Particle Swarm Optimization Algorithm for the Shortest Confidence Interval Problem

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1 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT Particle warm Optimizatio Algorithm for the hortest Cofece Iterval Problem hag Gao ad Zaiyue Zhag chool of Computer ciece ad Egieerig, Jiagsu Uiversity of ciece ad Techology, Zhejiag 003, Chia gao_shag@hotmail.com yzzjzzy@sia.com Cuge Cao Istitute of Computig Techology,The Chiese Academy of cieces,beijig 00080, Chia cgcao@ict.ac.c Abstract Based o the example of costructig a cofece iterval for variace, the otio ad costructio of the shortest cofece iterval are put forward. Furthermore, the particle swarm algorithm for this problem is preseted to solve this o-liear programmig problem. Compared with the cofece iterval calculated with traditioal method, it has distict advatage. The optimum results which is used to f the shortest cofece iterval of variace ad mea variace are give uder the cofece level 0.9 ad The shortest cofece iterval about Gamma distributio, Laplace distributio, Weibull distributio ad beta distributio are also discussed. Idex Terms mathematical statistics, cofece iterval, the shortest iterval, particle swarm algorithm I. INTRODUCTION The cofece iterval of ukow parameter represets the rage of the value ad the reliability about estimatio of parameter. Uder give cofece level,the legth of cofece iterval represets the precisio of the estimatio. It is very ecessary to research the problem how to costruct good radom variables to make the legth of cofece iterval as short as possible. I practical applicatios,people geerally get cofece iterval by probability symmetry. But the legth of this k of cofece iterval is ot the shortest. Although most itroductory textbooks i mathematical statistics discuss cofece itervals, the cocept of a shortest cofece iterval commads little or o attetio. Ramachadra (958 ad Tate ad Klett (959 specify shortest ubiased cofece itervals for the variace of a ormal distributio. I additio, Ramachadra (958 specifies the shortest ubiased cofece itervals for the ratio of two ormal variaces. Tate ad Klett (959 ad Guether (969 also specify the physically shortest itervals for the variace of a sigle ormal distributio. upported by Natioal Basic Research Program of Jiagsu Provice Uiversity (08KJB Correspodig author: hag Gao. The method of obtaiig such a iterval of biomial probability is preseted as well by Zieliski Wojciech (00. K.K. Feretios (006 clarify ad commet o methods of fig such itervals, ivestigate the relatioship betwee these types of itervals, poit out that cofece itervals with the shortest legth do ot always exist, eve whe the distributio of the pivotal quatity is symmetric; ad fially, ad give similar results whe the Bayesia approach is used. I this paper ad of the shortest cofece iterval about ormal distributio are give uder the cofece level 0.90 ad Fially, this paper gives the coditio that the shortest cofece iterval about Gamma distributio, Laplace distributio, Weibull distributio ad Beta distributio. II. THE HORTET CONFIDENCE INTERVAL A. The hortest Cofece Iterval Let X, X,, X be a radom sample from a distributio with probability desity fuctio f ( x;. I usig the stadard method for obtaiig a cofece iterval for, oe seeks a radom variable T X, X,, X ; T( whose distributio is ( epedet of. The the probability statemet P a T( b is coverted to P W W x, x,, x ad, after observig, the specific umbers w, w are calculated ad form the edpoits of the cofece iterval. For every T (, a ad b ca be w chose i differet ways, oe of which is make a miimum[5]. uch a iterval is the shortest iterval based upo T (. w 0 ACADEMY PUBLIHER doi:0.4304/jcp

2 80 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT 0 B. hortest Cofece Iterval of of Normal Distributio uppose X, X,, X are epedet ad etically distributed (i observatios of a ormal sample, say N (,. The, ( ~ ( Thus, we have ( P ( ( ( which gives the as level cofece iterval for ( (, ( ( Obviously, the iterval ( ca ot guaratee the shortest legth cofece iterval. The followig illustrate the cocept ad the method to f the shortest cofece iterval. ( P ( ( ( The ( ( P ( ( ( The cofece iterval is ( (, (3 ( ( ( With legth ( ( L ( ( [ ( ( ( ]( ( The problem of the shortest cofece iterval problem is to f some suitable which miimize L '. mi L' ( ( ( Programmig (4 is a ucostraied o-liear programmig problem about. The objective fuctio cotais percetile, so it is hard to solve by derivative methods. I geeral, we ca use direct search optimizatio methods, such as "radom method", "radom direct method", "simplex method", etc.. I this paper, we use particle swarm optimizatio to solve this problem. The ( is calculated by Matlab statistics * (4 ( toolbox. X= chiiv(p,v computes the iverse of the cumulative distributio fuctio with parameters specified by V for the correspodig probabilities i P. P ad V ca be vectors, matrices, or multimesioal arrays that have the same size. A scalar iput is expaded to a costat array with the same dimesios as the other iputs. III. THE PARTICLE WARM OPTIMIZATION ALGORITHM A. BasicPparticle warm Optimizatio (PO Algorithm I the particle swarm optimizatio (PO algorithm[7-9], the birds i a flock are symbolically represeted as particles. These particles ca be cosered as simple agets flyig through a problem space. A particle s locatio i the multi-dimesioal problem space represets oe solutio for the problem. Whe a particle moves to a ew locatio, a differet problem solutio is geerated. This solutio is evaluated by a fitess fuctio that proves a quatitative value of the solutio s utility. The velocity ad directio of each particle movig alog each dimesio of the problem space will be altered with each geeratio of movemet. I combiatio, the particle s persoal experiece, P ad its eighbors experiece, P gd ifluece the movemet of each particle through a problem space. The radom values rad ad rad are used for the sake of completeess, that is, to make sure that particles explore a we search space before covergig aroud the optimal solutio. The values of c ad c cotrol the weight balace of P ad P gd i decig the particle s ext movemet velocity. At every geeratio, the particle s ew locatio is computed by addig the particle s curret velocity, v, to its locatio, x. Mathematically, give a multi-dimesioal problem space, the ith particle chages its velocity ad locatio accordig to the followig equatios[7-9]: v c c 0 v rad x c rad ( p ( p gd x x (5 x v (6 where c 0 deotes the iertia weight factor; p is the locatio of the particle that experieces the best fitess value; P gd is the locatio of the particles that experiece a global best fitess value; c ad c are costats ad are kow as acceleratio coefficiets; d deotes the dimesio of the problem space; rad, rad are radom values i the rage of (0,. For equatio (, the first part represets the iertia of pervious velocity; the secod part is the cogitio part, which represets the private thikig by itself; the third part is the social part, which represets the cooperatio amog the particles. If the sum of acceleratios would cause the velocity v, o that dimesio to exceed v max,d,the v, is limited to v max,d. v max,d determies the resolutio with which regios betwee the preset positio ad the target positio are searched. 0 ACADEMY PUBLIHER

3 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT 0 8 B. PO Algorithm For The hortet Cofece Iterval Problem The PO algorithm for the shortest cofece iterval problem ca be described as follows: I Iitialize particle a et costats c 0, c ad c.. b Radomly iitialize particle positios. c Radomly iitialize particle velocities. II Do: a For each particle: Calculate fitess value (the L. If the fitess value is better tha the best fitess value P i history. 3 et curret value as the ew P. Ed b For each particle: F i the particle eighborhood, the particle with the best fitess P gd. Calculate particle velocity accordig to the velocity equatio (5. 3 Apply the velocity costrictio. 4 Update particle positio accordig to the positio equatio (6. 5 Apply the positio costrictio. Ed While maximum iteratios or miimum error criteria is ot attaied. C.Rresults Accordig to this algorithm usig Matlab laguage, the values of * at which is used to f the shortest with cofece iterval of 0. ad are show i table ad table. TABLE I. THE VALUE OF * WHICH I UED TO FIND THE HORTET CONFIDENCE INTERVAL OF WITH 0. * ( ' ( * * ( TABLE II THE VALUE OF * WHICH I UED TO FIND THE HORTET CONFIDENCE INTERVAL OF WITH * ( ( * * ( ACADEMY PUBLIHER

4 8 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT IV. HORTET CONFIDENCE INTERVAL OF The cofece iterval for is ( (, ( ( (7 with 0. ad ad table 4. are show i table3 TABLE III. THE VALUE OF * AT WHICH I UED TO FIND THE HORTET CONFIDENCE INTERVAL OF WITH 0. * ( ( * * ( The legth of is L [ ( ( ( ( ( ( ( ] ( The problem of the shortest cofece iterval problem is to f some suitable which miimize L '. mi L' (9 ( ( ( Use the above method recout, the values of * at which is used to f the shortest cofece iterval of ( ACADEMY PUBLIHER

5 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT 0 83 TABLE IV. THE VALUE OF * AT WHICH I UED TO FIND THE HORTET CONFIDENCE INTERVAL OF WITH * ( ( * * ( V. NUMERICAL EXAMPLE Example As a illustrative of the use of the above results, coser a simulated life test o 6 compoets from a orm distributio. The observed failure times are as follows: 506, 508, 499, 503, 504, 50, 497, 5, 54, 505, 403, 496, 506, 50, 509, 496 ( 0. From calculatig, we got 6, From chi-square critical value table, we got ( , ad ( , apply the formula (, the ( (, =[ , ] 0.05( 0.95( o from traditioal method, the legth of cofece iterval of is From table, we got * , ( ad ( ( * * Apply the formula (3, the ( (, =[ , ] *( ( *( Usig particle swarm optimizatio (PO algorithm, the legth of cofece iterval of is imilarly, the cofece iterval of is calculated by two methods. Apply the formula (7, the ( (, =[ , ] 0.05( 0.95( o from traditioal method, the legth of cofece iterval of is From table 3, we got * , ( ad ( ( * * Apply the formula (8, the ( (, =[ ,8.4658] *( ( *( Usig particle swarm optimizatio (PO algorithm, the legth of cofece iterval of is The compariso of the results are show i table 5. TABLE V COMPARION OF THE REULT WITH 0. Particle swarm Methods Ttraditioal method optimizatio (PO algorithm Cofece iterval of [ , ] [ , ] Legth of Cofece [ , ] [ , ] iterval of Legth of ACADEMY PUBLIHER

6 84 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT 0 ( From chi-square critical value table, we got 0.975( ad 0.05( , apply the formula (, the ( (, =[0.9905, 9.458] 0.05( 0.975( o from traditioal method, the legth of cofece iterval of is From table, we got * , ( ad ( ( * * Apply the formula (3, the ( (, =[ , ] *( ( *( Usig particle swarm optimizatio (PO algorithm, the legth of cofece iterval of is imilarly, the cofece iterval of is calculated by two methods. Apply the formula (7, the ( (, =[ , ] 0.05( 0.975( o from traditioal method, the legth of cofece iterval of is From table 4, we got * , ( ad ( ( * * Apply the formula (8, the ( (, =[ , *( ( *( ] Usig particle swarm optimizatio (PO algorithm, the legth of cofece iterval of is The compariso of the results are show i table 6. TABLE VI COMPARION OF THE REULT WITH Methods Ttraditioal method Particle swarm optimizatio (PO algorithm Cofece [ , ] [0.9905, 9.458] iterval of Legth of Cofece [ , ] [ , ] iterval of Legth of VI. DICUION A. The hortest Cofece Iterval For Gamma Distributios Above all, the miimum legth of cofece iterval for the variace of the ormal distributio are discussed. I fact, the shortest cofece iterval for other distributios ca use the results of table. Let X have a gamma distributio with parameters ad ad assume that is a kow iteger. Thus, the desity fuctio of X is f ( x x T( A sufficiet statistic for is e x has a chi-square distributio with freedom. The P Y X i i v Y ( ( ad Y degrees of ( Y P ( ( The cofece iterval of is With legth Y, ( L Y ( Y ( ( Y[ ] ( ( ( (0 The problem of the shortest cofece iterval problem is to f some suitable which miimize L '. mi L' ( ( ( ( Thus the solutio for ( * ( ad * ( is the same as table with 0. or the same as table with B. The hortest Cofece Iterval For Laplace Distributios Let X have the Laplace distributio with probability desity fuctio The Y x f ( x e, 0 X i i is a sufficiet statistic for ad Y has a chi-square distributio with v of freedom. P Y ( ( ( degrees 0 ACADEMY PUBLIHER

7 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT 0 85 The Y P ( ( The cofece iterval of is With legth Y, ( L Y ( ( Y ( Y[ ] ( ( ( ( The problem of the shortest cofece iterval problem is to f some suitable which miimize L '. mi L' ( ( ( (3 Thus the solutio for ( * ( ad * ( is also the same as table with 0. or the same as table with C. The hortest Cofece Iterval For Weibull Distributios Let X have the Weibull distributio with probability desity fuctio f x ( x x e, 0, x 0 X i i With 0 beig kow. The Y sufficiet statistic for ad Y is a has a chi-square distributio with v degrees of freedom as i Example 3. whe, it is expoetial distributio. The probability desity fuctio is f x ( x e, 0, 0 x (4 This coclusio iclude the shortest cofece iterval for expoetial distributio. D. The hortest Cofece Iterval For Beta Distributios The probability desity fuctio of the beta distributio is: ( f ( x x ( x, 0, 0 x ( ( where is the gamma fuctio. The beta desity fuctio ca take o differet shapes depedig o the values of the two parameters. Here, we discuss beta distributio with. The probability desity fuctio is:, 0 x The f ( x x, 0 Y i l is sufficiet for ad Y X i a chi-square distributio with freedom. P The has v degrees of ( Y ( ( P ( ( Y The cofece iterval of is With legth ( (, Y Y ( ( ( ( L Y Y [ ( ( (] Y ( Y (5 The problem of the shortest cofece iterval problem is to f some suitable which miimize L '. mi L' ( ( (6 ( Accordig to this algorithm usig Matlab laguage, the values of * at which is used to f the shortest cofece iterval of with 0. is show i table 6. TABLE VI THE VALUE OF * WHICH I UED TO FIND THE HORTET CONFIDENCE INTERVAL OF WITH 0. * ( ( * * ( ACADEMY PUBLIHER

8 86 JOURNAL OF COMPUTER, VOL. 7, NO. 8, AUGUT paper are used. The shortest cofece iterval of other distributio ca be obtaied by similar method. ACKNOWLEDGMENT This work was partially supported by Natioal Basic Research Program of Jiagsu Provice Uiversity (08KJB50003 ad the Ope Project Program of the tate Key Lab of CAD&CG. REFERENCE [] W.C. Guether, hortest cofece itervals. The America tatisticia, vol.3, o., pp.-5,969. [] K.V. Ramachadra, A test of variaces. Joural of the America tatistical Associatio, vol.53, pp , 958. [3] R.F. Tate ad G.W. Klett, Optimal cofece itervals forth variace of a ormal distributio. Joural of the America tatistical Associatio, vol.54, pp ,959. [4] Zieliski, Wojciech, The shortest Clopper-Pearso cofece iterval for biomial probability. Commuicatios i tatistics: imulatio ad Computatio, vol.39, o., pp.88-93, Jauary 00. [5] K.K. Feretios ad K.X. Karakostas, More o shortest ad equal tails cofece itervals. Commuicatios i tatistics - Theory ad Methods, vol.35, o.5, pp.8-89, 006. [6] J. Goodma, O the defiitio of the 'best' cofece iterval. Reliability egieerig, vol.7, o.4, pp.3-8, 984. [7] R. C. Eberhart ad J. Keedy. A New Optimizer Usig Particles warm Theory. Proc. 6th Iteratioal ymposium o Micro Machie ad Huma ciece, Nagoya, Japa, 995, pp [8] Y. H. hi ad R. C. Eberhart. A Modified Particle warm Optimizer. IEEE Iteratioal Coferece o Evolutioary Computatio, Achorage, Alaska, May 4-9, 998, pp [9]. Gao ad J. Y. Yag, warm Itelligece Algorithm ad Applicatios. Beijig: Chia Water Power Press, 006, pp.7-0(i Chiese. VII. CONCLUION The iterval estimatio of parameter is a basic form for statistical coclusio, which ca icate the possible scale for estimated geeral parameter i a certai extet depedability based o the distributio of pivot quatity. The theory of Neyma cofece iterval shows that the certai level of cofece esures the certai extet depedability, but the precisio is ofte scaled by the legth of iterva. The shortest cofece iterval problem is a o-liear programmig problem ad the particle swarm algorithm for this problem is preseted to solve it. The optimum results which is used to f the ad are give shortest cofece iterval of uder the sample sizes from 4 to 36 ad the cofece level 0.9. The shortest cofece iterval about Gamma distributio, Laplace distributio ad Weibull distributio ca use results also. The results of the shortest cofece iterval about Beta distributio are give also. Whe sample size is ot large, coclusio shows that the precisio of the iterval estimatio of parameter is remarkably icreasig if the data from Tables i this hag Gao was bor i 97, ad received his M.. degree i 996 ad Ph.D degree i 006. He ow works i school of computer sciece ad techology, Jiagsu Uiversity of ciece ad Techology. He is a associate professor ad He is egage maily i systems egieerig. Zaiyue Zhag was bor i 96, ad received his M.. degree i mathematics i 99 from the Departmet of Mathematics, Yagzhou Teachig College, ad the Ph.D. degree i mathematics i 995 from the Istitute of oftware, the Chiese Academy of cieces. Now he is a professor of Jiagsu Uiversity of ciece ad Techology. His curret research areas are recursio theory, kowledge represetatio ad kowledge reasoig. Cuge Cao was bor i 964, ad received his M.. degree i 989 ad Ph.D. degree i 993 both i mathematics from the Istitute of Mathematics, the Chiese Academy of cieces. Now he is a professor of the Istitute of Computig Techology, the Chiese Academy of cieces. His research area is large scale kowledge processig. 0 ACADEMY PUBLIHER