Linear Model Analysis of Observational Data in the Sense of Least Squares Criterion

Transcription

1 Aerca Joural of Matheatcal Aalyss, 05, Vol. 3, No., Avalable ole at Scece ad Educato Publshg DOI:0.69/aja-3--5 Lear Model Aalyss of Observatoal Data the Sese of Least Squares Crtero J.A.Basabra * Departet of Statstcs College of Scece for Grls, Kg Abdulazz Uversty, Jeddah, Saud Araba *Correspodg author: jbasabra@yahoo.co Receved May 05, 05; Revsed Jue 05, 05; Accepted Jue 4, 05 Abstract he preset paper s devoted for the followg goals: o develop a algorth for odel aalyss of observatoal data the sese of the least squares crtero wth full error aalyss.. By ths algorth oe coputes, all the solutos wth ther varaces, the varace of the ft, the average square dstace betwee the least square soluto ad the exact soluto, ad graphcal represetato betwee the row ad the ftted data. Matheatca odule of the algorth was establshed, through fve pots, ts purpose put - output eeded procedures ad the lst of the odule. By ths paper we have bee tred to produce a error cotrolled algorth of the least squares ethod for observatoal data. Keywords: statstcal data aalyss -least squares ethod, atheatca sulatos Cte hs Artcle: J.A.Basabra, Lear Model Aalyss of Observatoal Data the Sese of Least Squares Crtero. Aerca Joural of Matheatcal Aalyss, vol. 3, o. (05: do: 0.69/aja Itroducto Oe of the hallarks of oder astrophyscs s the extree rse the aout of data avalable ad the prevalece of ages ad other data electroc for partcularly that avalable to the publc va the Iteret. he preset stuato s arked cotrast to the vew gve by Kolb ad urer [] ore tha 5 years ago who laeted that cosology at that te there was a paucty of data. here are ow so ay people volved wth data reducto ad aalyss that the last decadal survey there was a call for the forato of a ew area of astrooy ad astrophyscs called astroforatcs [3]. Latest forato o ths rapdly eergg dscple ca be foud wth lstgs o the Iteret through a Google or slar search uder that ae. Astrooers ad astrophyscsts have always shared data ad t s eve truer today tha ever before. he electroc data revoluto for astrophyscs ad astrooy took real hold the pre-hubble spacecraft days wth the establshet of the NASA/ADC (Astroocal Data Ceter whe catalogs lstg data about a uber of stars ad other objects were dgtzed, recorded o CDs, ad dstrbuted at atoal ad teratoal eetgs for free. Although the least-squares ethod s the ost powerful techques that has bee devsed for the probles of data aalyss, t s at the sae te exceedgly crtcal. hs s because the least-squares ethod suffers fro the defcecy that, ts estato procedure does ot have detectg ad cotrollg techques for the sestvty of the soluto to the optzato crtero of the varace σ s u. As a result, there ay exst a stuato whch there are ay sgfcatly dfferet solutos that reduce the varace σ to a acceptable sall value. At ths stage we should pot out that ( the accuracy of the estators ad the accuracy of the ftted curve are to dstct probles; ad ( a accurate estator wll always produce sall varace, but sall varace does ot guaratee a accurate estator. hs could be see fro property 3 (Secto 3 of the least square soluto by otg that the lower bouds for the average square dstace betwee the exact ad the least-squares values s σ / λ whch ay be large eve f σ s very sall, depedg o the agtude of the u egevalue, λ, of the coeffcet atrx of the least-squares oral equatos. Uless detectg ad cotrollg the above etoed stuato, t s ot possble to ake a well-defed dcto about the aalyss of the observatoal data. Due to ths dffculty ad the portace of least squares for treatg the owadays credble ubers of the avalable data, the preset paper s devoted for the followg goals :o develop a algorth for odel aalyss of observatoal data the sese of the least squares crtero wth error aalyss.. By ths algorth oe coputes, all the solutos wth ther varaces, the varace o the ft, the average square dstace betwee the least square soluto a the exact soluto a graphcal represetato betwee the row ad the ftted data. Matheatca odule of the algorth s also gve.. Observatoal Data Let {x k};k =,,...,, be a sequeces of observatoal data pots assued free fro errors. Correspodg to

2 Aerca Joural of Matheatcal Aalyss 48 each x k we have a uber f k of soe fucto f(x at x k whch geerally wll be error. Deote f(x k, the true value at x k, by f k ad defe: δk = fk f k ; k =,,..., ( as the observatoal errors at dfferet data pots. he proble the aalyss of the observatoal data s to approxate or ft the data f k by soe fucto ˆp(x such a way that ˆp(x cotas or represets ost (f ot all the forato about f(x cotaed the data ad lttle (f ay of the errors. hs s accoplshed practce by selectg a fucto p(x p(x;c,c,,c, = ( whch depeds o the paraeters ;c,c,,c. (he superscrpt o c deotes the fact that they wll (geerally deped o. he fucto p(x ay be lear or o-lear the paraeters c. Sce our study s the lear odel aalyss of the observatoal data, the fucto p(x wll be selected as lear cobatos of the paraeters so that t takes the for: = p(x = c Φ (x. (3 he { Φ } ay, for exaple, be the set of ooals, expoetals, trgooetrc polyoals, or deed ay arbtrary set of suffcetly defed fuctoal values, provded oly that they are learly depedet of the values of x. Norally, s sall copared wth the uber, of data pots. I fact, the uber s ukow but for the ost approprate approxato to a gve data the uber should be : -large eough so that the forato about f(x the data ca well be represeted by a proper choce of the paraeters c, whle at the sae te should be, -too sall to avod the fttg of the observed data too closely the sese that the errors the observed data are retaed the approxato. Now the proble s to fd the best estate ˆp of the fucto p.e, to detere partcular values ˆ ˆ ˆ c,c,...,c for the paraeters c to obta the best approxato: pˆ p(x,c ˆ ˆ ˆ,c,,c = (4 Although there s agreeet that the best estate should gve sall devatos of the gve data fro the ftted curve-.e., that the odulus of the quatty :, ( εk = p x k,c,c,...,c f k ; k =,,..., (5 should be sall at each of the pots- there are dffereces of opo as to how ths to be acheved. Let ε deote a error vector coposed of the devatos ε k. Sce x k ad f k are regarded as fxed, the vector ε depeds oly o the paraeters c,c,...,c. Each coo ethod for dealg wth the copetg requreets that ε k are sall, correspods to the selecto of a or ε for the vector ε.soe exaples of ors the real - desoal space are : ( ( ( ε k= ε (6 k / k ε ε k= (7 ax ε k ε k. (8 he or s called Gershgor or, the Eucldea or ad theaxu oror, occasoally, the ufor or. For ay choce or ε,the best estate s oe satsfyg the requreet: ε = u. (9 Assug oe of the above ors, say the Q or s chose, the we ca detere (by solvg certa syste of equatos the estators ĉ of the best estate ˆp of p that satsfes Equato (9,.e., for the lear odel aalyss we have the best approxato: ˆ = p(x ˆ = c Φ (x, (0 to be the true fucto f(x over { x k} the sese of the Q or fttg for the gve Φ (x. Our best assupto s that for soe ukow value of, say N, the true fucto f(x ca be expressed as a fte lear cobato of the selected set of fuctos { Φ } the sese of the Q or fttg ;that s, we assue such that the Q or of N ˆ = f(x = c Φ (x ( ˆ k ( = f c Φ (x u, k =,,,; for soe = N ad for a gve Φ ( x. Equato (,wth the codto of Equato (, are the forulato of the lear odel aalyss of the observatoal data the sese of Q-or fttg. Now t reas to decde what s the ost effcet or that should be used the aalyss of observatoal data. hs decso costtutes very serous dffcultly because:

3 49 Aerca Joural of Matheatcal Aalyss - he l or ε l depeds greatly o the dstrbutos of the observatoal errors δ k. - here s a large varato the effectveess of varous ors ad o sgle or s good (or eve edocre. 3- he dstrbutos of the observatoal errors caot be detered wth ay precso. It s udoubtedly true that the Eucldea or s the ost effcet whe δ k. are orally dstrbuted, ad ths case the estators are the axu lkelhood estators. It s atural therefore, to cosder our study the assupto that δ k.are orally dstrbuted ad aalyzed the lear odel represetato of the observatoal data the sese of the Eucldea or-.e., the least squares crtero. Before startg the aalyss, however, we troduce addtoal codtos cocerg the character of the errors δ k ad assue that δ,δ,,δ k are ubased (wthout systeatc error,depedet ad orally dstrbuted wth zero ea ad varace σ. hese codtos dcate that: E(δ k = 0, E(δpδ q = E(δ pe(δ q; δkε N(0,σ;σ k = E(δ k = σ E(z beg the expectatos of z ad σ the ukow varace. I the case of weghted observatos, Equato (6 s replaced by: σ σk E(δ k, ωk = = (3 where ω k are the weghts of the observatos ( ω k = ω(x k. herefore the proble of ths secto s the applcato of the least-squares ethod for the lear odel represetato of the observatoal data. 3. Least-squares Approxato Below s a lst of vectors ad atrces used the preset secto: Sybol Order Eleet F f k F f k δ δ k Ĉ C Φ ω ĉ c Φ = Φ (x k k ωkk = ωk ; ωql = 0;q l. he traspose of a vector or a atrx wll be dcated by the superscrpt ''. 3.. Dervato of the Noral Equatos I atrx otato, the proble of the weghted leastsquares approxato s to detere the estators Ĉ such that ˆ e ( C = v ω v = u, (4 where the vector of the resduals v s defed as v = F Φ C ˆ. (5 Fro Equatos (4 ad (5 we have ; e ( Cˆ = ( F Φ Cˆ ω ( F Φ Cˆ ˆ = ( F C Φ ω ( F Φ Cˆ ˆ = ( F ω C Φω ( F Φ Cˆ that s e ( C ˆ = F ωf C ˆ ΦωF F ωφ C ˆ + C ˆ ΦωΦ C ˆ. (6 Sce F ωφ ˆ C s scalar the, ˆ ˆ ˆ F ωφ C= ( F ωφ C = [( Φ C.( F ω ] that s: ˆ ˆ F ωφ C= C Φω F, (7 sce ω = ω, (8 the ˆ ˆ F ωφ C= C ΦωF (9 Fro Equatos (6 ad (9 we get ; e ( C ˆ = F ωf C ˆ ΦωF + C ˆ ΦωΦ C ˆ. (0 Equato (0 could be wrtte as: e ( C ˆ = costat C ˆ b + C ˆ G C ˆ. ( costat F ωf b ΦωF G ΦωΦ ( he ecessary codtos for the u are e ( C ˆ = 0. Cˆ Cosequetly, fro Equato ( we get b + GCˆ = 0, (3 therefore GC ˆ = b (4 where the geeral eleets are gve by ad g j ωkφkφ jk k= = (5

4 Aerca Joural of Matheatcal Aalyss 50 b = ω Φ f. (6 k k k k= Equatos (4 are called the oral equatos of the weghted least-squares approxatos for the lear odel represetatos of the observatoal data. hese equatos represet a set of lear equatos ( ukows ĉ ; =,,,.he coeffcet atrx G s syetrc ad s postve defte (.e., all ege values λ ; =,,, are postve f all the fudaetal fuctos Φ (x are learly depedet for the arguets x ; that s, f the rows of Φ are learly depedet. It k should be oted that the oral Equatos (4 represet a relato betwee two exact quattes the sese that f the atrx G ad the rght-had sde b represet the observatoal data exactly, the there s a vector Ĉ whch satsfes exactly the least squares crtero. Of course, practce such a exact stuato o loger exsts. I the followg secto, soe portat propertes of Ĉ ( the sese descrbed above wll be gve. 3.. Soe Propertes of Ĉ Accordg to the codtos of Equatos (3ad (7 t was proved [] the followg propertes of Ĉ : e Property - he estators Ĉ by the paraeters C, obtaed by the ethod of least-squares are ubased;.e. E( C ˆ = C. Property 3 3-he varace covarace atrx Var( Cˆ of the ubased estators Ĉ s gve by Var( ˆ ~ C = σ G. Property 4 4-he average squared dstace betwee Ĉ ad C s E(L = ~ σ = λ Property 5 5-he average squared Eucldea or of Ĉ s E( C ˆ Cˆ = C C + ~ σ λ =

5 5 Aerca Joural of Matheatcal Aalyss Also t should be oted that: If H(u s a fucto of a easured quatty u, the the stadard error σ H the dh σ H = σ u du (7 where σ u s the stadard error of u If the precso s easured by the probable error e, the 4. Matheatca Sulato e = σ. (8 4.. Matheatca Module: Lear Model Purpose o ft observatoal data (x,y(say to the lear odel y = c Φ (x the sese of least-squares crtero wth ts error aalyss. Iput x: Lst of the depedet varbles of the data. y: Lst of the depedet varbles of the data. f : Lst of the basc fuctos used for the lear odel represetato of the data. I,I,I3 : Postve ubers each ε[0,] used to dsply color drawg accordg to the bult fucto Backgroug RGBColor[I,I,I3]. d : A charcter decatg the ae of the depedet varable, (x for exaple. dep: A charcter decatg the ae of the depedet varable, (y for exaple. Output he c s coeffcets ad ther probable errors. he probable error of the ft. he average square dstace betwee the exact ad the least squares solutos. Graphcal represetatos Needed procedures Noe Lst of the Module

6 Aerca Joural of Matheatcal Aalyss 5 We appled ths odule for ay probles for galactc structure, the results are very satsfactory ad wll be publshed soo astroocal jourals I cocludg the preset paper, a algorth for odel aalyss of observatoal data was deloped the sese of the least squares crtero wth full error aalyss.. By

7 53 Aerca Joural of Matheatcal Aalyss ths algorth oe coputes, all the solutos wth ther varaces, the varace of the ft, the average square dstace betwee the least square soluto ad the exact soluto, ad graphcal represetato betwee the row ad the ftted data. Matheatca odule of the algorth was establshed, through fve pots, ts purpose put - output - eeded procedures ad the lst of the odule. By ths paper we have bee tred to produce a error cotrolled algorth of the least squares ethod for observatoal data. Refereces [] Kolb, E.W., ad urer, M.S. 990, he Early Uverse. Froters Physcs. Redwood Cty, CA: Addso Wesley. Advaced Book Progra. [] Kopal, Z ad Sharaf, M.A. 980, Lear Aalyss of the Lght Curves of Eclpsg Varables, Astrophyscs ad Space Scece, 70, [3] Bore, J, M., ad 9 others: 009, Astroforatcs: A st Cetury Approach to Astrooy. ech. rept. Subtted to 00 Decadal Survey.