[SciPy-User] Orthogonal polynomials on the unit circle

josef.pktd@gmai... josef.pktd@gmai...
Sat Oct 27 10:34:26 CDT 2012


On Sat, Oct 27, 2012 at 10:35 AM, Charles R Harris
<charlesr.harris@gmail.com> wrote:
>
>
> On Fri, Oct 26, 2012 at 7:40 PM, <josef.pktd@gmail.com> wrote:
>>
>> http://en.wikipedia.org/wiki/Orthogonal_polynomials_on_the_unit_circle
>> with link to handbook
>>
>> application: goodness of fit for circular data
>>
>> http://onlinelibrary.wiley.com/doi/10.1111/j.1467-842X.2009.00558.x/abstract
>>
>> Are those available anywhere in python land?
>>
>
> Well, we have the trivial case: ϕ_n⁡(z)=z^n for the uniform measure. That
> reduces to the usual exp(2*pi*i*\theta) in angular coordinates when the
> weight is normalized. But I think you want more ;-)  I don't know of any
> collection of such functions for python.

I need to see if I can use this. In general, I would like other weight
functions
(Von Mises distribution in the density estimation example (?), like
hermite polynomials for the normal distribution).

I don't know much about the math of circular statistics and functions,
I just want to estimate distribution densities on a circle, and I
discovered that periodic or circular polynomials would be useful for
estimating seasonal/periodic effects. (the clock as a circle)
The ends don't match up with chebychev
https://picasaweb.google.com/106983885143680349926/Joepy#5747376116689698434

>
>> What's the difference between orthogonal polynomials on the unit
>> circle and periodic polynomials like Fourier series?
>
>
> It looks to be the weight. Also, the usual Fourier series include terms in
> 1/z which allows for real functions. I suspect there is some finagling that
> can be done to make things go back and forth, but I am unfamiliar with the
> topic. Hmm, Laurent polynomials on the unit circle might be more what you
> are looking for, see the reference at http://dlmf.nist.gov/18.33 .

Might we worth looking into, but this "finagling" usually turns out to
be very time consuming for me, where I don't have the background and
no pre-made recipes.

(Might be just finding the right coordinate system, or it might mean I
would have to look into complex random variables.)

Thank you,

Josef

>
> Chuck
>
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