From cberry at tajo.ucsd.edu  Wed Oct 10 20:03:22 2007
From: cberry at tajo.ucsd.edu (Charles C. Berry)
Date: Wed, 10 Oct 2007 11:03:22 -0700
Subject: [Rd] pt inaccurate when x is close to 0 (PR#9945)
In-Reply-To: <470D0788.4080307@stats.uwo.ca>
References: <20071010143513.11DD128800A8@mail.pubhealth.ku.dk>
	<470D0788.4080307@stats.uwo.ca>
Message-ID: <Pine.LNX.4.64.0710101036020.10430@tajo.ucsd.edu>

On Wed, 10 Oct 2007, Duncan Murdoch wrote:

> On 10/10/2007 10:35 AM, skylab.gupta at gmail.com wrote:
>> Full_Name: Skylab Gupta
>> Version: R version 2.5.1 (2007-06-27)
>> OS: Windows XP
>> Submission from: (NULL) (216.82.144.137)
>>
>>
>> Hello,
>>
>> I have been playing around with the statistical distributions in R. I think the
>> computations for the cumulative distribution function of the students t
>> distribution in R are not very accurate.
>>
>> For instance, the cdf of a students t distribution with 13 degrees of freedom at
>> 1e-4 is reported in R as "0.5000391350986764"; from Mathematica, it seems the
>> correct value is "0.50003913510150055", only about 9 accurate digits reported in
>> R.
>
> As Charles Berry told you when this was posted to R-help, it looks as
> though it is Mathematica that is inaccurate.  For example, I would
> expect this plot to be smooth, and it is not in either R or Mathematica,
> but R is at least monotone:
>
> # the Mathematica values
> plot(diff(z[80:100]), type='l')
>
> The R values
> plot(diff(pt(1e-4, df=80:100)), type='l')
>

Further, if one truly needs to get highly accurate values for

 	pt( near.zero, df )

recognize that dt(x, df ) is nearly quadratic around x==0 and dominated by 
a linear component for x > 0.

So, simple quadrature gets the area under the density for (0, near.zero] 
quite accurately. One knows that pt(0, df) is exactly 0.5, so this can be 
added to get the result.

This one point quadrature rule is accurate to better than 3e-14 for every 
df %in% 1:100 :

really.simple.values <- 0.5 +
 	sapply( 1:100, function(y) dt( 0.5e-04, y ) * 1e-04 )

Three point Gaussian quadrature (is overkill and) seems accurate up to 
machine precision.

Chuck

>>
>> I also did the following from within R:
>>
>> -------------
>> df<-seq(1,100,by=1)
>> y<-pt(1e-4,df)
>> z<-c(0.50003183098839799,0.50003535533895194,0.50003675525997071,0.50003749999985481,0.50003796066840744,0.50003827327749706,0.50003849914427922,0.50003866990364754,0.50003880349244212,0.50003891083995444,0.50003899897813187,0.50003907263208447,0.50003913510150055,0.50003918874627440,0.50003923531785055,0.50003927612461441,0.50003931217478748,0.50003934425324170,0.50003937297989520,0.50003939886014204,0.50003942229165621,0.50003944360703978,0.50003946308016112,0.50003948094039441,0.50003949738053710,0.50003951256485324,0.50003952663295181,0.50003953969680248,0.50003955185925653,0.50003956322006460,0.50003957385523301,0.50003958382054481,0.50003959318443636,0.50003960200394315,0.50003961032679112,0.50003961818144815,0.50003962562026172,0.50003963266089213,0.50003963934773465,0.50003964569404735,0.50003965173577758,0.50003965749688895,0.50003966298323521,0.50003966823056478,0.50003967322766096,0.50003967801868676,0.50003968260005904,0.50003968700228751,0.50003969121916547,0.
> 500
>>  03969526955183,0.50003969915340063,0.50003970290428668,0.50003970650705731,0.50003970997149927,0.50003971332909936,0.50003971654204993,0.50003971964040972,0.50003972264367180,0.50003972553808163,0.50003972835715427,0.50003973106835642,0.50003973370765664,0.50003973624942966,0.50003973868896101,0.50003974107556448,0.50003974338818691,0.50003974563557085,0.50003974781567961,0.50003974993203681,0.50003975199594708,0.50003975399737965,0.50003975593675354,0.50003975782715593,0.50003975966389691,0.50003976145762119,0.50003976321975063,0.50003976489560775,0.50003976655049909,0.50003976818673812,0.50003976975798736,0.50003977127434285,0.50003977277055756,0.50003977423495483,0.50003977566285773,0.50003977705769798,0.50003977841313474,0.50003977975147973,0.50003978102874791,0.50003978230822732,0.50003978356836509,0.50003978477872879,0.50003978596096421,0.50003978713049724,0.50003978827577344,0.50003978935715154,0.50003979045422919,0.50003979153680134,0.50003979256756137,0.500039793
> 589
>>  57851,0.50003979462027492)
>>
>> plot(df,(y-z)/z, type="s")
>> -------------
>>
>> In the above R code, df contains the 100 integers between 1-100, y contains the
>> cdf of the students t distribution computed at 1e-4 from R, for all the df
>> degrees of freedom; and z contains the correct values (to 17 decimal digits) of
>> the students t distribution cdf at 1e-4 computed from Mathematica; when I plot
>> the relative errors between the computed values from Mathematica and R, it seems
>> the relative errors are large; we get only about 10-12 digits of accuracy from R
>> rather than about 15 digits (all this assuming that the Mathematica computed
>> values are correct).
>
> It seems you are making a bad assumption.
>
> Duncan Murdoch
>
>
>
>  This happens for all values close to 0 where the cdf is
>> evaluated.
>>
>> I am working on Windows XP, and I installed a precompiled binary version of R.
>> The following information might also be useful:
>>
>> ---------------
>>> sessionInfo()
>> R version 2.5.1 (2007-06-27)
>> i386-pc-mingw32
>>
>> locale:
>> LC_COLLATE=English_United States.1252;LC_CTYPE=English_United
>> States.1252;LC_MONETARY=English_United
>> States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252
>>
>> attached base packages:
>> [1] "stats"     "graphics"  "grDevices" "utils"     "datasets"  "methods"
>> "base"
>>
>>> version
>> platform       i386-pc-mingw32
>> arch           i386
>> os             mingw32
>> system         i386, mingw32
>> status
>> major          2
>> minor          5.1
>> year           2007
>> month          06
>> day            27
>> svn rev        42083
>> language       R
>> version.string R version 2.5.1 (2007-06-27)
>> ---------------
>>
>> Is there a reason for this loss of accuracy, or am I missing something here?
>> Thanks.
>>
>> ______________________________________________
>> R-devel at r-project.org mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-devel
>
> ______________________________________________
> R-devel at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-devel
>

Charles C. Berry                            (858) 534-2098
                                             Dept of Family/Preventive Medicine
E mailto:cberry at tajo.ucsd.edu	            UC San Diego
http://famprevmed.ucsd.edu/faculty/cberry/  La Jolla, San Diego 92093-0901



[Rd] pt inaccurate when x is close to 0 (PR#9945)
Charles C. Berry