https://github.com/j-jith/orthopoly

Generate orthogonal polynomials for arbitrary probability density functions
https://github.com/j-jith/orthopoly

orthogonal-polynomials pce polynomial-chaos-expansion

Last synced: about 2 months ago
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Generate orthogonal polynomials for arbitrary probability density functions

• Host: GitHub
• URL: https://github.com/j-jith/orthopoly
• Owner: j-jith
• License: mit
• Created: 2017-12-09T15:04:02.000Z (about 7 years ago)
• Default Branch: master
• Last Pushed: 2017-12-09T15:07:16.000Z (about 7 years ago)
• Last Synced: 2023-10-19T18:54:04.795Z (about 1 year ago)
• Topics: orthogonal-polynomials, pce, polynomial-chaos-expansion
• Language: Python
• Homepage:
• Size: 146 KB
• Stars: 6
• Watchers: 2
• Forks: 2
• Open Issues: 1
• Metadata Files:
  • Readme: README.rst
  • License: LICENSE

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README

        
OrthoPoly

#########

**OrthoPoly** is a Python class for generating orthogonal polynomials with

respect to arbitrary probability density functions. The primary application of

this script is in performing *Arbitrary* Polynomial Chaos Expansion (PCE) in

uncertainty quantification studies.

Installation

============

Orthopoly is written in Python 3 and requires the following packages to be

present in your system:

- `numpy `_

- `scipy `_

Since OrthoPoly is a simple class, no special installation steps are

required. You can simply copy ``orthopoly.py`` to your working directory,

and start using it.

Usage

=====

OrthoPoly can be imported into your script as follows.

.. code:: python

   from orthopoly import OrthoPoly

To create an instance of OrthoPoly, you need to supply a probability density

function (pdf). The code below defines the pdf of a random variable created by

adding a uniform random variable and a normal random variable. This pdf is used

to create an instance of OrthoPoly, and orthogonal polynomials are generated

with respect to the supplied pdf.

.. code:: python

    def pdf(z, coeffs):

        mu, sigma, a, b = coeffs

        return 0.5/(b-a) * ( erf((z-a-mu)/sigma/sqrt(2)) - erf((z-b-mu)/sigma/sqrt(2)) )

    pp = OrthoPoly(pdf, margs=[0, 1, -1, 1])

    pp.gen_poly(5)

The function ``gen_poly()`` takes as an argument the largest order of the

polynomial to be generated, and populates the variable ``pp.poly`` with the

appropriate polynomials (which are ``numpy.polynomial.polynomial``).

For numerical integration with respect to these polynomials, the quadrature

points and weights can be obtained as follows.

.. code:: python

   points, weights = pp.get_quad_rule()

Numerical integration can also be performed directly using the ``quadrature()``

function. For instance, to compute the mean of the supplied pdf, one can do the

following.

.. code:: python

   mean = pp.quadrature(lambda x: x)

For more examples, please look at the ``__main__`` section of ``orthopoly.py``,

as well as the ``main.py`` script. ``main.py`` is a script which generates

orthogonal polynomials for the pdf of a random variable which is a sum of a

uniform and a normal random variable. It generates the polynomials, calculates

the quadrature points and weights, and writes them to an output file.

References

==========

To understand the mathematics behind OrthoPoly, please take a look at

`notes.pdf `_. For a more in-depth reading, consult

[gautschi1982]_ and [golub1969]_.

.. [gautschi1982] Gautschi W. On Generating Orthogonal Polynomials. SIAM J Sci

   and Stat Comput 1982;3:289–317. doi:10.1137/0903018.

.. [golub1969] Golub GH, Welsch JH. Calculation of Gauss Quadrature Rules.

   Mathematics of Computation 1969;23:221. doi:10.2307/2004418.