e4 = e1.join(e2)
#print('e4=', e4)
proba_e4 = e4.getSample(10000).computeMean()[0]
print("proba_e4 = %.3g" % proba_e4)
ott.assert_almost_equal(proba_e4, 0.75, 1e-2, 1e-2)
e5 = ot.IntersectionEvent([e1, e2])
#print('e5=', e5)
proba_e5 = e5.getSample(10000).computeMean()[0]
print("proba_e5 = %.3g" % proba_e5)
ott.assert_almost_equal(proba_e5, 0.25, 1e-2, 1e-2)
e6 = ot.UnionEvent([e1, e2])
#print('e6=', e6)
proba_e6 = e6.getSample(10000).computeMean()[0]
print("proba_e6 = %.3g" % proba_e6)
ott.assert_almost_equal(proba_e6, 0.75, 1e-2, 1e-2)
# intersection of intersection
e7 = ot.IntersectionEvent([e1, ot.IntersectionEvent([e1, e2])])
ott.assert_almost_equal(e7.getSample(10000).computeMean()[0], 0.25, 1e-2, 1e-2)
e8 = ot.IntersectionEvent([ot.IntersectionEvent([e1, e2]), e1])
ott.assert_almost_equal(e8.getSample(10000).computeMean()[0], 0.25, 1e-2, 1e-2)
# union of union
e9 = ot.UnionEvent([e1, ot.UnionEvent([e1, e2])])
myPolygon = ot.Polygon(data)
myPolygon.setColor('grey')
myPolygon.setEdgeColor('black')
myGraph.add(myPolygon)
view = otv.View(myGraph)
axes = view.getAxes()
_ = axes[0].set_xlim(-4.0, 4.0)
_ = axes[0].set_ylim(-4.0, 4.0)
# %%
# The probability of that event is :math:`P_{E_4} = 1/4`. A basic estimator is:
print("Probability of e4 : %.4f" % e4.getSample(10000).computeMean()[0])
# %%
# We define the union :math:`E_5 = E1 \bigcup E_2`. It is the whole plan without the lower right quadrant.
e5 = ot.UnionEvent([e1, e2])
# %%
# The restriction of the domain :math:`E_5` to :math:`[-4,4] \times [-4, 4]` is the grey area.
myGraph = ot.Graph(r'Representation of the event $E_5 = E_1 \bigcup E_2$',
r'$x_1$', r'$x_2$', True, '')
data = [[-4, -4], [0, -4], [0, 0], [4, 0], [4, 4], [-4, 4]]
myPolygon = ot.Polygon(data)
myPolygon.setColor('grey')
myPolygon.setEdgeColor('black')
myGraph.add(myPolygon)
view = otv.View(myGraph)
axes = view.getAxes()
_ = axes[0].set_xlim(-4.0, 4.0)
_ = axes[0].set_ylim(-4.0, 4.0)
solver.setMaximumIterationNumber(1000)
solver.setMaximumAbsoluteError(1.0e-3)
solver.setMaximumRelativeError(1.0e-3)
solver.setMaximumResidualError(1.0e-3)
solver.setMaximumConstraintError(1.0e-3)
for event in [e0, e3, e4, e5, e1, e2]:
algo = ot.FORM(solver, event, mean)
algo.run()
result = algo.getResult()
print('beta = %.6g' % result.getGeneralisedReliabilityIndex())
# system event in DNF form (union of intersections)
event = ot.UnionEvent([
ot.IntersectionEvent([e0, e3, e4, e5]),
ot.IntersectionEvent([e1, e3, e4, e5]),
ot.IntersectionEvent([e2, e3, e4, e5])
])
# sampling test
pf_sim = event.getSample(100000).computeMean()[0]
print('pf_sim = %.6g' % pf_sim)
ott.assert_almost_equal(pf_sim, 0.00384, 1e-4, 1e-4)
# system FORM
algo = ot.SystemFORM(solver, event, mean)
algo.run()
result = algo.getResult()
pf_sysform = result.getEventProbability()
print('pf_sysform = %.6g' % pf_sysform)
ott.assert_almost_equal(pf_sysform, 0.00418394, 1e-4, 1e-4)