we frstly ntroduce a new concept hestance degree of each dual hestant fuzzy element (DHFE) to these exstng dstance measures and then develop several novel dstance measures n whch both the values and the numbers of values of DHFE are taken nto account. The propertes of these new dstance measures are dscussed. Fnally, we apply our proposed dstance measures of DHFSs n pattern recognton makng to llustrate ther valdty and applcablty.. Introducton The concept of fuzzy set was ntroduced by Zadeh n 965 as an extenson of the classcal noton of sets (see [, 2]). In classcal set theory, the membershp of elements n a set s assessed n bnary terms accordng to a bvalent condton an element ether belongs to or does not belong to the set, by contract, fuzzy sets theory permts the gradual assessment of the membershp of elements n a set, ths s descrbed wth the ad of a membershp functon valued n the real unt nterval [0, ]. Snce ts orgnal defnton, several extensons have been proposed for fuzzy sets, among them, we can underlne, for ther relevance n ths paper, hestant fuzzy sets (HFSs) and dual hestance fuzzy sets (DHFSs). The HFSs defne the membershp of an element and the membershp degree may be a set of possble values rather than nter-values [0, ]. On the bass of HFSs, Zhu and Xu [3] ntroduced the defnton of DHFSs whch uses the membershp hestancy functon and the non-membershp hestancy functon to support a more exemplary and flexble access to assgn values for each element n the doman, t s a very useful tool to deal wth vagueness and ambguty n the pattern recognton problems under hestant fuzzy envronment. Consderng how to descrbe the relatonshp of two gven fuzzy sets, researchers proposed the concept of dstance measures whch are used for estmatng the degree of dstance between two fuzzy sets. The most wdely used dstance measures are the Hammng dstance, Eucldean dstance and Hausdorff metrc. Based on these researchers, Su and Xu have made some sgnfcant extensons for 8
these measures to deal wth DHFSs. Now, these dstance measures have been extensvely appled n some felds such as decson makng, pattern recognton, machne learnng and market predcton and so on. However, n the process of practcal applcaton, we found that these exstng dstance measures of DHFSs stll have some notable short comngs, the most obvous s that they can only cover the dvergence of the values but fal to consder the numbers of value of two gven DHFSs. However, the man characterstc of DHFSs s that they can descrbe the hestant stuatons flexbly, such a hestaton s depcted by a number of values of DHFSs beng greater than just one. Hence, t s very necessary to take nto account both the dfference of the values and that of the numbers when we study the dfference between the DHFSs. In ths paper, we referred to [4, 5], we propose some new dstance measures for DHFSs n ths paper by takng nto account the hestance of the hestant fuzzy sets and nvestgate ther applcaton n practcal pattern recognton. Now, we frstly revew the defnton of DHFSs and the propertes of ther dstance, and then lst several frequently-used dstance measures of DHFSs. Defnton.. Let X be a fxed set, then a dual hestant fuzzy set (DHFSs) H on X s descrbed as: H={ x, h(x), g(x) xx}. () n whch h(x) and g(x) are two sets of value n [0, ] denotng the possble membershp degrees and non-membershp degrees of the element xh to the set, respectvely. Defnton.2. Let A and B be two DHFSs on X={x, x 2,, x n }, then the dstance between A and B denoted as d(a, B), whch satsfy the followng propertes: () 0d(A, B); (2) d(a, B)=0 only f A=B; (3) d(a, B) =d(b, A). () Defnton.3. Let elements n d E (x)=(h E (x), g E (x)) n decreasng order, and let E be the th (j) largest value n h E (x) and E be the jth largest value n g E (x). Let l h (d E (x )) and l g (d E (x )) be the number of values n h E (x ) and g E (x ), respectvely. But n most case, l h (d E (x ))l g (d E (x )). To operate correctly, we should extend the shorter one makng both of them have the length by addng dfferent values. On the bass of abovng defntons, we can refer several exstng dstance measures for DHFSs now n [3]. Defnton.4. we defne a dual hestant normalzed Hammng dstances at frst : n x (j) (j) (k) (k) d(a,b) A (x) B (x) A (x) B (x) (2) nlx j k n whch 0. l x (#h x ) ( x ) and #h and are the numbers of the elements n h and g respectvely. Defnton.5. If we take the weght of each element nto account, the followng weghted dstance measures for DHFSs can be attaned ( 0,, n and 0 ): n x (j) (j) (k) (k) d 2(A,B) A (x ) B (x ) A (x ) B (x ) lx j k (3) 9
2. New dstance measures for DHFSs wth ther applcaton to pattern recognton In ths secton, we wll propose a smple but convncng example to reveal the dsadvantage of the above-mentoned dstance measures at frst. And then by ntroducng the defnton of hestance degree, we can get our modfed dstance measures. At last, we utlze the proposed dstance measures to a practcal pattern recognton example to prove ther valdty and superorty. Example 2.. Let X={x }, assume that there exst two patterns whch are presented by DHFSs A and B, A={{0.69, 0.75}, {0.37, 0.76}}, {{0.2, 0.53, 0.74}, {0.96}}, {{0.22, 0.3, 0.60}, {0.6}}, {{0.3, 0.67}, {0.44, 0.69}}, B={{0.34, 0.35}}. Now, there s a sample to be recognzed whch s represented by another DHFS H= {{0.34, 0.53}, {0.2, 0.54}}, {{0.6, 0.3, 0.52}, {0.53}}, {{0.09, 0.5, 0.39}, {0.42}}, {{0.23, 0.49}, {0.29, 0.5}}. Frstly, we should analyze ths queston by our above-mentoned knowledge, t s obvously that the dfference of the membershp values between A and H as well as that between B and H are almost the same, but the strucure of H and that of A s almost totally unform, whch s qute dfferent from that of B.what eles, the number of values of H s the same as that of A, but dfferent from that of B to a great extent. As we stressed before, a hestaton s depcted by a number of values of DHFSs beng greater than just one, so the number of values s equally mportant. So through dscusson, we thnk t s easy to understand that H should belong to the pattern A. However, by applyng the exstng dstance measure equaton (2), we can obtan d(a, H)=0.0584 and d(b, H)=0.0366, so we get the result that H should belongs to the pattren B, t s obvously contrast our analyss. The error s because the exstng dstance measures can only cover the dvergence of the values but fal to consder the numbers of value of two gven DHFSs. As we all known one small false step wll make a great dfference, so t s very necessary to take nto account both the dfference of the values and that of the numbers when we study the dfference between the DHFSs. In the followng, we propose some new dstance measures between DHFSs by takng nto account the hestance extent of each DHFE, whch can overcome the above-mentoned shortcomng. Before that, we frst ntroduce a new concept as follows: Defnton 2.. Let A be a DHFSs on X={x, x 2,, x n }, f A (x ) and g A (x ) are the membershp functon and non-membershp functon of A. l(f A (x )) and l(g A (x )) are the length of f A (x ) and g A (x ), respectvely. (h A(x )) 2 l(f A (x )) l(g A (x )) (4) n (h ) (h (x )) (5) A A n We call (h A(x )) the hestance degree of h A(x ), (h A) the hestance degree of h A.The value of (h A) reflects the degree of hestance for a decsonmaker to determnne the membershp for h A. In the followng, we present some new dstance measures whch nclude the value of (h A). Defnton 2.2. Let h A and h B be a DHFS on X={x, x 2,..., x n }.Then the normalzed Hammng dstance wth hestance degree between h A (X ) and h B (X ) can be redefned as( 0 ): n x (j) (j) (k) (k) d n (h A(x )) (h B(x )) A (x ) B (x ) A (x ) B (x ) 2n lx j k (6) 0
Defnton 2.3. A generalzed dual hestant weghted dstance wth hestance degree between h A(x ) and h B(x ) s gven as( 0 ): n x (j) (j) (k) (k) d 2n (h A(x )) (h B(x )) A (x ) B (x ) A (x ) B (x ) lx j k (7) Now we need to prove (6) satsfy the condtons of Defnton.2. () It s obvously that 0 d(a, B), because all the values of DHFSs are obtaned n the nterval [0,]; (2) Necessty: f d n (A,B) 0, then (j) (j) (h A(x )) (h B(x )) 0, (x ) (x ) 0, A B We can know h A(x) h B(x), so A=B. Suffcency: f A=B, then (j) (j) (h A(x )) (h B(x )) 0, (x ) (x ) 0, A B (x ) (x ) 0. (k) (k) A B (x ) (x ) 0. (k) (k) A B We can know d n (A,B) 0. (3) Obvously, d n (A, B) =d n (B, A). We can prove equaton (7) satsfy the condtons of Defnton.2. as well. Now, we reconsder Example 3. by applyng the above equaton (6), we can obtan that d(a, H)=0.0292 and d(b, H)=0.0704, so ths result s accord wth our analyss. 3. The applcaton n pattern recognton To valdate the proposed dstance measures n practcal applcaton, we present another example n ths secton: an avalable example quotng from [6]. The problem of buldng materals recognton s common n pattern recognton. Let each of metal materals be related to four attrbute ndces G j (j=, 2, 3, 4), let the weght vector of the T atttrbutes G j (j=, 2, 3, 4) be (0.40, 0.22, 0.8, 0.20), all data of other metal materal be expressed n Table. In order to recognze whch pattern a new metal materal B={{0.8, 0.2}, {0.8, 0.2}, {0.5, 0.2}, {0.7, 0.3}}. By applyng the above-mentoned weghted equaton, we can obtan table 2-4. Table. DHFSs for buldng materals A G G G 2 G 3 G 4 A {{0.5,0.6}{0.3}} {{0.2}{0.7,0.8}} {{0.3,0.4}{0.5,0.6}} {{0.5,0.60.7}{0.3}} A 2 {{0.8}{0.2}} {{0.6,0.7,0.8}{0.2}} {{0.,0.2}{0.3}} {{0.2}{0.6,0.7,0.8}} A 3 {{0.7,0.8}{0.2}} {{0.2,0.3,0.4{0.5}} {{0.4,0.5}{0.2}} {{0.2,0.4}{0.5,0.6}} A 4 {{0.3,0.4}{0.6}} {{0.4,0.5}{0.3,0.4}} {{0.3,0.4}{0.6}} {{0.4,0.5}{0.5}} A 5 {{0.7}{0.3}} {{0.4,0.5}{0.3,0.4}} {{0.3}{0.5,0.6,0.7}} {{0.5}{0.4,0.5}} Table 2. Dstances among A and B calculated by equaton (3) A A A 2 A 3 A 4 A 5 Rankng = 0.85 0.0680 0.0685 0.500 0.0990 A 4 > A > A 5 > A 3 > A 2 =2 0.0876 0.0496 0.0487 0.09 0.0723 A 4 > A > A 5 > A 2 > A 3 =4 0.0687 0.0433 0.044 0.0948 0.0637 A 4 > A > A 5 > A 2 > A 3 =6 0.0895 0.045 0.0394 0.099 0.067 A 4 > A > A 5 > A 2 > A 3
Table 3. Hestance degree of A and B A G G G 2 G 3 G 4 Hestance degree of A and B A 0.2500 0.2500 0.5000 0.3334 0.3334 A 2 0.5000 0.3334 0.2500 0.3334 0.3542 A 3 0.2500 0.3334 0.2500 0.5000 0.3334 A 4 0.2500 0.5000 0.2500 0.2500 0.325 A 5 0.5000 0.5000 0.3334 0.2500 0.3959 0. 0.9 0.3 0.7 0.5 0.5 0.7 0.3 0.9 0. Table 4. Dstances among A and B calculated by equaton (7) A A A 2 A 3 A 4 A 5 Rankng = 0.480 0.3535 0.3243 0.5255 0.4200 A 4 > A > A 5 > A 2 > A 3 =2 0.2288 0.900 0.586 0.987 0.795 A > A 4 > A 2 > A 5 > A 3 =4 0.663 0.389 0.080 0.248 0.228 A > A 2 > A 4 > A 5 > A 3 =6 0.530 0.274 0.0970 0.098 0.46 A > A 2 > A 5 > A 4 > A 3 = 0.4430 0.3535 0.3229 0.4765 0.4200 A 4 > A > A 5 > A 2 > A 3 =2 0.205 0.830 0.480 0.776 0.83 A > A 2 > A 5 > A 4 > A 3 =4 0.330 0.47 0.0880 0.008 0.048 A > A 2 > A 5 > A 4 > A 3 =6 0.20 0.006 0.0763 0.0862 0.093 A > A 2 > A 5 > A 4 > A 3 = 0.4058 0.3675 0.327 0.4275 0.4225 A > A 4 > A 5 > A 2 > A 3 =2 0.742 0.758 0.374 0.565 0.83 A 5 > A > A 2 > A 4 > A 3 =4 0.0996 0.0905 0.0678 0.0768 0.0868 A > A 2 > A 5 > A 4 > A 3 =6 0.0864 0.0738 0.0555 0.0626 0.068 A > A 2 > A 5 > A 4 > A 3 = 0.3670 0.3745 0.320 0.3785 0.4200 A 5 > A 4 > A 2 > A > A 3 =2 0.469 0.687 0.267 0.354 0.849 A 5 > A 2 > A > A 4 > A 3 =4 0.0662 0.0575 0.0476 0.0528 0.0688 A 5 > A > A 2 > A 4 > A 3 =6 0.053 0.0470 0.0347 0.0390 0.0449 A > A 2 > A 5 > A 4 > A 3 = 0.3290 0.385 0.387 0.3295 0.4200 A 5 > A 2 > A > A 4 > A 3 =2 0.96 0.66 0.06 0.43 0.867 A 5 > A 2 > A > A 4 > A 3 =4 0.0329 0.042 0.0275 0.0288 0.0508 A 5 > A 2 > A > A 4 > A 3 =6 0.098 0.0202 0.040 0.054 0.026 A 5 > A 2 > A > A 4 > A 3 It s clear n the table 2, when the values of are dfferent, the optmum metal materal s dfferent (A 2 or A 3 ), so the tradtonal equaton only consder the membershp values but cannot take nto account the hestance degree of each hestant fuzzy element. In the table 3, t s obvously that the hestance degree of A 3 and B s relatvely smaller than that of A 2 and B. We also can obtan that no matter how much the value of s, the mnmal dstance s the dstance among A 3 and B n tables 4.Based on the mnmum dstance prncple, t s easy to get the concluson that A 3 s the optmum metal materal. 4. Summary Ths paper present a new defnton of DHFSs based on the orgnal defnton by ntroducng the concept of hestance degree and nvestgated ther applcaton. We also apply our proposed new dstance measures of DHFSs n pattern recognton. Compared to the exstng defntons, the proposed defnton has a better dstncton to some degree. We also look forward to make some further development about DHFSs. Acknowledgements Ths research was fnancally supported by the Natonal Scence Foundaton of Chna (Grant No. 6573240, 6473239). 2
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