[Rd] Suggested change to integrate.Rd (was: Re: 0.5 != integrate(dnorm, 0, 20000) = 0)
John Nolan
jpnolan at american.edu
Wed Dec 8 05:08:33 CET 2010
R developers understand intimately how things work, and terse
descriptions are sufficient. However, most typical R users
would benefit from clearer documentation. In multiple places
I've found the R documentation to be correct and understandable
AFTER I've figured a function out.
And to be fair, this problem with integrate( ) isn't really R's
fault: the QUADPACK routines that R uses are very good algorithms,
but neither they nor any other package can handle all cases.
I would support reasonable changes in the documentation for
integrate( ). Just saying it "gives wrong answer without
warning on many systems" seems misleading (it works fine in
many cases) and it doesn't help a user understand how to use
integrate( ) correctly/carefully. IMO a simple example like
this one w/ dnorm would catch peoples attention and a couple
lines of explanation/warning would then make more sense.
John Nolan, American U
-----Spencer Graves <spencer.graves at structuremonitoring.com> wrote: -----
To: John Nolan <jpnolan at american.edu>
From: Spencer Graves <spencer.graves at structuremonitoring.com>
Date: 12/07/2010 07:58PM
Cc: pchausse at uwaterloo.ca, r-devel at r-project.org
Subject: Suggested change to integrate.Rd (was: Re: [Rd] 0.5 != integrate(dnorm,0,20000) = 0)
What do you think about changing the verbiage with that example
in "integrate.Rd" from "fails on many systems" to something like
"gives wrong answer without warning on many systems"?
If I had write access to the core R code, I'd change this
myself: I'm probably not the only user who might think that saying
something "fails" suggest it gives an error message. Many contributions
on this thread make it clear that it will never be possible to write an
integrate function that won't give a "wrong answer without warning" in
some cases.
Thanks,
Spencer
#############################
On 12/7/2010 7:02 AM, John Nolan wrote:
> Putting in Inf for the upper bound does not work in general:
> all 3 of the following should give 0.5
>
>> integrate( dnorm, 0, Inf )
> 0.5 with absolute error< 4.7e-05
>
>> integrate( dnorm, 0, Inf, sd=100000 )
> Error in integrate(dnorm, 0, Inf, sd = 1e+05) :
> the integral is probably divergent
>
>> integrate( dnorm, 0, Inf, sd=10000000 )
> 5.570087e-05 with absolute error< 0.00010
>
> Numerical quadrature methods look at a finite number of
> points, and you can find examples that will confuse any
> algorithm. Rather than hope a general method will solve
> all problems, users should look at their integrand and
> pick an appropriate region of integration.
>
> John Nolan, American U.
>
>
> -----r-devel-bounces at r-project.org wrote: -----
> To: r-devel at r-project.org
> From: Pierre Chausse
> Sent by: r-devel-bounces at r-project.org
> Date: 12/07/2010 09:46AM
> Subject: Re: [Rd] 0.5 != integrate(dnorm,0,20000) = 0
>
> The warning about "absolute error == 0" would not be sufficient
> because if you do
> > integrate(dnorm, 0, 5000)
> 2.326323e-06 with absolute error< 4.6e-06
>
> We get reasonable absolute error and wrong answer. For very high upper
> bound, it seems more stable to use "Inf". In that case, another
> .External is used which seems to be optimized for high or low bounds:
>
> > integrate(dnorm, 0,Inf)
> 0.5 with absolute error< 4.7e-05
>
>
> On 10-12-07 8:38 AM, John Nolan wrote:
>> I have wrestled with this problem before. I think correcting
>> the warning to "absolute error ~<= 0" is a good idea, and printing
>> a warning if subdivisions==1 is also helpful. Also, including
>> a simple example like the one that started this thread on the
>> help page for integrate might make the issue more clear to users.
>>
>> But min.subdivisions is probably not. On the example with dnorm( ),
>> I doubt 3 subdivisions would work. The problem isn't that
>> we aren't sudividing enough, the problem is that the integrand
>> is 0 (in double precision) on most of the region and the
>> algorithm isn't designed to handle this. There is no way to
>> determine how many subdivisions are necessary to get a reasonable
>> answer without a detailed analysis of the integrand.
>>
>> I've gotten useful results with integrands that are monotonic on
>> the tail with a "self-triming integration" routine
>> like the following:
>>
>>> right.trimmed.integrate<- function( f, lower, upper, epsilon=1e-100, min.width=1e-10, ... ) {
>> + # trim the region of integration on the right until f(x)> epsilon
>> +
>> + a<- lower; b<- upper
>> + while ( (b-a>min.width)&& (f(b)<epsilon) ) { b<- (a+b)/2 }
>> +
>> + return( integrate(f,a,b,...) ) }
>>
>>> right.trimmed.integrate( dnorm, 0, 20000 ) # test
>> 0.5 with absolute error< 9.2e-05
>>
>> This can be adapted to left trim or (left and right) trim, abs(f(x)-c)>epsilon,
>> etc. Setting the tolerances epsilon and min.width is an issue,
>> but an explicit discussion of these values could encourage people to
>> think about the problem in their specific case. And of course, none
>> of this guarantees a correct answer, especially if someone tries this
>> on non-monotonic integrands with complicated 0 sets. One could write
>> a somewhat more user-friendly version where the user has to specify
>> some property (or set of properties) of the integrand, e.g. "right-tail
>> decreasing to 0", etc. and have the algorithm try to do smart
>> trimming based on this. But perhaps this getting too involved.
>>
>> In the end, there is no general solution because any solution
>> depends on the specific nature of the integrand. Clearer messages,
>> warnings in suspicious cases like subdivisions==1, and a simple
>> example explaining what the issue is in the help page would help
>> some people.
>>
>> John
>>
>> ...........................................................................
>>
>> John P. Nolan
>> Math/Stat Department
>> 227 Gray Hall
>> American University
>> 4400 Massachusetts Avenue, NW
>> Washington, DC 20016-8050
>>
>> jpnolan at american.edu
>> 202.885.3140 voice
>> 202.885.3155 fax
>> http://academic2.american.edu/~jpnolan
>> ...........................................................................
>>
>> -----r-devel-bounces at r-project.org wrote: -----
>> To: r-devel at r-project.org, Prof Brian Ripley<ripley at stats.ox.ac.uk>
>> From: Martin Maechler
>> Sent by: r-devel-bounces at r-project.org
>> Date: 12/07/2010 03:29AM
>> Subject: Re: [Rd] 0.5 != integrate(dnorm,0,20000) = 0
>>
>>>>>>> Prof Brian Ripley<ripley at stats.ox.ac.uk>
>>>>>>> on Tue, 7 Dec 2010 07:41:16 +0000 (GMT) writes:
>> > On Mon, 6 Dec 2010, Spencer Graves wrote:
>> >> Hello:
>> >>
>> >>
>> >> The example "integrate(dnorm,0,20000)" says it "fails on many systems".
>> >> I just got 0 from it, when I should have gotten either an error or something
>> >> close to 0.5. I got this with R 2.12.0 under both Windows Vista_x64 and
>> >> Linux (Fedora 13); see the results from Windows below. I thought you might
>> >> want to know.
>>
>> > Well, isn't that exactly what the help page says happens? That
>> > example is part of a section entitled
>>
>> > ## integrate can fail if misused
>>
>> > and is part of the illustration of
>>
>> > 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.
>>
>> yes, of course,
>> and the issue has been known for ``ages'' ..
>> ..........
>> ..........
>> but it seems that too many useRs are not reading the help
>> page carefully, but only browse it quickly.
>> I think we (R developers) have to live with this fact
>> and should consider adapting to it a bit more, particularly in
>> this case (see below)
>>
>> >>
>> >> Thanks for all your work in creating and maintaining R.
>> >>
>> >>
>> >> Best Wishes,
>> >> Spencer Graves
>> >> ###############################
>>
>> >>
>> >> integrate(dnorm,0,20000) ## fails on many systems
>>
>> >> 0 with absolute error< 0
>>
>> and this is particularly unsatisfactory for another reason:
>>
>> "absolute error< 0"
>> is *always* incorrect, so I think we should change *some*thing.
>>
>> We could just use "<=" (and probably should in any case, or
>> "< ~= x" which would convey ``is less than about x'' which I
>> think is all we can say),
>> but could consider giving a different message when the integral
>> evaluates to 0 or, rather actually,
>> only when the error bound ('abs.error') is 0 *and* 'subdivisions == 1'
>> as the latter indicates that the algorithm treated the integrand
>> f(.) as if f() was a linear function.
>>
>> But in my quick experiments, even for linear (incl. constant)
>> functions, the 'abs.error' returned is never 0.
>>
>> If we want to be cautious,
>> such a warning could be made explicitly suppressable by an argument
>> .warn.if.doubtful = TRUE
>>
>> An additional possibility I'd like to try, is a new argument
>> 'min.subdivisions = 3' which specifies the *minimal* number
>> of subdivisions to be used in addition to the already present
>> 'subdivisions = 100' (= the maximum number of subintervals.)
>>
>> Martin Maechler,
>> ETH Zurich
>>
>> ______________________________________________
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