def _frank_PKTE(X):
# calculate empirical kendall's tau
ktau = multivariate_stats.kendalls_tau(X)
# inverse to find dependency parameter
alpha_hat = invcopulastat('Frank', 'kendall', ktau)
return alpha_hat
def optimalCopulaFamily(X, K=4, family_search=['Gaussian', 'Clayton', 'Gumbel', 'Frank']):
"""
This function, given a multivariate data set X, computes the best copula family which fits
the data, using the procedure described in the paper "Highly Efficient Learning of Mixed
Copula Networks," by Gal Elidan
X - the multivariate dataset for which we desire the copula. Must be a numpy array of
dimension [M x N], where M is the number of data points, and N is the dimensionality
of the dataset
K - the square root of the number of grid points (for now, we assume square gridding of the
unit cube)
family_search - a list of all the copula families to search. Currently, what is supported is
Gaussian, Clayton, Gumbel, and Frank. As more copula's are added, the default list will
be expanded.
"""
# compute the empirical Kendall's Tau
tau_hat = multivariate_stats.kendalls_tau(X)
# compute empirical multinomial signature
empirical_mnsig = empirical_copulamnsig(X, K)
empirical_mnsig = empirical_mnsig[0]['esig']
# replace any 0 values w/ smallest possible float value
empirical_mnsig[empirical_mnsig==0] = np.spacing(1)
# compute the multinomial signature for each of the copula families specified
# and simultaneously compute the kullback leibler divergence between the empirical
# and the computed, and store that info
distances = {}
for family in family_search:
# because the Clayton and Gumbel Copula's have restrictions for the valid values of
# Kendall's tau, we do checks here to ensure those restrictions are met, because there
# will be a certain variance associated with the tau_hat measurement
if(family.lower()=='clayton'):
# here we add some additional optimizatons as follows. We know that the Clayton copula
# captures only positive concordance. Like any estimator, tau_hat will have some variance
# associated with it. Thus, the optimization we make is as follows, if tau_hat is within
# a configurable amount less than 0, then we will set tau_hat to 0 and continue processing.
# However, if tau_hat is greater than that, we theoretically wouldn't have to test against
# the Clayton copula model, so we set the KL-divergence to be infinity to exclude
# this family from being selected
if(tau_hat<-0.05):
distances[family] = np.inf
continue
elif(tau_hat>=-0.05 and tau_hat<0):
tau_hat = 0
elif(tau_hat>=1):
tau_hat = 1 - np.spacing(1) # as close to 1 as possible in our precision
elif(family.lower()=='gumbel'):
# here we add some additional optimizatons as follows. We know that the Gumbel copula
# captures only positive concordance. Like any estimator, tau_hat will have some variance
# associated with it. Thus, the optimization we make is as follows, if tau_hat is within
# a configurable amount less than 0, then we will set tau_hat to 0 and continue processing.
# However, if tau_hat is greater than that, we theoretically wouldn't have to test against
# the Gumbel copula model, so we set the KL-divergence to be infinity to exclude
# this family from being selected
if(tau_hat<-0.05):
distances[family] = np.inf
continue
elif(tau_hat>=-0.05 and tau_hat<0):
tau_hat = 0
elif(tau_hat>=1):
tau_hat = 1 - np.spacing(1) # as close to 1 as possible in our precision
# any other copula families with restrictions can go here
mnsig = copulamnsig(family,K,'kendall',tau_hat)
# replace any 0 values w/ smallest possible float value
mnsig[mnsig==0] = np.spacing(1)
# compute KL divergence, see
# http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.stats.entropy.html
distances[family] = entropy(mnsig, empirical_mnsig)
# search for the minimum distance, that is the optimal copula family to use
minDistance = np.inf
for family, distance in distances.iteritems():
if distance<minDistance:
minDistance = distance
optimalFamily = family
depParams = invcopulastat(optimalFamily, 'kendall', tau_hat)
return (optimalFamily, depParams, tau_hat)
def optimalCopulaFamily(X,
K=4,
family_search=[
'Gaussian', 'Clayton', 'Gumbel', 'Frank'
]):
"""
This function, given a multivariate data set X, computes the best copula family which fits
the data, using the procedure described in the paper "Highly Efficient Learning of Mixed
Copula Networks," by Gal Elidan
X - the multivariate dataset for which we desire the copula. Must be a numpy array of
dimension [M x N], where M is the number of data points, and N is the dimensionality
of the dataset
K - the square root of the number of grid points (for now, we assume square gridding of the
unit cube)
family_search - a list of all the copula families to search. Currently, what is supported is
Gaussian, Clayton, Gumbel, and Frank. As more copula's are added, the default list will
be expanded.
"""
# compute the empirical Kendall's Tau
tau_hat = multivariate_stats.kendalls_tau(X)
# compute empirical multinomial signature
empirical_mnsig = empirical_copulamnsig(X, K)
empirical_mnsig = empirical_mnsig[0]['esig']
# replace any 0 values w/ smallest possible float value
empirical_mnsig[empirical_mnsig == 0] = np.spacing(1)
# compute the multinomial signature for each of the copula families specified
# and simultaneously compute the kullback leibler divergence between the empirical
# and the computed, and store that info
distances = {}
for family in family_search:
# because the Clayton and Gumbel Copula's have restrictions for the valid values of
# Kendall's tau, we do checks here to ensure those restrictions are met, because there
# will be a certain variance associated with the tau_hat measurement
if (family.lower() == 'clayton'):
# here we add some additional optimizatons as follows. We know that the Clayton copula
# captures only positive concordance. Like any estimator, tau_hat will have some variance
# associated with it. Thus, the optimization we make is as follows, if tau_hat is within
# a configurable amount less than 0, then we will set tau_hat to 0 and continue processing.
# However, if tau_hat is greater than that, we theoretically wouldn't have to test against
# the Clayton copula model, so we set the KL-divergence to be infinity to exclude
# this family from being selected
if (tau_hat < -0.05):
distances[family] = np.inf
continue
elif (tau_hat >= -0.05 and tau_hat < 0):
tau_hat = 0
elif (tau_hat >= 1):
tau_hat = 1 - np.spacing(
1) # as close to 1 as possible in our precision
elif (family.lower() == 'gumbel'):
# here we add some additional optimizatons as follows. We know that the Gumbel copula
# captures only positive concordance. Like any estimator, tau_hat will have some variance
# associated with it. Thus, the optimization we make is as follows, if tau_hat is within
# a configurable amount less than 0, then we will set tau_hat to 0 and continue processing.
# However, if tau_hat is greater than that, we theoretically wouldn't have to test against
# the Gumbel copula model, so we set the KL-divergence to be infinity to exclude
# this family from being selected
if (tau_hat < -0.05):
distances[family] = np.inf
continue
elif (tau_hat >= -0.05 and tau_hat < 0):
tau_hat = 0
elif (tau_hat >= 1):
tau_hat = 1 - np.spacing(
1) # as close to 1 as possible in our precision
# any other copula families with restrictions can go here
mnsig = copulamnsig(family, K, 'kendall', tau_hat)
# replace any 0 values w/ smallest possible float value
mnsig[mnsig == 0] = np.spacing(1)
# compute KL divergence, see
# http://docs.scipy.org/doc/scipy-dev/reference/generated/scipy.stats.entropy.html
distances[family] = entropy(mnsig, empirical_mnsig)
# search for the minimum distance, that is the optimal copula family to use
minDistance = np.inf
for family, distance in distances.iteritems():
if distance < minDistance:
minDistance = distance
optimalFamily = family
depParams = invcopulastat(optimalFamily, 'kendall', tau_hat)
return (optimalFamily, depParams, tau_hat)
