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.
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
>
>
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