[SciPy-User] orthogonal polynomials ?

[Charles R Harris]

On Mon, May 16, 2011 at 9:17 AM, Charles R Harris <charlesr.harris at gmail.com
> wrote:

>
>
> On Mon, May 16, 2011 at 9:11 AM, <josef.pktd at gmail.com> wrote:
>
>> On Sat, May 14, 2011 at 6:04 PM, Charles R Harris
>> <charlesr.harris at gmail.com> wrote:
>> >
>> >
>> > On Sat, May 14, 2011 at 2:26 PM, <josef.pktd at gmail.com> wrote:
>> >>
>> >> On Sat, May 14, 2011 at 4:11 PM, nicky van foreest <
>> vanforeest at gmail.com>
>> >> wrote:
>> >> > Hi,
>> >> >
>> >> > Might this be what you want:
>> >> >
>> >> > The first eleven probabilists' Hermite polynomials are:
>> >> >
>> >> > ...
>> >> >
>> >> > My chromium browser does not seem to paste pngs. Anyway, check
>> >> >
>> >> >
>> >> > http://en.wikipedia.org/wiki/Hermite_polynomials
>> >> >
>> >> > and you'll see that the first polynomial is 1, the second x, and so
>> >> > forth. From my courses on quantum mechanics I recall that these
>> >> > polynomials are, with respect to some weight function, orthogonal.
>> >>
>> >> Thanks, I haven't looked at that yet, we should add wikipedia to the
>> >> scipy.special docs.
>> >>
>> >> However, I would like to change the last part "with respect to some
>> >> weight function"
>> >> http://en.wikipedia.org/wiki/Hermite_polynomials#Orthogonality
>> >>
>> >> Instead of Gaussian weights I would like uniform weights on bounded
>> >> support. And I have never seen anything about changing the weight
>> >> function for the orthogonal basis of these kind of polynomials.
>> >>
>> >
>> > In numpy 1.6, you can use the Legendre polynomials. They are orthogonal
>> on
>> > [-1,1] as has been mentioned, but can be mapped to other domains. For
>> > example
>> >
>> > In [1]: from numpy.polynomial import Legendre as L
>> >
>> > In [2]: for i in range(5): plot(*L([0]*i + [1],
>> domain=[0,1]).linspace())
>> >    ...:
>> >
>> > produces the attached plots.
>>
>> I'm still on numpy 1.5 so this will have to wait a bit.
>>
>> > <snip>
>> >
>> > Chuck
>> >
>>
>>
>> as a first application for orthogonal polynomials I was trying to get
>> an estimate for a density, but I haven't figured out the weighting
>> yet.
>>
>> Fourier polynomials work better for this.
>>
>>
> You might want to try Chebyshev then, the Cheybyshev polynomialas are
> essentially cosines and will handle the ends better. Weighting might also
> help, as I expect the distribution of the errors are somewhat Poisson.
>
>
I should mention that all the polynomial fits will give you the same
results, but the Chebyshev fits are more numerically stable. The general
approach is to overfit, i.e., use more polynomials than needed and then
truncate the series resulting in a faux min/max approximation. Unlike power
series, the coefficients of the Cheybshev series will tend to decrease
rapidly at some point.

Chuck

> Chuck
>
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