3D FCRM MODELING IN MILES PER GALLON OF CAR

Transcription

1 3D FCRM Modelng n Mles Per Gallon of Car 3 4 3D FCRM MODELING IN MILES PER GALLON OF CAR Mohd Safullah Rusman Robah Adnan Efend N. Nasbo The new fuzzy c-regresson modelng (FcRM) are wdely used n order to ft swtchng regresson models. Mnmzaton of objectve functon yelds mmedate estmates for dfferent c regresson models. The functons of model, estmaton technque and results are dscussed n ths paper. A case study n mles per gallon (MPG) of dfferent cars usng the FcRM modelng was carred out. The 3D graph for sgnfcant ndependent varables for FcRM clusterng s shown n ths study. The comparson between multple lnear regresson and FcRM modelng were done. The mean square error (MSE) was used to fnd the better model. It was found that the FcRM modelng wth lower MSE to be the better model and has great capablty n predctng the dependent varable effectvely. 4. OVERVIEW For the past few years, new fuzzy modelng has become popular because of ther better explanaton of descrbng complex systems. The modfed verson of FCM called the fuzzy c- regresson model (FcRM) clusterng algorthm [] develops hyperplane-shaped clusters. The FcRM assumes that the nput output

2 36 Advances n Fundamental and Socal Scences data are drawn from c dfferent regresson models. Ths FcRM modelng s also known as a swtchng regresson modelng. The mnmzaton of the objectve functon n FcRM clusterng smultaneously yelds a fuzzy c-parttonng matrx of the data and the c regresson models. Ths new technque has been wdely used n engneerng, scence, medcne, economc and other felds. 4. DATA BACKGROUND A measure of fuel economy n automobles or mles per gallon (MPG) used smlarly n North Amerca and the Unted Kngdom, although the Imperal gallon used n the UK s about 0% larger than the U.S. gallon. A metrc term measures how many mles a vehcle can travel on one gallon of fuel. Most countres other than the US and UK use the SI (aka metrc) unts lter (0. Imperal gallon or 0.64 US lqud gallon) and km (0.6 statute mles). These can be combned to ether km/l (effcency) or L/00km (consumpton). MPG fgures are 0.09% hgher n the UK than n the U.S. for the same real fuel economy. The research on MPG done by Larson et. al n [] was about the cty and hghway MPG predcton models (a plot study). In ths research, n order to comply wth federal law, automoble manufacturers must dsplay wndow stckers on automobles leavng ther factores. Ths paper presents an nexpensve statstcal method of predctng cty and hghway mpg estmates whch could be possble alternatve methods. Another research was done by Jun Meng and angyn Lu n [3]. They use the data mnng theory to construct a BP (back propagaton) neural network model to predct MPG (mle per gallon). The sx ndependent varables are number of cylnders, dsplacement, horsepower, weght, acceleraton and model year. In ths paper, the data were ganed fom SPSS Software verson 0. The data were collected nvolvng 0 cars. The dependent varable s mles per gallon (MPG), whereas the ndependent varables are engne dsplacement n cubc nches,

3 3D FCRM Modelng n Mles Per Gallon of Car 3 horsepower, weght of cars n lbs., tme to accelerate from 0 to 60 mph, model year, country of orgn (USA, European and Japanese), number of cylnders and cylnder flter. The FCRM modelng proposed by Harthway and Bezdek n 993 develops hyper-plane-shaped clusters. Km et al. [4] successfully appled FcRM to construct fuzzy models n two phases algorthms whch s c clusters are frstly dentfed through the FCRM cluster and a supervsed learnng algorthm further adjusts the obtaned parameters to mprove the modelng accuracy. The number of clusters (rules), c, s fxed and assgned by the user. In [], for an unknown system, the approprate number of clusters (rules) s supposed to be unknown and could be ganed by usng Equaton (4.6). 4.3 METHODOLOGY Frstly, the analyss of nfluental and outler data should be done to the data to dscard unmportant data due to human error, machne error or envronment error. The analyss used lke Pearson standardzed resdual (outlers Y test) n [6] and [], Leverage (outlers test) and DFBETA (nfluental test) n []. However, there are no condtons needed n FcRM modelng. A swtchng regresson model s specfed by y = f ( x; θ ) + ε, c The optmal estmate of θ depends on assumptons made about the dstrbuton of random vectors orε. Generally, the ε are assumed to be ndependently generated from some pdf p(ε; η, σ) such as the Gaussan dstrbuton wth mean 0 and unknown standard devaton σ, p ( ε η ) σ ( ε; η, σ ) = e σ π

4 38 Advances n Fundamental and Socal Scences Based on the Hathaway and Bezdek algorthm n [] and [8]; () Fx the number of cluster c, c. Choose the termnaton tolerance δ > 0. Fx the weght, w, w > and ntalse (0) U M fc randomly. () Estmate θ,...,θ c smultaneously by modfyng the fuzzy c- means algorthm (FCM). If the regresson functons f ( x; θ ) are lnear n the parameters θ, the parameters can be obtaned as a soluton of the weghted least squares: θ = [ (3) Calculate the objectve functon: W]WYTb-TbbE c d w w[ U,{ θ}] = uj. y j f ( x j ; θ = j= ) (4) Make teratons n order to mnmze the objectve functon. ( l ) ( l ) Repeat for l =,,... untl U U < δ. The steps are descrbed as follows: (l) Step : Calculate model parameters θ to globally mnmze (4). ( l ) Step : Update U wth Ej = E j [ θ ], to satsfy: ( l) for E u = j > 0 j c E w j k= Ekj and c otherwse u = 0 f E > 0, and j j

5 3D FCRM Modelng n Mles Per Gallon of Car 39 u j [0, ] wth ( u j ucj ) = ( l ) ( l ) untl U U < δ In the FCRM clusterng algorthm, the number of clusters, c, s fxed and assgned by the user. In practce, the approprate number of clusters s usually decded wth the ad of the cluster valdty crteron lke the Bezdek s partton coeffcent [9] as follows, V PC = N c h= = ( μ ) N h. The optmal number c s chosen when V s closest to. PC Takag and Sugeno [0,] has ntroduced a fuzzy rule-based model and also called as an affne T-S fuzzy model [], descrbed as follows: R : IF x THEN s A y and x n s A = a x + + a n n x n + a 0 where R denotes the th IF-THEN rule =,,, c where c s the number of rules x m, m =,..., n, are ndvdual nput varables A are ndvdual antecedent fuzzy sets m a k, k =,..., n are consequent parameters a 0 denotes a constant y R s the output of each rule The output of the fuzzy model ŷ s nferred n [3,4] f the sngleton fuzzfer, the product fuzzy nference and the centre average defuzzfer are appled:

6 Advances n Fundamental and Socal Scences where yˆ = c = c w ( x) y = w ( x) w (x) = A n ( x ) A ( x )... An ( x n ) = Am ( x m ) m = denotes the degree of fulfllment of the antecedent, that s, the level of frng of the th rule. The consequent parameters can be found drectly from the FcRM program output. The antecedent fuzzy sets A are acheved m by projectng the membershp degrees n the fuzzy parttons matrx U onto the axes of ndvdual antecedent varable x m and then to approxmate t by a normal bell-shaped membershp functon. Hence, each antecedent fuzzy set A s calculated from m T the sampled nput data x h = [ x h,..., xnh] and the fuzzy partton matrx U = [ μ g ] as follows [,6] : z μ q Aq ( z) = exp σ q, wth the mean and standard devaton are respectvely gven as N N μhxqh μh xqh αq h= h= and σ N q N μq = = μ h h= h= ( ) μ h.

7 3D FCRM Modelng n Mles Per Gallon of Car NUMERICAL EAMPLES Frstly, the analyss of nfluental and outler data should be done to the data. The analyss used are Pearson standardzed resdual (outlers of Y), Leverage (outlers of ) and DFBETA (nfluental test). Nne data are dscarded due to mssng value of MPG. From the analyss of nfluental and outler data, t was found that seven set of data should be dscarded. The dependent varable s mles per gallon (MPG), whereas the sgnfcant ndependent varables are endsplacement ( x ), horsepower ( x ), weght ( x 3 ), year ( x 4 ), orgn ( x ), cylnderno ( x 6 ), cylnderflt ( x ) and accelleratont ( x 8 ). Fgure 4. shows the ndvdual scatter plots for MPG versus x to 8 x. The plots ndcate the negatve relatonshp between MPG versus endsplacement, horsepower, weght and cyclnder number. The postve relatonshp s shown between MPG versus cylnderflt whereas the scatter plot for MPG versus year, orgn and accelleratont ndcate no any relatonshp.

8 4 Advances n Fundamental and Socal Scences Fgure 4.. Scatter plot for MPG versus ndependent varables. The FcRM clusterng for the data were analysed by usng Matlab software. Table 4. shows the optmal value for the number of clusters s two for x to x snce the value 8 V s close to when PC c =. Table 4.. The value of V for Y versus x PC to x. 8 Cluster Number (c) 3 4 Y vs x Y vs x Y vs x Y vs x Y vs x Y vs x Y vs x Y vs x

9 3D FCRM Modelng n Mles Per Gallon of Car 43 Graphs n Fgure 4. ndcate the ndvdual cluster plot for x to x Y vs Y vs Y vs Y vs Y vs 6 Y vs Fgure 4.. Indvdual cluster plot.

10 44 Advances n Fundamental and Socal Scences The membershps for antecedent parameters are calculated usng formula (4.0). Hence, only two parameters are sgnfcant whch are orgn and cyclnderflt wth the smallest MSE. The number of clusters chosen s two snce ts V s the closest to as PC summarzed n Table 4.. Table 4.. The value of V for Y versus x PC and x. Number of clusters, c 3 4 V PC A sgnfcant affne T-S fuzzy model descrbed as follows: R : IF x THEN s A y and x s A =.8 x x R : IF x THEN s A y and x s A =.39 x +.4 x where the antecedent parameter s descrbed n detal n Table 4.3.

11 3D FCRM Modelng n Mles Per Gallon of Car 4 Table 4.3. Detals of the antecedent parameter. R = = A n x : μ : σ A n x : μ : σ A 3 n x 3 : μ : σ A 4 n x 4 : μ : σ A n x : μ : σ A 6 n x 6 : μ : σ A n x : μ : σ A 8 n x 8 : μ : σ

12 46 Advances n Fundamental and Socal Scences Fgure 4.3. Membershp functon plot for y versus x and x. Fgure 4.3 represents the membershp functon graph for MPG versus x and x wth the optmal two clusters. Fgure 4.4 shows the three-dmenson graph of clusterng for MPG versus x and x wth two clusters. The mddle plane s the multple lnear regresson plane for all data. The other two planes are the two clusterng planes wth two dfferent multple lnear regresson planes.

13 3D FCRM Modelng n Mles Per Gallon of Car Fgure 4.4. Three-dmenson FcRM clusterng graph for y versus x and x. In fndng the better model, the mean square error (MSE) s used as follow:. For Multple Lnear Regresson Model MSE N p. For FcRM Model MSE ( ˆ ) Y Y = ( ˆ ) Y Y N = where Y denotes the real data, Ŷ represents the predcted value of Y, N s the number of data, and p s the number of parameters.

14 48 Advances n Fundamental and Socal Scences The comparsons between these two model are summarzed n Table 4.4. Table 4.4. The comparsons between models. Model Varable Chosen MSE Multple Lnear Regresson,, 3, 4, Model, 6, (Sgnfcant Varable) FcRM Model,,,, 3 4, 6,, 8 (All Varables) and (Sgnfcant Varable) CONCLUSION A new model called FcRM has been proposed n analyzng a contnuous data where no assumptons are needed n the analyss. The mnmzaton of objectve functon yelds mmedate estmates for dfferent c regresson models. The comparson modelng between FcRM and multple lnear regresson modelng ndcate that FcRM modelng appeared to be the better model wth the lower MSE. Ths FcRM modelng could be proposed as one of the best model n analyzng manly n a complex system. Hence, the value of MPG could be predcted based on the varable of orgn and cylnderflt. The best cars that could ncrease MPG come from Japan, followed by Europe and Amerca. Cars wth cylnder flter have better MPG than cars wthout cylnder flter.

15 3D FCRM Modelng n Mles Per Gallon of Car REFERENCES [] Hathaway, R. J., and Bezdek, J. C.: Swtchng regresson models and fuzzy clusterng, IEEE Trans. Fuzzy Syst., 993,, (3), pp [] Larson, E. C., Foster, K. A., Hopfe, M. W. : Cty and hghway MPG predcton models (a plot study), Proceedngs Decson Scences Insttute 994 Annual Meetng, Part vol., 994, pp [3] Jun Meng and angyn Lu : MPG Predcton based on BP Neural Network, IEEE on Industral Electroncs and Applcatons, Internatonal Conference, 006, pp. -3. [4] Km, E., Park, M., J, S., and Park, M.: A new approach to fuzzy modelng, IEEE Trans. Fuzzy Syst., 99,, (3), pp [] Chuang, C.C., Su, S.F., and Chen, S.S.: Robust TSK fuzzy modelng for functon approxmaton wth outlers, IEEE Trans. Fuzzy Syst., 00, 9, (6), pp [6] Agrest, A. : An Introducton to Categorcal Data Analyss,John Wley & Sons, Inc, New York, 996. [] Noruss, M. J. : SPSS for Wndows Advanced Statstcs Release 6.0, SPSS Inc, USA, 993. [8] Abony, J. and Fel, B. : Cluster analyss for data mnng and system dentfcaton, Sprnger, USA, 00. [9] Bezdek, J.C. : Cluster valdty wth fuzzy set, J. Cybern., 94, 3, pp. 8. [0] Takag, T., and Sugeno, M.: Fuzzy dentfcaton of systems and ts applcatons to modelng and control, IEEE Trans. Syst., Man Cybernatc, 98,, pp [] Sugeno, M., and Kang, G.T.: Structure dentfcaton of fuzzy model, Fuzzy Sets Syst., 988, 8, (), pp. 33. [] Babuska, R.: Fuzzy modelng for control, Kluwer Academc Publshers, Boston, 988. [3] Wang, L.. : Adaptve fuzzy systems and control: desgn and stablty analyss, Prentce-Hall, NJ, 994.

16 0 Advances n Fundamental and Socal Scences [4] Wang, L..: A course n fuzzy systems and control, Prentce-Hall, NJ, 99. [] Hoppner, F., Klawonn, F., Kruse, R., and Runkler, T.: Fuzzy cluster analyss, methods for classfcaton, data analyss and mage recognton, John Wley & Sons, NJ, 999. [6] Km, E., Park, M., J, S., and Park, M.: A new approach to fuzzy modelng, IEEE Trans. Fuzzy Syst., 99,, (3), pp