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#### d|ensity#### p|robability (cumulative)#### q|uantile#### r|andom number generation######## Functions for ``d/p/q/r''.ptime <- proc.time()F <- FALSET <- TRUE###-- these are identical in ./arith-true.R ["fixme": use source(..)]opt.conformance <- 0Meps <- .Machine $ double.epsxMax <- .Machine $ double.xmaxoptions(rErr.eps = 1e-30)rErr <- function(approx, true, eps = .Options$rErr.eps){if(is.null(eps)) { eps <- 1e-30; options(rErr.eps = eps) }ifelse(Mod(true) >= eps,1 - approx / true, # relative errortrue - approx) # absolute error (e.g. when true=0)}## Short cut:All.eq <- function(x,y) all.equal.numeric(x,y, tolerance = 100*.Machine$double.eps)if(!interactive()).Random.seed <- c(0,rep(7654, 3))## The prefixes of ALL the PDQ & R functionsPDQRinteg <- c("binom", "geom", "hyper", "nbinom", "pois","signrank","wilcox")PDQR <- c(PDQRinteg, "beta", "cauchy", "chisq", "exp", "f", "gamma","lnorm", "logis", "norm", "t","unif","weibull")PQonly <- c("tukey")###--- Discrete Distributions --- Consistency Checks pZZ = cumsum(dZZ)##for(pre in PDQRinteg) { n <- paste("d",pre,sep=""); cat(n,": "); str(get(n))}##__ 1. Binomial __## Cumulative Binomial '==' Cumulative F :## Abramowitz & Stegun, p.945-6; 26.5.24 AND 26.5.28 :n0 <- 50; n1 <- 16; n2 <- 20; n3 <- 8for(n in rbinom(n1, size = 2*n0, p = .4)) {cat("n=",n,": ")for(p in c(0,1,rbeta(n2, 2,4))) {cat(".")for(k in rbinom(n3, size = n, prob = runif(1))) {## For X ~ Bin(n,p), compute 1 - P[X > k] = P[X <= k] in three ways:tst1 <- all.equal( pbinom(0:k, size = n, prob = p),cumsum(dbinom(0:k, size = n, prob = p)))tst <- all.equal(if(k==n || p==0) 1 elsepf((k+1)/(n-k)*(1-p)/p, df1=2*(n-k), df2=2*(k+1)),sum(dbinom(0:k, size = n, prob = p)))if(!(is.logical(tst1) && tst1) ||!(is.logical(tst) && tst ) ) {cat("n=", n,"; p =",format(p),". k =",k)if(!is.logical(tst1)) cat("; tst1=",tst1)if(!is.logical(tst )) cat("; tst=", tst)cat("\n")}}}cat("\n")}##__ 2. Geometric __for(pr in seq(0,1,len=15)) {print(All.eq((dg <- dgeom(0:10, pr)),pr * (1-pr)^(0:10)))print(All.eq(cumsum(dg), pgeom(0:10, pr)))}##__ 3. Hypergeometric __m <- 10; n <- 7for(k in 2:m) {x <- 0:(k+1)print(All.eq(phyper(x, m, n, k), cumsum(dhyper(x, m, n, k))))}##__ 4. Negative Binomial __## PR #842for(size in seq(0.8,2, by=.1))print(all.equal(cumsum(dnbinom(0:7, size, .5)),pnbinom(0:7, size, .5)))All.eq(pnbinom(c(1,3), .9, .5), c(0.777035760338812, 0.946945347071519))##__ 5. Poisson __all(dpois(0:5,0) == c(1, rep(0,5)))all(dpois(0:5,0, log=TRUE) == c(0, rep(-Inf, 5)))## Cumulative Poisson '==' Cumulative Chi^2 :## Abramowitz & Stegun, p.941 : 26.4.21 (26.4.2)n1 <- 20; n2 <- 16for(lambda in rexp(n1))for(k in rpois(n2, lambda)) {tst <- all.equal(1 - pchisq(2*lambda, 2*(1+ 0:k)),pp <- cumsum(dpois(0:k, lambda=lambda)), tol= 100*Meps)if(!(is.logical(tst) && tst))cat("lambda=", format(lambda),". k =",k, " --> tst=", tst,"\n")tst2 <- all.equal(pp, ppois(0:k, lambda=lambda), tol = 100*Meps)if(!(is.logical(tst2) && tst2))cat("lambda=", format(lambda),". k =",k, " --> tst2=", tst2,"\n")tst3 <- all.equal(1 - pp, ppois(0:k, lambda=lambda, lower.tail=FALSE))if(!(is.logical(tst3) && tst3))cat("lambda=", format(lambda),". k =",k, " --> tst3=", tst3,"\n")}##__ 6. SignRank __for(n in rpois(32, lam=8)) {x <- -1:(n + 4)eq <- All.eq(psignrank(x, n), cumsum(dsignrank(x, n)))if(!is.logical(eq) || !eq) print(eq)}##__ 7. Wilcoxon (symmetry & cumulative) __is.sym <- TRUEfor(n in rpois(5, lam=6))for(m in rpois(15, lam=8)) {x <- -1:(n*m + 1)fx <- dwilcox(x, n, m)Fx <- pwilcox(x, n, m)is.sym <- is.sym & all(fx == dwilcox(x, m, n))eq <- All.eq(Fx, cumsum(fx))if(!is.logical(eq) || !eq) print(eq)}is.sym###-------- Continuous Distributions ----------##--- Gamma (incl. central chi^2) Density :x <- round(rgamma(100, shape = 2),2)for(sh in round(rlnorm(30),2)) {Ga <- gamma(sh)for(sig in round(rlnorm(30),2)) {tst <- all.equal((d1 <- dgamma( x, shape = sh, scale = sig)),(d2 <- dgamma(x/sig, shape = sh, scale = 1) / sig),tol = 1e-14)## __ad interim__ was 1e-15if(!(is.logical(tst) && tst))cat("ERROR: dgamma() doesn't scale:",tst,"\n"," x =", formatC(x),"\n shape,scale=",formatC(c(sh, sig)),"\n")tst <- All.eq(d1, (d3 <- 1/(Ga * sig^sh) * x^(sh-1) * exp(-x/sig)))if(!(is.logical(tst) && tst))cat("NOT Equal:",tst,"\n x =", formatC(x),"\n shape,scale=",formatC(c(sh, sig)),"\n")}}pgamma(1,Inf,scale=Inf) == 0all(is.nan(c(pgamma(Inf,1,scale=Inf), pgamma(Inf,Inf,scale=1), pgamma(Inf,Inf,scale=Inf))))pgamma(Inf,1,scale=xMax) == 1 && pgamma(xMax,1,scale=Inf) == 0##-- non central Chi^2 :xB <- c(2000,1e6,1e50,Inf)for(df in c(0.1, 1, 10))for(ncp in c(0, 1, 10, 100)) stopifnot(pchisq(xB, df=df, ncp=ncp) == 1)all.equal(qchisq(0.025,31,ncp=1,lower.tail=FALSE),# inf.loop PR#87549.77662465615, tol= 1e-11)## p ~= 1 (<==> 1-p ~= 0) -- gave infinite loop in R <= 1.8.1 -- PR#6421psml <- 2^-(10:54)q0 <- qchisq(psml, df=1.2, ncp=10, lower.tail=FALSE)q1 <- qchisq(1 -psml, df=1.2, ncp=10)p0 <- pchisq(q0, df=1.2, ncp=10, lower.tail=FALSE)p1 <- pchisq(q1, df=1.2, ncp=10, lower.tail=FALSE)iO <- 1:10all.equal(q0[iO], q1[iO])all.equal(p0[iO], psml[iO], tol = 0.08)# bad tol (0.0744 on 386-Linux).##--- Beta (need more):## big a & b (PR #643)summary(a <- rlnorm(20, 5.5))summary(b <- rlnorm(20, 6.5))pab <- expand.grid(seq(0,1,by=.1), a, b)p <- pab[,1]; a <- pab[,2]; b <- pab[,3]all.equal(dbeta(p,a,b), exp(pab <- dbeta(p,a,b, log = TRUE)), tol = 1e-11)sample(pab, 50)##--- Normal (& Lognormal) :qnorm(0) == -Inf && qnorm(-Inf, log = TRUE) == -Infqnorm(1) == Inf && qnorm(0, log = TRUE) == Infis.nan(qnorm(1.1)) &&is.nan(qnorm(-.1)) # + warnx <- c(-Inf, -1e100, 1:6, 1e200, Inf)rbind(d.s0 =dnorm(x,3,s=0), p.s0 = pnorm(x,3,s=0),d.sI =dnorm(x,3,s=Inf), p.sI = pnorm(x,3,s=Inf))## 3 Test data from Wichura (1988) :all.equal(qnorm(c( 0.25, .001, 1e-20)),c(-0.6744897501960817, -3.090232306167814, -9.262340089798408),tol = 1e-15)# extreme tail -- available on log scale only:all.equal(qnorm(-1e5, log = TRUE), -447.1974945)z <- rnorm(1000); all.equal(pnorm(z), 1 - pnorm(-z), tol= 1e-15)z <- c(-Inf,Inf,NA,NaN, rt(1000, df=2))z.ok <- z > -37.5 | !is.finite(z)for(df in 1:10) if(!is.logical(all.equal(pt(z, df), 1 - pt(-z,df), tol= 1e-15)))cat("ERROR -- df = ", df, "\n")All.eq(pz <- pnorm(z), 1 - pnorm(z, lower=FALSE))All.eq(pz, pnorm(-z, lower=FALSE))All.eq(log(pz[z.ok]), pnorm(z[z.ok], log=TRUE))y <- seq(-70,0, by = 10)cbind(y, "log(pnorm(y))"= log(pnorm(y)), "pnorm(y, log=T)"= pnorm(y, log=TRUE))y <- c(1:15, seq(20,40, by=5))cbind(y, "log(pnorm(y))"= log(pnorm(y)), "pnorm(y, log=T)"= pnorm(y, log=TRUE),"log(pnorm(-y))"= log(pnorm(-y)), "pnorm(-y, log=T)"= pnorm(-y, log=TRUE))## Symmetry:y <- c(1:50,10^c(3:10,20,50,150,250))y <- c(-y,0,y)for(L in c(FALSE,TRUE))stopifnot(identical(pnorm(-y, log= L),pnorm(+y, log= L, lower=FALSE)))## Log normAll.eq(pz, plnorm(exp(z)))###========== p <-> q Inversion consistency =====================ok <- 1e-5 < pz & pz < 1 - 1e-5all.equal(z[ok], qnorm(pz[ok]), tol= 1e-12)###===== Random numbers -- first, just output:.Random.seed <- c(0, 17292, 29447, 24113)n <- 20## for(pre in PDQR) { n <- paste("r",pre,sep=""); cat(n,": "); str(get(n))}(Rbeta <- rbeta (n, shape1 = .8, shape2 = 2) )(Rbinom <- rbinom (n, size = 55, prob = pi/16) )(Rcauchy <- rcauchy (n, location = 12, scale = 2) )(Rchisq <- rchisq (n, df = 3) )(Rexp <- rexp (n, rate = 2) )(Rf <- rf (n, df1 = 12, df2 = 6) )(Rgamma <- rgamma (n, shape = 2, scale = 5) )(Rgeom <- rgeom (n, prob = pi/16) )(Rhyper <- rhyper (n, m = 40, n = 30, k = 20) )(Rlnorm <- rlnorm (n, meanlog = -1, sdlog = 3) )(Rlogis <- rlogis (n, location = 12, scale = 2) )(Rnbinom <- rnbinom (n, size = 7, prob = .01) )(Rnorm <- rnorm (n, mean = -1, sd = 3) )(Rpois <- rpois (n, lambda = 12) )(Rsignrank<- rsignrank(n, n = 47) )(Rt <- rt (n, df = 11) )## Rt2 below (to preserve the following random numbers!)(Runif <- runif (n, min = .2, max = 2) )(Rweibull <- rweibull (n, shape = 3, scale = 2) )(Rwilcox <- rwilcox (n, m = 13, n = 17) )(Rt2 <- rt (n, df = 1.01))(Pbeta <- pbeta (Rbeta, shape1 = .8, shape2 = 2) )(Pbinom <- pbinom (Rbinom, size = 55, prob = pi/16) )(Pcauchy <- pcauchy (Rcauchy, location = 12, scale = 2) )(Pchisq <- pchisq (Rchisq, df = 3) )(Pexp <- pexp (Rexp, rate = 2) )(Pf <- pf (Rf, df1 = 12, df2 = 6) )(Pgamma <- pgamma (Rgamma, shape = 2, scale = 5) )(Pgeom <- pgeom (Rgeom, prob = pi/16) )(Phyper <- phyper (Rhyper, m = 40, n = 30, k = 20) )(Plnorm <- plnorm (Rlnorm, meanlog = -1, sdlog = 3) )(Plogis <- plogis (Rlogis, location = 12, scale = 2) )(Pnbinom <- pnbinom (Rnbinom, size = 7, prob = .01) )(Pnorm <- pnorm (Rnorm, mean = -1, sd = 3) )(Ppois <- ppois (Rpois, lambda = 12) )(Psignrank<- psignrank(Rsignrank, n = 47) )(Pt <- pt (Rt, df = 11) )(Pt2 <- pt (Rt2, df = 1.01) )(Punif <- punif (Runif, min = .2, max = 2) )(Pweibull <- pweibull (Rweibull, shape = 3, scale = 2) )(Pwilcox <- pwilcox (Rwilcox, m = 13, n = 17) )dbeta (Rbeta, shape1 = .8, shape2 = 2)dbinom (Rbinom, size = 55, prob = pi/16)dcauchy (Rcauchy, location = 12, scale = 2)dchisq (Rchisq, df = 3)dexp (Rexp, rate = 2)df (Rf, df1 = 12, df2 = 6)dgamma (Rgamma, shape = 2, scale = 5)dgeom (Rgeom, prob = pi/16)dhyper (Rhyper, m = 40, n = 30, k = 20)dlnorm (Rlnorm, meanlog = -1, sdlog = 3)dlogis (Rlogis, location = 12, scale = 2)dnbinom (Rnbinom, size = 7, prob = .01)dnorm (Rnorm, mean = -1, sd = 3)dpois (Rpois, lambda = 12)dsignrank(Rsignrank, n = 47)dt (Rt, df = 11)dunif (Runif, min = .2, max = 2)dweibull (Rweibull, shape = 3, scale = 2)dwilcox (Rwilcox, m = 13, n = 17)## Check q*(p*(.)) = identityAll.eq(Rbeta, qbeta (Pbeta, shape1 = .8, shape2 = 2))All.eq(Rbinom, qbinom (Pbinom, size = 55, prob = pi/16))All.eq(Rcauchy, qcauchy (Pcauchy, location = 12, scale = 2))All.eq(Rchisq, qchisq (Pchisq, df = 3))All.eq(Rexp, qexp (Pexp, rate = 2))All.eq(Rf, qf (Pf, df1 = 12, df2 = 6))All.eq(Rgamma, qgamma (Pgamma, shape = 2, scale = 5))All.eq(Rgeom, qgeom (Pgeom, prob = pi/16))All.eq(Rhyper, qhyper (Phyper, m = 40, n = 30, k = 20))All.eq(Rlnorm, qlnorm (Plnorm, meanlog = -1, sdlog = 3))All.eq(Rlogis, qlogis (Plogis, location = 12, scale = 2))All.eq(Rnbinom, qnbinom (Pnbinom, size = 7, prob = .01))All.eq(Rnorm, qnorm (Pnorm, mean = -1, sd = 3))All.eq(Rpois, qpois (Ppois, lambda = 12))All.eq(Rsignrank, qsignrank(Psignrank, n = 47))All.eq(Rt, qt (Pt, df = 11))all.equal(Rt2, qt (Pt2, df = 1.01), tol = 1e-2)All.eq(Runif, qunif (Punif, min = .2, max = 2))All.eq(Rweibull, qweibull (Pweibull, shape = 3, scale = 2))All.eq(Rwilcox, qwilcox (Pwilcox, m = 13, n = 17))## Same with "upper tail":All.eq(Rbeta, qbeta (1- Pbeta, shape1 = .8, shape2 = 2, lower=F))All.eq(Rbinom, qbinom (1- Pbinom, size = 55, prob = pi/16, lower=F))All.eq(Rcauchy, qcauchy (1- Pcauchy, location = 12, scale = 2, lower=F))All.eq(Rchisq, qchisq (1- Pchisq, df = 3, lower=F))All.eq(Rexp, qexp (1- Pexp, rate = 2, lower=F))All.eq(Rf, qf (1- Pf, df1 = 12, df2 = 6, lower=F))All.eq(Rgamma, qgamma (1- Pgamma, shape = 2, scale = 5, lower=F))All.eq(Rgeom, qgeom (1- Pgeom, prob = pi/16, lower=F))All.eq(Rhyper, qhyper (1- Phyper, m = 40, n = 30, k = 20, lower=F))All.eq(Rlnorm, qlnorm (1- Plnorm, meanlog = -1, sdlog = 3, lower=F))All.eq(Rlogis, qlogis (1- Plogis, location = 12, scale = 2, lower=F))All.eq(Rnbinom, qnbinom (1- Pnbinom, size = 7, prob = .01, lower=F))All.eq(Rnorm, qnorm (1- Pnorm, mean = -1, sd = 3,lower=F))All.eq(Rpois, qpois (1- Ppois, lambda = 12, lower=F))All.eq(Rsignrank, qsignrank(1- Psignrank, n = 47, lower=F))All.eq(Rt, qt (1- Pt, df = 11, lower=F))all.equal(Rt2, qt (1- Pt2, df = 1.01, lower=F), tol = 1e-2)All.eq(Runif, qunif (1- Punif, min = .2, max = 2, lower=F))All.eq(Rweibull, qweibull (1- Pweibull, shape = 3, scale = 2, lower=F))All.eq(Rwilcox, qwilcox (1- Pwilcox, m = 13, n = 17, lower=F))## Check q*(p* ( log ), log) = identityAll.eq(Rbeta, qbeta (log(Pbeta), shape1 = .8, shape2 = 2, log=TRUE))All.eq(Rbinom, qbinom (log(Pbinom), size = 55, prob = pi/16, log=TRUE))All.eq(Rcauchy, qcauchy (log(Pcauchy), location = 12, scale = 2, log=TRUE))all.equal(Rchisq, qchisq (log(Pchisq), df = 3, log=TRUE),tol=1e-14)All.eq(Rexp, qexp (log(Pexp), rate = 2, log=TRUE))All.eq(Rf, qf (log(Pf), df1= 12, df2= 6, log=TRUE))All.eq(Rgamma, qgamma (log(Pgamma), shape = 2, scale = 5, log=TRUE))All.eq(Rgeom, qgeom (log(Pgeom), prob = pi/16, log=TRUE))All.eq(Rhyper, qhyper (log(Phyper), m = 40, n = 30, k = 20, log=TRUE))All.eq(Rlnorm, qlnorm (log(Plnorm), meanlog = -1, sdlog = 3, log=TRUE))All.eq(Rlogis, qlogis (log(Plogis), location = 12, scale = 2, log=TRUE))All.eq(Rnbinom, qnbinom (log(Pnbinom), size = 7, prob = .01, log=TRUE))All.eq(Rnorm, qnorm (log(Pnorm), mean = -1, sd = 3, log=TRUE))All.eq(Rpois, qpois (log(Ppois), lambda = 12, log=TRUE))All.eq(Rsignrank, qsignrank(log(Psignrank), n = 47, log=TRUE))All.eq(Rt, qt (log(Pt), df = 11, log=TRUE))all.equal(Rt2, qt (log(Pt2), df = 1.01, log=TRUE), tol = 1e-2)All.eq(Runif, qunif (log(Punif), min = .2, max = 2, log=TRUE))All.eq(Rweibull, qweibull (log(Pweibull), shape = 3, scale = 2, log=TRUE))All.eq(Rwilcox, qwilcox (log(Pwilcox), m = 13, n = 17, log=TRUE))## same q*(p* (log) log) with upper tail:All.eq(Rbeta, qbeta (log(1- Pbeta), shape1 = .8, shape2 = 2, lower=F, log=T))All.eq(Rbinom, qbinom (log(1- Pbinom), size = 55, prob = pi/16, lower=F, log=T))All.eq(Rcauchy, qcauchy (log(1- Pcauchy), location = 12, scale = 2, lower=F, log=T))All.eq(Rchisq, qchisq (log(1- Pchisq), df = 3, lower=F, log=T))All.eq(Rexp, qexp (log(1- Pexp), rate = 2, lower=F, log=T))All.eq(Rf, qf (log(1- Pf), df1 = 12, df2 = 6, lower=F, log=T))All.eq(Rgamma, qgamma (log(1- Pgamma), shape = 2, scale = 5, lower=F, log=T))All.eq(Rgeom, qgeom (log(1- Pgeom), prob = pi/16, lower=F, log=T))All.eq(Rhyper, qhyper (log(1- Phyper), m = 40, n = 30, k = 20, lower=F, log=T))All.eq(Rlnorm, qlnorm (log(1- Plnorm), meanlog = -1, sdlog = 3, lower=F, log=T))All.eq(Rlogis, qlogis (log(1- Plogis), location = 12, scale = 2, lower=F, log=T))All.eq(Rnbinom, qnbinom (log(1- Pnbinom), size = 7, prob = .01, lower=F, log=T))All.eq(Rnorm, qnorm (log(1- Pnorm), mean = -1, sd = 3, lower=F, log=T))All.eq(Rpois, qpois (log(1- Ppois), lambda = 12, lower=F, log=T))All.eq(Rsignrank, qsignrank(log(1- Psignrank), n = 47, lower=F, log=T))All.eq(Rt, qt (log(1- Pt ), df = 11, lower=F, log=T))all.equal(Rt2, qt (log(1- Pt2), df = 1.01, lower=F, log=T), tol = 1e-2)All.eq(Runif, qunif (log(1- Punif), min = .2, max = 2, lower=F, log=T))All.eq(Rweibull, qweibull (log(1- Pweibull), shape = 3, scale = 2, lower=F, log=T))All.eq(Rwilcox, qwilcox (log(1- Pwilcox), m = 13, n = 17, lower=F, log=T))## Check log( upper.tail ):All.eq(log(1 - Pbeta), pbeta (Rbeta, shape1 = .8, shape2 = 2, lower=F, log=T))All.eq(log(1 - Pbinom), pbinom (Rbinom, size = 55, prob = pi/16, lower=F, log=T))All.eq(log(1 - Pcauchy), pcauchy (Rcauchy, location = 12, scale = 2, lower=F, log=T))All.eq(log(1 - Pchisq), pchisq (Rchisq, df = 3, lower=F, log=T))All.eq(log(1 - Pexp), pexp (Rexp, rate = 2, lower=F, log=T))All.eq(log(1 - Pf), pf (Rf, df1 = 12, df2 = 6, lower=F, log=T))All.eq(log(1 - Pgamma), pgamma (Rgamma, shape = 2, scale = 5, lower=F, log=T))All.eq(log(1 - Pgeom), pgeom (Rgeom, prob = pi/16, lower=F, log=T))All.eq(log(1 - Phyper), phyper (Rhyper, m = 40, n = 30, k = 20, lower=F, log=T))All.eq(log(1 - Plnorm), plnorm (Rlnorm, meanlog = -1, sdlog = 3, lower=F, log=T))All.eq(log(1 - Plogis), plogis (Rlogis, location = 12, scale = 2, lower=F, log=T))All.eq(log(1 - Pnbinom), pnbinom (Rnbinom, size = 7, prob = .01, lower=F, log=T))All.eq(log(1 - Pnorm), pnorm (Rnorm, mean = -1, sd = 3, lower=F, log=T))All.eq(log(1 - Ppois), ppois (Rpois, lambda = 12, lower=F, log=T))All.eq(log(1 - Psignrank), psignrank(Rsignrank, n = 47, lower=F, log=T))All.eq(log(1 - Pt), pt (Rt, df = 11, lower=F, log=T))All.eq(log(1 - Pt2), pt (Rt2,df = 1.01, lower=F, log=T))All.eq(log(1 - Punif), punif (Runif, min = .2, max = 2, lower=F, log=T))All.eq(log(1 - Pweibull), pweibull (Rweibull, shape = 3, scale = 2, lower=F, log=T))All.eq(log(1 - Pwilcox), pwilcox (Rwilcox, m = 13, n = 17, lower=F, log=T))cat("Time elapsed: ", proc.time() - .ptime,"\n")