[Rd] unstable corner of parameter space for qbeta?

[J C Nash]

Given that a number of us are housebound, it might be a good time to try to
improve the approximation. It's not an area where I have much expertise, but in
looking at the qbeta.c code I see a lot of root-finding, where I do have some
background. However, I'm very reluctant to work alone on this, and will ask
interested others to email off-list. If there are others, I'll report back.

Ben: Do you have an idea of parameter region where approximation is poor?
I think that it would be smart to focus on that to start with.

Martin: On a separate precision matter, did you get my query early in year about double
length accumulation of inner products of vectors in Rmpfr? R-help more or
less implied that Rmpfr does NOT use extra length. I've been using David
Smith's FM Fortran where the DOT_PRODUCT does use double length, but it
would be nice to have that in R. My attempts to find "easy" workarounds have
not been successful, but I'll admit that other things took precedence.

Best,

John Nash



On 2020-03-26 4:02 a.m., Martin Maechler wrote:
>>>>>> Ben Bolker 
>>>>>>     on Wed, 25 Mar 2020 21:09:16 -0400 writes:
> 
>     > I've discovered an infelicity (I guess) in qbeta(): it's not a bug,
>     > since there's a clear warning about lack of convergence of the numerical
>     > algorithm ("full precision may not have been achieved").  I can work
>     > around this, but I'm curious why it happens and whether there's a better
>     > workaround -- it doesn't seem to be in a particularly extreme corner of
>     > parameter space. It happens, e.g., for 	these parameters:
> 
>     > phi <- 1.1
>     > i <- 0.01
>     > t <- 0.001
>     > shape1 = i/phi  ##  0.009090909
>     > shape2 = (1-i)/phi  ## 0.9
>     > qbeta(t,shape1,shape2)  ##  5.562685e-309
>     > ##  brute-force uniroot() version, see below
>     > Qbeta0(t,shape1,shape2)  ## 0.9262824
> 
>     > The qbeta code is pretty scary to read: the warning "full precision
>     > may not have been achieved" is triggered here:
> 
>     > https://github.com/wch/r-source/blob/f8d4d7d48051860cc695b99db9be9cf439aee743/src/nmath/qbeta.c#L530
> 
>     > Any thoughts?
> 
> Well,  qbeta() is mostly based on inverting pbeta()  and pbeta()
> has *several* "dangerous" corners in its parameter spaces
> {in some cases, it makes sense to look at the 4 different cases
>  log.p = TRUE/FALSE  //  lower.tail = TRUE/FALSE  separately ..}
> 
> pbeta() itself is based on the most complex numerical code in
> all of base R, i.e., src/nmath/toms708.c  and that algorithm
> (TOMS 708) had been sophisticated already when it was published,
> and it has been improved and tweaked several times since being
> part of R, notably for the log.p=TRUE case which had not been in
> the focus of the publication and its algorithm.
> [[ NB: part of this you can read when reading  help(pbeta)  to the end ! ]]
> 
> I've spent many "man weeks", or even "man months" on pbeta() and
> qbeta(), already and have dreamed to get a good student do a
> master's thesis about the problem and potential solutions I've
> looked into in the mean time.
> 
> My current gut feeling is that in some cases, new approximations
> are necessary (i.e. tweaking of current approximations is not
> going to help sufficiently).
> 
> Also not (in the R sources)  tests/p-qbeta-strict-tst.R
> a whole file of "regression tests" about  pbeta() and qbeta()
> {where part of the true values have been computed with my CRAN
> package Rmpfr (for high precision computation) with the
> Rmpfr::pbetaI() function which gives arbitrarily precise pbeta()
> values but only when  (a,b) are integers -- that's the "I" in pbetaI().
> 
> Yes, it's intriguing ... and I'll look into your special
> findings a bit later today.
> 
> 
>   > Should I report this on the bug list?
> 
> Yes, please.  Not all problem of pbeta() / qbeta() are part yet,
> of R's bugzilla data base,  and maybe this will help to draw
> more good applied mathematicians look into it.
> 
> 
> 
> Martin Maechler
> ETH Zurich and R Core team
> (I'd call myself the "dpq-hacker" within R core -- related to
>  my CRAN package 'DPQ')
> 
> 
>     > A more general illustration:
>     > http://www.math.mcmaster.ca/bolker/misc/qbeta.png
> 
>     > ===
>     > fun <- function(phi,i=0.01,t=0.001, f=qbeta) {
>     > f(t,shape1=i/phi,shape2=(1-i)/phi, lower.tail=FALSE)
>     > }
>     > ## brute-force beta quantile function
>     > Qbeta0 <- function(t,shape1,shape2,lower.tail=FALSE) {
>     > fn <- function(x) {pbeta(x,shape1,shape2,lower.tail=lower.tail)-t}
>     > uniroot(fn,interval=c(0,1))$root
>     > }
>     > Qbeta <- Vectorize(Qbeta0,c("t","shape1","shape2"))
>     > curve(fun,from=1,to=4)
>     > curve(fun(x,f=Qbeta),add=TRUE,col=2)
> 
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