Open Access Research on Piecewise Linear Fitting Method Based on Least Square Method in 3D Space Points

Transcription

1 Sen Orers for Reprnts to The Open Automaton an Control Systems Journal, 05, 7, Open Access Research on Pecewse Lnear Fttng Metho Base on Least Square Metho n 3D Space Ponts Lhong Xue * an Dongpng L Department of mathematcs, JNng Normal Unversty, Wulanchabu Cty, 0000, Inner Mongola Autonomous Regon, Chna Abstract: Because the nown ata cannot be completely foun to etermne the functon of the corresponng relatonshp, t s neee to escrbe the ata n tabular form. As the research goes on, people want to use some smple mathematcal functons to express the relatonshp between the ata an the ata. In the process of ata processng, the least square metho s one of the most common methos. Base on the least square metho, the pecewse lnear fttng algorthm s propose to replace the polynomal curve fttng. Ths paper frstly scusses the plane segmentaton metho to be extene to three mensonal screte pont; followe by the screte ponts n the Euclean space square pecewse lnear fttng metho to analyze the nherent law between the 3D ata, through the experment shows the metho s effectve. Keywors: Data processng, least square metho, lnear fttng, spatal scretzaton.. INTRODUCTION The applcaton of fttng s very extensve, such as n the fel of econometrcs can use the economc statstcal ata over the years to prect the tenency of the economc evelopment of the next stage, through the scovery of quanttatve relatonshp between some toxc ose an mortalty of anmals to rect the mecal wor, n sensng technology can be use for mult sensor measurement calbraton n the fel of mecal statstcs [, ]. The common curve fttng metho s the least square metho. Least square metho s a mathematcal optmzaton technque, whch s base on the square of the error an the best matchng of ata. Usng the least square metho, the unnown ata can be easly obtane, an the square of the error between the ata an the actual ata s obtane [3]. When usng the metho of least squares curve fttng, although can well fttng n an orerly manner, an a goo mathematcal theory an very hgh accuracy, but nee to calculate a large number of complex, an the obtane fttng functon s often hgher, whch further ncrease the complexty. In orer to smplfy the establshment of the mathematcal moel of the algorthm an reuce the amount of computaton, ths paper proposes a new use least square metho mplement pecewse lnear fttng metho, wth the observe orere pars of moelng [4-7]. The least square metho s a mathematcal optmzaton technque, whch s use to nfer the best functon matchng relatonshp between the screte ata by the square of the fference between the measure value an the actual value [8]. In engneerng technology an scentfc experment, the ata set s often obtane [9]. How to etermne ther functon curve accorng to the ata, ths s the problem of curve fttng n expermental ata processng. Curve fttng metho s the most commonly use metho, an the fttng of the pecewse curve s better than the general curve fttng effect [0-]. So far, a lot of practcal curve fttng methos have been put forwar. In 966 Yor D propose the use of least squares fttng lne [3]. In 987 Arun K S ftte the two mensonal pont set wth the least square metho [4]. In 999 Ftzgbbon A, Plu M, Fsher R B realze the ellpse rect least squares fttng [5].. SPACE LINEAR LEAST SQUARES FITTING PRINCIPLE Data fttng s an mportant metho of ata processng, an polynomal curve fttng s a more common metho of ata fttng. When the ata pont s too large, the polynomal orer s too low, the fttng precson an the effect s not very eal. In orer to mprove the fttng precson an effect, we nee to ncrease the number of curves, but the complexty of the orer s too hgh an the complexty of the calculaton an other aspects of the savantages. Therefore, t s ffcult to obtan goo fttng accuracy an effect by usng a polynomal curve fttng of the ata ponts. In orer to solve the above problems, the pecewse curve fttng s generally use. In the past, the fttng metho of pecewse curve fttng metho s manly ame at the measurement of ata n the fel of natural scence. Therefore, n fttng of these ata, the tratonal segmentaton metho s generally base on the subjectve experence of the ata segmentaton, an then fttng [6]. But for some practcal problems of ata, such as socal an economc lfe of a large amount of statstcal ata, the mechansm of these changes n ata s very complcate n general, often le the laws of physcs as a strct rules, so changes n uncertanty s very strong. Therefore, the /5 05 Bentham Open

2 576 The Open Automaton an Control Systems Journal, 05, Volume 7 Xue an L tratonal segmentaton curve fttng base on the subjectve experence of the ata s segmente nto a clearly nsuffcent. For ths problem, ths paper presents an objectve to segment the three curve fttng metho, whch can elmnate the efects of the ata from the subjectve experence [7]. Fnally, the test results show that the fttng effect s goo. Polynomal curve fttng s a more common metho of ata fttng. When the ata pont s too large, the polynomal orer s too low, the fttng precson an the effect s not very eal. In orer to mprove the fttng precson an effect, we nee to ncrease the number of curves, but the complexty of the orer s too hgh an the complexty of the calculaton an other aspects of the savantages. Therefore, t s ffcult to obtan goo fttng accuracy an effect by usng a polynomal curve fttng of the ata ponts. In orer to solve the above problems, the pecewse curve fttng s generally use, an local least squares fttng s performe on each nterval [8]. The tratonal pecewse curve fttng s base on the subjectve experence an the rawng ata to etermne the emprcal functon an the segmentaton pont. A fttng metho s propose to segment the pecewse nterval. The three ata ponts are etermne by each of the 4 ata ponts, but the nterval s too ense an not sutable for ense ata fttng. Later, t s propose that the global contnuous fttng metho of the polynomal bass functon s restrcte to pecewse nterval. In the next few years, the curve fttng metho of the global contnuous curve s put forwar, but the bass functon s restrcte to a polynomal. The metho propose by the researcher s base on the subjectve experence to etermne the emprcal functon an the segmentaton nterval, an then the fttng [9, 0]. Ths metho has the followng ponts: n the practcal applcaton of the metho of automatc segmentaton polynomal curve fttng. Prove several fferent emprcal functon, accorng to the fferent experence functon was ftte to the ata an the measure ata, the error varance, to automatcally etermne the better experence functon;. From the ata fttng metho many, such as logarthmc curve fttng, nverse functon curve fttng, quaratc curve fttng, cubc curve fttng, exponental curve fttng, exponental curve fttng. Frst commonly observe scatter agram to etermne the curve type, but scattere pont agram s correlaton between the coarse representatons, an sometmes scattere pont graph an several curves are very close, then establsh corresponng emprcal functon may be reasonable, but ue to the choce of fferent curves, a problem wth a number of fferent emprcal functon, how to select the optmal from these experences functon n a. Wth several functon fttng calculaton of the hstorcal ata value an the fttng values of the error square an mnmum for optmal functon of the methos may exst such a problem: mnmzng the sum of square errors, but error fluctuaton larger,.e. some pont error s very small, some error s relatvely large. In vew of ths stuaton, ths paper proposes a pecewse lnear fttng metho, wth several fferent functon fttng an the selecton of optmal experence functon an optmal functon to etermne the contons are as follows: for hstorcal ata pont error an error s negatve number fference less than aaptaton parameters []. The least squares metho s a mathematcal optmzaton technque It matches the optmal functon of the ata by mnmzng the square error of the square error. Usng the least square metho, the unnown ata can be easly obtane, an the square of the error between the ata an the actual ata s mnmum. The least square metho can be use n curve fttng. Other optmzaton problems can be expresse by the least square metho for mnmzng energy or maxmum entropy. In the lmte measurement seres, the accuracy of the measurement s to see the true value, whch maes the square an the mnmum of the error. The least square metho s a basc metho to measure the ajustment of fferent ns of observaton ata []. If n fferent precson multple observatons of one or more unnown quantty, n orer to etermne the unnown quanttes are the most relable value, the concept of measurement must be wth correctons to the mofe the sum of weghts postve multple by the square of the observaton value s mnmum. So calle least squares metho. The so-calle "weght " s a measure of the relatve relablty of the qualty of observaton results [3]. From the pont of vew, the least square metho s smlar to the nterpolaton metho, whch s the algorthm of processng ata. But from the creaton of the thought, the two are essentally fferent. The former sought a curve, an the observaton ata s "closest" to represent the observaton ata of the tren; the latter s the curve strctly by the gven observatonal ata, ts purpose s through ata from functon moel to approxmately escrbe the functon [4]. In the observatonal ata wth measurement error wll mae these observatons evate from the curve functon, as a result an observatonal ata to stay consstent nterpolaton metho s better than the metho of least squares curve more n lne wth the objectve realty. The basc ea of the least square metho s: gven a set of ata ( x,y )(,,...,n). The ntal equaton s F(x) an the equaton contans some unetermne coeffcents a n. Put ( x,y ) nto the equaton an calculate y. F( x) Conserng the error fluctuatons, to prevent the emergence of error of plus or mnus cancellaton, wll nee equaton square: e (y F( x)) The mnmum value of can be obtane by n. In orer to fn the best fttng functon of the group ata, the functon maes the error square an the mnmum. Extene to a certan pont n space. Stanar equaton for space lne: x x y y z z Can be smplfe to get: x ( z z0) + x0 az+ b e a () () (3)

3 Research on Pecewse Lnear Fttng Metho Base on Least Square Metho The Open Automaton an Control Systems Journal, 05, Volume y ( z z0) + y0 xz+ (4) Among them:,,, y z 0 0. a c b x0 z0 For lnear space, space straght lne s equvalent to two plane equaton of ntersecton, so on lnear fttng can be respectvely of the plane equaton fttng. The fttng equaton are obtane for the approxmate value an the actual value of the fference between the square an recore as: [ ( z b) ] Δ x x a+ [ ( z )] Δ y y a+ Accorng to the prncple of least square metho, the partal ervatve of n the formula s zero. abc,,, [ x ( az + b)] ( z ) 0 [ x ( az + b)] ( ) 0 [ y ( cz + )] ( z ) 0 [ y ( cz + )] ( ) 0 On the formula (.7), An the numercal value of s obtane. abc,,, x a z xz x z a z z z b y c z yz y z c z z z (5) (6) (7) (8) (9) (0) () 3. PRINCIPLE OF FITTING STRAIGHT LINE SEGMENT The functon relatonshp between the coornates of the representaton of the screte pont set n a contnuous curve or a relatvely flat surface. More broaly speang, the corresponng problem n space or hgh mensonal space s also a category. In the numercal analyss, the curve fttng s use to approxmate the screte ata, that s, the formula of the screte ata. In practce, group or screte pont ata s often varous physcal problems an statstcal problem was 3 relate to the amount of tmes of observaton value or expermental value, they are scattere, not only t s not easy to eal wth an cannot generally be exact an fully reflect the nherent law. Ths efect s beng mae up by an approprate analytcal expresson. The TLS metho has been wely use n the lne an the plane fttng, the SVD metho or teratve metho to establsh the coeffcent matrx contans error moel to solve the parameters, however, for the space of lnear fttng of the TLS metho oes not see more. At present, there are commonly use spatal straght lne fttng metho: on the Newton graent optmzaton algorthm for representng teraton algorthm, the algorthm s wth two ntal value, reach the accuracy requrements of the gven by an teratve metho, t nvolves a large number of equatons are solve, solvng process s relatvely cumbersome, practcablty s poor; non teratve algorthm, the metho s essentally obtane wth measurng pont stance an the mnmum plane, then three-mensonal problem nto a twomensonal problem, n plane usng least squares metho of fttng a straght lne. The metho taes nto account the error of y an x rectons, an the error of the z recton s neglecte, an the error s not tght n theory [5]. The metho of jugng the fttng lne s that the stance between the ata coornates an the corresponng fttng lne shoul be less than the fxe value of ε. In the mplementaton of the algorthm, wth mnmal space screte ata ponts of straght lne fttng, meet the contons for a ata pont ncreases when the stance between all of the selecte ata ponts an fttng straght lne less than ε. If the maxmum stance between the ata ponts an the maxmum stance of them excees the fxe value of ε, the fttng process of the segment s stoppe. When the ata ponts are not ae to the fxe value of ε, contnue to perform an ncrease n the operaton of the ata ponts untl the maxmum number of ponts s ae. When all of the above processes are performe, a spatal lne s ntene to be synthesze wth the most ata ponts [6-8]. For example, the space of any pont Px ( to the 0,y 0,z 0) space of a two lnear Ax an + D 0 Ax stance s + D 0 uuv uv ( Ax + D) n ( Ax + D) n uv uuv () n n uv uuv Among them: n. By the { A, B, C}, n { A, B, C} above ata to ft the parameters of the lne shoul be: A. The, B 0, C a, D b, A 0, B, C c, D resultng system can be brought nto the stance formula, the stance between the ata ponts an the fttng lne s obtane. ( y+ cz + ) + ( x az b) + ( c x + ay + bc a) a + c + (3)

4 578 The Open Automaton an Control Systems Journal, 05, Volume 7 Xue an L The obtane s the stance between the ata ponts to the fttng lne, an then the ata can be ftte by a pecewse lnear fttng by usng ths value. 4. COMPUTATIONAL PROCEDURE Through the analyss of the above fttng prncple, we can get the man steps: the frst step s to tae the expermental ata two ponts ( A( x, ).,y,z) Bx (,y,z) Mathematcal moel of fttng lne l by usng least square metho. Then a new pont Cx (,y,z ). The cosne of the angle between the lne BC an the lne s calculate: cosθ (4) If ther cosne value s smaller than a gven threshol, the pont nto the front, re calculate by least square metho to ft the lnear equaton untl greater than a gven value. An we got the frst fttng a lnear equaton. The secon step s to repeat the last pont of the straght lne to the next pont, untl all ponts are ftte, so that all the equatons of the pecewse lnear equaton are obtane. All processes as shown n Fg. (). 5. SIMULATED DATA EXPERIMENT Ten sets of ata were obtane from statstcs. Usng the above ata to raw the space lne Fg. (). CONCLUDING REMARKS l ax ( x) + ay ( y) + ( z z) a + c + ( x x ) + ( y y ) + ( z z ) The above example shows that the mprove least square metho for pecewse lnear fttng, n the processng of the expermental ata, can sgnfcantly smplfy the mathematcal moel establshment an reuce the computaton, can better fttng of the sequence pont, whch proves that the metho s feasble. On the other han, the metho can automatcally etermne the number of branchng, automatc search for brea pont poston, the overall optmal fttng, t s prove that the metho s effcent. But ths metho also has certan lmtatons, the ata of ranom changng fttng often fal to get the eal results, but to change relatvely slow ata can get better effect, especally a monotone or monotone approxmaton of ata. In the actual wor, the ata obtane from the test are often nonranom, so the applcaton of ths metho s qute we. The classcal least squares polynomal fttng as the founaton, through expanson of the orgnal ata, number of oman swtchng, ata expanson, a seres of ata reconstructon, to ensure the pecewse lnear fttng relablty an stablty, to reconstruct the ata wth fferent number of polynomal repeately teratve fttng, after each teraton of the ata ponts of large error correcton an then contnue the teratve, untl the fttng stablty. Fnally, through teraton error, number of teratons mofe the sze of ntegrate an aaptve to get the best fttng results. In ths paper, fferent theoretcal moels are use to verfy Fg. (). Fttng subsecton lnear program bloc agram. Fg. (). Spatal pecewse lnear fttng. the optmzaton algorthm, an the results of fferent moels are analyze, an the algorthm s correct an feasble. The treatment effect of the actual expermental ata to verfy the analyss, an raws the concluson that the optmzaton algorthm can be use for any type of ranom error of the test ata ftte better than toay's metho of conclusons. The savantage s that the teraton tme s longer, an the evaluaton of the optmal fttng results nees to fn a better jugment metho. Accorng to the prncple of least square metho s propose for spatal screte ponts by the metho of least squares pecewse lnear fttng, calculate the space screte ata fttng for all coeffcents of lnear space algorthm, an

5 Research on Pecewse Lnear Fttng Metho Base on Least Square Metho The Open Automaton an Control Systems Journal, 05, Volume Table. Table. Data sheets for any 0 groups. x y z x y z Equaton coeffcents for runnng programs. a b C puts forwar the pecewse lnear fttng metho to stngush, the basc prncple an then come to the least squares metho space segments of straght lne fttng. The valty of ths metho s verfe by the results of the program. CONFLICT OF INTEREST The authors confrm that ths artcle content has no conflct of nterest. ACKNOWLEDGEMENTS Ths wor s supporte by the Scentfc research project of JNng Normal Unversty (No. jsy0505). REFERENCES [] A. Chfa, A. Cohen, G. Mglorat, F. Noble an R. Tempone, Dscrete least squares polynomal approxmaton wth ranom evaluatons applcaton to parametrc an stochastc ellptc PDEs, ESAIM: Mathematcal Moellng an Numercal Analyss, vol. 49, no. 3, pp , 05. [] H.C. Gocoechea, A.C. Olver an R. Tauler, "Applcaton of the correlaton constrane multvarate curve resoluton alternatng leastsquares metho for analyte quanttaton n the presence of unexpecte nterferences usng frst-orer nstrumental ata", Analyst, vol. 35, no. 3, pp , 00. [3] O. Götz, K. Lehr-Gobbers an M. Krafft, Evaluaton of structural equaton moels usng the partal least squares (PLS) approach, Hanboo of partal least squares, Sprnger, Berln Heelberg, 00, pp [4] W.R. Schwartz, A. Kembhav, D. Harwoo an L.S. Davs, Human etecton usng partal least squares analyss, IEEE th Internatonal Conference on Computer Vson, pp. 4-3, 009. [5] H. Gavn, The Levenberg-Marquart Metho for Nonlnear Least Squares Curve-Fttng Problems, Department of Cvl an Envronmental Engneerng, Due Unversty, pp. -5, 0. [6] X. Tong, Y. Jn, an L. L, An mprove weghte total least squares metho wth applcatons n lnear fttng an coornate transformaton, Journal of Surveyng Engneerng, vol. 37, no. 4, pp. 0-8, 0. [7] F. Dng, P.X. Lu an G. Lu, Multnnovaton least-squares entfcaton for system moellng, IEEE Transactons on Systems, Man, an Cybernetcs, Part B: Cybernetcs, vol. 40, no. 3, pp , 00. [8] H. Han, L. Xe, F. Dng an X. Lu, Herarchcal least-squares base teratve entfcaton for multvarable systems wth movng average noses, Mathematcal an Computer Moellng, vol. 5, no. 9, pp. 3-0, 00. [9] J. Hesch an S. Roumelots, A rect least-squares (DLS) metho for PnP, In: IEEE Internatonal Conference on Computer Vson (ICCV), pp , 0. [0] T. Quan, X. Lu, an Q. Lu, Weghte least squares support vector machne local regon metho for nonlnear tme seres precton, Apple Soft Computng, vol. 0, no., pp , 00. [] S. Van Huffel, Total Least Squares an Errors-n-Varables Moelng: Analyss, Algorthms an Applcatons, Sprnger Scence & Busness Mea, Berln, 03. [] J.B. Lohmöller, Latent Varable Path Moelng wth Partal Least Squares, Sprnger Scence & Busness Mea, Berln, 03. [3] D. Yor, Least-squares fttng of a straght lne, Canaan Journal of Physcs, vol. 44, no. 5, pp , 966. [4] K.S. Arun, T.S. Huang an S.D. Blosten, Least-squares fttng of two 3-D pont sets, IEEE Transactons on Pattern Analyss an Machne Intellgence, vol. 5, pp , 987. [5] A. Ftzgbbon, M. Plu an R.B. Fsher, Drect least square fttng of ellpses, IEEE Transactons on Pattern Analyss an Machne Intellgence, vol., no. 5, pp , 999. [6] S.J. Km, K. Koh, M. Lustg, S. Boy an D. Gornevsy An nterorpont metho for large-scale l -regularze least squares, IEEE Journal of Selecte Topcs n Sgnal Processng, vol., no. 4, pp , 007. [7] K.M. Brownf an J.E. Denns, A new algorthm for nonlnear leastsquares curve fttng, Mathematcal Software, vol. 5, no. 3, p. 39, 04. [8] J. Shults, Quas least squares fttng, Wley Interscplnary Revews: Computatonal Statstcs, vol. 7, no. 3, pp , 05. [9] X. Zhuang, H. Zhu an C. Augare, "An mprove meshless Shepar an least squares metho possessng the elta property an requrng no sngular weght functon", Computatonal Mechancs, vol. 53, no., pp , 04. [0] T. Kuosmanen an A.L. Johnson, Data envelopment analyss as nonparametrc least-squares regresson, Operatons Research, vol. 58, no., pp , 00. [] Y. Lu, D. Wang an F. Dng, Least squares base teratve algorthms for entfyng Box Jenns moels wth fnte measurement ata, Dgtal Sgnal Processng, vol. 0, no. 5, pp , 00. [] I. Daubeches, R. DeVore, M. Fornaser an C.S. Güntür, Iteratvely reweghte least squares mnmzaton for sparse recovery, Communcatons on Pure an Apple Mathematcs, vol. 63, no., pp. -38, 00. [3] Z.-M. Zhang, S. Chen an Y.-Z. Lang, "Baselne correcton usng aaptve teratvely reweghte penalze least squares", Analyst, vol. 35, no. 5, pp , 00. [4] K.C. Toh an S. Yun, An accelerate proxmal graent algorthm for nuclear norm regularze lnear least squares problems, Pacfc Journal of Optmzaton, vol. 6, no. 5, pp , 00. [5] R. Chuaman, K. Vasuevan an C.S. Ramalngam, Real-tme estmaton of power system frequency usng nonlnear least squares, IEEE Transactons on Power Delvery, vol. 4, no. 3, pp. 0-08, 009. Receve: September 6, 04 Revse: December 3, 04 Accepte: December 3, 04 Xue an L et al.; Lcensee Bentham Open. Ths s an open access artcle lcense uner the terms of the ( whch permts unrestrcte, non-commercal use, strbuton an reproucton n any meum, prove the wor s properly cte.