Một số kiến thức cơ sở của phương pháp Monte Carlo

Cc kin thc c s ca phng php Monte Carlo

ng Nguyn Phng

[email protected]

Ngy 20 thng 5 nm 2014

Mc lc

1 M u 2

2 Lch s hnh thnh phng php Monte Carlo 3

3 Cc phng php Monte Carlo 5

4 C s ca phng php Monte Carlo 6

5 S ngu nhin 75.1 Cc loi s ngu nhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75.2 Nhng iu cn lu khi m phng s ngu nhin . . . . . . . . . . . . . . . . . 75.3 Phng php to s ngu nhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

6 Phn b xc sut 86.1 Bin ngu nhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86.2 Hm mt xc sut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96.3 Moment thng k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106.4 Lut s ln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116.5 nh l gii hn trung tm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

7 Ly mu phn b 137.1 Phn b mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137.2 Cc phng php ly mu vi phn b xc sut khng ng nht . . . . . . . . . 13

8 c lng Monte Carlo 178.1 c lng mu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178.2 chnh xc ca c lng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.3 Khong tin cy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Ti liu tham kho 19

1

ng Nguyn Phng Ti liu ni b NMTP

1 M u

Gia th k 20, s pht trin ca cc lnh vc quan trng nh vt l ht nhn, nguyn t, ccnghin cu v v tr, nng lng, ch to cc thit b phc tp i hi phi tin hnh cc biton ln phc tp, khng th gii c bng cc k thut c vo thi by gi. Cng vi s phttrin my tnh in t lm xut hin kh nng nhn c y cc m t nh lng cacc hin tng c nghin cu, v phm vi gii cc bi ton c m rng. Nhng yu t trn gp phn hnh thnh nn vic thc nghim my tnh (computing experiment).

Thc nghim my tnh thc cht l p dng my tnh gii cc bi ton, nghin cu cc ktcu hay cc qu trnh, thc hin tnh ton da trn m hnh ton hc v vt l bng cc tnhton nh lng ca i tng c nghin cu khi tng i cc tham s. N tri di trn rtnhiu lnh vc t vt l (computational physics), ha hc (computational chemistry) n sinhhc (computational biology),...

Mt trong nhng phng php thc nghim my tnh ph bin nht trn th gii hin nay lphng php Monte Carlo (Monte Carlo experiment hay Monte Carlo method)1. y l mt lpcc thut ton (computationl algorithm) s dng vic ly mu ngu nhin (random sampling) thu c cc kt qu s (numerical result). Phng php ny thng c s dng giiquyt cc bi ton c cu hnh phc tp, lin quan n nhiu bin s m khng th d dng giiquyt c bng cc thut ton tt nh (deterministic algorithm). C th ni hin nay mtphn ln cc sn phm ca c khoa hc c bn ln ng dng u da vo b ba thc nghim,l thuyt v Monte Carlo.

Phng php Monte Carlo c gng m hnh ho cc hin tng t nhin thng qua s m phngtrc tip cc l thuyt cn thit da theo yu cu ca h, chng hn nh m phng s tngtc ca nhng vt th ny vi nhng vt th khc hay l vi mi trng da trn cc mi quanh vt th vt th v vt th mi trng n gin. Li gii c xc nh bng cch lymu ngu nhin ca cc quan h, hay l cc tng tc vi m, cho n khi kt qu hi t. Dovy, cch thc hin li gii bao gm cc hnh ng hay php tnh c lp i lp li, c thc thc hin trn my tnh.

Cc l thuyt vi m cung cp ci nhn bn trong v cho php chng ta c th suy lun mt hv m s hot ng nh th no, phng php Monte Carlo khng th cnh tranh c trongvic ny. Trong vic khm ph cc tnh cht ca cc h v m, phng php Monte Carlo rtging vi mt ngi lm th nghim. Nu khng c s hng dn ca l thuyt, cc qu trnhkhm ph s tr nn rt kh khn v c th b sai lch. Tuy nhin khi bi ton tr nn qu phctp, k thut Monte Carlo tr nn thun li trong trng hp bi ton c phc tp tng cao.iu ny c th c m t nh trong Hnh 1. Chng ta c th d dng thy c khi phctp ca bi ton (chng hn nh phc tp ca hnh hc) tng, thi gian tnh ton cacc phng php tt nh s tng nhiu hn l phng php Monte Carlo.

Do nhng u im ca mnh, phng php Monte Carlo c ng dng trong rt nhiu lnhvc khc nhau

Trong khoa hc x hi: phn lung giao thng, nghin cu s pht trin dn s, nghincu th trng chng khon,...

Trong khoa hc t nhin: nghin cu s vn chuyn bc x, thit k l phn ng ht nhn,thit k v kh ht nhn, tnh liu bc x, sc ng hc lng t, nghin cu s chuynpha, tnh cc tch phn s (numerical integration),...

1Cn phn bit phng php Monte Carlo vi thut ton Monte Carlo (Monte Carlo algorithm) vn l mtthut ton lm vic vi cc bit ngu nhin

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Hnh 1: So snh thi gian gii quyt bi ton ca 2 phng php Monte Carlo v tt nh

2 Lch s hnh thnh phng php Monte Carlo

Tn gi ca phng php ny c t theo tn ca mt thnh ph Monaco, ni ni ting vicc sng bc, c l l do phng php ny da vo vic gieo cc s ngu nhin. Tuy nhin vicgieo s ngu nhin gii cc bi ton xut hin t rt lu ri.

Vo khong th k 18, ngi ta thc hin cc th nghim m trong h nm mt cy kimtrong mt mt cch ngu nhin ln trn mt mt phng c k cc ng thng song song v suy ra gi tr ca pi t vic m s im giao nhau gia cc cy kim v cc ng thng2.

2c bit n vi tn gi bi ton cy kim Buffon (Buffons needle problem), trong bi ton ny ngi ta thngu nhin cc cy kim c chiu di l ln trn mt mt sn c k cc ng thng song song cch nhau mt ont (vi l t) v tnh xem xc sut ca cy kim ct ngang ng thng l bao nhiu.

Gi x l khong cch t tm cy kim n ng thng gn nht v l gc to bi cy kim v ng thng,ta c hm mt xc sut (probability density function) ca x v nh sau

0 x t2

:2

tdx

0 theta pi2

:2

pid

Hm mt xc sut kp hp (joint probability density function)

4

tpidxd

iu kin cy kim ct ngang ng thng x l2

sin , xc sut cy kim ct ngang ng thng s thu

c bng cch ly tch phn hm mt xc sut kt hp pi/20

(l/2) sin 0

4

tpidxd =

2l

tpi

Ga s ta gieo N kim, trong c n kim ct cc ng thng

n

N=

2l

tpi

pi =2lN

tn

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Hnh 2: Minh ha bi ton tnh s pi vi cc cy kim v ng thng song song

Trong khong nhng nm 1930, Enrico Fermi s dng phng php Monte Carlo giiquyt cc bi ton khuch tn neutron nhng khng xut bn bt c cng trnh no v vn ny.

Phng php Monte Carlo ch c thc s s dng nh mt cng c nghin cu khi vic chto bom nguyn t c nghin cu trong sut thi k chin tranh th gii ln th hai. Cngvic ny i hi phi c s m phng trc tip cc vn mang tnh xc sut lin quan n skhuch tn neutron ngu nhin trong vt liu phn hch. Nm 1946, cc nh vt l ti Phng thnghim Los Alamos, dn u bi Nicholas Metropolis, John von Neumann v Stanislaw Ulam, xut vic ng dng cc phng php s ngu nhin trong tnh ton vn chuyn neutrontrong cc vt liu phn hch. Do tnh cht b mt ca cng vic, d n ny c t mtdanh Monte Carlo v y cng chnh l tn gi ca phng php ny v sau. Cc tnh tonMonte Carlo c vit bi John von Neumann v chy trn my tnh in t a mc ch utin trn th gii ENIAC (Electronic Numerical Integrator And Computer) (Hnh 3).

Hnh 3: My tnh in t ENIAC c t ti BRL building 328

Cc tng ca phng php ny c pht trin v h thng ha nh vo cc cng trnh caHarris v Herman Kahn vo nm 1948. Cng vo khong nm 1948, Fermi, Metropolis, v Ulamthu c c lng ca phng php Monte Carlo cho tr ring ca phng trnh Schrodinger.

Mi cho n nhng nm 1970, cc l thuyt mi pht trin v phc tp ca tnh ton btu cung cp cc tnh ton c chnh xc cao hn, nhng c s l lun thuyt phc cho vics dng v pht trin phng php Monte Carlo cho n tn ngy hm nay.

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3 Cc phng php Monte Carlo

Sau hn na th k pht trin t ph, phng php Monte Carlo gn nh c ng dngrng khp trn mi lnh vc ca khoa hc, cng ngh. Cng vi , rt nhiu bin th caphng php ny c xy dng nhm phc v cho cc nhu cu tnh ton c th. Bn thntc gi khng th thng k ni c bao nhiu phng php Monte Carlo ang c s dng hinnay, ch xin nu ra mt s phng php tiu biu

Assorted random model hay self-organized criticality (SOC): l thut ng c dng trongvt l m t mt h ng hc m c mt im ti hn nh l mt im thu ht. Do vy cchot ng v m ca chng c thc hin trn thang khng-thi gian c trng bt bin caim ti hn ca mt s chuyn pha (phase transition), nhng trong SOC cc im ny cdn ra m khng cn cc thng s a vo t dc gi tr chnh xc. N c ng dngnhiu trong cc lnh vc khc nhau chng hn nh a vt l, v tr hc, sinh hc, sinh thihc, kinh t, x hi hc,...

Phng php m phng Monte Carlo trc tip (Direct Simulation Monte Carlo DSMC): c a ra bi GS. Prof. Graeme Bird, y l phng php s dng k thut mphng xc sut gii cc phng trnh Boltzman m t cc dng kh long m trong qungng t do trung bnh ca phn t c cng bc (hoc ln hn) thang chiu di vt l c trngca h.

Phng php Monte Carlo ng lc (Dynamic Monte Carlo DMC): l phng phpm phng cc trng thi ca phn t bng cch so snh t l ca cc bc ring l vi cc sngu nhin. Phng php DMC thng dng kho st cc h khng cn bng chng hn nhcc phn ng, khuych tn,... Phng php ny c ng dng ch yu phn tch cc hotng ca cc cht b ht bm trn cc b mt. Phng php DMC rt ging vi phng phpKinetic Monte Carlo m ta s trnh by phn tip theo.

C rt nhiu phng php thng dng c s dng m phng DMC, gm c First ReactionMethod (FRM) v Random Seelection Method (RSM). D cho FRM v RSM u cho ra cc ktqu ging nhau vi cng mt m hnh, nhng cc ti nguyn my tnh li khc nhau ph thucvo h ng dng.

Phng php Monte Carlo ng hc (Kinetic Monte Carlo KMC): l mt phngphp Monte Carlo da trn s m phng my tnh m phng s tin trin theo thi gian camt vi qu trnh xy ra trong t nhin, in hnh l cc qu trnh m chng xut hin vi mtt l c cho trc. Vic hiu r cc t l ny l rt quan trng bi v chng l d liu u vocho thut ton KMC, t bn thn phng php khng th d on chng.

Phng php KMC cng rt ging vi phng php DMC, s khc bit chnh gia chng dngnh nm ch thut ng v lnh vc s dng: KMC c s dng ch yu trong vt l cnDMC th c s dng ch yu trong ho hc.

Phng php Monte Carlo lng t (Quantum Monte Carlo QMC): l phng phpm phng cc h lng t vi mc ch gii quyt cc bi ton nhiu vt th (many-body). QMCdng phng php Monte Carlo bng cch ny hay cch khc tnh ton cc tch phn nhiuchiu. QMC cho php m t mt cch trc tip cc hiu ng nhiu vt th trong hm sng, vi bt nh c th c gim vi thi gian m phng ko di.

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Phng php Quasi-Monte Carlo (Quasi-Monte Carlo): l mt phng php tnh tonmt tch phn (hay i khi l mt bi ton) m da trn c s l cc dy s c s nht qunthp. N tri ngc vi phng php Monte Carlo thng thng, c da trn cc dy s gingu nhin.

4 C s ca phng php Monte Carlo

Phng php Monte Carlo c xy dng da trn nn tng

Cc s ngu nhin (random numbers): y l nn tng quan trng, gp phn hnh thnhnn thng hiu ca phng php. Cc s ngu nhin khng ch c s dng trong vicm phng li cc hin tng ngu nhin xy ra trong thc t m cn c s dng lymu ngu nhin ca mt phn b no , chng hn nh trong tnh ton cc tch phns (numerical integration).

Lut s ln (law of large numbers): lut ny m bo rng khi ta chn ngu nhin ccgi tr (mu th) trong mt dy cc gi tr (qun th), kch thc dy mu th cng lnth cc c trng thng k (trung bnh, phng sai,...) ca mu th cng gn vi ccc trng thng k ca qun th. Lut s ln rt quan trng i vi phng php MonteCarlo v n m bo cho s n nh ca cc gi tr trung bnh ca cc bin ngu nhinkhi s php th ln.

nh l gii hn trung tm (central limit theorem): nh l ny pht biu rng di mts iu kin c th, trung bnh s hc ca mt lng ln cc php lp ca cc binngu nhin c lp (independent random variables) s c xp x theo phn b chun(normal distrbution). Do phng php Monte Carlo l mt chui cc php th c lpli nn nh l gii hn trung tm s gip chng ta d dng xp x c trung bnh vphng sai ca cc kt qu thu c t phng php.

Cc thnh phn chnh ca phng php m phng Monte Carlo (Hnh 4) gm c

Hm mt xc sut (probability density function PDF): mt h vt l (hay ton hc)phi c m t bng mt b cc PDF.

Ngun pht s ngu nhin (random number generator RNG): mt ngun pht cc sngu nhin ng nht phn b trong khong n v.

Quy lut ly mu (sampling rule): m t vic ly mu t mt hm phn b c th.

Ghi nhn (scoring hay tallying): d liu u ra phi c tch lu trong cc khong gitr ca i lng cn quan tm.

c lng sai s (error estimation): c lng sai s thng k (phng sai) theo s phpth v theo i lng quan tm.

Cc k thut gim phng sai (variance reduction technique): cc phng php nhm gimphng sai ca p s c c lng gim thi gian tnh ton ca m phng MonteCarlo.

Song song ho (parallelization) v vector ho (vectorization): cc thut ton cho phpphng php Monte Carlo c thc thi mt cch hiu qu trn mt cu trc my tnhhiu nng cao (high-performance).

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Hnh 4: Nguyn tc hot ng ca phng php Monte Carlo

5 S ngu nhin

Trong phng php m phng Monte Carlo, chng ta khng th no thiu c cc s ngunhin. Cc s ngu nhin c mt trong cc hin tng t nhin nh nhiu lon in t, phnr phng x,... gii mt bi ton bng phng php Monte Carlo iu quan trng nht lchng ta cn to ra cc s ngu nhin phn b u (uniform distribution) trn khong (0,1).

5.1 Cc loi s ngu nhin

C 3 loi s ngu nhin chnh

S ngu nhin thc (real random number): cc hin tng ngu nhin trong t nhin.

S gi ngu nhin (pseudo-random number): cc dy s xc nh m n vt qua ccc kim tra v tnh ngu nhin.

S gn ngu nhin (quasi-random number): cc im c s phn b tt (c s khng nhtqun thp).

5.2 Nhng iu cn lu khi m phng s ngu nhin

C hai iu chng ta cn lu khi m phng cc s ngu nhin

My tnh khng th to ra cc dy s ngu nhin tht s m ch l cc s gi ngu nhin.

Bn thn cc s khng phi l ngu nhin m ch c dy s mi c th c xem l ngunhin

Mt dy s ngu nhin tt phi hi t y cc yu t sau y

Chu k lp li phi di tc l vic gieo s ngu nhin phi to ra c nhiu s trc khilp li dy s c ca n cho khng c phn no ca dy b trng trong tnh ton.

Cc s c to ra phi hng ti phn b u, tc l mt dy s bt k gm vi trm sphi hng ti phn b ng nht trong ton vng kho st.

Cc s khng tng quan vi nhau, tc l cc s trong dy phi c lp v mt thng kvi cc s trc n.

Thut ton phi truy xut nhanh, tc l thi gian my tnh to ra s ngu nhin phi nh

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Cc s gi ngu nhin trong phng php Monte Carlo ch cn t ra mc ngu nhin, nghal tun theo phn b u hay theo phn b nh trc, khi s lng ca chng ln.

5.3 Phng php to s ngu nhin

to c mt dy s ngu nhin, chng ta c th dng nhiu phng php khc nhau. y ti xin trnh by mt phng php c dng ph bin nht. Phng php ny cs dng trong nhiu ngn ng lp trnh, chng hn nh C, Fortran,... chnh l phng phpng d tuyn tnh (linear congruential generator). Thut ton ca phng php ny nh sau

x0 = s gieo ban u, l s nguyn l < M (1)xn = axn1 + cmodM (2)n = xn/M (3)

y a v c l cc s nguyn v M thng l mt s nguyn c gi tr ln, s gieo ban u x0c th c t bi ngi dng trong qu trnh tnh ton.

Thc s y khng phi l mt thut ton to s ngu nhin tt nht nhng u im ca thutton ny l n gin, d s dng, tnh ton nhanh v dy s ngu nhin do n to ra l kh tt.

Ta c th thy rng trong dy s c to ra bi phng php ny mi s ch c th xut hinduy nht mt ln trc khi dy b lp li. Do chu k ca phng php ng d tuyn tnh(chiu di ca dy s cho n khi s u tin b lp li) M . c ngha l trong trng hptt nht th xn s ly tt c cc gi tr c trong on [0,M 1]. i vi phng php ng dtuyn tnh th chu k cc i s ph thuc vo di k t ca my tnh. V d: chu k ln nhti vi my 16 bit c chnh xc n (single precision) l 216 = 65536 i v vi chnh xckp (double precision) l 232 = 4.29 109.Ngoi ra cn mt s thut ton khc to dy s ngu nhin

Shift register : yn = yns + ynr mod 2 vi r > s

Additive lagged Fibonacci : zn = zns + znr mod 2k vi r > s Phng php kt hp (combined): wn = yn + zn mod p

Multiplicative lagged Fibonacci : xn = xns xnr mod 2k vi r > s Phng php ng d nghch o ngm (implicit inversive congruential): xn = axn1 +cmodM

Phng php ng d nghch o tng minh (explicit inversive congruential): xn =an+ cmodM

6 Phn b xc sut

6.1 Bin ngu nhin

Cc bin ngu nhin (random variable hay stochastic variable) l cc bin m gi tr m n nhnc mt cch ngu nhin. Mt bin ngu nhin c th bao gm mt tp hp cc gi tr mmi gi tr i km vi mt xc sut (probability) trong trng hp gi tr ri rc hoc mt hmmt xc sut (probability density function) trong trng hp gi tr lin tc (xem Hnh 5).

Gi s ta tin hnh php o mt bin ngu nhin x (trong thc nghim) hay gieo ngu nhingi tr ca bin ny (trong phng php Monte Carlo) N ln, ta s thu c mt tp hp ccgi tr ca bin nh sau {x1, x2, . . . , xn}.

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Hnh 5: Minh ha phn b xc sut ca bin ri rc v lin tc

Gi tr k vng (expected value hay expectation) hay cn gi l gi tr trung bnh (mean) cabin x (thng c k hiu l ) chnh l gi tr m ta k vng s thu c khi lp li N lnphp o vi N tin n v cc. Hay ni mt cch khc, gi tr k vng chnh l trung bnhc trng s (weight average) ca tt c cc gi tr kh d (possible values) ca bin x, trng sc dng y chnh l xc sut fi tng ng vi cc gi tr ca bin.

E(x) =

Ni=1

xifi

Ni=1

fi

(4)

Phng sai (variance) c dng nh gi mc phn tn ca tp hp gi tr thu c,gi tr ca phng sai bng 0 c ngha l tt c cc gi tr ca tp hp l ng nht. Phngsai thng c k hiu l 2.

V ar(x) = E[(x E(x))2

]=

Ni=1

(xi )2fiNi=1

fi

= E(x2) [E(x)]2 (5)

lch chun (standard deviation) k hiu l cn bc hai ca phng sai, c cng th nguynvi gi tr ca bin x nn thng c dng km vi gi tr trung bnh biu din kt qu thuc.

6.2 Hm mt xc sut

Hm mt xc sut (Probability Density Function PDF)3 ca mt bin ngu nhin lin tcl mt hm m t kh nng (xc sut) nhn mt gi tr ca bin .

Hm mt xc sut c xem nh l chun ha khi

+

f(x)dx = 1 (6)

Hm mt tch ly (cumulative density function hay cumulative distribution function CDF)3i khi cn c gi l hm phn b xc sut (probability distribution function) hay hm xc sut (probability

function), tuy nhin khng c quy nh no thng nht cho cc tn gi. Hm xc sut i khi cn c dng ch hm mt tch ly (cumulative distribution function)

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c tnh nh l tch phn ca hm mt xc sut (Hnh 6)

F (x) =

x

f(t)dt (7)

Hnh 6: So snh hai hm PDF v CDF, gi tr ca hm CDF ti v tr x chnh l tch phn cahm PDF t n x

Trong trng hp ta c hm g(x) vi x l bin ngu nhin vi mt f(x), gi tr trung bnhca hm g(x) s c tnh theo cng thc

E[g(x)] =

+

g(x)f(x)dx E[g(x)] =

Nk=1

gkfk (8)

6.3 Moment thng k

Cc moment thng k (statistical moment) l cc i lng thng c s dng nh gidng (shape) ca mt phn b chng hn nh v tr (location), mc phn tn (dipersion),...Moment thng k bc n ca mt phn b c nh ngha theo cng thc

+

(x )nf(x)dxNk=1

(x )nfk (9)

Moment bc 1 dng nh gi lch ca phn b ra khi k vng ca phn b (). Trongtrng hp phn b l tng, gi tr moment ny c gi tr bng 0. Trong thc t, ngi ta cng

10


xem gi tr trung bnh chnh l moment bc 1 ca phn b 4.

+

(x )f(x)dxNk=1

(x )fk (10)

Moment bc 2 c s dng nh gi phn tn ca phn b. Trong trng hp phnb ch c 1 gi tr duy nht, moment ny c gi tr bng 0. Trong thc t, ngi ta xem phngsai (2) l moment bc 2 ca phn b 5.

+

(x )2f(x)dxNk=1

(x )2fk (11)

Moment bc 3 c dng nh gi i xng (symmetry) ca phn b. Nu phn bl hon ton i xng, moment ny c gi tr bng 0. Nu phn b c ui di v theo chiudng ca trc ta , moment ny s c gi tr dng, trong trng hp ngc li s c gi trm (xem Hnh 7). Trong thc t, ngi ta thng hay s dng moment chun ha (normalisedmoment hay standardised moment)6 nh gi, moment chun ha bc 3 ca phn b cgi l skewness.

+

(x )3f(x)dx3

Nk=1

(x )3fk3

(12)

Moment bc 4 c dng nh gi phng (flatness) ca phn b. Tng t nhskewness, moment chun ha bc 4 (kurtosis) thng c s dng

+

(x )4f(x)dx4

Nk=1

(x )4fk4

(13)

Phn b c dng phng c gi l platykurtic, cn phn b c nh nhn c gi l leptokurtic,phn b chun (normal distribution) c kurtosis bng 3 v c xem l chun phn bit phng ca cc phn b (xem Hnh 7).

6.4 Lut s ln

Lut s ln (Law of Large Numbers LLN) m t kt qu thu c khi thc hin php o mts ln ln, theo gi tr trung bnh ca cc kt qu thu c s cng gn vi gi tr k vngkhi s php o cng ln (v d trong Hnh 8). Lut s ln c vai tr quan trng v n m bocho s n nh v mt lu di ca gi tr trung bnh ca cc s kin ngu nhin.

4c gi l moment th (raw moment) dng nh gi lch ca tr trung bnh ra khi gi tr 0. Khi

cng thc tnh moment bc 1 c vit li

+

xf(x)dx

5Cc moment loi ny c gi l moment trung tm (central moment) dng nh gi lch khi gi trtrung bnh, khc vi moment th dng nh gi lch khi gi tr 0.

6c tnh nh l t s gia moment trung tm v lch chun ca phn b.

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Hnh 7: Minh ha skewness v kurtosis

Hnh 8: Minh ha Lut s ln khi thc hin th nghim tung ng xu, th biu din t l phntrm xut hin mt nga (head) nhiu hn mt sp (tail) theo s ln tung

6.5 nh l gii hn trung tm

Theo nh l gii hn trung tm (Central Limit Theorem CLT), tng ca cc bin ngu nhinc lp (independent random variable) v phn phi ng nht (identically distribution) theocng mt phn phi xc sut, s hi t v mt bin ngu nhin no (Hnh 9).

Gi s ta c N tp hp cc bin ngu nhin c lp Xi (X1, X2, ..., XN ), mi tp hp u cphn b tng minh (arbitrary) vi xc sut P (x1, x2, ..., xN ) c tr trung bnh i, phng saihu hn 2i tng ng. Khi i lng c dng chun ha

Xnorm =

Ni=1

xi Ni=1

iNi=1

2i

(14)

s c mt hm phn b tch ly gii hn xp x theo phn b chun.

Mt cch n gin hn, nh l gii hn trung tm c th hiu nh l phn b ca vic ly mu

12


Hnh 9: Minh ha nh l gii hn trung tm, trung bnh ca cc phn b t X1 n X5 s cdng phn b chun

ngu nhin s tin v phn b chun khi kch c mu c tng ln, d cho phn b thc haycn gi l phn b qun th (population) ca bin khng phi l phn b chun.

Lu : in kin ca nh l gii hn trung tm l c tr trung bnh v phng sai ca phnb phi tn ti hu hn.

7 Ly mu phn b

7.1 Phn b mu

Phn b mu (sampling distribution) hay cn gi l phn b mu hu hn (finite-sample distri-bution) l phn b xc sut thng k ca cc gi tr trong mu ngu nhin c ly ra t mtphn b qun th. Phn b mu ph thuc vo cc yu t nh phn b ca bn thn qun th,cch thc ly mu, kch c mu,...

Gi s ta c mt qun th c phn b chun vi tr trung bnh v phng sai 2, c k hiul N(, 2). Sau chng ta ly cc mu c kch thc n cho trc t qun th ny v tnhton cc gi tr trung bnh xi cho mi mu c ly, cc gi tr ny c gi l cc gi tr trungbnh ca mu (sample mean) v phn b ca cc gi tr trung bnh ny c gi l phn bca cc gi tr trung bnh mu. Phn b ny s tun theo phn b chun N(, 2/n)7 do phnb qun th l phn b chun (mc d theo nh lut gii hn trung tm, nu kch thc mun ln, phn b trung bnh mu vn c th c xp x theo phn b chun d cho phnb qun th c l phn b chun hay khng). Trong trng hp kch thc mu nh, phn btrung bnh mu c cho trong Bng 1.

lch chun ca phn b trung bnh mu c gi l sai s chun (standard error)8, trongtrng hp cc mu c lp vi nhau ta c

x =n

(15)

vi l lch chun ca qun th v n l kch c mu.

7.2 Cc phng php ly mu vi phn b xc sut khng ng nht

Thng thng cc php gieo ngu nhin ly mu trong Monte Carlo u l cc php ly mung nht (uniform sampling), cc gi tr c ly mu vi xc sut nh nhau. Tuy nhin, trongrt nhiu trng hp tnh ton Monte Carlo, ta cn phi ly mu t mt bin c hm mt

7lu phn b chun ny khc vi phn b chun ca qun th8cn phn bit vi lch chun ca qun th vn c gi vi tn standard deviation

13


Bng 1: Mt s v d phn b mu ngu nhin c ly t qun th

Phn b qun th Phn b mu

Normal(, 2) X Normal(,2

n

)Bernoulli(p) nX Binomial(n, p)

Normal(1, 21) v Normal(2, 22) X1 X2 Normal

(1 2,

21n

+22n

)

xc sut khng ng nht (non-uniform) chng hn nh ly mu bin x trong khong [a, b] vihm mt f(x) c phn b nh trong Hnh 5. C rt nhiu phng php thc hin vicly mu ny, di y l mt s phng php thng dng nht

Phng php bin i (transformation method) hay cn gi l phng php bin i ngc(inverse transform method). Phng php ny thng c p dng trong nhng trng hphm phn b f(x) c dng n gin, ta c th thc hin mt php bin i x(t) v mt phnb t ng nht, ta c cng thc bo ton xc sut

P (x)dx = P (t)dt (16)

vi P (x) = f(x) v P (t) = 1 (phn b ng nht) ta thu c

f(x) =

dtdx = |t(x)| (17)

V d: Ly mu bin ngu nhin x c hm mt xc sut f(x) = aeax trong khong [0,)Ta c dtdx

= f(x) = aeax nn t = eax hay x = ln(t)aKhi x = 0 th t = 1 v x = th t = 0, do ta c th thu c bin x bng cch gieo ngunhin bin t trong khong (0, 1) v p dng cng thc

x = ln(1 t)a

Phng php chp nhn loi b (acceptance-rejection method) thng c s dngtrong nhng trng hp hm f(x) c dng phc tp, khng th d dng ly mu c bng ccphng php khc nh phng php bin i. Trong phng php ny, ta s i tm mt hmphn b cng c (instrumental distribution) g(x) c th d dng ly mu bng cc phng phpkhc nh ly mu phn b ng u hay phng php bin i.

Gi C l chn trn ca gi tr cc i t s f(x)/g(x)

C max(f(x)

g(x)

)(18)

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Hm f(x) s c ly mu bng cch ly mu hm g(x) v gi li s im ly mu vi xc sut

P =f(x)

C g(x)(19)

Hnh 10: Minh ha phng php chp nhn loi b. ng mu xanh l phn b f(x) m tamun ly mu, ng gch on l phn b Cg(x), ng mu v mu xanh l l xc sutloi b v chp nhn gi tr gieo ngu nhin thu c

Cch thc tin hnh phng php ny nh sau

1. Gieo mt cp s ngu nhin (xi, yi) trong khong xmin < x < xmax v 0 < y < Cg(x).

2. Nu y f(x) th chp nhn gi tr x, v loi b trong trng hp ngc li3. Lp li cho n khi thu c N gi tr x

Nhc im ca phng php ny l khng phi lc no ta cng c th xc nh c gi trca C mt cch d dng, v vic la chn gi tr C s quyt nh hiu sut ly mu ca phngphp cao hay thp.

Ly mu theo trng s (importance sampling) trong k thut ny, ta s i ly mu t mtphn b khc thay v ly mu trc tip t phn b cn quan tm. K thut ny thng cxem l mt k thut gim phng sai trong ly mu Monte Carlo.

Ta bit rng k vng ca bin x c tnh theo cng thc

E(x) =

x f(x)dx (20)

Thay v ly mu bin x t phn b f(x), ta s i ly mu theo mt phn b g(x) n gin hn,khi k vng ca x c tnh li theo cng thc

E(x) =

xf(x)

g(x)g(x)dx (21)

iu ny tng ng vi vic ly mu bin x.w(x) theo phn b g(x) vi w(x) = f(x)/g(x)c gi l trng s ly mu (importance sampling weight).

15


Phng php Metropolis-Hastings (Metropolis-Hastings method) hay cn c gi lphng phpMarkov Chain Monte Carlo (MCMC), thng c p dng cho vic ly mu t ccphn b nhiu chiu (multi-dimensional distribution). Mc ch ca phng php Metropolis-Hastings l to ra mt tp hp cc trng thi (gi tr) da trn hm mt xc sut f(x) cho trc . lm c iu ny chng ta s to ra mt qu trnh Markov (Markov process)m qu trnh ny s tin dn v mt phn b cn bng (stationary distribution) pi(x).

Mt chui Markov (Markov chain) l mt chui ngu nhin cc gi tr x1, x2, ..., xN vi c iml xc sut ca gi tr sau (x) ch ph thuc vo gi tr trc n (x) v c c trng bi xcsut dch chuyn (transition probability) P (x x). Phn b ny s tin v phn b cn bngkhi c hai iu kin sau c tha

Phn b cn bng pi(x) phi tn ti. iu kin ca n c xy dng da trn nguynl cn bng chi tit (detailed balance), i hi rng chuyn dch x x l c th nghcho c (chuyn dch theo c hai chiu)

P (x)P (x x) = P (x)P (x x) (22)

Phn b cn bng l duy nht (unique), iu ny c m bo bi tnh cht ca qutrnh Markov9.

Ta c th vit li cng thc (22)

P (x x)P (x x) =

P (x)P (x)

(23)

Gi phn b xut (proposal distribution) g(x x) l xc sut iu kin (conditional proba-bility) thu c x t x cho trc; v phn b chp nhn (acceptance distribution) A(x x)l xc sut iu kin chp nhn x. Xc sut dch chuyn l tch ca hai xc sut ny

P (x x) = g(x x)A(x x) (24)

Thay (24) vo (23) ta cA(x x)A(x x) =

P (x)P (x)

g(x x)g(x x) (25)

Bc k tip l chn gi tr chp nhn ph hp vi nguyn l cn bng chi tit, thng thngngi ta chn

A(x x) = min(

1,P (x)P (x)

g(x x)g(x x)

)(26)

ngha l ta s chn dch chuyn x x nu A(x x) ln hn 1 v loi b n nu nh hn 1.Cc bc tin hnh nh sau

1. Chn ngu nhin 1 gi tr x ban u

2. Chn ngu nhin gi tr x tng ng vi phn b g(x x)10

3. Chp nhn gi tr mi x da vo gi tr ca A(x x)9c gi l ergodicity i hi rng mi trng thi phi (1)phi tun hon (aperiodic), h thng khng th tr

li trng thi c trong mt khong c nh; (2)c kh nng quay tr li, k vng ca s bc dch chuyn trli trng thi c phi l hu hn

10Cn phi lu mt iu l hm phn b g(x x) l do ngi dng t la chn vo iu chnh ty thucvo tng trng hp c th.

16


4. Lp li bc 2 cho n khi N gi tr ca x c to ra

Cc phng php khc ngoi nhng phng php va k trn, ta cng cn nhiu cch lymu khc nh

i vi cc phn b ri rc, ta c cc phng php ly mu tuyn tnh (linear sampling),theo cu trc cy (decision tree), phng php mng (array method),...

i vi cc phn b lin tc, ta c thut ton Ziggurat (Ziggurat algorithm), Gibbs (Gibbssampling), HMC (Hamiltonian Monte Carlo),...

8 c lng Monte Carlo

8.1 c lng mu

Trung bnh mu (sample mean) l gi tr c lng ca trung bnh qun th (populationmean) da trn mt mu c chn ngu nhin t qun th ny. c lng trung bnhca mu ta s dng cng thc

x =1

N

Ni=1

xi (27)

vi xi l cc gi tr trong mu v N l kch thc mu11.

Gi tr x ny s phn b quanh gi tr trung bnh ca qun th vi

x = E(x) = (28)

2x = V ar(x) =2

N(29)

D cho phn b ca x l ng nht vi gi tr trung bnh ca qun th nhng phng sai snh hn nhiu nu kch thc ca mu l ln.

Phng sai mu (sample variance) thng c k hiu l S2 hay S2N c xc nh bicng thc12

S2 =1

N

Ni=1

(xi x)2 (30)

Nu ta xem x nh l mt c lng ca trung bnh qun th vi

E(x) = (31)

th S2 cng c xem nh l mt c lng ca phng sai qun th , tuy nhin y li l11gi tr x cn c xem l trung bnh khng trng s ca cc gi tr, ngc li vi l trung bnh c trng

s, xem cng thc (4)12Cn lu phn bit gia cc i lng 2 (phng sai ca qun th), 2x (phng sai ca phn b trung bnh

mu) v S2 (phng sai mu)

17


mt c lng b chch (biased estimator)

E[S2] = E

[1

N

Ni=1

(xi x)2]

= E

[1

N

Ni=1

((xi ) (x ))2]

= E

[1

N

Ni=1

(xi )2 (x )2]

= 2 E [(x )2] = N 1N

2 < 2 (32)

iu ny c ngha l E(S2) 6= 2, k vng ca S2 khng phi l phng sai 2 ca qun th13. hiu chnh cho s chch ny, chng ta thay th S2 bng

s2 =1

N 1Ni=1

(xi x)2 (33)

T s gia phng sai cha hiu chnh trn phng sai hiu chnh (S/s)2 = N/(N 1) cgi l h s hiu chnh Bessel (Bessels correction).

8.2 chnh xc ca c lng

Sai s (error) hay cn gi l bt nh (uncertainty) th hin khng chnh xc ca mtc lng so vi gi tr thc ca n. Sai s thng hay c chia lm hai loi l sai s ngu nhin(random error) hay cn gi l sai s thng k (statistical error) v sai s h thng (systematicerror).

2total = 2statistical +

2systematic (34)

Sai s ngu nhin lin quan n kch thc hu hn ca mu, trong khi sai s h thng lilin quan n vic mu thhu c khng i din y cc tnh cht ca qun th (v nhiul do nh sai s thit b, con ngi,...). Sai s h thng thng kh c nh lng tuy nhintrong mt s trng hp c th ta cng c th c lng c gi tr ca n.

chnh xc (accuracy) dng nh gi gn (closeness) hay chch (bias) ca gitr trung bnh c lng so vi gi tr thc ca i lng vt l, i khi cn c m t bi sais h thng (systematic error). Trong Monte Carlo, ta khng th c lng chnh xc nymt cch trc tip c.

Cc nhn t chnh nh hng ln chnh xc gm c

chnh xc ca code (m hnh vt l,...)

M hnh bi ton (hnh hc, ngun,...)

Li do ngi s dng13L do l v trung bnh mu x l mt c lng bnh phng cc tiu tuyn tnh (linear least squares) ca ,

gi tr ca x c chn sao cho tng

(xi x)2 t gi tr nh nht. Do vy, khi a thm s hng vo trongtng, gi tr ca tng ch c th tng ln, c bit khi 6= x ta c

1

N

Ni=1

(xi x)2 < 1N

Ni=1

(xi )2

18


tp trung (precision) l bt nh ca ca cc thng ging thng k trong vic ly mu.Mi tng quan gia chnh xc v tp trung c mnh ha trong Hnh 11 v Hnh 12.

Hnh 11: Minh ha chnh xc v tp trung ca mt phn b c lng

Hnh 12: Minh ha cc mc ca chnh xc v tp trung

8.3 Khong tin cy

Khong tin cy (Confidence Interval CI) l mt khong gi tr m c th cha trong n gitr ca tham s cn c lng (unknown parameter). rng ca khong tin cy cho chng tathng tin v bt nh ca php tnh c lng tham s14.

Cc khong tin cy thng dng i vi phn b chun nh sau (Hnh 13)15

P (xn 1 < < xn + 1) = 68% (35)P (xn 2 < < xn + 2) = 95% (36)P (xn 3 < < xn + 3) = 99% (37)

14Nhiu ngi cho rng xc sut c cho bi khong tin cy chnh l xc sut m gi tr trung bnh ca qunth ri vo trong khong tin cy , suy ngh ny l khng ng. Gi tr trung bnh ca qun th l mt hng s,n khng thay i, do xc sut gi tr trung bnh qun th ri vo trong khong tin cy ch l mt trong 2gi tr 0 hoc 1.

15Cc gi tr xc sut nh 68%, 95%, 99% c cho bi khong tin cy tnh theo ch ng trong trng hpphn b mt chiu (1-dimension)

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Hnh 13: Minh ha khong tin cy ca phn b chun

Ti liu

[1] J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Methuen & Co Ltd (1975).

[2] A.F. Bielajew, Fundamentals of the Monte Carlo Method for Neutral and Charged ParticleTransport, National Reseach Council of Canada (2001).

[3] Malvin H. Kalos, Paula A. Whitlock, Monte Carlo Methods, WILEY-VCH Verlag GmbH& Co. KGaA (2004).

[4] http://en.wikipedia.org/wiki/Monte_Carlo_method

[5] http://www.inference.phy.cam.ac.uk/tcs27/talks/sampling.html#0

[6] https://quanto.inria.fr/pdf_html/mc_standard_doc/

[7] http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm

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M uLich s hnh thnh phuong php Monte CarloCc phuong php Monte CarloCo s cua phuong php Monte CarloS ngu nhinCc loai s ngu nhinNhng iu cn luu khi m phong s ngu nhinPhuong php tao s ngu nhin

Phn b xc sutBin ngu nhinHm mt xc sutMoment thng kLut s lninh l gii han trung tm

Ly mu phn bPhn b muCc phuong php ly mu vi phn b xc sut khng ng nht

Uc lung Monte CarloUc lung mu chnh xc cua uc lungKhoang tin cy

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