# [R] Solved (??) Behaviour of integrate (was 'Poisson-lognormal probab ility calculations')

k.jewell at campden.co.uk k.jewell at campden.co.uk
Mon Feb 18 13:25:26 CET 2008

```Hi Again,

I think I've solved my problem, but please tell me if you think I'm wrong,
or you can see a better way!

A plot of the integrand showed a very sharp peak, so I was running into the
integrand "feature" mentioned in the note. I resolved it by limiting the
range of integration as shown here:
--------------------------------------------------
function (x, meanlog = 0, sdlog = 1, sdlim=6, rel.tol =
.Machine\$double.eps^0.5,...) {
#+
# K. Jewell Feb 2008
# Based on VGAM::dpolono, modified to cover wider range of x
# Some simplification, but major change is to avoid integration problems by
#  limiting range of integration to sdlim (default = 6) standard deviations
each side
#  of mean of Poisson or lognormal. Also increase default precision of
integration.
#  For x in 0:170 (working range of VGAM::dpolono) and default parameters,
max deviation
#  from VGAM::dpolono is 3.243314e-05 of VGAM::dpolono result
#-
require(stats)
mapply(function(x, meanlog, sdlog, ...){
lower <- min(max(0, x-sdlim*sqrt(x)), exp(meanlog-sdlim*sdlog))
upper <- max(x+sdlim*sqrt(x), exp(meanlog+sdlim*sdlog))
integrate(function(t, x, meanlog, sdlog)
exp(dpois(x,t,TRUE) + dlnorm(t, meanlog, sdlog, TRUE)),
lower = lower, upper = upper, x = x, meanlog = meanlog, sdlog
= sdlog,
rel.tol = rel.tol, ...)\$value
},
x, meanlog, sdlog, ...
)
}
---------------------------------------------------------------

Best regards,

Keith Jewell
mailto:k.jewell at campden.co.uk telephone (direct) +44 (0)1386 842055

> -----Original Message-----
> From: Jewell, Keith
> Sent: 15 February 2008 16:48
> To: 'r-help at r-project.org'
> Subject: Behaviour of integrate (was 'Poisson-lognormal probability
> calculations')
>
> Hi again,
>
> Adding further information to my own query, this function gets to the
> core of the problem, which I think lies in the behaviour of
> 'integrate'.
> -------------------------------------
> function (x, meanlog = 0, sdlog = 1, ...) {
>     require(stats)
>     integrand <- function(t, x, meanlog, sdlog) dpois(x,t)*dlnorm(t,
> meanlog, sdlog)
>     mapply(function(x, meanlog, sdlog, ...)
> #                (1/gamma(x+1))*
>          integrate(function(t, x, meanlog, sdlog)
> #                   gamma(x+1)*
>               integrand(t, x, meanlog, sdlog),
>               lower = 0, upper = Inf, x = x, meanlog = meanlog, sdlog =
> sdlog, ...)\$value,
>            x, meanlog, sdlog, ...
>            )
>                                            }
> ----------------------------------------
> Mathematically, the presence or not of the two commented lines should
> make no difference; they multiply the integrand by a constant (with
> respect to the integration), then divide the result by the same
> constant. In practice they make a big difference! I guess they're
> altering the behaviour of the 'integrate'.
>
> I'd have thought the presence of the lines would worsen the behaviour.
> Without the lines the integrand is reasonably small, the integral is <
> 1. With the lines the limit on the integral is x!,  leading to "non-
> finite function values" for x much > 170, even if we use logs to get
> around the limit on gamma(x).
>
> In fact with the lines the plot of function(x) v. x looks reasonable
> (but I don't know if the values are correct!!), but without the lines
> it looks silly, I just don't believe it!
>
> I thought the problem might relate to the note in ?integrate "If the
> function is approximately constant (in particular, zero) over nearly
> all its range it is possible that the result and error estimate may be
> seriously wrong.". I wouldn't really expect multiplication by a large
> constant to fix such errors (??), but the lognormal distribution is
> skew so it might be considered "approximately constant ... over nearly
> all its range". Even
> though ...
> > integrate(dlnorm, 0, Inf)
> 1 with absolute error < 2.5e-07
> ... suggests that this is not the source of the problem,  I tried
> changing variables to integrate over a normally distributed variable:
> ----------------
> function (x, meanlog = 0, sdlog = 1, ...) {
>     require(stats)
>     integrand <- function(t, x, meanlog, sdlog)
> dpois(x,exp(t))*dnorm(t, meanlog, sdlog)
>     mapply(function(x, meanlog, sdlog, ...)
>  #              (1/gamma(x+1))*
>          integrate(function(t, x, meanlog, sdlog)
>  #                  gamma(x+1)*
>               integrand(t, x, meanlog, sdlog),
>               lower = -Inf, upper = Inf, x = x, meanlog = meanlog,
> sdlog = sdlog, ...)\$value,
>            x, meanlog, sdlog, ...
>            )
>                                            }
> -----------------------------
> Still no better; with the constants the values look reasonably smooth
> (but are they correct??), without the constants the values are silly.
>
> I've tried reducing rel.tol and increasing subdivisions, they change
> the behaviour a little but don't "fix" it, and I still get the marked
> difference between the presence and absence of those lines (and I'm
> increasingly unsure whether either answer is correct!).
>
> I'm still trying. but I really think I'm going nowhere. Has anyone any
> ideas?
>
>
> Keith Jewell
> mailto:k.jewell at campden.co.uk telephone (direct) +44 (0)1386 842055
> > -----Original Message-----
> > From: Jewell, Keith
> > Sent: 15 February 2008 11:16
> > To: 'r-help at r-project.org'
> > Subject: Poisson-lognormal probability calculations
> >
> > Hi,
> >
> > just for the record, although I don't think it's relevant (!)
> > -------------------------------------
> > > sessionInfo()
> > R version 2.6.0 (2007-10-03)
> > i386-pc-mingw32
> >
> > locale:
> > LC_COLLATE=English_United Kingdom.1252;LC_CTYPE=English_United
> > Kingdom.1252;LC_MONETARY=English_United
> > Kingdom.1252;LC_NUMERIC=C;LC_TIME=English_United Kingdom.1252
> >
> > attached base packages:
> > [1] stats4    splines   stats     graphics  grDevices utils
> > datasets
> > methods   base
> >
> > other attached packages:
> > --------------------------------
> >
> > I'm having some problems with a Poisson-lognormal density (mass?)
> > function.
> >
> > VGAM has the dpolono function, but that doesn't work for x-values
> over
> > 170, and I need to go to *much* bigger numbers. It fails first
> because
> > of gamma overflow, then because of non-finite integrand.
> > ---------------------
> > > VGAM::dpolono(170)
> > [1] 4.808781e-09
> > > VGAM::dpolono(171)
> > [1] 0
> > Warning message:
> > In VGAM::dpolono(171) : value out of range in 'gammafn'
> > > VGAM::dpolono(172)
> > [1] 0
> > Warning message:
> > In VGAM::dpolono(172) : value out of range in 'gammafn'
> > > VGAM::dpolono(173)
> > [1] 0
> > Warning message:
> > In VGAM::dpolono(173) : value out of range in 'gammafn'
> > > VGAM::dpolono(174)
> > Error in integrate(f = integrand, lower = -Inf, upper = Inf, x =
> x[i],
> > :
> >   non-finite function value
> > -------------
> >
> > I tidied up a little (to my eyes only, no offence intended to the
> > estimable authors of VGAM) and avoided the gamma overflow by using
> logs
> > - OK so far, it agrees almost perfectly with VGAM::dpolono for x up
> to
> > 170, and now extends the range up to 173 (wow!!).
> > -------------
> > dApolono <-
> > function (x, meanlog = 0, sdlog = 1, ...) {
> >     require(stats)
> >     integrand <- function(t, x, meanlog, sdlog) exp(-t+x*log(t)-
> > log(sdlog*t*sqrt(2*pi))-0.5*((log(t)-meanlog)/sdlog)^2)
> >     mapply(function(x, meanlog, sdlog, ...){
> >        temp <- try(
> >          integrate(f = integrand, lower = 0, upper = Inf, x = x,
> > meanlog = meanlog, sdlog = sdlog, ...)
> >                   )
> >          ifelse(inherits(temp, "try-error"), NA, exp(log(temp\$value)-
> > lgamma(x+1)))
> >                                       }
> >               , x, meanlog, sdlog, ...
> >            )
> >                                            }
> > plot(log(dApolono(0:173)))
> > dApolono(174)
> > -----------------------------
> >
> > Addressing the non-finite integrand, I noticed that the gamma(x+1)
> > divisor was outside the integrand so that as x gets bigger the
> > integrand gets bigger, only to be finally divided by an increasing
> > gamma(x+1). I reasoned that putting the divisor inside the integral
> > might incur a substantial performance hit, but would keep the
> integrand
> > at reasonable values.
> >
> > --------------------------------------
> > dApolono <-
> > function (x, meanlog = 0, sdlog = 1, ...) {
> >     require(stats)
> >     integrand <- function(t, x, meanlog, sdlog) exp(-t+x*log(t)-
> > log(sdlog*t*sqrt(2*pi))-0.5*((log(t)-meanlog)/sdlog)^2-lgamma(x+1))
> >     mapply(function(x, meanlog, sdlog, ...){
> >        temp <- try(
> >          integrate(f = integrand, lower = 0, upper = Inf, x = x,
> > meanlog = meanlog, sdlog = sdlog, ...)
> >                   )
> >          ifelse(inherits(temp, "try-error"), NA, temp\$value)
> >                                       }
> >               , x, meanlog, sdlog, ...
> >            )
> >                                            }
> > plot(log(dApolono(0:173)))
> > dApolono(174)
> > dApolono(1E3)
> > -------------------------
> >
> > This avoids the non-finite integrand and gives me answers at much
> > higher x, but now at least some of the answers are wrong (not to say
> > silly), even in the range where the other versions worked. I've tried
> > other variations including changing the variable of integration to
> > log(t) and integrating dpois()*dlnorm(), but I can't fix it; if the
> > factorial (=gamma) is inside the integrand I get silly answers, if
> it's
> > outside I get non-finite integrand.
> >
> > I'm tearing my hair. Can anyone suggest where I may be going wrong?
> Any
> > suggestions at all will be appreciated.
> >
> >
> > Keith Jewell
> > mailto:k.jewell at campden.co.uk telephone (direct) +44 (0)1386 842055
> >

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