Compuer and Informaion Science; Vol. 5, No. 5; 0 ISSN 93-8989 E-ISSN 93-8997 Published by Canadian Cener of Science and Educaion Air Traffic Forecas Empirical Research Based on he MCMC Mehod Jian-bo Wang, Chong-jun Fan & Lei Bai Business School, Universiy of Shanghai for Science and Technology, Shanghai 00093, China Correspondence: Chong-jun Fan, Business School, Universiy of Shanghai for Science and Technology, Shanghai 00093, China. Tel: 86-36-97-4830. E-mail: wangjianbai@homail.com Received: June, 0 Acceped: July, 0 Online Published: July 7, 0 doi:0.5539/cis.v5n5p50 URL: hp://dx.doi.org/0.5539/cis.v5n5p50 Absrac Airpor air raffic is one of he mos imporan and hardes one among all he airpor daa forecass. In his paper, he Markov Chain Mone Carlo (MCMC) mehod of applied heory of saisics has been inroduced ino he aviaion secor, and he discussion on airpor air raffic forecas has been conduced aking Shanghai airpor as he applicaion background. MCMC mehod is designed o solve he problem parameers expecaions where some baysian inference can no be direcly calculaed. This paper conducs he ieraive compuaion using WinBUGS, EViews is used o esimae he iniial ieraion parameer of he regression model, and i is on his basis ha he regression equaion under he MCMC mehod is obained. Keywords: MCMC, airpor, raffic daa, muliple linear regression. Inroducion In Bayesian saisical learning, he esimaion of overall parameer ofen needs high-dimensional probabiliy disribuion funcion o do he inegral. Under normal circumsances, here is no clear expression of his inegral, so i is very difficul o do i. In his case, he Markov Chain Mone Carlo (MCMC) mehod is a simple and effecive calculaion mehod. The basic idea of MCMC mehod is o consruc a Markov Chain, making is saionary disribuion he poserior disribuion of he parameers o be esimaed, hus generaing he samples of poserior disribuion hrough his Markov Chain, and doing he Mone Carlo inegraion of he samples of Markov Chain when hey reach saionary disribuion. For he reason ha he join disribuion of he parameers and variables in many ime series models are complex and high dimensional, i is impossible o exrac samples direcly he disribuion. MCMC can generae samples he join densiy of parameers, wihou having o know he specific forms of expression of densiy. The mos commonly used MCMC algorihm is Gibbs sampling, which is more accessible o gain he condiional disribuion densiy hrough simulaion, and i adops ieraive procedure o consruc he Markov Chain. Under he simulaion of large sample, i is expeced o converge o he arge disribuion (Lin & Han, 007; Sephen, 998; Lin & Han, 005). WinBUGS (Lunn, Spiegelhaler, Thomas, & Bes, 009) is a kind of specialized saisical sofware based on MCMC mehod, in which all unknown parameers are regarded as random variables in order o find ou he soluion of his kind of probabiliy model hen. In his paper, basing on he regression model, prereamen is done o he iniial value of he MCMC model, and empirical sudy is done regarding he relaionship beween air raffic, GDP growh rae and populaion growh (David, 00). The GDP means he money value of all he final goods and services produced in he domesic erriory of a counry in a year s ime. In his paper, we conduc hree pars, including MCMC mehod descripion, Empirical research on air raffic forecas and conclusions.. MCMC Mehod and Gibbs Sampling The MCMC mehod is a general simulaion mehod for sampling poserior disribuions and compuing poserior quaniies of ineres. MCMC mehods sample succeed a arge disribuion, and each sample depends on he previous one, hence he noion of he Markov Chain. A Markov Chain is a sequence of random variables,,,, in which he random variable depends on all s previous only hrough is immediae predecessor. The Markov Chain applied o sampling as a mechanism ha raverses randomly hrough a arge disribuion wihou having any memory of where i has been. Where i moves nex is enirely dependen on where i is now. 50
www.ccsene.org/cis Compuer and Informaion Science Vol. 5, No. 5; 0 Mone Carlo, as in Mone Carlo inegraion, is mainly used o approximae an expecaion by using he Markov Chain samples. In he simples version n g( ) p( ) d g( ) () s n where g (.) is a funcion of ineres and are samples p( ) on is suppor S. This approximaes he expeced value of g( ). The Gibbs sampler is a special case of he Meropolis sampler in which he proposal disribuions exacly mach he poserior condiional disribuions and proposals are acceped 00% of he ime. The main feaures of he Gibbs sampling, a Markov Chain sample, is exraced a series of condiional disribuions (George & Edward, 99; Gelfand & Smih, 990). Suppose ( )' is he parameer vecor, py ( ) is he likelihood, and ( ) is he prior disribuion. The full poserior condiional disribuion of ( i j, i j is proporional o he join poserior densiy; ha is, ( i j, i j p( y ) ( ) () For insance, he one-dimensional condiional disribuion of given * j, j j k is compued as he following:, j k p( y (,,, ) (,,, )') (3) * * * * * j j k k The Gibbs sampler works as follows: (0) 0 (0) Se 0 and choose an arbirary iniial value of { }. Generae each componen of as follows: ( ) draw () () ( ) draw ( ) ( ) ( ), 3... ( ) draw ( ) ( ) k k,, k Se. If T he number of desired samples, reurn o sep. When Gibbs sampling ieraes enough number of seps, we can believe he sample is subjec o he arge disribuion, or converge o he arge disribuion. The choice of he iniial value will affec he convergence rae of MCMC. 3. Empirical Research on Air Traffic Forecas I is raher imporan o choose proper facors ahead of modeling and esimaion. The following pars include facors analysis and parameers esimaion. 3. Influencing Facors Analysis Air raffic forecas has always been a difficul and key poin of he airpor aviaion forecass. There are many aspecs of facors ha influence air raffic, in his paper, macro-economic facor and demographic impac facor are used o analyze he air raffic. For economic facor, GDP is an imporan sandard o measure he comprehensive level of he economic developmen; populaion growh rae index is used as demographic facor. A he same ime, because he daa of GDP and populaion growh rae are raher consecuive, he reference value of he daa is quie high. We conduc a linear regression of air raffic growh based on GDP growh rae and populaion growh rae. There a binary regression model is buil as follows: Y 0 X X (4) Where 0,, are parameers o be esimaed, X and X are know vecors sand for GDP growh and populaion growh. Y sands for Airpor raffic volume growh. This aricle uses Eviews simulaion, according o daa released by Naional Bureau of Saisic 990 o 009, and hen Equaion (4) would have he form: Y 3.07 0.5453X 3.77X (5) 5
www.ccsene.org/cis Compuer and Informaion Science Vol. 5, No. 5; 0 3. Parameers Esimaion Based on he MCMC Mehod WinBUGS is a special package based on MCMC mehod, which can easily conduc Gibbs sampling on many models and disribuions. Before implemening Bayesian analysis, programmers need o se prior disribuion and iniial values of variables. The analysis processes include modeling, daa loading, iniial values and ieraive analysis. In his case, he priors are defined as Regression equaion: muy[] i X [] i X [] i (6) Prior disribuion of he parameers: 0 0 ~ N (0,.0 E 6) ; ~ N (0,.0 E 6) ; ~ N (0,.0 E 6) ; ~ Gamma(0.00,0.00) ; Yi ~ N( muy[ i], ). The iniial parameer value: 0 3.07; 0.5453; 3.77;. In order o ensure he convergence of he parameers, we firs conduc 5000 Gibbs pre-ieraions, and hen discard he iniial ieraion. We consruc he model using a graphical inerface Doodle (Figure ). Meanwhile, we se up hree Markov Chains in WinBUGS, and a he same ime ener he iniial values. This is because, for he convergence parameers, (and) hree ieraions resuls of he Markov Chain end o be overlapping. I is conducive o he observaion of he resuls. Figure. Bayesian modeling in WinBUGS Ieraions sar 500 o 60000. The saisical resuls show as in Table, which produces summary saisics for he variable, pooling over he chains seleced. In his case, we choose.5% and 97.5% as he confidence / inerval. The quaniy repored in he MC error column gives an esimae of / N, he Mone Carlo sandard error of he mean. During he 95% Confidence inervals, he average value of α0, α, α and τ are he inerval (-4.3, 0.4), (-0.383,.4), (-7.6, 4.69), (0.0044, 0.00946) respecively. Table. Poserior esimae saisics for moniored parameers wih 60000 ieraions node mean sd MC error.50% median 97.50% sar sample α 0 3.054 8.653 0.03505-4.3 3.06 0.4 500 60000 α 0.546 0.3488 0.00483-0.383 0.5455.4 500 60000 α 3.756 0.56 0.0496-7.6 3.755 4.69 500 60000 τ 0.009543 0.00338.53E-05 0.0044 0.00946 0.073 500 60000 Before drawing our conclusion, we need o have a judgmen on densiy esimaion and convergence saisics of he parameers. The following plos a smoohed Poserior Kernel Densiy Esimae for he variable if i is coninuous or a hisogram if i is discree (Figure ). 5
www.ccsene.org/cis Compuer and Informaion Science Vol. 5, No. 5; 0 Figure. Poserior Kernel Densiy Esimaion for moniored parameers wih 60000 ieraions In his case, we are running hree chains simulaneously. The race and hisory plos show each chain in a differen color. As he plos show (Figure 3), all he chains appear o be overlapping one anoher, herefor, we can be reasonably confiden ha convergence has been achieved. Figure 3. Trace plos for moniored parameers hree Chains ieraions 53
www.ccsene.org/cis Compuer and Informaion Science Vol. 5, No. 5; 0 The following plos (Figure 4) show he Gelman-Rubin convergence saisics. The widh of he cenral 80% inerval of he pooled runs is green, he average widh of he 80% inervals wihin he individual runs is blue, and heir raio R (= pooled / wihin) is red. R would generally be expeced o be greaer han if he saring values are suiably over-dispersed. The proper one should be concerned boh wih convergence of R o, and wih convergence of boh he pooled and wihin inerval widhs o sabiliy. Figure 4. Parameers convergence diagnosics According o he daa of Table, as well as Poserior Kernel Densiy Esimaion, Trace Plos, Parameers Convergence Diagnosics, we believe ha parameers esimaion based on MCMC mehod is valid. The linear regression model equaion is shown as in Equaion (7). Y 3.054 0.546X 3.756X (7) 4. Conclusions Air raffic forecas has always been a problem roubling airpor senior decision-makers o make scienific decisions, bu he use of convenional forecasing mehods ofen canno ge saisfacory resuls. In his paper, MCMC simulaion mehods and WinBUGS sofware applicaions based on Gibbs sampling are used o conduc he volailiy analysis of he airpor air raffic daa. The analyical mehod proposed in his paper can provide new ideas and mehods for he scienific decision-making of he airpor senior decision-makers. We hope ha, based on he fundamenal predicion research of he MCMC mehod, airpors can have a deeper level of sudy. References David, P. M. S. (00). Acuarial modeling wih MCMC and BUGS. Norh American Acuarial Journal, 5(), 96-5. Gelfand, A. E., & Smih, A. F. M. (990). Sampling-Based Approaches o Calculaing Marginal Densiies. J. American Saisical Associaion, 85, 398-409. hp://dx.doi.org/0.080/06459.990.04763 George, C., & Edward, I. G. (99). Explaining he Gibbs sampler. The American Saisician, 46, 67-74. Lin, J., & Han, Y. (005). Exponenial Regression Model Based on MCMC Mehod and Is Applicaion. Operaions Research and Managemen Science, 4(4), 95-00. Lin, J., & Han, Y. (007). Gamma Process Priors Model Based on MCMC Simulaion and Is Applicaion in Reliabiliy. Journal of Sysem Simulaion, 9(), 5099-50. Lunn, D., Spiegelhaler, D., Thomas, A., & Bes, N. (009). The BUGS projec: Evoluion, criique and fuure direcions. Saisics in Medicine, 8, 3049-3067. hp://dx.doi.org/0.00/sim.3680 Sephen, P. B. (998). Series D (The Saisician): Chain Mone Carlo Mehod and Is Applicaion. Journal of he Royal Saisical Sociey, 47(), 69-00. 54