Some Applications of the Resampling Methods in Computational Physics

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1 Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 Some Applcatos of the Resampl Methods Computatoal Physcs Sotraq Marko #, Lorec Ekoom * # Physcs Departmet, Uversty of Korca, Albaa, * Mathematcs Departmet, Uversty of Korca, Albaa Abstract The statstcal methods are be very mportat for estmat the ukow parameters computatoal physcs Betwee them we ca meto the resampl methods: the jackkfe ad bootstrap estmate I our work we have doe ther deas ad have show the results of some applcatos physcs problems of parameter estmato Keywords Jackkfe, bootstrap, bas, computatoal physcs, Bder rato, Loretza I INTRODUCTION The physcas use varous statstcal methods ther works But the books o these topcs usually fall to oe of two camps At oe etreme, the books for physcas do t dscuss all that s eeded ad rarely prove the results that they quote At the other etreme, the books for mathematcas presumably prove everyth but are wrtte the style of lemmas, proofs ad ufamlar otato whch s tmdat to physcas For the ecepto, there are some works whch fd a ood mddle roup [], [6], [0] I the follow we have treated some applcato of the resampl methods physc problems Let us see the statstcal problem of the mea estmato of a radom varable X Let suppose that,,, are depedet observatos from the radom varable X We ote the mea of the radom varable X by µ ad the sample mea by, where radom varable X the -th observato, we propose the statstc X X to be a estmator for the ukow parameter µ Sce X µ If we deote X, =,,, the ISSN: Pae 54 E, the statstc () s a ubased estmator for the ukow parameter µ If we deote the varace of the radom varable X by, we have Var(X) We ca use Var( X) to measure the ucertaty the sample mea or the error bar estmate Hece our estmate of µ s () Furthermore, X coveres to µ We ca say that X become more ad more accurate as the umber of observatos crease Now let suppose that we wat to estmate ot µ, but some fucto of µ e µ I Secto we have doe stadard statstcal methods for the estmato of µ ad have aalyzed the bas of the estmato ad ts order I Secto 3 we have show the dea of the jackkfe ad bootstrap estmatos ad have ve some cosderato about the jackkfe ad bootstrap estmato of the bas We have stressed that the bas order of jackkfe ad bootstrap estmate s stead of stadard estmatos I Secto 4 we have show some smulatos cases ad have doe a comparso betwee the jackkfe ad bootstrap methods wth stadard methods ()

2 form Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 II THE BIAS ORDER IN STANDARD METHODS I the above codtos, we compute some statstc of terest, say θˆ,, θˆ ad defe the bas the Bas Eθˆ - µ, (3) where E θˆ s the mea of θˆ Now let us aalyze the estmato of the ukow parameter µ A poor way to estmate µ would be from However, ths s really a estmate for the mea of (X), rather tha µ we evaluate the dfferece θˆ X (4) E(X) µ For the () we have µ µ µ µ µ But, eeral E(X) µ ()!! (5) We ca see that the bas s equal to µ varx E() - µ O()! (6) So, the bas does ot vash for If s a lear fucto the 0 ad Bas=0 Thus, there s o bas f s a lear fucto We take a better result f we chae (5) () wth I ths case µ µ µ µ ( ) µ!! (7) ad µ varx E( ) - µ O! (8) Now, the bas s of order rather tha the order Ths bas ca usually be elected because t s smaller tha Let statstcal error () of order To decrease the bas order we use the jackkfe ad the bootstrap methods I the follow secto we have ve the dea of the jackkfe ad the bootstrap estmatos III THE BIAS ESTIMATE WITH RESAMPLING METHODS The resampl methods are ett a mportat space may statstcal problems of estmat ukow parameters ad dstrbutos It depeds o two reasos The frst reaso s the use of computers ad software ad the secod reaso s that these methods ot ask ay codto about the dstrbutos Let us see the dea of the jackkfe ad bootstrap estmate A The Jackkfe Estmate of the Bas We defe the observato mea whe we have removed the observato the form - =,, I smlar way we defe =,, or, =,, The jackkfe estmate of µ j s the averae of,, ISSN: Pae 55

3 ad Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 J Let us aalyze the jackkfe estmate We have The ( µ j µ µ, =,, (0) - ) µ µ µ - E( The bas of the jackkfe estmate s E J j µ ) - µ µ µ - µ - µ - j µ - µ! () We see that hh order terms are at order The bas vashes for ad t s of the same order wth the estmate Sometmes we wat to estmate drectly the bas To do t, let us see Queoulle s bas estmate [9] Ths method s based o sequetally delet pots ad recomput the statstc θˆ Deot bas (8) of the X θˆ θˆ (,,,,, ) ad θˆ θˆ, Queoulle s estmate of bas s Bas J θˆ - θˆ, () lead to the bas-corrected jackkfe estmate of µ ~ θ θˆ - BasJ θˆ - -θˆ For may commo statstcs, clud most mamum lkelhood estmates, a a Eθˆ µ, (3) where a, a, do ot deped upo [8] After some calculatos we have ~ a Eθ θ - a - 3 (4) We see that the bas order of ~ θ s O, compared to O for the oral estmator (3) The jackkfe estmate of varace s [] θˆ θˆ varj (5) Suppose θˆ X, where s some cely behaved fucto (dervatve ests cotuously) The a frst order Taylor epasos ves θˆ So, substtut ths epresso to (5), we have (9) ISSN: Pae 56

4 Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 var (6) B The Bootstrap Estmate of the Bas The bootstrap [3] s coceptually a smple techque The bootstrap, lke the jackkfe, s a resampl of data pots Whereas jackkfe cosders ew data sets, each cota all the oral data pots mus, bootstrap uses B data sets each cota pots obtaed by radom sampl wth the same probablty of the oral set of pots * Let us ote such data set by,, X * * * X ad calculate θˆ (X,,X * ) Ths s the bootstrap estmate for the ukow parameter Idepedetly we repeat ths procedure a lare umber B of tmes obta bootstrap * replcatos θˆ,, θˆ*b The bootstrap estmate for the bas s B BasB θˆ*b θˆ (7) B b ad the bootstrap estmate for the varace s B varb (θˆ*b θˆ) (8) B b The order of the bas the bootstrap estmate s the same wth the order of the jackkfe estmate bas For the quadratc fuctoal we have BasJ BasB (9) IV EXAMPLES I ths Secto we have show the results of some applcatos of resampl methods Computatoal Physcs ) Eample 4 π Let us suppose we wat to compute cos(e(x)), where ε ad ε s a Gaussa radom varable wth 3 mea zero ad stadard devato uty We took a sample =000 The jackkfe estmate of the mea was 0496, wth a error estmate of 0080 whch s cosstet wth the eact value of 05 For comparso cos was also equal to 0496 to ths precso Us equato () to et a less based estmate of the mea, ad us the full precso of the umbers the computer ves 0499 The dfferece betwee 0496 ad 0499 s completely umportat sce the error bar of 0080 s very much larer Wth B=00 data sets we foud Ths result s cosstet wth the eact value of 05 ad very close to the jackkfe result ) Eample 4 Jackkfe ad bootstrap ca be used to compute error bars for qute eeral fuctos of the data set For eample, oe ca use the jackkfe ad bootstrap resampl schemes to estmate parameters descrb the shape of the dstrbuto from the data set A eample, whch s also used may researches phase trastos (where t s called the Bder rato ),, defed by 4 EX EX E X EX (0) ISSN: Pae 57

5 Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 Sce the total power of X the umerator ad deomator s the same, the kurtoss depeds oly o the shape of the dstrbuto ad ot o ts overall scale It takes the value 3 for a Gaussa dstrbuto I the eample we took =000 pots from a Gaussa dstrbuto ad computed the kurtoss us the jackkfe method For the Gaussa dstrbuto we kow that EX=0 For each of the jackkfe data sets we computed ad obtaed a averae ad error bar us () above The result s whch s cosstet wth the eact value of 3 The kurtoss s foud for each B=00 bootstrap samples, ad the mea ad error obtaed from (7) The result s whch arees well wth the jackkfe estmate ad s cosstet wth the eact value of 3 3) Eample 43 Lear reresso: where Cosder the lear reresso model y β ε, =,,, () ε are depedet radom varables wth detcally ukow dstrbuto F ad ε 0 E, s a kow p vector of covarates whe y β ε, where β s a p vector of ukow parameters The statstc of terest s the least squares estmate of β the form T βˆ X X X T, () where Y y,, T, T Y y X,,X cov s T ˆ X X, (3) where ˆ εˆ ad εˆ be the estmated resdual y βˆ The multvarate verso of Tukey s formula s covj βˆ βˆ βˆ βˆ (4) If all the εˆ are detcal value ths s about the same as the stadard aswer, but otherwse the two formulas are qute dfferet I quadratc ft of 50 pots wth Gaussa ose to 3-+, the follow results were obtaed X The usual estmate of βˆ TABLE I THE ESTIMATE RESULTS OF THE LINEAR REGRESSION WITH STANDARD AND REAMPLING METHODS Least square ft Jackkfe ft Bootstrap ft Ft parameters Error bars Covarace matr ) Eample 4 The smple meda For most problems, the jackkfe ad bootstrap ve smlar results However, there s at least oe class of problems where the jackkfe approach s usatsfactory, because the data set are too smlar to each other, whle the bootstrap method works We ca ote that the jackkfe estmate of the varace fals the case of the sample meda A estmator for the sample meda s m f m- From the formula (3) we have (5) f m m m ISSN: Pae 58

6 Iteratoal Joural of Mathematcs Treds ad Techoloy Volume 6 February 04 - varj m m 4 (6) Stadard theory [7] shows that f the dstrbuto F of the radom varable X has a desty fucto f the var, (7) 4f µ Y where f µ s the desty at the sample meda µ, µ f s assumed >0 ad Y s a radom varable wth epectato ad varace 0 The true varace oes to the lmt [5] var (8) 4f µ I ths case, the jackkfe estmate s ot eve a cosstet estmator of the varace sample meda From the other had, the bootstrap estmate of the varace oes well for the sample meda The bootstrap estmate of stadard devato s show to be asymptotcally cosstet for the true stadard devato [4] I the bootstrap estmate, the B data samples are sfcatly dfferet from each other, so the error the meda ca be estmated As a eample, we took =00 data pots eerated from the postve half of a Loretza 0 f π (9) 0 0 Note that ths s a very broad dstrbuto for whch eve the mea s ot defed However, the meda s defed ad a stadard teral ves the value Iclud all 00 values of we have, we foud the meda to be 0963 Us the bootstrap wth B=5000 data sets we foud We see that the overall averae ad the mea of the bootstraps are very close, ad the result arees wth the eact value of wth the error bar V CONCLUSIONS I ths paper we have ve some applcatos of jackkfe ad bootstrap Computatoal Physcs We have show the effcecy ad faclty of these methods estmato whe the dstrbutos are ukow or asymmetrc We ca see t Eamples 4, 4 ad 43 But some cases the resampl methods do ot work We ca see t Eample 44, whe the bootstrap method works, meawhle the jackkfe method does ot work REFERENCES [] V Ambeaokar ad M Troyer M Estmat errors relably Mote Carlo smulatos of the Ehrefest model, Am J Phys Vol 78, 50, 009 [] G Bohm ad G Zech G (00) Itroducto to statstcs ad data aalyss for physcsts Verla Deutches Electroe Sychrotro, 00 [3] B Efro Bootstrap methods: aother look at the jackkfe A Statst, Vol 7, pp -6, 979 [4] B Efro The jackkfe, the bootstrap ad other resampl plas SIAM, 98 [5] M Kedall ad A Stuart The advaced theory of statstcs Grff Lodo, 958 [6] W H Press, S A Teukolsky, W T Wetterl ad B P Flaery Numercal recpes C, ed Ed Cambrde Uversty Press, 99 [7] R Pyke Spacs, J Roy Statst Soc Ser B Vol 7, pp , 965 [8] W Schucay, H Gray ad O Owe O bas reducto estmato JASA, Vol 66, pp , 97 [9] M Queoulle Appromate tests of correlato tme seres J Roy Statst Soc Ser B, Vol, pp -84, 949 [0] J R Taylor The study of ucertates physcal measuremets Uversty Scece Books, Sausalto, Calfora, 997 [] J Tukey Bas ad cofdece ot qute lare samples Abstract A Math Statst, Vol 9, pp 64, 958 ISSN: Pae 59

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