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orthopoly.py
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orthopoly.py
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from __future__ import print_function, division
import scipy.integrate as integrate
import numpy as np
import numpy.polynomial.polynomial as nppoly
from scipy.special import erf
from math import sqrt
class OrthoPoly(object):
def __init__(self, measure, **kwargs):
self.measure = measure
self.measure_args = kwargs.get('margs', None)
self.poly = []
self.jacobi = None
self.order = None
self.epsrel = kwargs.get('epsrel', 1e-6)
self.intlims = kwargs.get('intlims', [-np.inf, np.inf])
def gen_poly(self, n):
self.order = n
# zeroth polynomial
self.poly = [nppoly.polyone]
alpha = [self.get_alpha(self.poly[0])]
beta = [1.]
# first polynomial
self.poly.append(nppoly.polymulx(self.poly[0]))
self.poly[1] = nppoly.polyadd(self.poly[1], -alpha[0]*self.poly[0])
alpha.append(self.get_alpha(self.poly[1]))
beta.append(self.get_beta(self.poly[1], self.poly[0]))
# reccurence relation for other polynomials
for i in range(2, n+1):
p_i = nppoly.polymulx(self.poly[i-1])
p_i = nppoly.polyadd(p_i, -alpha[i-1] * self.poly[i-1])
p_i = nppoly.polyadd(p_i, -beta[i-1] * self.poly[i-2])
self.poly.append(p_i)
alpha.append(self.get_alpha(self.poly[i]))
beta.append(self.get_beta(self.poly[i], self.poly[i-1]))
# normalise polynomials
for i in range(len(self.poly)):
self.poly[i] = self.poly[i] / np.prod(beta[:i])
# create Jacobi matrix
self.jacobi = (np.diag(np.sqrt(beta[1:]), -1)
+ np.diag(alpha, 0)
+ np.diag(np.sqrt(beta[1:]), 1))
def eval(self, x, **kwargs):
n = kwargs.get('i', None)
if n == None:
y = np.zeros((len(x), self.order))
for i in range(self.order):
y[:, i] = nppoly.polyval(x, self.poly[i])
return y
else:
return nppoly.polyval(x, self.poly[n])
def integrate(self, p, lim1, lim2):
if self.measure_args:
return integrate.quad(lambda x: nppoly.polyval(x, p) *
self.measure(x, self.measure_args), lim1, lim2)[0]
else:
return integrate.quad(lambda x: nppoly.polyval(x, p) *
self.measure(x), lim1, lim2)[0]
def get_alpha(self, p):
p2 = nppoly.polypow(p, 2)
xp2 = nppoly.polymulx(p2)
return (self.integrate(xp2, self.intlims[0], self.intlims[1]) /
self.integrate(p2, self.intlims[0], self.intlims[1]))
def get_beta(self, p, p0):
p2 = nppoly.polypow(p, 2)
p02 = nppoly.polypow(p0, 2)
return (self.integrate(p2, self.intlims[0], self.intlims[1]) /
self.integrate(p02, self.intlims[0], self.intlims[1]))
def get_quad_rule(self):
if self.jacobi.any():
S, U = np.linalg.eigh(self.jacobi)
locs = S
weights = U[0, :]**2
return locs, weights
else:
print('Run gen_poly() first!')
return None
def quadrature(self, func, **kwargs):
args = kwargs.get('args', None)
order = kwargs.get('order', self.order)
if not len(self.poly):
self.gen_poly(order)
x, w = self.get_quad_rule()
I = 0
if args:
for i in range(order+1):
I = I + func(x[i], args)*w[i]
else:
for i in range(order+1):
I = I + func(x[i])*w[i]
return I
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)) )
if __name__ == '__main__':
coeffs = [0, 1, -1, 1]
pp = OrthoPoly(pdf, margs=coeffs)
pp.gen_poly(20)
import matplotlib.pyplot as plt
xx = np.linspace(-1, 1, 50)
for p_i in pp.poly:
plt.plot(xx, nppoly.polyval(xx, p_i))
plt.show()
mean = [pp.quadrature(lambda x: x),
integrate.quad(lambda x, cc: x*pdf(x, cc), -np.inf, np.inf, args=coeffs)[0]]
print('Mean:')
print(mean)
var = [pp.quadrature(lambda x: (x-mean[0])**2),
integrate.quad(lambda x, cc: (x-mean[1])**2 * pdf(x, cc), -np.inf, np.inf, args=coeffs)[0]]
std = np.sqrt(var)
print('Std. dev.:')
print(std)
skew = [pp.quadrature(lambda x: ((x-mean[0])/var[0])**3),
integrate.quad(lambda x, cc: ((x-mean[1])/var[1])**3 * pdf(x, cc), -np.inf, np.inf, args=coeffs)[0]]
print('Skew:')
print(skew)