Greatly enrich my mind.
Thank you!
David
2013/10/16 Spencer Graves <spencer.gra...@structuremonitoring.com>
> On 10/15/2013 5:37 PM, Marino David wrote:
>
> Hi Spencer:
>
> Thanks for your interpretation again and again. Your statement does
> enable me to have a good understanding of Gaussian quadrature.
>
> This sos package you recommended is greatly powerful. From now on, I will
> use the sos package to find something helpful before I do some research.
>
> Yes, I want to compute the expection of functions whos variables follow
> common distriubtions (e.g. F, t, Beta, Gamma, etc.). Via searching
> referrences, I know that Gaussian quadrature based on orthogonal
> polynomials is fast method for integration.
>
>
>
> Often, the most expensive part of any data analysis is the time of
> the person doing it. For that reason, it's often wise to first try the
> quickest thing available. Just for fun, I performed the following tests on
> the F distribution:
>
>
> f. <- Fd(1,1)
> sinF <- sin(f.)
> E(sinF)
> # 0.2093619
>
>
> This didn't work for round(sinF). For that, we can try the
> following:
>
>
> rf. <- r(sinF)(10000)
> mean(round(rf.))
> # 0.197
>
>
> If I were only doing this once, I wouldn't mess with quadrature. If
> the options like "integrate" or the above take too much compute time, then
> I'll consider other options, like numerical integration with quadrature.
> However, orthogonal polynomials for a continuous distribution won't work
> properly with a discontinuous transformation like round ;-)
>
>
> Hope this helps.
> Spencer
>
>
> David
>
>
> 2013/10/14 Spencer Graves <spencer.gra...@structuremonitoring.com>
>
>> David:
>>
>>
>> What you have is close, but I perceive some problems:
>>
>>
>> integral{from -Inf to Inf of f(x)t(x)dx} = integral{from 0 to
>> Inf of (f(-x)+f(x))t(x)dx}, because Student's t distribution is symmetric.
>>
>>
>> Now do the change of variables x = sqrt(z), so dx =
>> 0.5*dz/sqrt(z). Then t(x)dx = 0.5*t(sqrt(z))dz/sqrt(z). Play with this
>> last expression a bit, and you should get it into the form of g(z)dz, where
>> g(z) = the density for the F distribution.
>>
>>
>> Next transform the F distribution to a beta distribution on [0, 1],
>> NOT a beta distribution on [-1, 1]. There are Jacobi polynomials on [0, 1]
>> you can use. Or further transform the interval [0, 1] to [-1, 1].
>>
>>
>> Did you look at the literature search results I sent you using
>> findFn{sos}? When I need to do something new in statistics, the first
>> thing I do is a literature search like I described, ending with using the
>> installPackages and writeFindFn2xls functions, as described in the sos
>> vignette. That rarely takes more than a minute or two. The
>> writeFindFn2xls function should create an Excel file in your working
>> directory, which you can find with getwd(). Open that. The first sheet is
>> a summary of the different packages. This gives you a list of different
>> packages. You can then use that to prioritize your further study.
>>
>>
>> Two more comments:
>>
>>
>> 1. It is conceptually quite simple to write an algorithm to
>> compute polynomials that are orthonormal relative to any distribution. The
>> Wikipedia article on "Orthogonal polynomials" gives a set of linear
>> equations that must be solved to create them.
>>
>>
>> 2. Why do you want orthogonal polynomials? To obtain a very
>> fast algorithm for computing the expected values of a certain class of
>> functions? If no, have you considered doing without orthogonal polynomials
>> and just computing the expected value of whatever function you want using
>> the distr package to compute the distribution of f(X) and E{distrEx} to
>> compute the expected value?
>>
>>
>> Best Wishes,
>> Spencer
>>
>>
>> p.s. Could you please post a summary of this exchange to R-help, so
>> someone else with a similar question a year from now can find it? Thanks.
>>
>>
>>
>> On 10/13/2013 10:12 PM, Marino David wrote:
>>
>> Hi Spencer:
>>
>> I still have trouble in understanding your response to email
>> about Gaussian quadrature. I tried to describe it in detail. See attachment.
>>
>> Thank you!
>>
>> David
>>
>>
>>
>> --
>> Spencer Graves, PE, PhD
>> President and Chief Technology Officer
>> Structure Inspection and Monitoring, Inc.
>> 751 Emerson Ct.
>> San José, CA 95126
>> ph: 408-655-4567
>> web: www.structuremonitoring.com
>>
>>
>
>
> --
> Spencer Graves, PE, PhD
> President and Chief Technology Officer
> Structure Inspection and Monitoring, Inc.
> 751 Emerson Ct.
> San José, CA 95126
> ph: 408-655-4567
> web: www.structuremonitoring.com
>
>
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