[SciPy-User] orthonormal polynomials (cont.)

[josef.pktd at gmail.com]

On Tue, Jun 14, 2011 at 4:40 PM, Charles R Harris
<charlesr.harris at gmail.com> wrote:
>
>
> On Tue, Jun 14, 2011 at 12:10 PM, nicky van foreest <vanforeest at gmail.com>
> wrote:
>>
>> Hi,
>>
>> Without understanding the details... I recall from numerical recipes
>> in C that Gram Schmidt is a very risky recipe. I don't know whether
>> this advice also pertains to fitting polynomials, however,

I read some warnings about the stability of Gram Schmidt, but the idea
is that if I can choose the right weight function, then we should need
only a few polynomials. So, I guess in that case numerical stability
wouldn't be very relevant.

>>
>> Nicky
>>
>> On 14 June 2011 18:58,  <josef.pktd at gmail.com> wrote:
>> > (I'm continuing the story with orthogonal polynomial density
>> > estimation, and found a nice new paper
>> > http://www.informaworld.com/smpp/content~db=all~content=a933669464 )
>> >
>> > Last time I managed to get orthonormal polynomials out of scipy with
>> > weight 1, and it worked well for density estimation.
>> >
>> > Now, I would like to construct my own orthonormal polynomials for
>> > arbitrary weights. (The weights represent a base density around which
>> > we make the polynomial expansion).
>> >
>> > The reference refers to Gram-Schmidt or Emmerson recurrence.
>> >
>> > Is there a reasonably easy way to get the polynomial coefficients for
>> > this with numscipython?
>> >
>
> What do you mean by 'polynomial'? If you want the values of a set of
> polynomials orthonormal on a given set of points, you want the 'q' in a qr
> factorization of a (row) weighted Vandermonde matrix.  However, I would
> suggest using a weighted chebvander instead for numerical stability.

Following your suggestion last time to use QR, I had figured out how
to get the orthonormal basis for a given set of points.
Now, I would like to get the functional version (not just for a given
set of points), that is an orthonormal polynomial basis like Hermite,
Legendre, Laguerre and Jacobi, only for any kind of weight function,
where the weight function is chosen depending on the data.

In the paper they use a mixture of 3 normal distributions as weight
function in one example.

I have no idea if I can approximate the orthonormal polynomial basis
just on a predefined set of points, and use QR, since it needs to be
defined for a continuous domain.

>
> You can also solve for the three term recursion (Emerson?), but that is more
> work.

(I would prefer not to do more work, I would like to apply it and not
fight with numerical problems. :)

Thanks,

Josef

>
> Chuck
>
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