Suites of Tests. DIEHARD TESTS (Marsaglia, 1985) See
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1 Sutes of Tests DIEHARD TESTS (Marsagla, 985 See NIST Test sute- 6 tests on the sequences of bts Test U0 Includes the above tests.
2 Numercal Recpes Provdes C or Fortran Code for a number of numercal routnes ncludng random number generators
3 Increasng perod of generatorshufflng
4 Eample of shufflng
5 Sums or dfferences mod(m (see Fshman Sect. 7.3
6 Generatng non-unform random varates.
7 The nverse transform
8 Inverse transform for a Contnuous Dstrbuton Suppose we wsh to generate a random varable from a dstrbuton wth probablty densty functon f ( /, 0 < < Then the c.d.f. s F( Inverse / 4, for 0 < <.. F ( U U so s U where U s U[0,] the requred generator.
9 Eample: Inverse transform for a dscrete dstrbuton Consder the dstrbuton wth probablty functon P[ ] /0, for,,3,4. Then the nverse transform method results n the generator F ( U 3 4 U <.. U <.3.3 U <.6.6 U
10 Graph of c.d.f
11 Eponental dstrbuton generator.
12 j Geometrc Dstrbuton Generator Geometrc( f ( The F ( Then, F(j - ( cumulatve solvng < U p p ( log( - U log( - p Dstrbut p F(j p, for dstrbut on functon s [ ], where [] for an nteger gves where on has,,... j denotes probablt y functon denotes such that the celng. nteger part.
13 Bnomal generators Bnomal(n,p, n small -nverse transform or use n I( U < p Bnomal(n,p, n large, p small nf{ ; + G > n}, where G acceptance-rejecton log(- U log(- p geometrc( p
14 Cauchy
15 What f ncrements n a tme seres returns have a Cauchy dstrbuton???? plot(0+cumsum(.00./tan(p*rand(,0000
16 Generators n Matlab and R(Splus
17 Acceptance Rejecton
18 Acceptance-rejecton (cont
19 Acceptance-rejecton Suppose a probablty densty f( has c.d.f. that s NOT easly nverted. We cannot use nverse transform. How do we use the above theorem to generate from f(? Generate a pont at random (unformly under the graph of f( and output the coordnate (how? Pck the pont at random unformly n some larger regon and use t only f t s under f(
20 Eample Generate U[a,b] Generate Y U[0,c] If Y<f(, then accept (.e. output otherwse generate new values of,y and contnue untl one s accepted.
21 Acceptance-Rejecton: Dscrete Random varables Want to generate such that P[ ] We are able easly to generatey wth P[ Y ] q. p Suppose there s some constant c such that c. q Method GenerateY, say Accept ths value (.e. put wth probablty Otherwse generate new Y. p. p / cq.
22 Eample: acceptance rejecton for dscrete random varables E.g. generate r.v. wth probablty functon P[ k p q ] c k 00k,,,...,00 where Here q 9.3 for all 00 y<-celng(00*runf(00000; <-y[runf(00000</y]; hst(, freq FALSE, breaks 50, p k,...,00 and, p cq.
23 When to use acceptance-rejecton (dscrete or contnuous The c.d.f. F( s ether hard to obtan or hard to nvert. We know the p.d.f. f( We can fnd another p.d.f. g( such that It s easy (e.g. by nverse transform to generate from random varables wth p.d.f. g(. Ideally cg( close to f( (.e. c close to f all ( cg ( for and some c.
24 where ( ( have we 0, ( gven by p.d.f. that wth Notce,0 ( 6 45 ( from p.d.f. r.v. a Generate 4 / < < < < c cg f g f Eample
25 Effcency of Acceptance- Rejecton Algorthm Probablty that we accept on each teraton(area under f(/(area under cg( /c Probablty we need to generate n ponts before generatng one accepted n ( c c Ths s a geometrc dstrbuton wth probablty p/c Epected number of tmes through loop to generate one s c.
26 Eample. Gamma(3/, Dstrbuton Want to generate gamma random varable by acceptance rejecton. f ( Γ (3/ / e, > 0 Try domnatng wth an eponental dstrbuton g( λe λ, > 0. f ( cg( f c ma( f ( / g(. For effcent method, we may choose λ so that c ma( f ( / g( s small. The best value of λ /3.(gves an eponental wth the same mean, 3/, as ths gamma dstrbuton.
27 Bvarate normal p.d.f. f (, y ep( π + y
28 The Bo-Muller transformaton for normal random varables
29 The Bo-Muller transformaton for normal random varables (graph
30 Bo-Muller generator Generate R ln( U and Θ π U, whereu, U ndep U[0,]. Put R cos( Θ, Y R sn( Θ Then (, Y ndependent N(0,
31 Bo-Muller & lattce Structure Can use nverse transform to machne precson: P.J. Acklam (003
32 . Generate. If 3. Let R Y Z Z Z Marsagla Polar Normal generator + Z RZ RZ + + Z Z Z, Z >, GO TO. ln( Z,, ndep. U[-,], Then (, Y are ndep N(0, + Z
33 Generatng Dependent Normal ( Put ndependent N(0,., Generate. where,, but wth correlaton(,,, Normal(, Suppose we want Y Y ρ ρ σ µ σ µ ρ ρ σ µ < <
34 Obtan the sngular value decomposton of the covarance matr so that Σ AA' for some matr A Defne AY + µ wherey`s vector of ndependent N(0, Then s mvnorm( µ,aa' Σ rmvnorm <- functon(n, meanrep(0, nrow(sgma, sgmadag(length(mean{ f(nrow(sgma! ncol(sgma{stop("sgma must be a square matr" } f(length(mean! nrow(sgma{ stop("mean and sgma have nonconformng sze" } ev <- egen(sgma, sym TRUE$values f (!all(ev > -sqrt(.machne$double.eps * abs(ev[] warnng("sgma s numercally not postve defnte" sgsvd <- svd(sgma retval <- t(sgsvd$v %*% (t(sgsvd$u * sqrt(sgsvd$d retval <- matr(rnorm(n * ncol(sgma, nrow n %*% retval retval <- sweep(retval,, mean, "+" retval} (R code from
35 The lognormal dstrbuton
36 Composton: for mtures of dstrbutons. Suppose a c.d.f. takes the form n j ( for and wth c.d.f.'s > 0 wth (. Frst choose random I,,... n wth probabltes p p F p F and then generate from the c.d.f. F p I (.
37 Eample: Composton 4 / / set 0.4, If., Generate (, ( (, ( U else U U U U F F F f < + < < +
38 λ Posson(. Use Posson process; for small nf{ n; n + U < e λ or nverse transform } λ For large use acceptance-rejecton λ
39 Mean squared error for medan of Cauchy n R n <- 0 nsm < t<-proc.tme([] thetahat <- double(nsm for ( n :nsm { <- rcauchy(n thetahat[] <- medan( } mean(thetahat^ cat("calculaton took", (proc.tme([]-t, "seconds.\n" hst(thetahat, freq FALSE, breaks 00 curve(dnorm(, sd sqrt(mean(thetahat^, add TRUE curve(dnorm(, sd sqrt( / (4 * n * dcauchy(0^, add TRUE, col "red" smulatons of a medan from a Cauchy sample of sze 0. The red curve s the normal appromaton wth se estmated usng the cauchy densty; the black curve uses the standard devaton of the estmates.
40 Gamma( dstrbuton 0 0, 0,, ( ( / > > Γ β α β α α β α e f β α, freedom of wth n degrees - squared ch called s Gamma,, / f and Ep(/ s then, If var(, (, Gam( s., Gam s then, are ndependen t Gam( If β α β α β α β α β α β α β α n E c c ( n n : Propertes
41 Some gamma p.d.f.
42 Gamma tmes Beta
43 Gamma dstrbuton generators (cont when α >, agan use acceptance rejecton (domnate wth a functon lke g( cp ( c + p p for a sutable constant c or f α small (e.g..6 add ndependen t Gamma(0.6, + Gamma(,
44 Gamma dstrbuton generators. when α β α s ln( U a small,. When α <, β, use acceptance - rejecton wth eα g( e + α e e + α and c eαγ( α α U,, nteger are U[0,] <
45 Beta Dstrbuton for a and b small
46 Logstc Ft to Normal f logt vs. Probt b e ( + e ( a/ b ( ( a/ b
47 Symmetrc stable laws (specal cases. Normal dstrbuton (alpha. Cauchy dstrbuton (alpha 3. For most normal-lke data,.5<alpha< Lke the normal dstrbuton, the sum of ndependent stable(alpha random varables s stable(alpha.
48 Symmetrc Stable dstrbutons f c ( παc k 0 ( k k Γ( α + (k! ( y c k In the case c generate usng
49 Symmetrc stable random walk, alpha.7
50 Smulatng the Law of large numbers. z<-:0000 <-cumsum(rnorm(0000,0,/z; <-cumsum(rstable(0000,.7/z; # must defne rstable 3<-cumsum(rstable(0000,.3/z; 4<-cumsum(rt(0000,/z; plot(z,, l,col red lnes(z,,col green lnes(z,3,col blue lnes(z,4, col black"
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