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"""
Estimate a multivariate distribution
====================================
"""
# %%
# In this example we are going to estimate a joint distribution from a multivariate sample by fitting marginals and finding a set of copulas.
#
# While the estimation of marginals is quite straightforward, the estimation of the dependency structure takes several steps:
#
# - find the dependent components
# - estimate a copula on each dependent bloc
# - assemble the estimated copulas
#
# %%
import openturns as ot
import math as m
# %%
# Generate some multivariate data to estimate, with correlation
cop1 = ot.AliMikhailHaqCopula(0.6)
cop2 = ot.ClaytonCopula(2.5)
copula = ot.BlockIndependentCopula([cop1, cop2])
marginals = [
ot.Uniform(5.0, 6.0),
ot.Arcsine(),
ot.Normal(-40.0, 3.0),
ot.Triangular(100.0, 150.0, 300.0),
]
distribution = ot.JointDistribution(marginals, copula)
sample = distribution.getSample(10000).getMarginal([0, 2, 3, 1])
# %%
# Estimate marginals
dimension = sample.getDimension()
marginalFactories = []
for factory in ot.DistributionFactory.GetContinuousUniVariateFactories():
if str(factory).startswith("Histogram"):
# ~ non-parametric
continue
marginalFactories.append(factory)
estimated_marginals = [
ot.FittingTest.BestModelBIC(sample.getMarginal(i), marginalFactories)[0]
for i in range(dimension)
]
estimated_marginals
# %%
# Find connected components of a graph defined from its adjacency matrix
# %%
def find_neighbours(head, covariance, to_visit, visited):
visited[head] = 1
to_visit.remove(head)
current_component = [head]
for i in to_visit:
# If i is connected to head and has not yet been visited
if covariance[head, i] > 0:
# Add i to the current component
component = find_neighbours(i, covariance, to_visit, visited)
current_component += component
return current_component
def connected_components(covariance):
N = covariance.getDimension()
to_visit = list(range(N))
visited = [0] * N
all_components = []
for head in range(N):
if visited[head] == 0:
component = find_neighbours(head, covariance, to_visit, visited)
all_components.append(sorted(component))
return all_components
# %%
# Estimate the copula
# -------------------
#
# First find the dependent components : we compute the `Spearman` correlation
# %%
C = sample.computeSpearmanCorrelation()
print(C)
# %%
# We filter and consider only significantly non-zero correlations.
# %%
epsilon = 2.0 / m.sqrt(sample.getSize())
for j in range(dimension):
for i in range(j):
C[i, j] = 1.0 if abs(C[i, j]) > epsilon else 0.0
print(C)
# %%
# Note that we can apply the `HypothesisTest.Spearman` test. As the null hypothesis of the test is the `independence`, we must take the complementary of the binary measure as follow:
#
# >>> M = ot.SymmetricMatrix(dimension)
# >>> for i in range(dimension):
# >>> M[i,i] = 1
# >>> for j in range(i):
# >>> M[i, j] = 1 - ot.HypothesisTest.Spearman(sample[:,i], sample[:,j]).getBinaryQualityMeasure()
#
# %%
# Now we find the independent blocs:
# %%
blocs = connected_components(C)
blocs
# %%
# For each dependent block, we estimate the most accurate parameteric copula.
#
# To do this, we first need to transform the sample in such a way as to keep the copula intact but make all marginal samples follow the uniform distribution on [0,1].
# %%
copula_sample = ot.Sample(sample.getSize(), sample.getDimension())
copula_sample.setDescription(sample.getDescription())
for index in range(sample.getDimension()):
copula_sample[:, index] = estimated_marginals[index].computeCDF(sample[:, index])
# %%
copulaFactories = []
for factory in ot.DistributionFactory.GetContinuousMultiVariateFactories():
if not factory.build().isCopula():
continue
if factory.getImplementation().getClassName() == "BernsteinCopulaFactory":
continue
copulaFactories.append(factory)
estimated_copulas = [
ot.FittingTest.BestModelBIC(copula_sample.getMarginal(bloc), copulaFactories)[0]
for bloc in blocs
]
estimated_copulas
# %%
# Finally we assemble the copula
# %%
estimated_copula_perm = ot.BlockIndependentCopula(estimated_copulas)
# %%
# Take care of the order of each bloc vs the order of original components !
# %%
permutation = []
for bloc in blocs:
permutation.extend(bloc)
inverse_permutation = [-1] * dimension
for i in range(dimension):
inverse_permutation[permutation[i]] = i
estimated_copula = estimated_copula_perm.getMarginal(inverse_permutation)
estimated_copula
# %%
# We build joint distribution from marginal distributions and dependency structure:
# %%
estimated_distribution = ot.JointDistribution(estimated_marginals, estimated_copula)
estimated_distribution
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