_qmc_quad(func, a, b, *, n_points=1024, n_estimates=8, qrng=None, log=False, args=None)
Values of the integrand at each of the n_points points of a QMC sample are used to produce an estimate of the integral. This estimate is drawn from a population of possible estimates of the integral, the value of which we obtain depends on the particular points at which the integral was evaluated. We perform this process n_estimates times, each time evaluating the integrand at different scrambled QMC points, effectively drawing i.i.d. random samples from the population of integral estimates. The sample mean m of these integral estimates is an unbiased estimator of the true value of the integral, and the standard error of the mean s of these estimates may be used to generate confidence intervals using the t distribution with n_estimates - 1 degrees of freedom. Perhaps counter-intuitively, increasing n_points while keeping the total number of function evaluation points n_points * n_estimates fixed tends to reduce the actual error, whereas increasing n_estimates tends to decrease the error estimate.
The integrand. Must accept a single arguments x, an array which specifies the point at which to evaluate the integrand. For efficiency, the function should be vectorized to compute the integrand for each element an array of shape (n_points, n), where n is number of variables.
One-dimensional arrays specifying the lower and upper integration limits, respectively, of each of the n variables.
One QMC sample of n_points (default: 256) points will be generated by qrng, and n_estimates (default: 8) statistically independent estimates of the integral will be produced. The total number of points at which the integrand func will be evaluated is n_points * n_estimates. See Notes for details.
An instance of the QMCEngine from which to sample QMC points. The QMCEngine must be initialized to a number of dimensions corresponding with the number of variables x0, ..., xn passed to func. The provided QMCEngine is used to produce the first integral estimate. If n_estimates is greater than one, additional QMCEngines are spawned from the first (with scrambling enabled, if it is an option.) If a QMCEngine is not provided, the default scipy.stats.qmc.Halton will be initialized with the number of dimensions determine from a.
When set to True, func returns the log of the integrand, and the result object contains the log of the integral.
A result object with attributes:
integral
integral
standard_error :
The error estimate. See Notes for interpretation.
Compute an integral in N-dimensions using Quasi-Monte Carlo quadrature.
import numpy as np
from scipy import stats
dim = 8
mean = np.zeros(dim)
cov = np.eye(dim)
def func(x):
return stats.multivariate_normal.pdf(x, mean, cov)
from scipy.integrate import qmc_quad
a = np.zeros(dim)
b = np.ones(dim)
rng = np.random.default_rng()
qrng = stats.qmc.Halton(d=dim, seed=rng)
n_estimates = 8
res = qmc_quad(func, a, b, n_estimates=n_estimates, qrng=qrng)
res.integral, res.standard_error
t = stats.t(df=n_estimates-1, loc=res.integral,
scale=res.standard_error)
t.interval(0.99)
stats.multivariate_normal.cdf(b, mean, cov, lower_limit=a)
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