.. _sphere_sampling: Sphere sampling method ---------------------- Within the context of the First and Second Order of the Reliability Method, the Strong Maximum Test helps to check whether the design point computed is: - the true design point, which means a global maximum point, - a strong design point, which means that there is no other local maximum verifying the event and associated to a value near the global maximum. The Strong Maximum Test samples a sphere in the standard space. the gaussian random sampling technique used is described hereafter. #. sampling of points in :math:\Rset^N according to a radial distribution: we generate :math:N independent standard normal samples, #. projection of the points onto :math:\cS^{*} : we map the points different from the origin using the transformation :math:M\longmapsto m such as :math:\displaystyle\mathbf{Om}=R\frac{\mathbf{OM}}{\|\mathbf{OM}\|} where :math:R is the radius of the sphere of interest. This transformation does not depend on the angular coordinates. Thus, the generated points follow a uniform distribution on :math:\cS^{*}. .. plot:: from mpl_toolkits.mplot3d import Axes3D import matplotlib.pyplot as plt import openturns as ot x = ot.Normal(3).getSample(1000) for i in range(len(x)): x[i] /= x[i].norm() xs, ys, zs = map(lambda j: x.getMarginal(j).asPoint(), range(x.getDimension())) fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.scatter(xs, ys, zs, marker='.') .. topic:: API: - See :class:~openturns.StrongMaximumTest - See :class:~openturns.FORM .. topic:: References: - Luban, Marshall, Staunton, 1988, "An efficient method for generating a uniform distribution of points within a hypersphere," Computer in Physics, 2(6), 55.