7659, Eglad, UK Joural of Iformato ad Computg Scece Vol. 2, No. 3, 2007, pp. 9-96 Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto Xaopg Hu +, Jam He ad Hogsheg Ly School of Ecoomcs ad Maagemet, Southeast Uversty, Najg 20096 ( Receved 30 Jue 2006, accepted 4 August 4 2006 Abstract. Radom umbers of multvarate oormal dstrbuto are strogly requested by the area of theoretc research ad applcato practce. A ew algorthm of geeratg multvarate oormal dstrbuto radom umbers s gve based o the Copula fucto, ad theoretc aalyss suggests that the algorthm s sutable to be feasble. Furthermore, smulato shows that the emprcal dstrbuto whch s formed by radom umbers geeratg from the proposed algorthm ca well approach the orgal dstrbuto. Keywords: Multvarate Noormal Dstrbuto,Radom Number, Copula,Algorthm. Itroducto There are oly a few methods of geeratg radom umbers from multvarate oormal dstrbuto, ad such radom umbers are strogly requested by the area of multvarate aalyss ad statstcal modelg. Nagahara(2004 stated that the Pearso dstrbuto system could represet wde class of dstrbutos wth varous sewess ad urtoss. The Pearso system cluded some well-ow dstrbutos, for example, gamma, beta, t-dstrbuto, etc. Geeratg radom umbers from the Pearso dstrbuto system was gve that paper. I ths paper, a ew method of geeratg multvarate oormal dstrbuto radom umbers s proposed based o the Copula fucto. Frstly, every margal dstrbuto s obtaed from the multvarate dstrbuto. Secodly, the copula fucto of the multvarate dstrbuto s gaed accordg to the Slar s theory. Thrdly, uform dstrbuto radom umber vector betwee 0 ad are geerated usg the Bayesa codtoal probablty formula. lastly, every compoet of the uform dstrbuto radom umber vector obtaed 3th step s trasformed by correspodg to the quas-verses fucto of the margal dstrbuto fucto. Coget theoretcal aalyss shows that the proposed geerator s sutable to be a relable ad effcet multvarate oormal dstrbuto radom umbers geerator whch ca be used wdely multvarate aalyss ad statstcal modelg. The outle of the paper s as follows. A bref revew of Copula ad Noto are troduced Secto 2. A algorthm of geeratg multvarate oormal dstrbutos by usg the Copula method s show Secto 3.The smulato for a cocrete example are appled Secto 4. The cocluso s show Secto 5. 2. A Bref Revew of Copulas ad Noto Nelse defes copulas as fuctos that jo or couple multvarate dstrbuto fuctos to ther oedmesoal margal dstrbuto fuctos (Nelse, 999, page 5. Copulas cota all the formato about the depedece structure of a vector of radom varables. They ca capture olear depedece amog radom varables, whle correlato s oly a lear measure of depedece. I partcular, copulas cota formato about the jot behavor of the radom varables the tals of the dstrbuto, whch should be of prmary terest a study of cotago of facal crses. Moreover, copulas are able to capture tal behavor wthout the eed of usg dscreto to defe extreme outcomes. We ow assume that we are usg the creasg fucto defto of a Copula, ad the relatoshp + Correspodg author. Tel.: +86-025-8379 3035; fax: +86-025-8379 3035. E-mal address:hxpj@63.com. Publshed by World Academc Press, World Academc Uo
92 X. Hu, et al: Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto betwee the Copula ad jot probablty dstrbuto fucto ca be descrbed by theorem ad 2. Theorem. (Slar,959: Let F( z, z,..., z be the jot dstrbuto wth margs F( z, ad let 2 F ( U be quas-verses, the there exsts a copula fucto If the If the F F, U,..., U F( F ( U, F ( U,..., F ( U s cotuous, the C s uque. 2 2 2 s ot cotuous, there are some techcaltes that relate to what are called sub-copulas ad the rage of the correspodg varables. If Copula ad margal dstrbuto fuctos are ow, the multvarate jot probablty dstrbuto fucto wll be solved by theorem 2. Theorem 2 (Slar,959: Let, U,..., U be a Copula,ad assume that F( z are dstrbuto 2 fuctos. The there exsts a jot dstrbuto fucto ad the F( z are the margal dstrbuto fuctos. F( z, z2,..., z gve by ( ( F( z, z2,..., z C( F( z, F2 z2,..., F z (2 Therefore, f F s a cotuous multvarate dstrbuto fucto, Slar s theorem suggests that t s possble to separate the uvarate margs from the depedece structure. The depedece structure s represeted by the copula. Ths ca be see eve more clearly f we assume the F s are dfferetable, ad C ad F are -tmes dfferetable. The, dervg both sdes to get the desty of F, we get: (,,..., (,,..., F z z z C U U U... zz..., z..., z z 2 2 2 2 where U F( z (,2,...,, that s, the desty of F has bee expressed as the product of the copula desty ad the uvarate margal destes. I ths sese, we state that the copula has all the formato about the depedece structure. Cosequetly, a copula s essece a multvarate dstrbuto whose margal dstrbutos are U (0,, whch s uform dstrbuto o the terval (0,. Copulas allow oe to model the margal dstrbutos ad the depedece structure of a multvarate radom varable separately. For more dscussos o the theory of copulas ad specfc examples of copulas, see Nelse (998. 3. Algorthm for geeratg radom umber Now, we gve the dea, that s, how the radom umber vector( x,, x s geerated by the Cumulated Desty Fucto F( X,, X. Frstly, obta the margal dstrbuto F of a varable X from F( X,, X. Secodly, get the Copula fucto,, U accordg to the theorem 2, ad the geerate from Copula fucto,, U radom umber vector ( u,, u whose margal dstrbuto follows U (0,. Fally, versely trasform the margal dstrbuto F of a varable X, ad ga x F ( u.cosequetly, we have the radom umber vector ( x,, x from the exsted F( X,, X. I the above-metoed aalyss, the most sgfcat dsposal les wth geeratg radom umber vector ( u,, u whose the margal dstrbuto followsu (0, from Copula fucto,, U. The followg s the algorthm to ga the radom umber vector ( x,, x from the Cumulated Desty Fucto F( X,, X. ( (3 The quas-verses defto of a fucto ( f x s as follow: x f { x f( x y}. JIC emal for cotrbuto: edtor@jc.org.u
Joural of Iformato ad Computg Scece, 2 (2007 3, pp 9-96 93 where Algorthm: Step. Geerate U (0, radom umber U, let 2 Step 2. Geerate depedet U (0, radom umber p, the U FICCDF, p U U U (,..., s gve by,, U, U,,, U FICCDF, ( p U,..., U d 0,, fm( U,, U,, U, U,,,,, f ( U,, U m,, U,,, fm( U,, U,, Step 3. Let +. f <, go to step 2; else,stop. Theorem 3. The radom umber vector U ( U, U2,..., U gve by the aforemetoed algorthm follows the jot probablty dstrbuto, U2,..., U. Proof: For, by all appearace, the theory s vald. For 2 U, varables ad p follow U (0,,, U2, U2 p FICCDF,2 f ( U where So get the codtoal desty fucto of : 2 m U f ( U U 2 The two dmesos jot desty fucto of ad U s U 2 2, U2 U U 2 2 2, U2, U2 f( U, U2 f ( U2 U f( U Cosequetly, ( U, U2 follows, U2. For, suppose ( U,, U follows,, U For +, ( U,, U follows,, U ad p follows U (0, p F + ICCDF, + 2 2,, U, U+,, fm( U,, U f ( U,, U f( U U,, U m +,, U,,,,, U, U+,, f ( U,, U + m U JIC emal for subscrpto: publshg@wau.org.u
94 X. Hu, et al: Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto f( U,, U f( U U,, U f ( U,, U + + m +,, U, U+ fm( U,, U,, f ( U,, U + m,, U, U+,, + Cosequetly, radom umber vector ( U,, U + follows,, U +.Accordg to the prcple of ducto, the theorem s vald. If Copula belogs to the Ellptcal Copula famly, for example, ormal (or t Copula, multvarate radom umbers may be geerated by other smpler method as well. Radom vectors from these copulas ca be geerated by creatg radom vectors from the multvarate Ellptcal dstrbuto, the trasformg them to uform margal usg the Ellptcal CDF. A bloc dagram of the proposed procedure geeratg M groups of radom umber vectors s show Fgure. U(,j ~U(0,, j p ~U(0, U(,j+F ICCDF (p,u,,u j, jj+ j>dm(copula Yes + j>m stop Yes 4. Smulato procedure bloc dagram I ths secto, we use Clayto Copula gve by Nelse (998 to chec computatos as a cocrete applcato example. The expresso of Clayto Copula s α α α CUV (, ( U + V α [, ]\{} 0 JIC emal for cotrbuto: edtor@jc.org.u
Joural of Iformato ad Computg Scece, 2 (2007 3, pp 9-96 95 I ths smulato, we edue α 2, ad defe a ew resdua fucto: Z( UV, CUV (, F ( UV, where CUV (, s theoretc dstrbuto fucto, ad Femp( U, V s emprcal dstrbuto fucto. Accordg to theorem (Glveo-Catell, whe the quatty of observed samples s of greatess, the emprcal dstrbuto fucto F ( emp U, V ca well approxmate the real dstrbuto fucto. Now, we set the parameters. Smulato umbers are 0000, ad the square [0,] [0,] s parttoed to the grds by step 0.05. The fgure 2 s the resdual fgure. emp Fgure 2 resdual fgure I fgure 2, the mea ad maxmal resdual value s respectvely the lesser value: 0.0022, 0.074. It s show that the theoretc dstrbuto fucto CUV (, ca be well approached by the emprcal dstrbuto fucto F ( emp U, V. For the scearos of dfferet parameter value ofα ad some of other Copulas, we acheve the smlar result. So, radom umbers geerated by the proposed algorthm are assuredly sampled from the theoretc dstrbuto fucto CUV (,. If margal dstrbutos do t follow U (0,, for example, follow gamma dstrbuto, t s ecessary to versely trasform radom umbers geerated by the Copula the Algorthm. Trasform form s gve by x v GammaCDF( u 5. Cocluso I ths paper we have proposed a Copula-based algorthm to geerate radom umbers from multvarate oormal dstrbuto. The Copula approach allowed us to costruct algorthm by two stages. At oe stage, geerate radom umbers wth margal dstrbuto U (0, from Copula fucto correspodg to cumulatve dstrbuto fucto. At the other stage, trasform the radom umbers from the frst stage by mplemetg the quas-verse fucto of margal dstrbuto. Theoretc proof suggests that the proposed algorthm s sutable to be relable. Furthermore, smulato shows that the emprcal dstrbuto whch s formed by radom umbers geeratg from the proposed algorthm ca well approach the orgal dstrbuto. I cocluso, the copula-based algorthm s foud to perform well geeratg radom umbers from multvarate oormal dstrbuto. 6. Acowledgemets Hu Xaopg acowledges the support of The Natoal Natural Scece Foudato of Cha, grat N7037035. The author would also le to tha Academca Wu Qua ad colleagues at the school of ecoomc &maagemet,southeast uversty for ther terest to the problems studed here ad for ther helpful commets. The author thas also Professor Wu Guagmo for terestg dscussos ad the revewers for ther suggestos that allow to mprove the paper. JIC emal for subscrpto: publshg@wau.org.u
96 7. Refereces X. Hu, et al: Geeratg Multvarate Noormal Dstrbuto Radom Numbers Based o Copula Fucto [] R. B. Nelse. A Itroducto to Copulas. NewYor: Sprger-Verlag. 998. [2] E. W. Frees, & E. Valdez. Uderstadg relatoshps usg copulas. North Amerca Actuaral Joural. 2002, 2(: -25. [3] Y. Nagahara. A method of smulatg multvarate oormal dstrbutos by the Pearso dstrbuto system ad estmato. Computatoal Statstcs & Data Aalyss. JIC emal for cotrbuto: edtor@jc.org.u