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.

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.

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.

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.

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.