Models of Parametric Membership Function

Transcription

1 Journal of Automation and Information Sciences, 3(11) 200 Fuzzy ldentification on the Basis of Regression Models of Parametric Membership Function S.D. Shtovba Candidate of technical sciences, docent of Vinnitsa National Technical University. ABSTRACT A new structure of fuzzy regression model is suggested. Here, a fuzzy number with parametric membership function is set into correspondence to each point of factor space. Dependence of parameters of this membership function on affecting factors is described by well-defined regression models. Regression coefficients are determined by a fut y learning sampling. Key words: data processing, regression analysis, regression model, fuzzy number, fiuzy estimate, membership function, learning sampling. 3 rssn O 200 by Begell House Inc.

2 Introduction One of the most popular methods of data processing is the regression analysis. However, it is not suitable for applied identification problems, where information of the studied dependency,.inputsouput" contains fuzzy estimates of the type "low", "medium", "very high" etc. In this paper we study the problem of constructing a fvzy regression model by data sampling with well-defined inputs and fuzzy output. Fuzzy regression was for the first time described in paper tl]. It represents some furzy function, relating inputs and output of the studied dependence. Parameters of this function, coefficients of regression, are given by fu2ry numbers. For the current input vector the fuzzy value at the output of regression model is calculated by the Zadeh generalization principle. In paper [l] the problem of identification of finry coefficients of regression model was reduced to a problem of linear programming. It consists in finding such parameters of the membership functions, which minimize total fuzziness of fiizzy coefficients. Here, for each set of data, cr-section of fuzzy output of regression model should include the cr-section of the corresponding furry number from learning sampling. Fulfillment of this condition should be ensured for all cr,-levels above the prescribed beforehand threshold value. The main deficiency of the approach [] consists in high sensitivity of regression coefficients to data spikes. Besides that, the goal function in a fuzr identification problem is not interpreted as some index of similarity of the desired and the actual behavior of model. unlike conventibnal regression analysis. The author of paper [3] suggested to select fut y coefficients of regression in such a way, that distance between fuzry terms, the model output and data from the learning sampling, would be minimized. For that different techniques are used 13,1. The corresponding optimization problem becomes nonlinear, therefore for its solution besides gradient methods also genetic algorithms are used [5]. In this paper we suggest a new structure of fuzzy regression model. Instead of approximation of dependence "inputs-output" by a function with furry coefficients, we put a fut y number with parametric membership function into correspondence to each point of factor space. Dependence of parameters of this membership function on affecting factors is described by well-defined models by means of the conventional regression analysis of data sampling. 1. New structure of fuzzy regression model We consider mapping of vector X = (xt, x2,..., xn) of well-defined numeric values of affecting factors into a fut y value / of response function: v (x1, x2,..., xn) + V = I uy0)/ y, t where vv?) is the membership function of fuzzy number V oncarrier ly, VJ. Assume, that within the whole factor space the sought fuzzy number V can be described by a parametric membership function of one type. Denote this membership function by mf (y, Z), where Z is the vector of parameters of membership function. Dependence Z = -f (X,P) is described by a system of regression models with coefficients P, each of which relates affecting factors with one parameter of membershipfunctionofthe flnry number. Thus, mf (-y, Z)=mf (y, -f(x,p)). JI

3 Let, for instance, fuzzy number V beprescribed by a Gaussian membership function.\ ( 0-r)'-) (1) p(y)=e*pl -31, \. 2c" ) where parameters of the membership function are b and c, the coordinate of maximum and the concentration coefficient (Z - (b, c)). Then in use of linear regression models dependence of these parameters on factors X = (xt, x2t"';xn) is written as follows: b = bo + b1x1 + b2x2+ "' + bnxn, c = c0 * Qx1 * c2x2 + "'+ c7xx77; where (bo, bt,..., bn, cg, c1,..., cn) = P are regression coefficients' 2. Formulation of problem of fazzy regression analysis Let us define afi;zzy learning sampling as Mpairs of data (xr,',vr),, =ffi, (2) ;_v where X, =(xr1,xr2,...,xrn) is the input vector in the r-th row of sampliug' Vr = is the IvV,O) I t t corresponding output in the form of afuzzy number'. Let us formuiate the proble^ if n;p1r1 regression analysis by fuzzy sampling (2) as search of such coefficients P, which ensure.llirmse (v,,f(p,x,))2 - min, (3) lm r=t where F1f, Xr1 is a finry number with membership function mf (y,.f (X, P))' obtained for input vector x, by a system of regression models with coefficients p; RMSE is the distance between two fuzzynumbers, corresponding to the desired and the actual behavior of the model at the point x'' Distance between rwo ftzzynumbers 7 and.f with membership functio1' tr70) and F;0) in interval ly, Vl is defined as RMSE (A, B) = v't' / \,-.r12dy l\vzrv) - PE\Y ) v () v-y Formula () enables us to calculate carrier. If fuzzy numbers are prescribed transformed to the form distance between arbitrary finzy numbers on a continuous on a discrete carrier {yt,yz,..-,!k}, then formula () is 3

4 ,vt) RMSE (A, 1 - tu7u 1) - p; (t i))2 (s) Problem (3) can be solved by methods of nonlinear optimization. 3. Test problem One can find data of 392 experiments of time y of acceleration to speed 0 mph on the number of cylinders xr and thrust-to-weight ratio of an automobile (ratio of power to weight of the automobil e) xz. By these data we form fuzzy learning and test samplings in the following way. In experimental data, affecting factors take values: x1 e{3,,5,,}; x2e10.020; For forming fivry samplings let us round values of factor x2 to millesimals. Then x2 e e {0.021; 0.022;...;0.051;0.05; 0.073}. Cartesian product x1xx2 consists of 5x33:15 points, from them for 27 pairs (\, xz) there exist not less, than three different values of output variable y in experimental data. For these 27 pafts, using ideas of potential of point from mining clusterization [7], let us "balculate degrees of membership by distribution of values of output variable. Potential of point is the number indicating, how dense experimental data is located in its vicinity. The higher is this potential, the closer to the cluster center is the point. Potential of point y; Q =0 is calculated as follows [7]: Poti v _L -s j=l exp(-b2 (yi - t)2), where B > 0 is the coefficient of cluster fuzziness, v is the number of points. Before the use of this formula, let us project data onto a unit segment. Degrees of membership of afuzzy set / is calculated frompotentials as follows: DOt' ttt}i) ' = -- r*l-. mar (pot 7 ) j=l'v Then detemined degrees of membership are approximated by two-sided Gaussian curve: lsmf (1, b, cr), uor) = j Lgmf (y, b, cz), if if y<b, v2.b, where gmf is the Gaussian membership function (1). We include 20 pairs of data into the learning sampling and 7 ones into the test sampling. We list fi,nzy learning sampling in Table l, and Table 2 contains fiuzy test sampling. 39

5 Table I x1 c s r. t7. r t.r L s s s TT r.7 1. r t r I.r t Table 2 x1 v c r TI t.l t t Example of fuzry regression analysis For a test problem let us construct linear regression models, which will relate factors 'r1 and x2 wrth parameters of Gaussian membership function of finzytime! of acceleration run of automobile' In the result ofsolvingproblem(3)weobtainthefollowingmodels: b = x 1-22.x2, c x x2 ' () 0

6 on test sampling they ensure test discrepancy RMSE Plots of the membership function (Figure l) confirm acceptable quality of identification. For comparison, let us construct by the same data atraditionalttjzzy regression model of the form y=a0*a1x1*a2x2. (7) Let us describe firzy coefficien do,dt,dz by a two-sided Gaussian membership function. The optimal by (3) frrzzy coefficients of this model are shown in Figure 2. with these coefficients, discrepancy of model (7) on finzy test sampling is RMSE (Figure 3), which is worseo than in the previous case * Desiredfitzzyvalues 10 t ls Figure I c"*" Output of fiuzy model Figure 2 p * Desiredfuzzy values ***" Output of fiazzy model t Figure 3 l

7 5. Fast faz:zy linear regression analysis Assume, that fuzzy numbers in the learning sampling (2) are given by parameter membership functions of the same type. Denote parameters of this membership function by 21,22,..., zk. Hence, the learning sampling(2) canbe written in the form,-ffi. () Assume, that dependence of parameters of membership function of fuzzy number I on affecting factors (x1, x2,..., xn) is described by linear models zl = ol0 * a1fi1 * a12x ctlnxnt ZZ = o20 * a21x1 * a22x Q2nXnt zk = ako * a\ * ap2x2 + "'+ dknxn' '" Then finding coefficients of these models is reduced to the conventional procedure of linear regression analysis, which should be performed exactly ft times by sampling (). As example, let us present the constructed by Table 1 linear regression dependencies of parameters of trvo-sided Gaussian membership function (3) on factors x1 and x2: b = x2, cr x x2, cz = 3.L x1 +15.Ix2. these models onfiizzy test sampling (Table 2) ensure discrepancy RMSE : Synthesis of fuzzy,sets of the second type by fuzry regression models If values of affecting factors are given by fiizzy numbers, then in including them to regression model of the type () we obtain parameters of membership function in the form of finzy numbers. Applying the generalization principle to analytic formulae of membership functions with fiizzy parameters, we get fiizzy sets of the second fype. For example, let the thrust-to-weight ratio of automobile be described by a fvzzy number with Gaussian membership function (1) under parameters b = 0.03; c = (Figure, a)- Then for a four-cylinder automobile (xr :) by formulae () we get the following finry values of parameters of Gaussian membership function of time of acceleration of automobile (Figure, b): V - (2.232;2.\0.0r U (2.232; 2.233)0.s U (2.233;2.233)i f, = 1ts.+s: 1.39)s.sr U (to.i2;r7.73)0.5 U (1.92;1.92)r. 2

8 Substituting the found fuzzy parameters into formula () bv cr-level generalization principle we obtain the value of time of acceleration in the form of afuzzy number of the second type. In Figure, c, it is shown as aggregate of cr-cuts of degrees of membership. 2.23t c t t7 I c l Figure + C{,=l a r - r a=0.3 Cr=0.5 a=0.7 cr=0.9 Conclusions We suggested a new structure of flzy regression model, by which afuzzy number with parametric membership function is set into correspondence to each point of factor space. Dependence of parameters of this membership function on affecting factors is described by well-defined regression models. Regression coefficients are determined by minimization of the total distance between fu2ry numbers, the result of simulation and the value of response function in the learning sampling. The suggested approach simplifies procedure of regression analysis of data samplings with fazry values. In doing so, as computer experiments demonstrate approximation accuracy does not decrease. Besides, according to the suggested model the response function is calculated by well-defined numbers without application of the generalization principle, which significantly reduces computation complexity comparing to the traditionalfuzza regression model. After substitution to the suggested regression models of fuzzy values of factors, the response function is obtained in the form of a f,uzy set of the second type. We demonstrated, that for fuzzy learning samplings with output data in the form of parametric membership functions of the same type, the problem of syntheses of fazzy model "inputs--output" is 3

9 reduced to a conventional multifactor regression analysis. Such regression analysis should be performed separately for each parameter of the membership function. The membership function usually has from two to four parameters, therefore computational complexity for fuzzy regression analysis will be only 2-. timeshigher, than for the well-defined analysis' The described advantages of the suggested structure of fi:u,zy regression model enable it to compete with other methods of processing finzy data samplings in engineering, medicine, policy, sociology' political science, sport and other fields' References 1. Tanaka H., uejima s., Asai K., Linear regression analysis with fuzzy model, IEEE Trans' systems Man Cybernet.,192,12, No., ' 2. ZadehL.,Fvzzy sets,information and control,195,no., ' 3. Diamond P.,Fuzry least squares,informationscl., 19,, No' 3,lI-I57 '. papadopoulos., Sirpi M., Similarities and distances in fuzzy regression modeling, soft computing, 200' ' No' ' 55-51' nrr R Generic arsorithms-based.lys is,ibid.,2002, 5. Aliev R., Fazlollahi B., Vahidov R., Genetic algorithms-based fiury regresslon ana, No., MpG data base of ucl machine leaming repgsitory, 7. Yager R., Filev D., Essentials of fuzzy modeling and control, John wiley & sons, New York' 199'