# Estimate a multivariate distributionΒΆ

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
In [1]:

from __future__ import print_function
import openturns as ot
import math as m

In [2]:

# generate some multivariate data to estimate, with correlation
cop1 = ot.AliMikhailHaqCopula(0.6)
cop2 = ot.ClaytonCopula(2.5)
copula = ot.ComposedCopula([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.ComposedDistribution(marginals, copula)
sample = distribution.getSample(10000).getMarginal([0, 2, 3, 1])

In [3]:

# 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) for i in range(dimension)]
estimated_marginals

Out[3]:

[class=Uniform name=Uniform dimension=1 a=5.00008 b=6,
class=Normal name=Normal dimension=1 mean=class=Point name=Unnamed dimension=1 values=[-39.9413] sigma=class=Point name=Unnamed dimension=1 values=[3.04363] correlationMatrix=class=CorrelationMatrix dimension=1 implementation=class=MatrixImplementation name=Unnamed rows=1 columns=1 values=[1],
class=Trapezoidal name=Trapezoidal dimension=1 a=99.9716 b=149.173 c=152.366 d=299.9 h=0.00984637,
class=Beta name=Beta dimension=1 r=0.49567 t=0.993342 a=-1.0002 b=1.0002]

In [4]:

# find connected components of a graph defined from its adjacency matrix

def find_neighbours(head, covariance, to_visit, visited):
N = covariance.getDimension()
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

In [10]:

# estimate copula

# 1. find dependent components

# compute correlation
C = sample.computeSpearmanCorrelation()
print(C)
# consider only significantly non-zero correlations
epsilon = 1.0 / 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)
# find independent blocs
blocs = connected_components(C)
blocs

[[  1           0.00263067  0.00518309  0.23953    ]
[  0.00263067  1           0.736523   -0.00576173 ]
[  0.00518309  0.736523    1          -0.0083771  ]
[  0.23953    -0.00576173 -0.0083771   1          ]]
[[ 1 0 0 1 ]
[ 0 1 1 0 ]
[ 0 1 1 0 ]
[ 1 0 0 1 ]]

Out[10]:

[[0, 3], [1, 2]]

In [6]:

# 2. estimate the copulas on each dependent blocs
copulaFactories = []
for factory in ot.DistributionFactory.GetContinuousMultiVariateFactories():
if not factory.build().isCopula():
continue
copulaFactories.append(factory)
estimated_copulas = [ot.FittingTest.BestModelBIC(sample.getMarginal(bloc), copulaFactories) for bloc in blocs]
estimated_copulas

Out[6]:

[class=AliMikhailHaqCopula name=AliMikhailHaqCopula dimension=2 theta=0.601685,
class=ClaytonCopula name=ClaytonCopula dimension=2 theta=2.45478]

In [7]:

# 3. assemble the copula
estimated_copula_perm = ot.ComposedCopula(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

Out[7]:

class=MarginalDistribution name=MarginalDistribution dimension=4 distribution=class=ComposedCopula name=ComposedCopula dimension=4 copula[0]=class=AliMikhailHaqCopula name=AliMikhailHaqCopula dimension=2 theta=0.601685 copula[1]=class=ClaytonCopula name=ClaytonCopula dimension=2 theta=2.45478 indices=[0,2,3,1]

In [8]:

# Build joint distribution from marginal distributions and dependency structure
estimated_distribution = ot.ComposedDistribution(estimated_marginals, estimated_copula)
estimated_distribution

Out[8]:


ComposedDistribution(Uniform(a = 5.00008, b = 6), Normal(mu = -39.9413, sigma = 3.04363), Trapezoidal(a = 99.9716, b = 149.173, c = 152.366, d = 299.9), Beta(r = 0.49567, t = 0.993342, a = -1.0002, b = 1.0002), MarginalDistribution(distribution=ComposedCopula(AliMikhailHaqCopula(theta = 0.601685), ClaytonCopula(theta = 2.45478)), indices=[0,2,3,1]))