R help - Nov 2011 - how to use quadrature to integrate some complicated functions

Hello to all,

I am having trouble with intregrating a complicated uni-dimensional function
of the following form

Phi(x-a_1)*Phi(x-a_2)*...*Phi(x-a_{n-1})*phi(x-a_n).

Here n is about 5000, Phi is the cumulative distribution function of
standard normal, 
phi is the density function of standard normal, and x ranges over
(-infty,infty).

My idea is to to use quadrature to handle this integral. But since Phi has
not cloaed form,
I don't know how to do this effeciently. I appreciate very much if someone
has any ideas about it.

Thanks!

Jeff

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Have you tried wrapping it in a function and using integrate()? R is pretty good
at handling numerical integration. If integrate() isn't good for you, can
you say more as to why?

Michael


On Nov 6, 2011, at 4:15 PM, JeffND <Zuofeng.Shang.5 at nd.edu> wrote:
> Hello to all,
> 
> I am having trouble with intregrating a complicated uni-dimensional
function
> of the following form
> 
> Phi(x-a_1)*Phi(x-a_2)*...*Phi(x-a_{n-1})*phi(x-a_n).
> 
> Here n is about 5000, Phi is the cumulative distribution function of
> standard normal, 
> phi is the density function of standard normal, and x ranges over
> (-infty,infty).
> 
> My idea is to to use quadrature to handle this integral. But since Phi has
> not cloaed form,
> I don't know how to do this effeciently. I appreciate very much if
someone
> has any ideas about it.
> 
> Thanks!
> 
> Jeff
> 
> --
> View this message in context:
http://r.789695.n4.nabble.com/how-to-use-quadrature-to-integrate-some-complicated-functions-tp3996765p3996765.html
> Sent from the R help mailing list archive at Nabble.com.
> 
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide
http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

It's polite to include the whole list in your replies so the threads
get archived properly.

Yes, underflow is occasionally a problem: hence the common use of
log-likelihood in MLE and other applications. Might that help you out
here? You can get log-likelihoods directly from pnorm and dnorm with
the log = TRUE option.

Michael

On Tue, Nov 8, 2011 at 11:35 AM, Zuofeng Shang <Zuofeng.Shang.5 at nd.edu>
wrote:
> Hi Michael , Thanks for your suggestion!
>
> Integrate() possibly workx for my problem. However my function is a product
> of
> a sequence with values between zero and one, which could be extremely small
> when the length of the sequence is large. Note: my integrand is like
>
> Phi(x-a_1)*Phi(x-a_2)*...*Phi(x-a_{n-1})*phi(x-a_n).
>
> So when "n" is very small, the above product will be very close
to zero.
>
> So my concern is whether integrate() can handle accurately such integrand.
>
>
> Thanks a lot for your time!
> Best wishes,
> Zuofeng
>
>
>
> ? 2011/11/8 8:43, R. Michael Weylandt <michael.weylandt at gmail.com>
??:
>>
>> Have you tried wrapping it in a function and using integrate()? R is
>> pretty good at handling numerical integration. If integrate() isn't
good for
>> you, can you say more as to why?
>>
>> Michael
>>
>>
>> On Nov 6, 2011, at 4:15 PM, JeffND<Zuofeng.Shang.5 at nd.edu>
?wrote:
>>
>>> Hello to all,
>>>
>>> I am having trouble with intregrating a complicated uni-dimensional
>>> function
>>> of the following form
>>>
>>> Phi(x-a_1)*Phi(x-a_2)*...*Phi(x-a_{n-1})*phi(x-a_n).
>>>
>>> Here n is about 5000, Phi is the cumulative distribution function
of
>>> standard normal,
>>> phi is the density function of standard normal, and x ranges over
>>> (-infty,infty).
>>>
>>> My idea is to to use quadrature to handle this integral. But since
Phi
>>> has
>>> not cloaed form,
>>> I don't know how to do this effeciently. I appreciate very much
if
>>> someone
>>> has any ideas about it.
>>>
>>> Thanks!
>>>
>>> Jeff
>>>
>>> --
>>> View this message in context:
>>>
http://r.789695.n4.nabble.com/how-to-use-quadrature-to-integrate-some-complicated-functions-tp3996765p3996765.html
>>> Sent from the R help mailing list archive at Nabble.com.
>>>
>>> ______________________________________________
>>> R-help at r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>
>
> --
>
>
>