# [Rd] Calculation of e^{z^2/2} for a normal deviate z

Serguei Sokol @oko| @end|ng |rom |n@@-tou|ou@e@|r
Mon Jun 24 10:00:43 CEST 2019

On 22/06/2019 00:58, jing hua zhao wrote:
> Hi Peter, Rui, Chrstophe and Gabriel,
>
> Thanks for your inputs --  the use of qnorm(., log=TRUE) is a good point
Another approach could be simply to note that a function defined as
f(p)=exp(-z(p)^2/2) is regular around p=0 with f(0)=0.
It has roughly the shape of p*(2-p) for p \in [0; 1]. So we can
calculate let say f(10^-10) with sufficient precision using Rmpfr and
then use a linear approximation for p from [0, 10^-10]. After that a
simple inverse gives us e^(z*z/2).

Serguei.

>   in line with pnorm with which we devised log(p)  as
>
> log(2) + pnorm(-abs(z), lower.tail = TRUE, log.p = TRUE)
>
> that could do really really well for large z compared to Rmpfr. Maybe I am asking too much since
>
> z <-20000
>> Rmpfr::format(2*pnorm(mpfr(-abs(z),100),lower.tail=TRUE,log.p=FALSE))
> [1] "1.660579603192917090365313727164e-86858901"
>
> already gives a rarely seen small p value. I gather I also need a multiple precision exp() and their sum since exp(z^2/2) is also a Bayes Factor so I  get log(x_i )/sum_i log(x_i) instead. To this point, I am obliged to clarify, see https://statgen.github.io/gwas-credible-sets/method/locuszoom-credible-sets.pdf.
>
> I agree many feel geneticists go to far with small p values which I would have difficulty to argue againston the other hand it is also expected to see these in a non-genetic context. For instance the Framingham study was established in 1948 just got $34m for six years on phenotypewide association which we would be interesting to see. > > Best wishes, > > > Jing Hua > > > ________________________________ > From: peter dalgaard <pdalgd using gmail.com> > Sent: 21 June 2019 16:24 > To: jing hua zhao > Cc: Rui Barradas; r-devel using r-project.org > Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z > > You may want to look into using the log option to qnorm > > e.g., in round figures: > >> log(1e-300) > [1] -690.7755 >> qnorm(-691, log=TRUE) > [1] -37.05315 >> exp(37^2/2) > [1] 1.881797e+297 >> exp(-37^2/2) > [1] 5.314068e-298 > > Notice that floating point representation cuts out at 1e+/-308 or so. If you want to go outside that range, you may need explicit manipulation of the log values. qnorm() itself seems quite happy with much smaller values: > >> qnorm(-5000, log=TRUE) > [1] -99.94475 > > -pd > >> On 21 Jun 2019, at 17:11 , jing hua zhao <jinghuazhao using hotmail.com> wrote: >> >> Dear Rui, >> >> Thanks for your quick reply -- this allows me to see the bottom of this. I was hoping we could have a handle of those p in genmoics such as 1e-300 or smaller. >> >> Best wishes, >> >> >> Jing Hua >> >> ________________________________ >> From: Rui Barradas <ruipbarradas using sapo.pt> >> Sent: 21 June 2019 15:03 >> To: jing hua zhao; r-devel using r-project.org >> Subject: Re: [Rd] Calculation of e^{z^2/2} for a normal deviate z >> >> Hello, >> >> Well, try it: >> >> p <- .Machine$double.eps^seq(0.5, 1, by = 0.05)
>> z <- qnorm(p/2)
>>
>> pnorm(z)
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>> p/2
>> # [1] 7.450581e-09 1.228888e-09 2.026908e-10 3.343152e-11 5.514145e-12
>> # [6] 9.094947e-13 1.500107e-13 2.474254e-14 4.080996e-15 6.731134e-16
>> #[11] 1.110223e-16
>>
>> exp(z*z/2)
>> # [1] 9.184907e+06 5.301421e+07 3.073154e+08 1.787931e+09 1.043417e+10
>> # [6] 6.105491e+10 3.580873e+11 2.104460e+12 1.239008e+13 7.306423e+13
>> #[11] 4.314798e+14
>>
>>
>> p is the smallest possible such that 1 + p != 1 and I couldn't find
>> anything to worry about.
>>
>>
>> R version 3.6.0 (2019-04-26)
>> Platform: x86_64-pc-linux-gnu (64-bit)
>> Running under: Ubuntu 19.04
>>
>> Matrix products: default
>> BLAS:   /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.8.0
>> LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.8.0
>>
>> locale:
>>   [1] LC_CTYPE=pt_PT.UTF-8       LC_NUMERIC=C
>>   [3] LC_TIME=pt_PT.UTF-8        LC_COLLATE=pt_PT.UTF-8
>>   [5] LC_MONETARY=pt_PT.UTF-8    LC_MESSAGES=pt_PT.UTF-8
>>   [7] LC_PAPER=pt_PT.UTF-8       LC_NAME=C
>>   [9] LC_ADDRESS=C               LC_TELEPHONE=C
>> [11] LC_MEASUREMENT=pt_PT.UTF-8 LC_IDENTIFICATION=C
>>
>> attached base packages:
>> [1] stats     graphics  grDevices utils     datasets  methods
>> [7] base
>>
>> other attached packages:
>>
>> [many packages loaded]
>>
>>
>> Hope this helps,
>>
>> Rui Barradas
>>
>> �s 15:24 de 21/06/19, jing hua zhao escreveu:
>>> Dear R-developers,
>>>
>>> I am keen to calculate exp(z*z/2) with z=qnorm(p/2) and p is very small. I wonder if anyone has experience with this?
>>>
>>> Thanks very much in advance,
>>>
>>>
>>> Jing Hua
>>>
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>>>
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> --
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
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