import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if (ot.GeneralizedPareto().__class__.__name__ == 'ComposedDistribution'):
correlation = ot.CorrelationMatrix(2)
correlation[1, 0] = 0.25
aCopula = ot.NormalCopula(correlation)
marginals = [ot.Normal(1.0, 2.0), ot.Normal(2.0, 3.0)]
distribution = ot.ComposedDistribution(marginals, aCopula)
elif (ot.GeneralizedPareto().__class__.__name__ ==
'CumulativeDistributionNetwork'):
distribution = ot.CumulativeDistributionNetwork(
[ot.Normal(2), ot.Dirichlet([0.5, 1.0, 1.5])],
ot.BipartiteGraph([[0, 1], [0, 1]]))
else:
distribution = ot.GeneralizedPareto()
dimension = distribution.getDimension()
if dimension <= 2:
if distribution.getDimension() == 1:
distribution.setDescription(['$x$'])
pdf_graph = distribution.drawPDF()
cdf_graph = distribution.drawCDF()
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=False)
View(cdf_graph, figure=fig, axes=[cdf_axis], add_legend=False)
else:
distribution.setDescription(['$x_1$', '$x_2$'])
pdf_graph = distribution.drawPDF()
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
pdf_graph = ot.Graph('PDF graph', 'x', 'PDF', True, 'topleft')
cdf_graph = ot.Graph('CDF graph', 'x', 'CDF', True, 'topleft')
palette = ot.Drawable.BuildDefaultPalette(10)
for i, p in enumerate([-1.0, 0.0, 1.0]):
distribution = ot.GeneralizedPareto(1.0, p, 0.0)
pdf_curve = distribution.drawPDF().getDrawable(0)
cdf_curve = distribution.drawCDF().getDrawable(0)
pdf_curve.setColor(palette[i])
cdf_curve.setColor(palette[i])
pdf_curve.setLegend('xi={}'.format(p))
cdf_curve.setLegend('xi={}'.format(p))
pdf_graph.add(pdf_curve)
cdf_graph.add(cdf_curve)
fig = plt.figure(figsize=(10, 4))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=True)
View(cdf_graph, figure=fig, axes=[cdf_axis], add_legend=True)
fig.suptitle('GeneralizedPareto(1,xi,0)')
# Various estimators are provided by the GPD factory. Please refer to the :class:`~openturns.GeneralizedParetoFactory` class documentation for more information.
# The selection is based on the sample size and compared to the `GeneralizedParetoFactory-SmallSize` key of the :class:`~openturns.ResourceMap`:
#
# %%
print(
ot.ResourceMap.GetAsUnsignedInteger("GeneralizedParetoFactory-SmallSize"))
# %%
# The small sample case
# ^^^^^^^^^^^^^^^^^^^^^
#
# In this case the default estimator is based on a probability weighted method of moments, with a fallback on the exponential regression method.
# %%
myDist = ot.GeneralizedPareto(1.0, 0.0, 0.0)
N = 17
sample = myDist.getSample(N)
# %%
# We build our experiment sample of size `N`.
myFittedDist = ot.GeneralizedParetoFactory().buildAsGeneralizedPareto(sample)
print(myFittedDist)
# %%
# We draw the fitted distribution as well as an histogram to visualize the fit:
graph = myFittedDist.drawPDF()
graph.add(ot.HistogramFactory().build(sample).drawPDF())
graph.setTitle("Generalized Pareto distribution fitting on a sample")
graph.setColors(["black", "red"])
graph.setLegends(["GPD fitting", "histogram"])
# %% # Define a distribution # --------------------- # %% import openturns as ot import openturns.viewer as viewer from matplotlib import pylab as plt ot.Log.Show(ot.Log.NONE) # %% sigma = 1.4 xi = 0.5 u = 0.1 distribution = ot.GeneralizedPareto(sigma, xi, u) # %% # Draw the survival of a distribution # ----------------------------------- # %% # The `computeCDF` and `computeSurvivalFunction` computes the CDF :math:`F` and survival :math:`S` of a distribution. # %% p1 = distribution.computeCDF(10.) p1 # %% p2 = distribution.computeSurvivalFunction(10.) p2
graphPDF = myDistribution.drawPDF() graphPDF.setTitle(r"PDF of the GEV with parameters $\mu = 0.0$, $\sigma = 1.0$ and $\xi = 0.0$ ") view = otv.View(graphPDF) graphCDF = myDistribution.drawCDF() graphCDF.setTitle(r"CDF of the GEV with parameters $\mu = 0.0$, $\sigma = 1.0$ and $\xi = 0.0$ ") view = otv.View(graphCDF) # %% # The Generalized Pareto Distribution (GPD) # ----------------------------------------- # # In this paragraph we turn to the definition of a :class:`~openturns.GeneralizedPareto` distribution. # For instance we build a generalized Pareto distribution with parameters :math:`\sigma = 1.0`, :math:`\xi = 0.0` and :math:`u = 0.0` : # myGPD = ot.GeneralizedPareto(1.0, 0.0, 0.0) # %% # We draw its PDF and CDF : graphPDF = myGPD.drawPDF() graphPDF.setTitle(r"PDF of the GPD with parameters $\sigma = 1.0$, $\xi = 0.0$ and $u = 0.0$ ") view = otv.View(graphPDF) #%% graphCDF = myGPD.drawCDF() graphCDF.setTitle(r"CDF of the GPD with parameters $\sigma = 1.0$, $\xi = 0.0$ and $u = 0.0$ ") view = otv.View(graphCDF) # %% # Display all figures plt.show()
x_obs = y_obs
RWMHsampler = ot.RandomWalkMetropolisHastings(prior, initialState,
instrumental)
RWMHsampler.setLikelihood(conditional, y_obs, model, x_obs)
print("Log-likelihood of thetaTrue = {!r}".format(
RWMHsampler.computeLogLikelihood(thetaTrue)))
# produces an error with current master branch
real_504 = RWMHsampler.getRealization()
print("With 504 observations, getRealization() produces {!r}".format(
real_504[0]))
ott.assert_almost_equal(real_504[0], 2.0)
# this example throws in ot 1.19 as it tries to evaluate the likelihood outside the support of the prior
# see MetropolisHastingsImplementation::computeLogPosterior
obs = ot.TruncatedNormal(0.5, 0.5, 0.0, 10.0).getSample(50)
likelihood = ot.GeneralizedPareto()
prior = ot.ComposedDistribution(
[ot.LogUniform(-1.40, 4.0),
ot.Normal(), ot.Normal()])
proposals = [
ot.Uniform(-prior.getMarginal(k).getStandardDeviation()[0],
+prior.getMarginal(k).getStandardDeviation()[0])
for k in range(prior.getDimension())
]
initialState = prior.getMean()
mh_coll = [
ot.RandomWalkMetropolisHastings(prior, initialState, proposals[i], [i])
for i in range(2)
]
for mh in mh_coll:
mh.setLikelihood(likelihood, obs)
import openturns as ot
from matplotlib import pyplot as plt
from openturns.viewer import View
if ot.GeneralizedPareto().__class__.__name__ == 'ComposedDistribution':
correlation = ot.CorrelationMatrix(2)
correlation[1, 0] = 0.25
aCopula = ot.NormalCopula(correlation)
marginals = [ot.Normal(1.0, 2.0), ot.Normal(2.0, 3.0)]
distribution = ot.ComposedDistribution(marginals, aCopula)
elif ot.GeneralizedPareto().__class__.__name__ == 'CumulativeDistributionNetwork':
distribution = ot.CumulativeDistributionNetwork([ot.Normal(2),ot.Dirichlet([0.5, 1.0, 1.5])], ot.BipartiteGraph([[0,1], [0,1]]))
elif ot.GeneralizedPareto().__class__.__name__ == 'Histogram':
distribution = ot.Histogram([-1.0, 0.5, 1.0, 2.0], [0.45, 0.4, 0.15])
else:
distribution = ot.GeneralizedPareto()
dimension = distribution.getDimension()
if dimension == 1:
distribution.setDescription(['$x$'])
pdf_graph = distribution.drawPDF()
cdf_graph = distribution.drawCDF()
fig = plt.figure(figsize=(10, 4))
plt.suptitle(str(distribution))
pdf_axis = fig.add_subplot(121)
cdf_axis = fig.add_subplot(122)
View(pdf_graph, figure=fig, axes=[pdf_axis], add_legend=False)
View(cdf_graph, figure=fig, axes=[cdf_axis], add_legend=False)
elif dimension == 2:
distribution.setDescription(['$x_1$', '$x_2$'])
pdf_graph = distribution.drawPDF()
fig = plt.figure(figsize=(10, 5))
plt.suptitle(str(distribution))
