A simple piecewise cubic spline method for approximation of highly nonlinear data

Transcription

1 Vol., No., 9 () Natral Scence A sple pecewse cbc splne ethod for approaton of hghly nonlnear Mehd Zaan Cvl Engneerng Departent, Faclty of Techncal and Engneerng, Yaso Unversty, Yaso, Iran; ahd@al.y.ac.r Receved Septeber ; revsed October ; accepted October ABSTRACT Approaton ethods are sed n the analyss and predcton of, especally laboratory, n engneerng proects. These ethods are sally lnear and are obtaned by leastsqareerror approaches. There are any probles n whch lnear odels cannot be appled. Becase of that there are logarthc, eponental and polynoal crvefttng odels. These nonlnear odels have a lted applcaton n engneerng probles. The varaton of ost s sch that the nonlnearty cannot be approated by the above approaches. These ethods are also not applcable when there s a large aont of. For these reasons, a ethod of pecewse cbc splne approaton has been developed. The odel presented here s capable of followng the local nonnforty of n order to obtan a good ft of a crve to the. There s C contnty at the lts of the pecewse eleents. The odel s tested and eaned wth for probles here. The reslts show that the odel can approate hghly nonlnear effcently. Keywords: Slaton; Data analyss; Approaton; Cbc Splne; Optzaton; Crve Fttng. INTRODUCTION There are any cases n engneerng actvtes n whch one s confronted wth laboratory or feld. Sppose there are n pars of vales (, y ), where these vales are obtaned fro eperental, feld or statstcal analyss. The a s to fnd a fncton f() that predcts acceptable vales for the dependent varable y wth respect to the ndependent varable. There are always soe errors n the eperental analyss, and n the senstvty and calbraton of easrng nstrents, and therefore the vales of y are not eact. Also, soetes there est ltple vales for y for each vale of. In sch statons, f the crve of f() passes between and near to the ponts, t s ore accrate and soother than when t passes throgh all the ponts eactly. Ths, n those cases an approate analyss s ore stable for predctng and assgnng the y vales. If the crve governng f() were nstead to nterpolate sch, t wold have a snsodal or nsoothed zgzag for that wold not be acceptable and wold not be sable for analyss. Most of the pblcatons abot ths sbect are related anly to lnear or sple nonlnear odels []. Soe ore advanced ethods n ths feld are based on Bsplne and Bézer crves []. These ethods are sed ost for copter graphc. Becase of the wavness and the snsodal fors of ther crves, they are not applcable to engneerng probles.. FORMULATION Forlaton of Proble If the varaton of the follows a sple nonlnear crve, a sngle cbc splne fncton can be appled n the for of the followng eqaton: S abc d. () The paraeters a, b, c and d are obtaned by the ethod of leastsqareerror approaton fro the solton of the followng syste of eqatons: n y a n y b n c y d n y, () where n s the nber of pars of vales n the nterval [a, b]. In the case of the nonnfor varaton of that occrs n ost engneerng probles, a sngle cbc splne crve cannot be appled. Instead, t s nec Copyrght ScRes.

2 M. Zaan / Natral Scence () 9 essary to se pecewse cbc splne crves to approate sch. Sppose the dstrbton of s as n Fgre, for eaple. The doan of the proble s dvded nto n eleents. The nterval for the eleent s [, + ]. The cbc splne eqaton for ths eleent s S a b c d. () sbect to the constrants, At the edges of each eleent, C contnty ests: S S. () S S The paraeters a, b, c and d are obtaned fro Eqs. below, together wth the contnty condtons of Eq., by nzaton of the s of sqared errors (SSE):,, () SSE y Y y S n SSE y a b c d () d a b c d c a b c The nzaton of Eq. by the se of Lagrange ltplers and s epressed as follows: () n SSE y S d h a h b h c d c h a h b c To obtan the governng paraeters, the dervatves of Eq. are calclated so that, SSE SSE SSE SSE SSE SSE,,,,,. (9) a b c d The above eqatons are transfored to a lnear syste of eqatons for each eleent: y a b y c y d y h ahbc h ahbhcd S S S l S n λ λ λ λ l λ l+ λ n λ n λ n+ Fgre. Behavor of a dstrbton of nonnfor. Sn (). () The lnear syste of Eq. s a local syste, and all of these systes st be pt together to ake p a global lnear syste of eqatons wth denson n n. The paraeters for each splne fncton are calclated by solvng the global lnear syste. If there s a large nber of ponts, the denson of the governng lnear syste of eqatons ncreases drastcally and ts solton becoes teconsng. Therefore, n order to decrease the nber of calclaton operatons, we can apply the followng approach. If the contnty condtons of Eq. are nserted nto Eq., the followng eqaton reslts, S a b S S The vales of S and S () n the above Copyrght ScRes.

3 M. Zaan / Natral Scence () 9 eqaton can be calclated nercally, takng accont of the for of the varaton of the. Ths, the for paraeters of Eq. are redced to two. Ths decreases the nber of related calclaton operatons by a large aont. Then, by optzng the s of sqared errors, the paraeters a and b can be obtaned fro the followng eqaton: b a a f a a a a a a, () f n whch a a () a and Eq. (see the botto of the page). The bondary condtons for Eq. are s and s. For probles were selected for verfcaton and eanaton of the odel presented here.. PROBLEMS.. Proble Abot pars of vales, y were generated fro the followng eqaton, ln f sn ln r, () where r s a nfor rando nber between and. For splcty and to redce the nber of calclatons, two cbc splne eleents were sed. Fgre llstrates the dstrbton for the nterval [,]. As can be seen, there s a good relatonshp between the pecewse cbc splne crves and the set of. Therefore the odel that we have developed can satsfactorly approate ths set of... Proble Ths eaple conssted of pars of vales on the nterval [, ]. These were generated fro the followng eqaton:.. Nonnfor eleents were appled here becase of the coplety and heterogenety of the dstrbton. The locatons of the eleent bondares are lsted n Table. Fgre llstrates the dstrbton and the calclated pecewse splne crve. As can be observed fro the graph, there s a satsfactorly close relatonshp between the and the crve... Proble For ths proble, abot pars of vales were generated fro the followng eqaton [] on the nterval [,] wth nonnfor f e r. () Fgre. Coparson between real and odel. f() odel Fgre. Coparson between splne crves and real. Table. Characterstcs of the splne crve eleents odel f() f S S y f S S y () Copyrght ScRes.

4 M. Zaan / Natral Scence () 9 f sn r. () The above eqaton s hghly nonlnear and ts varaton s heterogeneos. A cbc splne crve consstng of eght peces was developed for ths eaple, based on the characterstcs of the eleents lsted n Table. Fgre shows a coparson between the splne crve and the for ths test proble. The fgre shows that there s a very close sperposton between the odel and the governng... Proble Ths proble s n the feld of grondwater engneerng. Thes solton for the noneqlbr partal dfferental eqaton for nsteady radal flow fro confned aqfer to ppng well s: Q e h h d πt () Thes approated the above eqaton by the followng seres: Q h h.ln (9) πt.!.! The vale of brackets eqals W() well fncton. The for approaton have logarth vale of W() respect to logarth vale of [9]. A cbc splne crve consstng of fve peces was developed for ths proble, based on the characterstcs of the eleents lsted n Table. Fgre llstrates the dstrbton for the Table. Characterstcs of the splne crve eleents f() odel. Fgre. Coparson between real and odel. Table. Characterstcs of the splne crve eleents. 9 Well fncton W() 9 Fgre. Coparson between real and odel. W() nterval [, ]. Accordng to the graph, there s an acceptable close relatonshp between the and the cbc splne crve.. RESULTS AND CONCLUSIONS The odel presented above for the approaton of wth hghly nonlnear varaton can be appled effcently to varos engneerng probles. Becase the odel s forlaton s straghtforward and eplct, there s no need to establsh and solve any lnear syste of eqatons. Therefore; the nber of calclaton operatons related to the generaton of pecewse cbc crves s notceably less than that for other classcal ethods of approaton. I sggest that ths forlaton and odel cold be etended to twodensonal approaton analyss. For ths prpose, ore attenton st be pad to defnng the forlaton accrately, especally the satsfacton of the contnty condtons at the bondares between eleents or patches. REFERENCES [] De Boor, C. (9) A practcal gde to splnes. Sprnger Verlag, Berln,. do:./9 [] Conte, S.D. and De Boor, C. (9) Eleentary nercal analyss: An algorth approach. rd Edton, McGraw Hll, Ackland,. [] Lobo, N.A. (99) Crve fttng sng splne sectons of dfferent orders. Proceedngs of the st Internatonal Matheatca Sypos, Sothapton, Jly,. [] Zaan, M. (9) Three sple splne ethods for ap Copyrght ScRes.

5 M. Zaan / Natral Scence () 9 proaton and nterpolaton of. Conteporary Engneerng Scences Jornal,,. [] Lehann, T.M., Gonner, C. and Sptzer, K. () B splne nterpolaton n edcal age processng. IEEE Transactons on Medcal Iagng,,. do:.9/.99 [] Saloon, D. () Crves and srfaces for copter graphcs. Sprnger, Northrdge,. [] Zaan, M. (9) An nvestgaton of Bsplne and Bezer ethods for nterpolaton of. Conteporary Engneerng Scences Jornal,,. [] Brden, R.L. and Fars, J.D. (99) Nercal analyss. th Edton, PWSKent, Boston,. [9] Todd, D.K. (9) Grondwater hydrology. nd Edton, Wley, New York,. Copyrght ScRes.