.. _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.