#### History(RCS): /u/maechler/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qbeta-dist.R,v #### Testing qbeta(.), pbeta(.), qt(.), ..... #### ---------------- source(system.file(package="DPQ", "test-tools.R", mustWork=TRUE))# ../inst/test-tools.R ## => showProc.time(), .. {from Matrix}, ## list_() , loadList() , readRDS_() , save2RDS() (doExtras <- DPQ:::doExtras()) ## save directory (to read from): (sdir <- system.file("safe", package="DPQ")) if(!dev.interactive(orNone=TRUE)) pdf("qbeta-dist.pdf") .O.P. <- par(no.readonly=TRUE) op <- options(nwarnings = 1e5) fcat <- function(..., f.dig= 4, f.wid = f.dig +5, f.flag = ' ', nl = TRUE, file = "", sep = " ", fill = FALSE, labels = NULL, append = FALSE) { ## Purpose: Formatted CAT -- for printing 'tables' ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 12 May 97 l <- unlist(lapply(list(...), formatC, wid= f.wid, digits= f.dig, flag= f.flag)) cat(l, if(nl)"\n", file=file, sep=sep, fill=fill,labels=labels, append=append) } format01prec <- function(x, digits = getOption("digits"), width = digits + 2, eps = 1e-6, ..., FUN = function(x,...) formatC(x, flag='-',...)) { ## Purpose: format numbers in [0,1] with "precise" result, ## using "1-.." if necessary. ## ------------------------------------------------------------------------- ## Arguments: x: numbers in [0,1]; (still works if not) ## digits, width: number of digits, width to use with 'FUN' ## eps: Use '1-' iff x in (1-eps, 1] -- 1e-6 is OPTIMAL ## ------------------------------------------------------------------------- ## Author: Martin Maechler, Date: 14 May 97, 18:07 if(as.integer(digits) < 4) stop('digits must be >= 4') if(eps < 0 || eps > .1) stop('eps must be in [0, .1]') i.swap <- is.na(x) | (1-eps < x & x <= 1) #-- Use "1- ." if <~ 1, normal 'FUN' otherwise r <- character(length(x)) if(any(i.swap)) r[i.swap] <- paste("1-", FUN(1-x[i.swap], digits=digits - 5, width=width-2, ...), sep='')# -5: '1-' + 4 for exponent -1 for '0' (in other case) if(any(!i.swap)) r[!i.swap] <- FUN(x[!i.swap], digits=digits, width=width,...) attributes(r) <- attributes(x) r } ###---------------------- short pre-millennium examples: ---------------- ## = ./beta-test-0.R, Jul 9, 1997 (!) showProc.time() qbeta(1e-3, 1,10^(1:10)) qbeta(1e-5, 1,10^(1:10)) qbeta(3e-2, 1,10^(1:10)) qbeta(2e-2, 1,10^(1:10)) ##-- quite problematic ones : cbind(qbeta(2e-2, 1, 10^(0:32))) cbind(qbeta(.2, 1, 10^(0:32))) cbind(qbeta(.2, .1, 10^(0:32))) cbind(qbeta(.1, 1e-1, 10^(0:32)))# "good" cbind(qbeta(.1, 1e-2, 10^(0:32)))# look good cbind(qbeta(.1, 1e-2, 10^(0:64)))# look good plot(qbeta(1/8, 2^-7, 10^(0:64)), log="y")# look good plot(qbeta(1/8, 2^-8, 10^(0:64)), log="y")# look good (down to 10^-298) all(qbeta(1/8, 2^-9, 10^(0:64)) == 0)# TRUE, but ... all(qbeta(1/8, 2^-10, 10^(0:64)) == 0)# TRUE, but should be positive (in principle) stopifnot(qbeta(.1, 1e-3, 10^(0:32)) == 0) # and we cannot do better: curve(pbeta(x, 1e-3, 1 ), 1e-310, log="x", col=2, xaxt="n")# to give nicer axis: sfsmisc::eaxis(1); axis(1, at=1); abline(h=1,v=1, lty=2) curve(pbeta(x, 1e-3, 1000), add=TRUE, col=3) curve(pbeta(x, 1e-3, 1e10), add=TRUE, col=4) curve(pbeta(x, 1e-3, 1e20), add=TRUE, col=5) curve(pbeta(x, 1e-3, 1e50), add=TRUE, col=6) curve(pbeta(x, 1e-3,1e100), add=TRUE, col=5) curve(pbeta(x, 1e-3,1e200), add=TRUE, col=4)## ==> Warnings already curve(pbeta(x, 1e-3,1e300), add=TRUE, col=2)## ==> WARNINGS: ## 1: In pbeta(x, ...) : bgrat(a=1e+300, b=0.001, x=1) *no* convergence: NOTIFY R-core! summary(warnings()) # ---------------- ^^^^^^^^^^^^^^ ## = ./beta-qbeta-mintst.R, July 31, 1997 (!) 1 - qbeta(.05, 5e10, 1/2) #-- infinite loop in 0.49; 0.50-a1 ##-- 3.8416e-11 with MM's qbeta(.) qbeta(.95, 1/2, 5e10) # the "same" (with swap_tail) showProc.time() ###----------------------- qt(p, df --> Inf) --> qnorm(p) ----------------- p.<- 5* 10^-c(6:8,10,15,20,100,300)#- well ... p <- p. #- try things below with this 'EXTREME' p p <- c(.975, .995, .9995, 1 - 5e-6) df1 <- c(1:5,10^c(1:21,25,10*(3:10))) df0 <- df1[df1< 10^30] ## df0 works for R 0.49 -patch2- df0f<- unique(sort(c((1:30)/10, df0))) ## system.time() gives 0.0 nowadays ... sys.time <- function(exp) system.time(for(i in 1:5000) exp) dig <- 18 dig <- 12 dig <- 8 cat(" df : CPU[s] | p=", formatC( p , form='e', dig=5, wid=dig+6),"\n") for(df in df0) cat(formatC(df,w=6),":", formatC(sys.time(qq <- qt(p, df))[1],form='f'), "|", formatC(qq, form='f', dig=dig,wid=dig+6),"\n") cat(" _Inf_ : --- |", formatC(qnorm(p), form='f', dig=dig,wid=dig+6),"\n") ### Use F-distribution : `` F_{1,nu} == (t_{nu}) ^2 '' ### Therefore: qt(p,n) == sqrt(qf(2*p-1, 1,n)) for p >= 1/2 ### ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -- was true for ORIGINAL in R 0.49 ##for(df in df0f) { ##-- ~ TRUE for (0.49) qt/qf which worked with qbeta if(any(p < 1/2)) stop("must have p >= 1/2 for these examples!") for(df in df0f) { cat(formatC(df,w = 6),":", all.equal( sqrt(qf(2*p-1, 1,df)), qt(p,df)),"\n") } ## R 3.0.3 : all TRUE, but ## 1e+06 : Mean relative difference: 3.240032e-06 ## 1e+07 : Mean relative difference: 3.240022e-07 ## 1e+08 : Mean relative difference: 3.240021e-08 ## ==> Applying pt(*) ==> (R 3.5.1) shows that qt() is *better* than qf(.) equivalence ## It seems, qt(.) is perfect p <- 1 - 1/64 # exact rational df <- 2^seq(10,20, by=1/32) plot(df, qt(p, df=df)- qnorm(p), log="xy", type="l") plot(df[-1], -diff(qt(p, df=df)), log="xy", type="l") plot(df[-(1:2)], diff(qt(p, df=df), diff=2), log="xy", type="l") all.equal( sqrt(qf(2*p-1, 1,df)), qt(p,df)) ## -> 3.15636e-07 (R 3.0.3, 64bit) ## looking closely: plot(df, qf(2*p-1, 1,df) - qt(p,df)^2, type="b") ## shows clear jump at df = 400'000 df <- 1000*seq(390, 410, by=1/16) plot(df, qf(2*p-1, 1,df) - qt(p,df)^2, type="l") ## ~/R/D/r-devel/R/src/nmath/qf.c : it uses qchisq() for df1 <= df2, df2 > 400000 ## ==> FIXME: qbeta() is *clearly* better here (than "df = Inf" !) ## ===== qf() is *clearly* improvable showProc.time() ### Really, the problem is qbeta (.): ## --------- ## qt (for x > 1/2) __used__ to be defined as qt(x,n) below, but not longer !! ## qt(x,n) := sqrt((1 / qbeta( 2*(1-x), n/2, 1/2) - 1) * n) ##--- with DEBUG, shows already BIG problems (needs *much* CPU time for the large df's if(FALSE) # is too large and slow for(p in 2^-(30:2)) { cat("\n\np = ", formatC(p),"\n===========\n") cat(sprintf("%6s : %5.3s | %11s %11s | %s\n", "df","cpu","qbeta","qt", "all.equal(sqrt((1/qb - 1)*.), qt(.))")) for(df in c(1:5,10^c(1:21,25,10*(3:10)))) { cpu <- system.time(qb <- qbeta(2 * p, df/2, 1/2))[1] qt. <- qt(p,df, lower.tail=FALSE) cat(sprintf("%6g : %5.3g | %11g %11g | %s\n", df, cpu, qb, qt., all.equal(sqrt(df*(1/qb - 1)), qt.))) } } library(DPQ) showProc.time() ##----------- Test the 'qnormAppr' function as used in qbeta(.) : p <- (0:256)/256 matplot(p, cbind(qnorm(p), qnormAppr(p)), type="l", ylab="") legend("topleft", c("qnorm(p)", "qnormAppr(p)"), col=1:2, lty=1:2, bty="n") abline(h=0,v=(0:2)/2, lty=2, col="gray60") ## absolute error: curve(qnorm(x) - qnormAppr(x), col=2, lwd=2, n=1001, 0, 1) abline(h=0,v=(0:2)/2, lty=2, col="gray60") ##==> Really make use of symmetry --> use qnormUappr(): matplot(p, cbind(qnorm(p), qnormUappr(p)), type="l", ylab="") abline(h=0,v=(0:2)/2, lty=2, col="gray60") ## absolute error: c.1 <- rgb(1,0,0); c.2 <- adjustcolor("black",.5) curve(qnorm(x) - qnormUappr(x), col=c.1, lwd=2, n=1001, 0, 1) curve(qnorm(x) - qnormAppr (x), col=c.2, lwd=4, n=1001, add=TRUE, lty=3) legend(.1, .0025, paste("qnorm(.) -",c("qnormUappr(.)", " qnormAppr (.)")), col=c(c.1,c.2), lwd=c(2,3), lty=c(1,3), bty="n") abline(h=0,v=(0:2)/2, lty=2, col="gray60") showProc.time() ## From R's source /tests/d-p-q-r-tests.R : options(rErr.eps = 1e-30) # {*not* to be reset via options(op)} 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 error true - approx) # absolute error (e.g. when true=0) } ## The relative error of this approximation is quite Asymmetric: mainly < 0 x <- c(10^(-9:-4),seq(1e-3, .6, len = 1+2^11)) plot(1-x, rErr(qnormAppr(1-x), qnorm(1-x)), type = 'l', col = 'red', main = "Rel. Error of 'qnormAppr(1-x)'") abline(h = 0, col = 'gray') ## Note: qnorm(.) used to use AS 111 (26)1977, but has been changed to ## ---- AS 241 (37)1988 which is more accurate {and a bit slower} ## ## Much more sensical in modern R {but looks identical, for 1-x > 1/2 }: lines(1-x, rErr(qnormUappr(x, lower=FALSE), qnorm(x, lower=FALSE)), lwd=3, col = adjustcolor("skyblue", 1/2)) curve(qnorm(x) - qnormUappr(x), col=2, lwd=2, n=1001, .44, 1) abline(h=0, col=adjustcolor(1,.5)); axis(1, at=.44) ## Warning message: ## In qnormUappr(x) : p[.] < 1/2 & lower tail is inaccurate curve(qnorm(x) - qnormAppr(x), lwd=5, n=1001, col=adjustcolor(3,1/3), add=TRUE) ##par(new=TRUE) ## 1/10 * rel.error(): cblu <- adjustcolor("blue4",.5) curve(rErr(qnormUappr(x), qnorm(x)) / 10, n=1001, col = cblu, lwd=3, lty=2, add=TRUE) ## axis(4, col=cblu, col.axis=cblu) showProc.time() ###-------- Testing my own qbeta.R(.) [[and qbetaAppr(.) used there] qbeta (.05 , 10/2, 1/2) #-== 0.6682436 qbeta (.05 , 100/2, 1/2) #-== 0.9621292 qbeta (.05 ,1000/2, 1/2) #-== 0.996164 ###=========== FIXME: More qbetaAppr*() versions ###============================================= ## and use lower.tail=TRUE/FALSE *and* log.p=FALSE/TRUE ## and determine regions where qbetaAppr*() are very accurate ((so, say 1 Newton step then is perfect!)) qb.names <- paste0("qbeta", c('', '.R', 'Appr', 'Appr.1')) qf <- lapply(setNames(,qb.names), get) str(qf) ## for (qn in qb.names) cat(qn,":", deparse(args(get(qn))),"\n\n") p.set2 <- c(1:5,10,20,50,100,500,1000, 10^c(4:20,25,10*3:10)) p.set2 <- c(1:5,10,20,50,100,500,1000, 10^c(4:18,20,30,100)) ## TODO: still too expensive p.set2 <- c(1e-10,1e-5,1e-3,.1,.3,.5, .8, 1,10, 10^c(6:18,20,30,100)) q.set2 <- c(1e-4, 1/2, 1, 3/2, 2, 5, 1e4) pn2 <- paste("p=",formatC(p.set2,wid = 1),sep = '') qn2 <- paste("q=",formatC(q.set2,wid = 1),sep = '') sfil1 <- file.path(sdir, "tests_qbeta-d-ssR.rds") if(!doExtras && file.exists(sfil1)) { ssR_l <- readRDS_(sfil1) str(ssR_l) loadList(ssR_l) } else { ## do run the simulation [always if(doExtras)] : ra <- array(dim = c(length(q.set2), length(qb.names), length(p.set2)), dimnames = list(qn2, qb.names, pn2)) ind_ct <- names(system.time(1))[ c(1,3) ] # "user" and "elapsed" cta <- array(dim = c(length(ind_ct), dim(ra)), dimnames = c(list(ind_ct), dimnames(ra))) for(qq in q.set2) { cat("\nq=",qq,"\n======\n") for (p in p.set2) { cat(sprintf("p=%-7g : >> ",p)) for (qn in qb.names) { cat(" ",qn,": ", sep = '') st <- system.time( r <- qf[[qn]](.05, p, qq))[ind_ct] cta[, qn2[qq == q.set2], qn, pn2[p == p.set2]] <- st ra [ qn2[qq == q.set2], qn, pn2[p == p.set2]] <- r }; cat("\n") } showProc.time() ## -- similar for all q } print(summary(warnings())) # many warnings from qbeta() inaccuracies save2RDS(list_(ra, cta), file = sfil1) }## {run simulations} ------------------------------------------------------- ## Timings : 1e4*apply(cta, 2:3, mean) 1e4*apply(cta, c(2,4), mean) noquote(format01prec(aperm(ra))) qbeta.R(.05 , 10/2, 1/2, verbose=2) qbeta.R(.05 , 100/2, 1/2, verbose=2) qbeta.R(.05 ,1000/2, 1/2, verbose=2) showProc.time()#----------------------------------------------------------------------------- DEBUG <- TRUE ## FIXME ## This is a "version" of qnormUappr() -- ## but that has "regular" ## ## lower.tail=TRUE, log.p=FALSE ## --- MM(2019) to MM(2008): I think this is a bad idea ## ---> ../TODO: Use *argument* lower.tail=FALSE for all qnormU*() : "U" means "Upper tail" !!!! qnormU <- function(u, lu) { if(missing(u)) qnorm(lu, lower.tail=FALSE,log.p=TRUE) else qnorm(u, lower.tail=FALSE) } nln <- "\n----------------\n" for(a in (10^(2:8))/2) cat("p=", a, qbeta.R(.05, a, 1/2, low.bnd = 1e-9, qnormU = qnormU), nln) showProc.time() for(a in (10^(2:16))/2) cat("p=",a, qb <- qbeta.R(.05, a, 1/2, low.bnd = 0), 1-qb,nln) ## ---- -- SENSATIONAL ! -------- a <- 5e13; cat("p=",a, qb <- qbeta.R(.05, a, 1/2, low.bnd = 0), 1-qb,nln) #-- problem: 0 outer it. a <- 5e14; cat("p=",a, qb <- qbeta.R(.05, a, 1/2, low.bnd = 0), 1-qb,nln) #-- problem: 19 outer it a <- 5e15; cat("p=",a, qb <- qbeta.R(.05, a, 1/2, low.bnd = 0), 1-qb,nln) #-- the last one "working" a <- 5e16; cat("p=",a, qb <- qbeta.R(.05, a, 1/2, low.bnd = 0), 1-qb,nln) # still fails: init xinbta =1 for(a in (10^(2:16))/2) cat("p=",a, qb <- qbeta.R(.001, a, 1/2, low.bnd = 0), 1-qb,nln) ## ---- ## BOTH p & q are BIG: --- sometimes LONG loop ! for(a in (10^(2:15))/2) cat("p=",a, qb <- qbeta.R(.05, a, a/2, low.bnd = 0), 1-qb,nln) ## ---- !WOW! for(a in (10^(2:15))/2) cat("p=",a, qb <- qbeta.R(.05, a, a/2, low.bnd = 0, qnormU = qnormU, f.acu = function(...)1e-26),1-qb,nln) options(op)# revert to usual showProc.time() ## When using a new qbeta(.) C-function with low.bnd=0.0, up.bnd = 1.0 ##--- STILL FAILS at 10^10 ... why ? ... op <- options(warn=1) # immediate {there are only 2, now} for(a in (10^(2:16))/2) cat("p=",a, qb <- qbeta(.05, a, 1/2), 1-qb, " relE{pb, .05): ", formatC(pbeta(qb, a, 1/2)/.05 - 1), nln) options(op) showProc.time() ## Had tests for qbeta() built on AS 109 --- but now R *does* use AS 109 ! ## This will probably all be irrelevant now, see also ## ==> ~/R/MM/NUMERICS/dpq-functions/beta-gamma-etc/qbeta109.R pr.val <- c(.2,.4, 10^(-1:-5)); pr.val <- sort(c(pr.val, 1/2, 1-pr.val)) p.seq <- sort(outer(c(1,3),10^(3:16))) p.seq <- c(2^(-15:6),10^(2:18)) p.seq <- c(1e5^c(-4:-1), 10^c(-2:2,10:18)) ## q = 1 ---------- know exact formula ... for(x in pr.val) { cat("-----------\nx=", format01prec(x, digits = 6),":\n~~~~~~~~~~\n") for(p in p.seq) { t0 <- proc.time()[1] qb <- qbeta(x, p, 1) # takes forever C1 <- (t1 <- proc.time()[1]) - t0 pb <- pbeta(x^(1/p), p, 1) C2 <- (t2 <- proc.time()[1]) - t1 fcat( ## NO Debug: paste0(formatC(p,w = 9), "|",formatC(C1, digits = 2, wid = 5),"|"), ## In any case: format01prec(qb, digits = 10), qb - x^(1/p), qb /( x^(1/p))-1, pb - x ##, f.dig=3, f.wid=8 # << Debug ) } } summary(warnings()) showProc.time() ## q = 1/2 (T-distrib.) ## qt is defined as ## qt(x,n) := - sqrt((1 / qbeta( 2*x , n/2, 1/2) - 1) * n) ## for x <= 1/2 ## qt(x,n) := sqrt((1 / qbeta( 2*(1-x), n/2, 1/2) - 1) * n) ## for x > 1/2 for(x in pr.val) { cat("-----------\nx=", format01prec(x, digits = 6),":\n~~~~~~~~~~\n") for(p in p.seq[p.seq >= 1/2]) { qb0 <- 1/(1+ qt(x/2, df = 2*p)^2/(2*p)) t0 <- proc.time()[1] qb <- qbeta(x, p, 1/2) # takes forever C1 <- (t1 <- proc.time()[1]) - t0 pb <- pbeta(qb, p, 1/2) C2 <- (t2 <- proc.time()[1]) - t1 ## NO Debug: fcat(p,paste("|",t2-t0,"|",sep = ''),format01prec(c(qb,qb0),digits = 10), ## Debug: fcat(format01prec(qb,digits=10), paste("|",formatC(qb0-qb,dig = 8,w = 12)), qb/qb0 - 1, pb/x - 1, f.wid = 12) } } summary(warnings()) showProc.time() ### q = p !! for(x in pr.val) { cat("-----------\nx=", format01prec(x, digits = 6),":\n~~~~~~~~~~\n") for(p in p.seq[p.seq >= 1/2]) { #-- otherwise, qt(.) is not defined y <- 1/(1+ qt(1-x, 2*p)^2/(2*p)) qb0 <- 1/2 + ifelse(x < 1/2, -1, 1)* sqrt(1-y)/2 #-- should be = qbeta(..) t0 <- proc.time()[1] qb <- qbeta(x, p,p) C1 <- (t1 <- proc.time()[1]) - t0 pb <- pbeta(qb, p,p) C2 <- (t2 <- proc.time()[1]) - t1 ## NO Debug: fcat(p,paste("|",t2-t0,"|",sep = ''),format01prec(qb,digits = 10), paste("|",formatC(qb0-qb,dig = 8,w = 12)), qb/qb0 - 1, pb/x - 1, f.wid = 12) ## Debug: ##fcat(format01prec(qb,digits=10,flag='-'),paste("|",formatC(qb0-qb,dig=8,w=12)), ## qb/qb0 - 1, pb/x - 1, f.wid=12) } } summary(warnings()) showProc.time() ###---- 2008-11-12 : another example, not sure if covered above: ### Letting a --> 0 --- Distribution only defined for a > 0 --- ## curve(qbeta(.95, x, 20), 1e-300, 0.1, n=1001)# a -> 0 : qbeta() -> 0 ## now, "zoom in logarithmically": curve(qbeta(.95, x, 20), 1e-7, 0.1, log="xy", n=1001, yaxt="n") if(require("sfsmisc")) { eaxis(2, at= c(10^-c(314,309), axTicks(2, nintLog=15))) } else { axis(2) warning("could not use the cool eaxis() function from CRAN package 'sfsmisc' as that is not yet installed.") } ## -> clearly should go to 0; ## now does ... not continually, as with pbeta() warning about bgrat() underflow ## now without any warnings and continually: not 0 but staying at unique(qbeta(.95, 10^-(100:5), 20)) ## 5.5626846462680034577e-309 ## limit : qbeta(.95, 0, 20)# gave '1' instead of 0 --> now 0 {as for dbeta(*, a=0, *) ## Variation on the above (a --> 0) : ## --------- ## this seems ok; is "the" behavior at large" x. <- seq(0,1,len=1001); plot(x., pbeta(x., 1e-20,20), type="l") ## but if you log-zoom in , ## the precision problem is already in pbeta(): ----- fixed on 2009-10-16 (in Skafidia, Greece) ##-- TOMS 708: 'a0 small .. -> apser() curve(pbeta(exp(x), 1e-16, 10, log.p=TRUE), -10, 0, n=1000) curve(pbeta(exp(x), 1e-16, 10, log.p=TRUE), -900, 0, n=1000) ## _jumps_ to 0 - because exp() *must* [no bug in pbeta !] ## but this is somewhat similar, *not* using apser(), but L20, then ## a) (x < 0.29): bup() + L131 bgrat() & w = 1-w1 ~= 1 -- now fixed ## b) x >=0.29: bpser() xpb <- curve(pbeta(exp(x), 2e-15, 10, log.p=TRUE), -21, 0, n=1000) xpb <- within(as.data.frame(xpb), {exp.x <- exp(x); bp <- exp(x) >= 0.29}) ## unfinished bps <- bpser(2e-15, 10, x = with(xpb, exp.x[bp]), log.p=TRUE, verbose=TRUE) u <- seq(-16, 1.25, length=501) op <- mult.fig(mfrow=c(3,5), marP = c(0,-1.5,-1,-.8), main = expression(pbeta(e^u, alpha, beta, ~~ log=='TRUE')))$old.par for(b in c(.5, 2, 5)) for(a in c(.1, .2, .5, 1, 2)/100) { plot(u, (qp <- pbeta(exp(u), a,b, log.p=TRUE)), ylab = NA, type="l", main = substitute(list(alpha == a, beta == b), list(a=a,b=b))) } par(op) summary(warnings()) showProc.time() ###---- Another (or the same?) set of problems: ## From: Martin Maechler ## Sender: r-devel-bounces@r-project.org ## To: leydold@statistik.wu-wien.ac.at ## Cc: r-devel@r-project.org ## Subject: Re: [Rd] Buglet in qbeta? ## Date: Wed, 7 Oct 2009 16:04:47 +0200 ## >>>>> "JL" == Josef Leydold ## >>>>> on Wed, 7 Oct 2009 14:48:26 +0200 writes: ## JL> Hi, ## JL> I sometimes play around with extreme parameters for distributions and ## JL> found that qbeta is not always monotone as the following example shows. ## JL> I don't know whether this is serious enough to submit a bug report (as ## JL> this example is near to the limitations of floating point arithmetic). ## well, it's not near enough the limits to *not* warrant a bug ## report! ## I "know" (I have saved a collection of ..) about several ## accuracy problems of pbeta() / qbeta(), most cases indeed ## "borderline" (at the limits of FP arithmetic) but this one may ## well be a new one. ## A bit more succint than the dump of your numbers is a simple ## graphic p <- (1:100)/100; qp <- qbeta(p, 0.01, 5) plot(p,qp, type="o", log = "y") # now fine ## or, already suggesting a fix to the bug: plot(p,qp, type="o", log = "xy")## now fine ## which is definitely a bug... though not a terrible one. ## ... ## more experiments -- adapt to log-scale: if(!exists("lseq", mode="function"))# package "sfsmisc" lseq <- function (from, to, length) exp(seq(log(from), log(to), length.out = length)) p <- lseq(.005,1, len = 500) ; qp <- qbeta(p, 0.01,5) ## (no warnings) plot(p,qp, ylab = "qbeta(p, .01, 5)", type="l", log = "xy") ## p --> 0 even closer -- next problem {but that's easier forgivable ..} p <- lseq(.0005,1, len = 500) a <- .01 ; b <- 5 ## nomore (gave 36 warnings "qbeta: ..precision not reached ..") plot(p, (qp <- qbeta(p, a,b)), ylab = expression(qbeta(p, alpha, beta)), type="l", log = "xy", main = substitute(list(alpha == a, beta == b), list(a=a,b=b))) ## Seems almost independent of beta =: b op <- mult.fig(mfrow = c(3,5), marP = c(0,-.8,-1,-.8), main = quote('qbeta(p, , .) vs p for '~ {p %->% 0}))$old.par for(b in c(.5, 2, 5)) for(a in c(.1, .2, .5, 1, 2)/100) { # last one, a = 0.02 does not underflow plot(p, (qp <- qbeta(p, a,b)), ylab = expression(qbeta(p, alpha, beta)), col=2, type="l", log = "xy", main = substitute(list(alpha == a, beta == b), list(a=a,b=b))) } par(op) summary(warnings()) showProc.time() ## x <- qbeta((0:100)/100,0.01,5) ## x ## .... ## order(x) ## 1 2 3 .... 50 51 55 68 72 56 59 ... ## pbeta(x,0.01,5) ## ... ## >> version ## JL> _ ## JL> platform x86_64-unknown-linux-gnu ## JL> arch x86_64 ## JL> os linux-gnu ## JL> system x86_64, linux-gnu ## JL> status Under development (unstable) ## JL> major 2 ## JL> minor 11.0 ## JL> year 2009 ## JL> month 10 ## JL> day 07 ## JL> svn rev 49963 ## JL> language R ## JL> version.string R version 2.11.0 Under development (unstable) (2009-10-07 ## JL> r49963) ## JL> p.s. there are similar results for R-2.9.2 in Windows (with ## JL> different round-off errors). ## --> only vary a == alpha == shape1: p.qbeta.p0 <- function(p1 = .005, p2 = 1, beta = 1, mark.p = 0.5, alphas = c(.1, .15, .2, .3, .4, .5, .8, 1, 2)/100, n = 500, verbose = getOption("verbose")) { ## Purpose: Plot qbeta() investigations for small alpha & p --> 0 ## ---------------------------------------------------------------------- ## Arguments: ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 7 Oct 2009, 22:46 stopifnot(exprs = { require("sfsmisc") # lseq(), mult.fig() require("DPQ") # qbetaAppr() .. p1 < p2 beta > 0 alphas > 0 ; diff(alphas) > 0 n == round(n) ; n > 2 }) p <- lseq(p1, p2, len = n) tit <- expression(qbeta(p, alpha, beta) * " for small"~alpha~ " and " * {p %down% 0}) cols <- adjustcolor(1:3, 1/2); lwds <- c(2,4,2); ltys <- c(2,1,4) op <- mult.fig(length(alphas), main = tit, marP = c(-1,-1,-1,-.6))$old.par on.exit(par(op)) for(a in alphas) { matplot(p, cbind(qbetaAppr (p, a,beta, verbose=verbose), qbeta (p, a,beta), qbetaAppr.3(p, a,beta)), col = cols, lty = ltys, lwd = lwds, ylab = "qbeta", type= "l", log = "xy", main = substitute(list(alpha == a, beta == b), list(a=a, b=beta))) if(match(a, alphas) == length(alphas) %/% 2) legend("left", c("qbetaAppr", "qbeta", "qbeta.a..3"), col = cols, lty = ltys, lwd = lwds, inset = .02, bty="n") if(is.numeric(mark.p) && length(mark.p)) { abline( v = mark.p, lty = 3, lwd = 2, col = "light blue") axis(1, at = mark.p, line=-1, lwd = 2, col = "light blue") } } invisible(p) } p.qbeta.p0(.05, mark=NULL, verbose=2) p.qbeta.p0(.0005) p.qbeta.p0(.00002) ## but for beta <~ 0.5, the approximation seems problematic p.qbeta.p0(beta = 0.8) p.qbeta.p0(beta = 0.4)# approx. does not go up to 1 ## but then, the "doc" says to use it for x < 1/2 p.qbeta.p0(beta = 0.2) p.qbeta.p0(beta = 0.1)## -> first plot, alpha = .001 is "catastrophe" << still not ok p.qbeta.p0(.00001, beta = .0001)# here, the approx. is completely hopeless ## larger alphas : another problem: the *approximation* is bad! p.qbeta.p0(.001, 0.5, n=2000, alpha= c(.2, .5, 1, 2)) showProc.time() p <- lseq(.0001, 1, len = 2000) b <- 1 tit <- expression(qb == qbeta(p, alpha, beta) * " for small"~alpha~" and " * {p %down% 0}) aset <- c(.1, .15, .2, .3, .4, .5, .8, 1, 2)/100 op <- mult.fig(length(aset), main = tit, marP = c(-1,-1,-1,-.6))$old.par c2 <- adjustcolor(2, 1/2) for(a in aset) { qpA <- qbetaAppr(p, a,b) qp <- qbeta (p, a,b) plot (p, qpA, ylab = "qb", type="l", log = "xy", ylim=pmax(1e-20, range(qp, qpA)), main = substitute(list(alpha == a, beta == b), list(a=a,b=b))) lines(p, qp, col = c2, lwd=3) legend("bottom", c("qbetaAppr()", "qbeta()"), lty=1, lwd=c(1,3), col=c(palette()[1], c2), bty="n") } par(op) ## Ok, let's look at pbeta() "here" curve(pbeta(x, 0.01, 5, log.p=TRUE), 0, 1, n=1000) # fine (but not close enough to x = 0) ## log-zoom: curve(pbeta(x, 0.01, 5, log.p=TRUE), 1e-100, 1, n=10000, log="x") # fine curve(pbeta(x, 0.01, 5, log.p=TRUE), 1e-250, 1, n=10000, log="x") # fine ## it does *not* seem to be the problem of pbeta ## Ok, the "best" qbeta() is the one matching pbeta: p. <- lseq(1e-3, 1, length=200) qb <- qbeta (p., 0.02, 0.01) qb3 <- qbetaAppr.3(p., 0.02, 0.01) ## qbetaAppr.3() is good for *small* p., but otherwise breaks down *before* ## ------------ *qbeta()* cbind(p., pbeta(qb,0.02, 0.01), pbeta(qb3,0.02, 0.01)) showProc.time() ## Excursion: In order to check if the extreme-left tail approximation is ok, ## we need to compute log(p*beta(p,q)) accurately for small p ## This suffers from "cancellation" and is for *EXPERIMENTATION* ## --> c12pBeta() further below is for "production" c12.pBeta <- function(q, c2.only=TRUE) { ## Purpose: Compute 1st and 2nd coefficient of p*Beta(p,q)= 1 + c_1*p + c_2*p^2 + .. ## ---------------------------------------------------------------------- ## Arguments: q -- can be a vector ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 30 Oct 2009 stopifnot(q > 0) psi1 <- digamma(1)# == - (Euler's)gamma = -0.5772157 psiD1 <- trigamma(1) c1 <- psi1 - digamma(q) ## Note: psi(1)^2 - psi(q)^2 = (psi(1) - psi(q))*(psi(1)+psi(q)) = ## and (trigamma(1) - trigamma(q) + c1*(psi1+psi(q)))/2 - psi(q) * c1 ##.... c2 <- (psiD1 - trigamma(q) + c1^2)/2 ## Note: cancellation for q -> 0 !! ---> c12pBeta() below ## digamma(q) = psi (q) ~= -1/q + psi(1) + q*psi'(1) + q^2/2*psi''(1) +... ## trigamma(q) = psi'(q) ~= 1/q^2 + psi'(1) + q*psi''(1) ... ## ==> 2*c_2 = psi'(1) - psi'(q) + (psi(q) - psi(1))^2 = ## = - 1/q^2 -q*psi''(1) -.. + (-1/q + q*psi'(1) + q^2/2*psi''(1)+..)^2 = ## = -q*psi''(1) -2*psi'(1) - q*psi''(1) = 2*(-psi'(1)-q*psi''(1))+ O(q^2) ## ==> c_2(q) = -(psi'(1) + q*psi''(1)) + O(q^2) ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ if(c2.only) c2 else cbind(c1 = c1, c2 = c2) } c12.pBeta(1e-5,FALSE) summary(warnings()) showProc.time() ## c1 c2 ## [1,] 1e+05 -1.644912 trigamma(1) ## [1] 1.644934 ## which confirms the expansion of c_2(q) above curve(c12.pBeta, .001, 100) ##-> curve(c12.pBeta, 1, 1000, log="xy") ## dominated by -digamma(q)^2 / 2 ## Cancellation problem when q -> 0 as well ... probably not really relevant ## *but* fixed in c12pBeta() below: curve(c12.pBeta, 5e-9, 100, log="x") tit.pBeta <- function() { title("c12.pBeta(x) and its (log-)linear approximation") mtext(substitute(c[1] + c[2]*x == C1 + C2*x, setNames(as.list(cq12), c("C1", "C2"))), col=2) } cq12 <- -psigamma(1, 1:2) ## == c( -digamma(1), -trigamma(1) ) curve(cq12[1] + cq12[2]*x, add=TRUE, col=2, n=1000) tit.pBeta() ## a bit zoomed: curve(c12.pBeta, 1e-8, .2, log="x") curve(cq12[1] + cq12[2]*x, add=TRUE, col="red", n=1000); tit.pBeta() ## difference only: curve(c12.pBeta(x) - (cq12[1] + cq12[2]*x), 1e-8, .2, log="x", ylim = .01*c(-1,1)); abline(h=0, lty=3, col="gray60") ## | . - . | log-scaled: curve(abs(c12.pBeta(x) - (cq12[1] + cq12[2]*x)), 1e-8, .2, log="xy") abline(v= 4e-4, col="lightblue") ##==> use q = 4e-4 as cutoff qq <- lseq(1e-6, 1e-2, length= 64) 1 - (cq12[1] + cq12[2]*qq) / c12.pBeta(qq) plot(qq, 1 - c12.pBeta(qq)/(cq12[1] + cq12[2]*qq), type="o", log="x") plot(qq, abs(1 - c12.pBeta(qq)/(cq12[1] + cq12[2]*qq)), type="o", log="xy") showProc.time() ## Compute log(p*beta(p,q)) accurately for small p c12pBeta <- function(q, c2.only=FALSE, cutOff.q = 4e-4) { ## Purpose: Compute 1st and 2nd coefficient of p*Beta(p,q) = 1 + c_1*p + c_2*p^2 + .. ## NB: The Taylor expansion of log(p*Beta(p,q)) = c_1*p + d_2*p^2 + .. ## is simpler: c_1=d_1, and d_j = \psi^(j)(1) - \psi^(j)(q) = psigamma(1,j) - psigamma(q,j) ## ---------------------------------------------------------------------- ## Arguments: q -- can be a vector ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 30 Oct 2009 stopifnot(q > 0, cutOff.q >= 0) psi1 <- digamma(1) # == - (Euler's)gamma = -0.5772157 psiD1 <- trigamma(1)# == pi^2 / 6 ## vectorize in q: r <- q if(!c2.only) c1 <- psi1 - digamma(q) if(any(sml <- q < cutOff.q)) { ## To avoid cancellation, use Taylor expansion for small q: ## digamma(q)= psi (q) ~= -1/q + psi(1) + q*psi'(1)+q^2/2*psi''(1)+.. ## trigamma(q)= psi'(q) ~= 1/q^2+ psi'(1)+ q*psi''(1) ... ## ==> 2*c_2 = psi'(1) - psi'(q) + (psi(q) - psi(1))^2 = ## = - 1/q^2 -q*psi''(1) -.. + (-1/q + q*psi'(1) + q^2/2*psi''(1)+..)^2= ## = -q*psi''(1) -2*psi'(1) - q*psi''(1) = 2*(-psi'(1)-q*psi''(1))+ .. ## ==> c_2(q) = -(psi'(1) + q*psi''(1)) + O(q^2) ## ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ r[sml] <- -(psiD1 + psigamma(1,2) * q[sml]) } if(!all(sml)) { ## regular case: i.ok <- !sml q <- q[i.ok] c1. <- if(c2.only) psi1 - digamma(q) else c1[i.ok] ## Note: psi(1)^2 - psi(q)^2 = (psi(1) - psi(q))*(psi(1)+psi(q)) = ## and (trigamma(1) - trigamma(q) + c1*(psi1+psi(q)))/2 - psi(q) * c1 ##.... r[i.ok] <- (psiD1 - trigamma(q) + c1.^2)/2 } if(c2.only) r else cbind(c1 = c1, c2 = r) } tit <- quote(c[2](q) ~~~ "from Beta expansion" ~~ p*Beta(p,q) %~~% 1 + c[1]*p + c[2]*p^2) doX <- function(cutOff.q = eval(formals(c12pBeta)$cutOff.q)) { eaxis(1); py <- par("usr")[3:4] abline(h=0, v=1, lty=3) text(1, py[1]+0.95*diff(py), quote(q==1), adj=-.005) abline(v = cutOff.q, col=2, lty=2, lwd=2) text(cutOff.q, py[1]+0.9*diff(py), paste0("cutOff.q=",formatC(cutOff.q)), col=2, adj=-0.05) } curve(c12pBeta(x,TRUE), 5e-9, 100, log="x", n=1025, xaxt='n', xlab=quote(q), main = tit); doX()#--> no cancellation problems showProc.time() ## zoom in curve(c12pBeta(x,TRUE), 3e-4, 5e-4, log="x", n=1025, xaxt='n', xlab=quote(q), main = tit); doX()#--> no break.. showProc.time() ## zoom out curve(c12pBeta(x,TRUE), 1e-100, 1e110, log="x", n=1025, xaxt='n', xlab=quote(q), main = tit); doX()# .. neither problems .. curve(c12pBeta(x,TRUE), 1e-100, 1e110, log="x", n=1025, xaxt='n', ylim = c(-2,1), xlab=quote(q), main = tit); doX()# .. neither problems .. summary(warnings()) showProc.time() ## In order for the first oder approximation p*beta(p,q) = 1 + c_1*p to be ok, ## that c_2*p^2 << 1, i.e. p^2 < eps.c / c_2 <=> p < sqrt(eps.c / c_2) ## ===================== ## --> draw the cutoff line below ( --> 'p.cut') p.pBeta <- function(q, p.min=1e-20, p.max=1e-6, n=1000, do.ord2=TRUE, log = "xy", ylim=NULL) { ## Purpose: Plot log(p * beta(p,q)) for small p -- to visualize cancellation ## ---------------------------------------------------------------------- ## Arguments: ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 30 Oct 2009, 09:16 stopifnot(0 < p.min, p.min < p.max, n > 100, n == round(n), is.character(log), length(log) == 1, length(q) == 1, 0 < q) sgn <- if(q > 1) -1 else 1 curve(sgn*log(x*beta(x,q)), p.min, p.max, n=n, log=log, ylim=ylim, col="gray", lwd=3, ylab="", xlab = expression(p), axes = FALSE, frame.plot = TRUE, # rather use eaxis() below main = substitute("Accurately computing "~ log(p %.% B(p,q)) ~ " for "~ {p %->% 0} *", "~ (q == Q), list(Q=format(q,digits=15)))) eaxis(1); eaxis(2) curve(sgn*(log(x)+lbeta(x,q)), add = TRUE, col="red3", n=n) ## Here comes the 1st order cancellation solution c1 <- sgn*( digamma(1) - digamma(q)) c2 <- sgn*(trigamma(1) - trigamma(q))/2 colc1 <- adjustcolor("skyblue", 1/2) c1y <- curve(c1 * x, add = TRUE, col=colc1, lwd=2, n=n)$y ## the 2nd order approx. if(do.ord2) { colc2 <- "#60C07064" # opaque (!) c12y <- curve(x*(c1 + c2* x), add = TRUE, col=colc2, lwd=3, n=n)$y } ## rather: collog1 <- adjustcolor("forestgreen", 1/2) lc1y <- curve(sgn*log1p(sgn*c1*x), add = TRUE, col=collog1, lty=5, lwd=3, n=n)$y ## the cutoff, from where 1st order should be perfect: p.cut <- sqrt(.Machine$double.eps / abs(c2)) txt.cut <- "cutoff(1st order approx.)" cat(txt.cut,": ", formatC(p.cut),"\n", sep="") ## FIXME: These *badly* fail, if we do NOT have log = "xy" : ux <- par("xaxp")[1:2] # e.g. [1e-20, 1e-6] uy <- par("yaxp")[1:2] in.x <- function(f) { stopifnot(0 <= f, f <= 1); 10^c(c(1-f, f) %*% log10(ux)) } in.y <- function(f) { stopifnot(0 <= f, f <= 1); 10^c(c(1-f, f) %*% log10(uy)) } if(ux[1] <= p.cut & p.cut <= ux[2]) { abline(v = p.cut, col = "blue", lwd=2, lty=2) text(p.cut, in.y(.96), txt.cut, col="blue") } else if(p.cut > ux[2]) text(in.x(.98), in.y(0.5), paste("-->", txt.cut), col = "blue", lwd=2, lty=2) else if(p.cut < ux[1]) text(in.x(.02), in.y(0.5), paste(txt.cut, "<--"), col = "blue", lwd=2, lty=2) leg <- if(sgn == -1) expression(-log(p*B(p,q)), -(log(p) + log(B(p,q))), -(psi(1)-psi(q))*p, -log1p(c[1]*p)) else expression(log(p*B(p,q)), log(p) + log(B(p,q)), (psi(1)-psi(q))*p, log1p(c[1](q)*p)) if(do.ord2) { leg <- leg[c(1:4,4)]; leg[[4]] <- quote(p*(c[1](q) + c[2](q)* p)) } legend("topleft", legend= leg, inset=.01, lwd= c(3,1,2,if(do.ord2)3, 3), col= c("gray","red3", colc1, if(do.ord2)colc2, collog1)) mtext(R.version.string, side=4, cex = 3/5, adj=0) } showProc.time() p.pBeta(q = 1.1) ##q should not matter much; q very close to 1 will be another challenge ## zoom in: p.pBeta(1.1, p.max = 1e-12) p.pBeta( 11, p.max = 1e-11)## log(..) is not much worse here p.pBeta(101, p.max = 1e-11)## !! interestingly, here log(..) is not worse p.pBeta(1001, p.max = 1e-11)## both versions are identical showProc.time() ## q -> 1 p.pBeta(1.001, p.max = 1e-10) p.pBeta(1+1e-5, 1e-15, 1e-8)# problems start earlier! p.pBeta(1+1e-7, 1e-11, 1e-6)# problems start even earlier! p.pBeta(1+1e-9, 1e-9, 1e-4)# problems start even earlier! ## showProc.time() ## q < 1 p.pBeta(0.9) p.pBeta(0.1, p.max = 1e-8) p.pBeta(0.01) p.pBeta(0.0001) p.pBeta(0.0001,1e-23, 1e-4) p.pBeta(0.0001,1e-12, 1e-2) p.pBeta(0.0001,1e-6, 1e-3) p.pBeta(0.0001,1e-6, 1e-1) summary(warnings()) showProc.time() if(!require("Rmpfr")) quit("no") ##--===============---====------------ Rmpfr needed from here ----------------------------- p.err.pBeta <- function(q, p.min=1e-12, p.max=1e-6, n=1000, kind = c("relErr", "absErr", "error")) { ## Purpose: Plot log(p * beta(p,q)) for small p -- to visualize cancellation ## ---------------------------------------------------------------------- ## Arguments: ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 30 Oct 2009, 09:16 stopifnot(0 < p.min, p.min < p.max, n > 100, n == round(n), length(q) == 1, 0 < q) c12 <- c12pBeta(q) d2 <- (trigamma(1) - trigamma(q))/2 c1 <- c12[1] ## == digamma(1) - digamma(q) c2 <- c12[2] p <- lseq(p.min, p.max, length=n) T0 <- log(p*beta(p,q))# which we know has its problems, too T1 <- c1*p T2 <- p*(c1 + d2*p) T3 <- log1p(T1) T4 <- log1p(p*(c1 + c2*p)) ## Better: compute "true value" very accurately: stopifnot(require("Rmpfr")) p. <- mpfr(p, prec=200) tt <- log(p.*beta(p.,q)) p.cut <- sqrt(.Machine$double.eps / abs(c2)) cat("cutoff :", formatC(p.cut),"\n") err.mat <- as(tt - cbind(T0, T1, T3, T2, T4), "matrix")## classic numeric matrix leg <- expression(T - log(p*beta(p,q)) ~ " (double)", T - c[1]*p, # T1 T - log1p(c[1]*p), # T3 T - (c[1]*p+d[2]*p^2), # T2 T - log1p(c[1]*p+c[2]*p^2))# T4 stopifnot(length(leg) == ncol(err.mat)) main <- paste("log(p*beta(p,q)) approx. errors, q = ", formatC(q)) ## `col` below is from ## dput(RColorBrewer::brewer.pal(length(leg), "Dark2")) col <- c("#1B9E77", "#D95F02", "#E7298A", "#7570B3", "#66A61E") ## put the color which is "most gray" (i.e equal R,G,B values) first: nc <- length(col) cm <- col2rgb(col) i1 <- which.min(colSums(sweep(cm, 2, colMeans(cm))^2)) if(i1 > 1) col <- col[c(i1:nc, 1:(i1-1))] m.matplot <- function(x,y, log, main, ..., LEG, xy.leg) { log.. <- strsplit(log, "")[[1]] log.x <- any("x" == log..) log.y <- any("y" == log..) matplot(x,y, log=log, ..., xlab = quote(p), type = "l", col=col, main=main, xaxt = if(log.x) "n" else "s", yaxt = if(log.y) "n" else "s") if(log.x) eaxis(1, nintLog = 15) if(log.y) eaxis(2, nintLog = 20) # "FIXME" eaxis(): default nintLog=10 does not cover well legend(xy.leg, LEG, lty=1:5, col=col, inset=.01) } kind <- match.arg(kind) switch(kind, "error" = m.matplot(p, err.mat, log="x", main=main, ylab = "log(p*beta(p,q)) - 'approx.'", LEG = leg, xy.leg = "left"), "absErr" = { m.matplot(p, abs(err.mat), ## <- absolute -- log scale log="xy", main=main, ylab = "|log(p*beta(p,q)) - 'approx.'|", LEG = as.expression(lapply(leg, function(.) substitute(abs(EXPR), list(EXPR= .)))), xy.leg = "topleft") }, "relErr" = { m.matplot(p, abs(err.mat/as(tt,"numeric")), ## <- RELATIVE errors log="xy", main = paste( "log(p*beta(p,q)) RELATIVE approx. errors, q = ", formatC(q)), ylab = "|1 - 'approx'/log(p*beta(p,q))|", LEG = as.expression(lapply(leg, function(.) substitute(abs(EXPR), list(EXPR= .)))), xy.leg = "topleft") }, stop("invalid 'kind' : ", kind) )# end{switch} abline(v = p.cut, col = "blue", lwd=2, lty=2) mtext(R.version.string, side=4, cex = 3/5, adj=0) } ## with q >> 1, the log1p() approx. are *worse* (see more below) ! p.err.pBeta(21, 1e-18) # cutoff : 5.527e-09 p.err.pBeta(21, , 1e-3) summary(warnings()) showProc.time() if(doExtras) { # each plot is somewhat large & expensive ..------------------- ## hmm, with q >> 1, the log1p() approx. are *worse* .. p.err.pBeta(q = 2.1, kind="absErr") p.err.pBeta(q = 2.1, 1e-13, kind="absErr") p.err.pBeta(1.1, 1e-13, 0.1)## log1p() still very slightly *worse* ## here, q < 1 and log1p() are already (slightly) better p.err.pBeta(0.8) p.err.pBeta(0.8, 1e-10, 1e-2) p.err.pBeta(0.1) # log1p(<2 trms>) is best; zoom out (to guess "cutoff_2") p.err.pBeta(0.1, 1e-15, 1e-3) p.err.pBeta(0.001,, 1e-2)## Here, log1p(<2 terms>) is clearly best ## reminding of the test qbeta(1e-300, 2^-12, 2^-10): p.err.pBeta(2^-10, p.max = 2^-10); abline(v=2^-12, lty=3, col="gray20") ## --> ok, the (*, 2^-12, 2^-10) is *not* yet critical ##---> q < 1 ==> the log1p() versions are superior ##---> q > 1 ==> the direct versions are superior ## in both cases: The quadratic term is an order of magnitude better ## the above p.cut is "order-of-magnitude" correct, but not really numerically ok! print(summary(warnings())) showProc.time() }# only if(doExtras) ---------------------------------------------------------------------- pBeta <- function(p, q) { ## Purpose: Compute log(p * beta(p,q)) accurately also for small p ## ---------------------------------------------------------------------- ## Arguments: p: (vector) q: scalar ## ---------------------------------------------------------------------- ## Author: Martin Maechler, Date: 31 Oct 2009, 18:38 stopifnot(0 < p, 0 < q, length(q) == 1, (n <- length(p)) > 0) if(q == 1) return(0*p) c12 <- c12pBeta(q) d2 <- (trigamma(1) - trigamma(q))/2 c1 <- c12[1] ## == digamma(1) - digamma(q) c2 <- c12[2] ## the cutoff from where the approximation is "perfect" p.cut <- sqrt(.Machine$double.eps / abs(d2)) r <- p if(any(sml <- p <= p.cut)) { p. <- p[sml] r[sml] <- if(q < 1) log1p(p.*(c1 + c2*p.)) else p.*(c1 + d2*p.) } if(any(ok <- !sml)) { p <- p[ok] r[ok] <- log(p * beta(p, q)) } r } ### More on log(p * Beta(p,q)) for p << 1 ### ================== ## 1. Slightly nicer Gamma(.) plot than in example(Gamma) : ---------- op <- options(warn = -1) x <- sort(c(seq(-3, 4, length.out = 201), outer(0:-3, (-1:1)*1e-6, "+"))) plot(x, gamma(x), ylim = c(-20,20), col = "red", type = "l", lwd = 2, main = expression(Gamma(x))) abline(h = 0, v = -3:0, lty = 3, col = "midnightblue") ## Warning .. in gamma(x) : NaNs produced axis(4, at=factorial(1:3), las=2, cex.axis=3/4) abline(h=1, lty=4, col="thistle") for(t in c("p","h")) points(1:4, gamma(1:4), pch=4, type=t, lty=2) ## Now try, comparing with Rmpfr exact values: if(!require("Rmpfr")) { message("Cannot attach 'Rmpfr' package"); q("no") } a. <- 1/256 # is exact! b. <- 1/2 # " " a <- mpfr(a., 512) b <- mpfr(b., 512) (aBa1 <- log(a*beta(a,b))) ## 0.0053902550661545715269612410527830537772316302960547912531493485401255362864762019.... (aBa2 <- log(a)+lbeta(a,b))# ## 0.0053902550661545715269612410527830537772316302960547912531493485401255362864762019.... ## how close? asNumeric((aBa1-aBa2)*2/(aBa1+aBa2)) # -3.426748e-152; (NB 512 bits ~= 154.1 digits) ## Double precison accuracy dput(log(a. * beta(a.,b.))) ## 0.00539025506615481 dput(log(a.)+lbeta(a.,b.))# (to my surprise!) slightly worse ## 0.00539025506615509 asNumeric( log(a. * beta(a.,b.))/(log(a)+lbeta(a,b)) - 1) # 4.504144e-14 asNumeric((log(a.)+lbeta(a.,b.))/(log(a)+lbeta(a,b)) - 1) # 9.653358e-14 psigamma(1) # -0.5772157 = Euler's gamma ## Use my approx: a.* (psigamma(1) - psigamma(b.)) ## 0.005415212 a.* (psigamma(1) - psigamma(b.) + a./2*(psigamma(1, d=1) - psigamma(b., d=1))) ## 0.005390113 {clearly better, still only ~ 4 correct digits showProc.time()