Unfortunately, no one has yet learned how to generate quasi-random numbers like this.

Transcription

1 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Daddy, Where do Random Numbers Really Come From? In any of the MC procedures we have prevously dscussed we need many random numbers. A sequence of truly random numbers s, by ts nature, unpredctable and therefore not reproducble. A sequence of truly random numbers can only be generated by a random physcal process, e.g. radoactve decay, thermal nose n a resstor, cosmc ray arrval tmes, etc. If such a source of random numbers were used for MC calculatons there would be no problem the theory we have developed would work fne. In practce, t s etremely dffcult to construct a physcal devce that s fast enough to delver random numbers from a random physcal process as rapdly as modern computers demand them durng a calculaton. Several alternatves est. For eample, some years ago Argonne Natonal Laboratory created a data fle wth bt random numbers constructed from a U-35 α-partcle source as follows: Usng a hgh-resoluton onzaton counter, t was found that n a Δt = 0 msec countng tme nterval, on average α-partcles were detected. When the count was odd, a 1-bt was recorded, and when the count was even, a 0-bt was recorded. Then the bts were strung together as 31-bt sequences of (unsgned) ntegers. Bas removal n ths data set was accomplshed as follows: Suppose one has a sequence of 0 s and 1 s whch s random, but the probabltes P(0) and P(1) are not both precsely equal to 1/. Dvde the sequence nto pars, e.g Ignore 00 and 11 pars, so the above wll become: For the remanng pars, drop the frst bt. Ths wll yeld: Now the probablty of fndng a 0 or 1 n the fnal sequence, P (0) and P (1) are: P (0) = P(1) P(0) (snce t came from a (10) par) P (1) = P(0) P(1) (snce t came from a (01) par) Thus, P (0) = P (1),.e. 0 s and 1 s are ndeed equally lkely n the fnal sequence. Unfortunately ths procedure s very wasteful, snce about half of the bts are thrown out rght away, and half of the remanng ones are not used. The common choce n modern applcatons s a source of Quas-random (or pseudorandom ) numbers. Ths s what s produced by random number generators on a computer. The numbers are generated accordng to a strct mathematcal procedure leadng to a sequence whch s predctable (n advance) and certanly reproducble. Thus random numbers on a computer are not at all random n the mathematcal sense... however, they are supposed to be ndstngushable from a sequence generated truly randomly. That s, f one ddn t know that the numbers were generated by a strct mathematcal procedure but one was just gven the sequence, then one would be unable to tell that a formula was used rather than a physcal process. Unfortunately, no one has yet learned how to generate quas-random numbers lke ths. P598AEM Lecture Notes 7 1

2 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede One can tell the dfference! However, ths does not prevent people from usng sequences of such numbers as f they were truly random (or even truly quas-random) and gnorng the fact that, theoretcally, they re on shaky ground For a long tme the most wdely used random # generator was the multplcatve congruental : ( ) r = ar 1 + b mod m where a and b are constants and mod m means take the remander after dvson by m..e. the th random # s the remander when {a( 1) th random # + b} s dvded by m. Snce takng the operaton of takng the remander commutes wth the dvson operaton, we can alternatvely work wth a pseudo-random nteger sequence, and then dvde the sequence of ntegers by m. For eample, f a = 13, b = 0, m = 31 and r 0 = 1, then ths pseudo-random nteger sequence begns wth: 1, 13, 14, 7, 10, 6, 16,, 7, 9, 5, 3, the net random # s r = mod 31 = = 8. ( ) The frst 30 terms n ths sequence are a permutaton of the ntegers 1 30, but then the random # sequence repeats tself,.e. t has perod m 1. If the pseudo-random nteger sequence such as the one above s then scaled by dvdng by m, the result are floatng-pont # s that are unformly dstrbuted on the nterval [0,1]. In our above eample: 0.033, , , , 0.36, , , Obvously, one doesn t want a sequence of random # s wth a short perod, so one chooses m to be a very large nteger, along wth choces for a and b. For eample, n the 1960 s, Scentfc Subroutne Package (SSP) on IBM manframe computers had a multplcatve congruental random # subroutne RND (or RANDU) for use wth 3-bt computaton, wth 16 a = + 3 = 65539, b = 0 and that the multplcaton operaton 1 31 m = = Because of the choce 16 a = + 3, note ar can be carred out by a shft operaton + addton operaton, whch was motvated by speed consderatons on such computers of that era. It can be shown that for ths choce of IBM s random # parameters that: 31 ( = ) r 6r 9r mod for all e. there were etremely hgh correlatons between 3 sequental random # s n IBM s code! Over the years, collectvely we have learned that the success of a random # generator s not related to ts complety. In fact, an arbtrary nventon of great complety s: a.) probably very hard to analyze as far as ts propertes are concerned, and b.) lkely to be a poor generator, gvng very un-random lke sequences. Durng the 1960 s, many random # generators were tred and tested. By the md-1970 s, only the multplcatve congruental type were serously used. P598AEM Lecture Notes 7

3 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Normally m s chosen as t, where t s the number of bts used to represent an nteger on the machne. Snce a and r 1 are t-bt ntegers, ther product has t bts. After b s added (often b = 0) the lowest t bts are kept as r. Fnally, r s dvded by m to gve a floatng pont number between 0.0 and 1.0. One property of most random # generators, ncludng the multplcatve congruental ones, s ther perod. Obvously, once any specfc number occurs a second tme, the sequence followng t s the same thereafter. For multplcatve congruental random # generators, the perod s dependent on a, b, and r 0, the ntal, or seed random number whch must be chosen before the random # sequence s generated. The mamum perod can be shown to be of length (m/4). For typcal 3-bt computers today, t = 31 (the remanng bt s used for a sgn) and t = , whch lmts the mamum length of a random # sequence to about 500 mllon. Testng Random # Generators: In the past, t was dangerous to depend on the random # generators suppled wth the subroutne/ functon lbrares of some computer systems. For eample, as mentoned prevously RANDU (dstrbuted by IBM wth the 360), was found (rght away) to be poor, but that was already too late! Even 10-0 years later (.e. n the 1970 s 80 s) people were stll usng RANDU! Eeeeek!!! In the dark ages (1960 s) there was no deep understandng of quas-random # generators and so the tests were not very specfc. Typcally, just apparent randomness was tested. For eample, consder the sample mean of a sequence of N random # s: r are a set of N random numbers drawn from ( 0,1) If the σ = 1 1 N. and that r { } r 1 N r N = 1 = U then we know that r ˆ = 1= 0.5 So we check the generator to see f r s wthn σ of 0.5, whch t should be 95% of the tme. If the random # generator were to fal ths test, we would say that t fals the test at the 5% level. Note ths actually proves nothng, snce for true random # s, we epect them to fal 5% of the tme! If the random # generator passes ths test, we mght go on to look at σ r, whch we fnd by generatng a long sequence of random numbers and dvdng t up nto groups of sze N. In realty we use batteres of more complcated tests. Then we assume that, havng passed all of these tests, the generator wll also pass the net test. The net test s the requrement that t be random enough to work properly n your problem. It s not known, n general, why t should pass ths test, but t sn t known, ether, why t should not. A very dfferent knd of test was dscovered n 196 by a hgh-energy physcst (Joe Lach) who was tryng to understand why the random # generator that came wth IBM s 709 gave larger fluctuatons than epected. He dsplayed the random # s on a CRT. (In those days, ths was not a common procedure!) When he dvded hs random # sequence nto pars and plotted them as (,y) coordnates of ponts, he saw nothng strange, just a unform densty of ponts. Then he dvded t nto trplets, consderng each as the (,y,z) of a pont n 3-D space. When he plotted the (,y) coordnates for all trplets for whch z < 0.1 he saw the ponts fallng nto a set of slanted bands wth no (,y) values n between, as shown n the fgure below: P598AEM Lecture Notes 7 3

4 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede 1.0 y Ths phenomenon was later shown by George Marsagla to be a defect (.e. a feature) of all multplcatve congruental random # generators. In Random Numbers Fall Manly n the Planes, (PNAS, Vol 61, 1968) he showed that f successve d-tuples from such a random # generator are taken as coordnates of ponts n d-dmensonal space, all of the ponts wll le on a certan fnte number of parallel hyperplanes. (The orgnal dscovery saw 3-D planes projected onto -D. They are tlted and thus appear as bands n -D.) The mamum number of hyperplanes depends on d and the number of bts used for nteger representaton on the partcular computer beng used: t The mamum number of hyperplanes = ( d! ) 1 d. Ths gves: t d = 3 d = 4 d = 6 d = Moral: Never trust a random # generator on a 16 bt machne, unless t uses double-precson!!! Even 3 bts may be dangerous, e.g. for carryng out a 10-dmensonal ntegral. Warnng: Mamum number s just that. For an arbtrary choce of the parameters a and b n r = ( ar 1 + b) mod m t can be smaller. As a result of Marsagla s work, and other efforts at the same tme (early 70 s) we now understand multplcatve congruental random # generators very well. Stll, t s not safe to assume that the people who supply them wth computers understand them. One should only trust those wth pedgrees, e.g. those from CERN, SLAC, ANL, ORNL, etc. Often these have been desgned to gve better random # s. For eample, t has been shown that by usng a random # generator such as r = ( ar 1+ br ) mod m where a and b are chosen carefully one can ncrease the number of hyperplanes by a factor of t / d. P598AEM Lecture Notes 7 4

5 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Snce the 70 s, consderable progress has been made on developng several new types of random # generators that don t have the above-mentoned problems that multplcatve congruental random # generators suffer from. One such random # generator s RANLUX, whch has a sold theoretcal bass n chaos theory, based on the work of M. Lüscher, whch n turn s based on work by G. Marsagla and A. Zaman, regardng so-called lagged Fbonacc sequence generators. These generators and other modern random # generators are effcent and have very long repeat perods of up to ~ and pass etensve DIEHARD or TESTU01 batteres of tests, for whch the most commonly avalable multplcatve congruental random # generators fal, wth repeat perods of less than 3 ~ The nterested reader s referred to the followng papers: 1.) Unform Random Number Generators: A Revew, Perre L Ecuyer, Proc Wnter Smulaton Conference, IEEE Press, Dec. 1997, ) A revew of pseudorandom number generators, F. James, Comp. Phys. Comm. 60, (1990). 3.) RANLUX: A Fortran mplementaton of the hgh-qualty pseudorandom number generator of Lüscher, F. James, Comp. Phys. Comm. 79, 111 (1994). 4.) A portable hgh-qualty random number generator for lattce feld theory smulatons, M. Lüscher, Comp. Phys. Comm (1994). 5.) Mamally Equdstrbuted Combned Tausworthe Generators, Perre L Ecuyer, Mathematcs of Computaton 65, 13 (1996). 6.) Tables of Mamally Equdstrbuted Combned LFSR Generators, Perre L Ecuyer, Mathematcs of Computaton 65, 5 (1999). 7.) Mersenne Twster: A 63-Dmensonally Equdstrbuted Unform Pseudo-Random Number Generator, M. Matsumoto and T. Nshmura, ACM Transactons on Modelng and Computer Smulatons, Vol. 8, No. 1, January 1998, ) Some dffcult-to-pass tests of randomness, G. Marsagla and W.W. Tsang, Journal of Statstcal Software, Volume 7, 00, Issue 3. 9.) Some portable very-long-perod random number generators, G. Marsaglla and A. Zaman, Computers n Physcs, Vol. 8, No. 1, Jan/Feb ) A New Class of Random Number Generators, G. Marsaglla and A. Zaman, The Annals of Appled Probablty 1991, Vol. 1, No. 3, ) W. H. Press, et al., Numercal Recpes, 3 rd Ed., Cambrdge Unversty Press, New York, ) D.E. Knuth, The Art of Computer Programmng, Vol. II, 3 rd Ed., Addson-Wesley, Readng, Mass The Marsagla Random Number CDROM ncludng the Dehard Battery of Tests of Randomness s avalable on the web at the followng URL: Info on the TESTU01 test sute s avalable on the web at the followng URL: Info on Deharder, another random # generator test sute s avalable at the followng URL: Hgh-qualty random # generators are essental n today s world of ever-ncreasng compute power for use n cphers & encrypton.e. secure communcatons and fnancal transactons e.g. on the nternet and elsewhere P598AEM Lecture Notes 7 5

6 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Quas-Monte Carlo: If we n use sequences of quas/pseudo-random numbers for Monte Carlo, then we are factually dong Quas-MC, but n practce we stll call t Monte Carlo. A theoretcal bass ests that justfes the use of Quas-MC, based on the work of Weyl. The bottom lne s that t works and typcally shares the propertes of a true MC, such as 1 N convergence. Non-Unform Random Numbers: We frequently need to generate random # s wth P.D.F. s other than U ( 0,1).We have already U a b, as = a+ ( b a) r, however often we need non-unform random seen how to generate (, ) # s. A long tme ago (n early lectures), we saw a trck for generatng N ( 0,1) from r whch are 1 U ( 0,1) : r 6 s appromately Gaussan, wth epectaton value zero and varance one. = 1 However, t has no tals beyond ± 6σ (.e. here, ± 6 ). Gaussan Random Number Generaton: If we defne (the Bo-Muller transformaton): ( π ) ( π ) z = lnu cos u 1 1 z = lnu sn u 1 where ( u1, u) are ndependent and drawn from ( 0,1) z each s dstrbuted as N ( 0,1),.e. (, ) = Random Trangular P.D.F. U. Then z 1 and z are ndependent and z g z1 z e e π π We also saw earler that f u and v are ndependent and drawn from U ( 0,1) then u + v has a trangular P.D.F. Random Square-Root P.D.F. We can also show that f we generate ( uv, ) pars but keep only the larger one (call t ), then s dstrbuted as. A General Method for Non-Unform Random Numbers: Suppose we are gven some P.D.F. g( y ) and we wsh to generate a sequence of random # s dstrbuted accordng to g( y ). Suppose we have avalable a sequence of random # s, dstrbuted as U ( 0,1). Then: g ( y) dy = f ( ) d = 1d P598AEM Lecture Notes 7 6

7 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede It turns out to be useful here to use the Cumulatve Dstrbuton Functon G( y) whch s related to the P.D.F. by: g( y) = dg( y) dy Recall that G( y ) s the probablty that the random varable wll be have a value y, and that y G( y). In general, we can fnd G( y ) from g( y ) by ntegraton: ( ) ( ) 0 1 G y = g y dy n.b. here stands for the smallest value of y allowed by the P.D.F., whch mght n fact be. Let us rewrte the ntegral n terms of. The smallest allowed value of s 0, and let correspond to the chosen y: y ( ) = ( ) = ( ) G y g y dy f d But s stll some random number dstrbuted accordng to U ( 0,1), call t r. Thus: y ( ) ( ) G y = g y dy = r 1 If we can nvert ths,.e. solve for y as a functon of r, then we re done, that s: y = G ( r) wll have the desred P.D.F. y ( ) ( ) G y g y dy = = 0 ( ) 1 d = g y dy y Here, les between 0 and 1 n an nterval d. (Both d s are the same.) We project onto the functon G(y). g( y) The correspondng y s more lkely to be n a narrow nterval dy n a regon where G(y) s rapdly varyng (.e. near the mamum of g(y)) rather than where g(y) s small, where G(y) s slowly varyng. y( ) y P598AEM Lecture Notes 7 7

8 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Random Eponental Dstrbuton: Let us dstrbute random numbers eponentally: f ( ) α ce 0 = 0 otherwse α α Here: F ( ) = f ( ) d = ce d = ( 1 e ) In order to normalze ths functon, we requre F ( ) = 1, or c 0 αe α 0 0 otherwse c α f ( ) = and ( ) 1 = α. F = e α In order to generate random numbers dstrbuted n ths way, just pck an then set r = F( ) and solve for : r = 1 e α or: 1 r e α Note that snce r s drawn from ( 0,1) we can nstead use: ( ln r) α = or: = ( ln ( 1 ) ) r α r from ( 0,1) U, U, then so s 1 r, and hence wth no loss of generalty = (and save a bt of computer tme). t Eponental dstrbutons are often encountered as lfetmes of unstable states (, so that α = 1 τ ) or n the absorpton of radaton by matter ( e λ where λ = mean free path ), etc. Unform cosθ Dstrbuton: Another eample: f ( θ ) csn θ for 0 θ π = 0 elsewhere e τ f ( θ ) θ θ Now F( θ ) = f ( θ ) dθ = csnθ dθ = c( 1 cosθ) 0 Demandng F ( π ) = 1 gves c = 1. f 1 ( θ ) = θ and: F ( θ ) = 1 ( θ ) So f sn 1 r s drawn from U ( 0,1), then we set r ( θ ) θ = 1 cos. 1 cos P598AEM Lecture Notes 7 8

9 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Ths gves: cos 1 r θ 1 = cos s where 1 θ = or: θ cos ( 1 r) s s drawn from U ( 1,1) =. Agan, wth no loss of generalty we can set. Ths also follows drectly from the followng: We can approach ths same problem slghtly dfferently: ( ) 1 1 f θ dθ = snθdθ = dcosθ Ths means that a snθ dstrbuton n the varable θ corresponds to a Unform U 1,1 we wll get the same angle dstrbuton as throwng θ accordng to snθ. dstrbuton n the varable cosθ over the same regon. So f we draw cosθ from ( ) Random Isotropc Drecton n 3-D: We can apply the above dea to the problem of generatng a random unt vector,.e. a vector whose magntude s 1 but whch can pont n any drecton on the unt sphere. dω da Then the number of resultng vectors pontng nto any da should be proportonal to da. But da = r dω= 1d Ω on the unt sphere, and dω= snθdθdϕ = dcosθ dϕ. However, θ and ϕ are ndependent random varables and hence we can throw them ndependently by choosng random cosθ drawn from U ( 1,1) and ϕ drawn from ( 0, ) Then ths wll gve the desred dstrbuton, unform n dω..e. eplctly, choose: cosθ = 1 r ϕ = πs where r and Then the unt vector vˆ ( snθ cos ϕ,snθ sn ϕ,cosθ ) s are drawn from ( 0,1) U. U π. = wll be dstrbuted randomly over the unt sphere. Ths method won t always work for a general g( y) because: y It may not be possble to ntegrate and get ( ) ( ) (e.g. n terms of elementary functons) G y = g y dy n a smple form 1 It may not be possble to nvert and solve r = F( ) for = F ( r) P598AEM Lecture Notes 7 9

10 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Random Unform Dstrbuton on a -D Dsk: As an another eample, let us randomly, unformly populate the nteror of a semcrcle. f ( ) (0,1 + y = 1 ( 1,0 (1,0 Specfcally, we want the number of s generated n the nterval + d to be proportonal to 1. Thus, the P.D.F. we want to generate accordng to ths wll be: f ( ) = 0 otherwse The correspondng Cumulatve Dstrbuton Functon s: If we set F( ) 1 ( ) = ( ) = 1 = 1 + sn F f d c d c 1 1 = r, a random #, we wll unable to nvert ths and solve for as a functon of r. In such cases we can use the Von Neumann MC Acceptance-Rejecton Method, a knd of ht-or-mss MC technque. Here, for eample, we: (1) Pck a random # U 1,1 and a random # f = 1. drawn from ( ) () We then calculate ( ) (3) If y f ( ) 1 y drawn from ( 0,1) U. < = we keep ; otherwse we reject and go to step (1) and repeat f ( ) f ( ) f ma = 1 = 1 = + 1 P598AEM Lecture Notes 7 10

11 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Step (1) unformly populates the outer dotted rectangle. By rejectng s for whch the correspondng y s outsde the red sem-crcle, we end up wth hts,.e. s such that the number of kept s between + d s proportonal to f ( ) d, beng large where f ( ) large and small where f ( ) s small,.e. we randomly populate the nteror of the sem-crcle by rejectng ponts that fall outsde of the sem-crcle. We now generalze the MC ht-or-mss (aka the acceptance-rejecton ) method: Suppose we have a general/arbtrary functon f ( ) n the range a b, and wsh to generate random # s as f the functon were a P.D.F. between a b, as shown n the fgure below: f ( ) fma ( ) s Fnd/determne the value ma f such that f ( ) fma (1) Pck a random # drawn from (, ) () We then evaluate f ( ). (3) If y f ( ) < we keep b Note that ths technque also gves ( ) successes, snce: a n the range a b. Then: U a b and a random # y drawn from ( 0,1) U. ; otherwse we reject and go to step (1) and repeat f d f we keep track of the # of tres and the # of a # of successes area under f = = # of tres area under rectangle b ( ) f ( ) d a ( b a) f ma It s also straghtforward to generalze ths method to the case of a mult-dmensonal functon. The MC acceptance-rejecton method wll be most effcent (.e. fewest # of tres) when f ma s as small as possble, however ths smallest value may not be smple to fnd n the more general case where the functon f s a very complcated mult-dmensonal functon. b P598AEM Lecture Notes 7 11

12 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Dependng on the detals of the shape of f ( ), one can mprove the effcency of the above acceptance-rejecton algorthm by usng an (elementary) mportance-samplng technque. In the fgure shown below, the f ( ) s such that we can use two rectangular boes to mprove the effcency/speed of ths algorthm: fma 1 ( ) f ( ) fma ( ) a Random Numbers from Hstogrammed Dstrbutons: b We can also generate random # s that follow dstrbutons that cannot be descrbed analytcally by smple contnuous functons. For eample, suppose we wsh to generate random s accordng to a hstogram: Σ bns The gven hstogram s on the left, ts assocated Cumulatve hstogram on the rght. P598AEM Lecture Notes 7 1

13 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Snce the sum of the contents of all bns s 7, we throw a random number r accordng to U ( 0, 7) and use a table: If: Keep: 0 r 5 = 0 5< r 14 = 1 14 < r 17 = 17 < r 1 = 3 1 < r 7 = 4 We can use an analogous procedure f we are gven a set of dscrete probabltes. For eample, suppose we know that: P( ) = P1 P( ) = P where: P1+ P + P3 = 1 P = P ( ) Then we obtan random # p drawn from U ( 0,1). If 0 < p P1 we select the -nterval If P1 < p P1+ P we select the -nterval Otherwse we select the -nterval Then we must decde separately (we are not gven any nformaton here) how to pck a partcular value of n one of these ntervals. We could e.g. pck = 0.15 each tme we landed U 0.0,0.15, or... n the frst regon, or we could pck from ( ) The same procedure can be used when smulatng a physcal process. At some nstant of tme or at a pont n 3-D space, a choce must be made between dfferent outcomes wth dfferent probabltes. For eample, a neutron, havng arrved at some pont n the matter t s traversng, can at that pont undergo an elastc scatterng, be absorbed by a nucleus, etc., each wth dfferent probabltes (whch depend on the materal and the energy of the neutron). MC technques are commonly used to do ntegrals, smulate physcal processes to answer physcs questons, etc. In hgh energy physcs they are frequently used to smulate data whch resembles as closely as possble the epected sgnals from real detectors. These are then used as nput to the programs that reduce the sgnals (e.g. obtan dgtzed tmes, pulse heghts, etc.) to meanngful quanttes, and (perhaps) ft them to theoretcal models whch etract basc parameters. Snce the MC nput s completely understood, the data analyss programs can be checked to see f they work as they are ntended to. A less common MC applcaton s to the Propagaton of Large Errors. Here large means that we cannot use standard appromatons for combnng errors and we must calculate, eactly, the P.D.F. of the result of some combnaton of measurements, whch may be very dffcult to do analytcally. P598AEM Lecture Notes 7 13

14 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede An nterestng eample s provded by a trtum β -decay eperment (no longer taken serously!) whch was the frst to clam evdence n support of neutrno oscllatons. Theory had shown that a quantty R could be measured and that the (electron) neutrno was not stable (.e. t oscllated) f R 0.4. For ths eperment t turned out that: R = a d kd ( b c) a 1 ke e The numercal values of the constants (and ther uncertantes), as measured by ths eperment were: a = 3.84 ± 1.33 b = 74.0 ± 4.0 c = 9.5 ± 3.0 d = 0.11 ± e = 0.30 ± 0.00 k = 0.89 If one assumes that the lnear appromaton n the Taylor seres epanson for error R = 0.191± , whch s a propagaton s vald, then one obtans: ( ) ( ) σ effect. If we assume that all uncertantes are Gaussan/normally-dstrbuted, then the probablty that a measurement wll be more than 3.1 standard devatons from ts epectaton value (.e. a value of R > 0.4) s 0.001, or 0.1%. The authors thus clamed that ther epermental result was strong evdence for neutrno oscllatons P598AEM Lecture Notes 7 14

15 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede However, note that the parameters a and c have very large uncertantes (on the order of 30%) and so e.g. σ a a 1 n the Taylor seres epanson for error propagaton s not satsfed. Recall from P598AEM Lect. Notes 5 that the famlar formula for the standard devatons of products y and quotents /y requres ths to be so! In such a case, a more correct procedure would be (assumng that the P.D.F.s of the measurements are ndeed purely Gaussan) to randomly generate a, b, c, d, e ( = 1, N), each accordng to ts own P.D.F. (usng the central value and standard devaton measured n the eperment) and then calculate the correspondng value of R for each set of a, b, c, d, e. If we actually count the # of entres beyond R = 0.4 n the above hstogram, we dscover that about 4% of the R values eceed 0.4. Thus, there s n fact a 4% (not 0.1%!!!) chance that a stable neutrno wll yeld the results of the eperment. Ths s why the frst clam of evdence for neutrno oscllatons obtaned from measurements of endpont of the trtum β -decay spectrum was not taken too serously The use of MC Propagaton of Large Errors technques s potentally even more powerful than that descrbed above. In the above eample, suppose e.g. some (or even all) of the uncertantes on the a-e parameters n the R-epresson are not eactly Gaussan/normallydstrbuted but n fact have tals on them. By makng a sem-log plot of each quantty a-e, there should be clear evdence for a Gaussan, parabola-shaped core to dstrbuton for small σ -values, however for large σ - values, low-level tals of the dstrbuton may arse e.g. from one or more unaccounted-for physcs source(s), possbly havng a Gaussan-type parabola-shape to t. In fact, a double- Gaussan LSQ ft of the uncertanty dstrbuton for a common mean μ can easly be carred out of the form: (, ) + g (, ) α g μ σ β μ σ where we epect σ < σ 1 and thus the normalzaton constants α > β. If double-gaussan LSQ ft results accurately descrbed each of the above eample s a-e varables, then the LSQ ft parameters could be used drectly n an enhanced verson of the above MC smulaton of R to obtan an mproved determnaton of the overall uncertanty on R and more mportantly, a P.D.F. for R from whch an mproved estmate of the probablty for eceedng R > 0.4 could thus be obtaned. Furthermore, wth the use of MC Propagaton of Large Errors technques, asymmetrcal errors and/or non-gaussan uncertantes assocated wth the a-e varables can also easly be ncorporated n the MC smulaton even usng e.g. actual hstograms of the epermentally measured uncertantes on the a-e parameters as MC P.D.F. s of the same. Addtonally, correlatons between any/all of the a-e varables can also be accurately modeled va MC smulaton as well. Thus, t can be seen that the use of MC Propagaton of Large Errors technques n prncple s accurate to all orders n the (nfnte) Taylor seres epanson that we use - only n the lnear regme (e.g. σ a 1) for error propagaton. a Potentally, MC Propagaton of Large Errors technques are very powerful, ndeed!!! P598AEM Lecture Notes 7 15

16 Fall 01 Analyss of Epermental Measurements B. Esensten/rev. S. Errede Lots of useful MC technques such as the ones dscussed above and more (e.g. trcks of the trade!) are contaned n A Thrd Monte Carlo Sampler, LASL Informal Report LA-971-MS, Los Alamos Natonal Laboratory, by C.J. Everett and E.D. Cashwell (1983). Ths document e.g. descrbes technques for generatng pars or trplets, etc. of correlated Gaussans, and more! See also e.g. the Monte Carlo Technques secton of the Revew of Partcle Physcs, avalable on the web at the followng URL: The followng books also have much useful/helpful nformaton on varous MC technques: W. H. Press, et al., Numercal Recpes, 3 rd Ed., Cambrdge Unversty Press, New York, 007. D.E. Knuth, The Art of Computer Programmng, Vol. II, 3 rd Ed., Addson-Wesley, Readng, MA P598AEM Lecture Notes 7 16