Suggested change to integrate.Rd (was: Re: 0.5 != integrate(dnorm, 0, 20000) = 0)

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