def visualizeMNSig():
# some tests on the copula multinomial signature
K = 4
M = 1000
N = 3
tauVec = np.arange(-0.9,0.95,0.05)
# the families to test against and pick optimal copula
families = ['Gaussian', 'Clayton', 'Gumbel', 'Frank']
helmAccuracyResults = testHELM_parametric(K,M,N,tauVec,families)
resultsAggregate = {}
for family in families:
famResults = {}
for tau in tauVec:
mnsig = copulamnsig(family,K,'kendall',tau)
famResults[tau] = mnsig
resultsAggregate[family] = famResults
# visualize the results
for tau in tauVec:
# we would also like to visualize this copula on the side, to try to
# understand what may be a better way todo model selection
try:
r = invcopulastat('Gaussian', 'kendall', tau)
except ValueError:
r = -1
Rho = np.empty((N,N))
for jj in range(0,N):
for kk in range(0,N):
if(jj==kk):
Rho[jj][kk] = 1
else:
Rho[jj][kk] = r
try:
alpha_clayton = invcopulastat('Clayton', 'kendall', tau)
except ValueError:
alpha_clayton = -1
try:
alpha_gumbel = invcopulastat('Gumbel', 'kendall', tau)
except ValueError:
alpha_gumbel = -1
try:
alpha_frank = invcopulastat('Frank', 'kendall', tau)
except ValueError:
alpha_frank = -1
if(r!=-1):
try:
U_gauss = copularnd('Gaussian', M, Rho)
except ValueError:
U_gauss = np.zeros((M,N))
if(alpha_clayton!=-1):
try:
U_clayton = copularnd('Clayton', M, N, alpha_clayton)
except ValueError:
U_clayton = np.zeros((M,N))
if(alpha_frank!=-1):
try:
U_frank = copularnd('Frank', M, N, alpha_frank)
except ValueError:
U_frank = np.zeros((M,N))
if(alpha_gumbel!=-1):
try:
U_gumbel = copularnd('Gumbel', M, N, alpha_gumbel)
except ValueError:
U_gumbel = np.zeros((M,N))
# get each family's MN signature and plot it
plt.figure(figsize=(30,20))
plt.subplot(231)
if(np.sum(resultsAggregate['Gaussian'][tau])>0):
plt.plot(np.arange(1,K*K+1), resultsAggregate['Gaussian'][tau], 'b.-', label='Gaussian Copula')
if(np.sum(resultsAggregate['Clayton'][tau])>0):
plt.plot(np.arange(1,K*K+1), resultsAggregate['Clayton'][tau], 'g.-', label='Clayton Copula')
if(np.sum(resultsAggregate['Gumbel'][tau])>0):
plt.plot(np.arange(1,K*K+1), resultsAggregate['Gumbel'][tau], 'r.-', label='Gumbel Copula')
if(np.sum(resultsAggregate['Frank'][tau])>0):
plt.plot(np.arange(1,K*K+1), resultsAggregate['Frank'][tau], 'k.-', label='Frank Copula')
plt.title(r'Copula Multinomial Signature $\tau$=' + "{0:.2f}".format(tau) + ' K=' + str(K))
plt.legend()
plt.grid()
plt.subplot(232)
if(r!=-1):
plt.scatter(U_gauss[:,0], U_gauss[:,1])
plt.grid()
plt.title(r'Gaussian Copula, $\rho$=' + "{0:.2f}".format(r) + r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(233)
if(alpha_clayton!=-1):
plt.scatter(U_clayton[:,0], U_clayton[:,1])
plt.grid()
plt.title(r'Clayton Copula, $\alpha$=' + "{0:.2f}".format(alpha_clayton) + r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(235)
if(alpha_frank!=-1):
plt.scatter(U_frank[:,0], U_frank[:,1])
plt.grid()
plt.title(r'Frank Copula, $\alpha$=' + "{0:.2f}".format(alpha_frank) + r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(236)
if(alpha_gumbel!=-1):
plt.scatter(U_gumbel[:,0], U_gumbel[:,1])
plt.grid()
plt.title(r'Gumbel Copula, $\alpha$=' + "{0:.2f}".format(alpha_gumbel) + r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(234)
# index manually to ensure accuracy
cla = np.array([helmAccuracyResults['Clayton'][tau]['clayton'],
helmAccuracyResults['Gaussian'][tau]['clayton'],
helmAccuracyResults['Gumbel'][tau]['clayton'],
helmAccuracyResults['Frank'][tau]['clayton']])
gau = np.array([helmAccuracyResults['Clayton'][tau]['gaussian'],
helmAccuracyResults['Gaussian'][tau]['gaussian'],
helmAccuracyResults['Gumbel'][tau]['gaussian'],
helmAccuracyResults['Frank'][tau]['gaussian']])
gum = np.array([helmAccuracyResults['Clayton'][tau]['gumbel'],
helmAccuracyResults['Gaussian'][tau]['gumbel'],
helmAccuracyResults['Gumbel'][tau]['gumbel'],
helmAccuracyResults['Frank'][tau]['gumbel']])
fra = np.array([helmAccuracyResults['Clayton'][tau]['frank'],
helmAccuracyResults['Gaussian'][tau]['frank'],
helmAccuracyResults['Gumbel'][tau]['frank'],
helmAccuracyResults['Frank'][tau]['frank']])
ind = np.arange(4)
width = 0.2
p1 = plt.bar(ind,cla,width,color='b')
p2 = plt.bar(ind,gau,width,color='g',bottom=cla)
p3 = plt.bar(ind,gum,width,color='k',bottom=cla+gau)
p4 = plt.bar(ind,fra,width,color='r',bottom=cla+gau+gum)
plt.xticks(ind+width/2.,('Clayton', 'Gaussian', 'Gumbel', 'Frank'))
plt.legend( (p1[0], p2[0], p3[0], p4[0]), ('Clayton', 'Gaussian', 'Gumbel', 'Frank') )
plt.grid()
plt.savefig(os.path.join('figures/HELM_performance/',
'HELM_DIM_' + str(N) + '_tau_' + "{0:.2f}".format(tau) + ' _K_' + str(K) + '.png'))
plt.close()
def visualizeMNSig():
# some tests on the copula multinomial signature
K = 4
M = 1000
N = 3
tauVec = np.arange(-0.9, 0.95, 0.05)
# the families to test against and pick optimal copula
families = ['Gaussian', 'Clayton', 'Gumbel', 'Frank']
helmAccuracyResults = testHELM_parametric(K, M, N, tauVec, families)
resultsAggregate = {}
for family in families:
famResults = {}
for tau in tauVec:
mnsig = copulamnsig(family, K, 'kendall', tau)
famResults[tau] = mnsig
resultsAggregate[family] = famResults
# visualize the results
for tau in tauVec:
# we would also like to visualize this copula on the side, to try to
# understand what may be a better way todo model selection
try:
r = invcopulastat('Gaussian', 'kendall', tau)
except ValueError:
r = -1
Rho = np.empty((N, N))
for jj in range(0, N):
for kk in range(0, N):
if (jj == kk):
Rho[jj][kk] = 1
else:
Rho[jj][kk] = r
try:
alpha_clayton = invcopulastat('Clayton', 'kendall', tau)
except ValueError:
alpha_clayton = -1
try:
alpha_gumbel = invcopulastat('Gumbel', 'kendall', tau)
except ValueError:
alpha_gumbel = -1
try:
alpha_frank = invcopulastat('Frank', 'kendall', tau)
except ValueError:
alpha_frank = -1
if (r != -1):
try:
U_gauss = copularnd('Gaussian', M, Rho)
except ValueError:
U_gauss = np.zeros((M, N))
if (alpha_clayton != -1):
try:
U_clayton = copularnd('Clayton', M, N, alpha_clayton)
except ValueError:
U_clayton = np.zeros((M, N))
if (alpha_frank != -1):
try:
U_frank = copularnd('Frank', M, N, alpha_frank)
except ValueError:
U_frank = np.zeros((M, N))
if (alpha_gumbel != -1):
try:
U_gumbel = copularnd('Gumbel', M, N, alpha_gumbel)
except ValueError:
U_gumbel = np.zeros((M, N))
# get each family's MN signature and plot it
plt.figure(figsize=(30, 20))
plt.subplot(231)
if (np.sum(resultsAggregate['Gaussian'][tau]) > 0):
plt.plot(np.arange(1, K * K + 1),
resultsAggregate['Gaussian'][tau],
'b.-',
label='Gaussian Copula')
if (np.sum(resultsAggregate['Clayton'][tau]) > 0):
plt.plot(np.arange(1, K * K + 1),
resultsAggregate['Clayton'][tau],
'g.-',
label='Clayton Copula')
if (np.sum(resultsAggregate['Gumbel'][tau]) > 0):
plt.plot(np.arange(1, K * K + 1),
resultsAggregate['Gumbel'][tau],
'r.-',
label='Gumbel Copula')
if (np.sum(resultsAggregate['Frank'][tau]) > 0):
plt.plot(np.arange(1, K * K + 1),
resultsAggregate['Frank'][tau],
'k.-',
label='Frank Copula')
plt.title(r'Copula Multinomial Signature $\tau$=' +
"{0:.2f}".format(tau) + ' K=' + str(K))
plt.legend()
plt.grid()
plt.subplot(232)
if (r != -1):
plt.scatter(U_gauss[:, 0], U_gauss[:, 1])
plt.grid()
plt.title(r'Gaussian Copula, $\rho$=' + "{0:.2f}".format(r) +
r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(233)
if (alpha_clayton != -1):
plt.scatter(U_clayton[:, 0], U_clayton[:, 1])
plt.grid()
plt.title(r'Clayton Copula, $\alpha$=' +
"{0:.2f}".format(alpha_clayton) + r' $\tau$=' +
"{0:.2f}".format(tau))
plt.subplot(235)
if (alpha_frank != -1):
plt.scatter(U_frank[:, 0], U_frank[:, 1])
plt.grid()
plt.title(r'Frank Copula, $\alpha$=' + "{0:.2f}".format(alpha_frank) +
r' $\tau$=' + "{0:.2f}".format(tau))
plt.subplot(236)
if (alpha_gumbel != -1):
plt.scatter(U_gumbel[:, 0], U_gumbel[:, 1])
plt.grid()
plt.title(r'Gumbel Copula, $\alpha$=' +
"{0:.2f}".format(alpha_gumbel) + r' $\tau$=' +
"{0:.2f}".format(tau))
plt.subplot(234)
# index manually to ensure accuracy
cla = np.array([
helmAccuracyResults['Clayton'][tau]['clayton'],
helmAccuracyResults['Gaussian'][tau]['clayton'],
helmAccuracyResults['Gumbel'][tau]['clayton'],
helmAccuracyResults['Frank'][tau]['clayton']
])
gau = np.array([
helmAccuracyResults['Clayton'][tau]['gaussian'],
helmAccuracyResults['Gaussian'][tau]['gaussian'],
helmAccuracyResults['Gumbel'][tau]['gaussian'],
helmAccuracyResults['Frank'][tau]['gaussian']
])
gum = np.array([
helmAccuracyResults['Clayton'][tau]['gumbel'],
helmAccuracyResults['Gaussian'][tau]['gumbel'],
helmAccuracyResults['Gumbel'][tau]['gumbel'],
helmAccuracyResults['Frank'][tau]['gumbel']
])
fra = np.array([
helmAccuracyResults['Clayton'][tau]['frank'],
helmAccuracyResults['Gaussian'][tau]['frank'],
helmAccuracyResults['Gumbel'][tau]['frank'],
helmAccuracyResults['Frank'][tau]['frank']
])
ind = np.arange(4)
width = 0.2
p1 = plt.bar(ind, cla, width, color='b')
p2 = plt.bar(ind, gau, width, color='g', bottom=cla)
p3 = plt.bar(ind, gum, width, color='k', bottom=cla + gau)
p4 = plt.bar(ind, fra, width, color='r', bottom=cla + gau + gum)
plt.xticks(ind + width / 2.,
('Clayton', 'Gaussian', 'Gumbel', 'Frank'))
plt.legend((p1[0], p2[0], p3[0], p4[0]),
('Clayton', 'Gaussian', 'Gumbel', 'Frank'))
plt.grid()
plt.savefig(
os.path.join(
'figures/HELM_performance/', 'HELM_DIM_' + str(N) + '_tau_' +
"{0:.2f}".format(tau) + ' _K_' + str(K) + '.png'))
plt.close()
def testHELM(tau, M, N, familyToTest, numMCSims, copulaFamiliesToTest):
results = {}
for fam in copulaFamiliesToTest:
results[fam.lower()] = 0
for ii in range(0,numMCSims):
# generate samples of the requested copula with tau same as the
# empirical signature we calculated above
if(familyToTest.lower()=='gaussian'):
r = invcopulastat(familyToTest, 'kendall', tau)
Rho = np.empty((N,N))
for jj in range(0,N):
for kk in range(0,N):
if(jj==kk):
Rho[jj][kk] = 1
else:
Rho[jj][kk] = r
try:
U = copularnd(familyToTest, M, Rho)
except ValueError:
# copularnd will throw a ValueError if Rho is not a positive semidefinite matrix
return results # return 0, which will then be ignored by tests
else: # assume Clayton, Frank, or Gumbel
try:
alpha = invcopulastat(familyToTest, 'kendall', tau)
U = copularnd(familyToTest, M, N, alpha)
except ValueError:
continue
lst = []
for jj in range(0,N):
U_conditioned = U[:,jj]
# if there are any 1's, condition it
U_conditioned[U_conditioned==1] = 0.99
if(jj%2==0):
lst.append(norm.ppf(U_conditioned))
else:
lst.append(expon.ppf(U_conditioned))
# combine X and Y into the joint distribution w/ the copula
X = np.vstack(lst)
X = X.T
ret = optimalCopulaFamily(X, family_search=copulaFamiliesToTest)
ret_family = ret[0].lower()
# aggregate results
results[ret_family] = results[ret_family] + 1.0
# display some progress
sys.stdout.write("\rComputing " + str(familyToTest) + " Copula (DIM=%d) (tau=%f)-- %d%%" % (N,tau,ii+1))
sys.stdout.flush()
sys.stdout.write("\r")
# convert results to percentage
for fam in copulaFamiliesToTest:
results[fam.lower()] = results[fam.lower()]/float(numMCSims) * 100
return results
def testHELM(tau, M, N, familyToTest, numMCSims, copulaFamiliesToTest):
results = {}
for fam in copulaFamiliesToTest:
results[fam.lower()] = 0
for ii in range(0, numMCSims):
# generate samples of the requested copula with tau same as the
# empirical signature we calculated above
if (familyToTest.lower() == 'gaussian'):
r = invcopulastat(familyToTest, 'kendall', tau)
Rho = np.empty((N, N))
for jj in range(0, N):
for kk in range(0, N):
if (jj == kk):
Rho[jj][kk] = 1
else:
Rho[jj][kk] = r
try:
U = copularnd(familyToTest, M, Rho)
except ValueError:
# copularnd will throw a ValueError if Rho is not a positive semidefinite matrix
return results # return 0, which will then be ignored by tests
else: # assume Clayton, Frank, or Gumbel
try:
alpha = invcopulastat(familyToTest, 'kendall', tau)
U = copularnd(familyToTest, M, N, alpha)
except ValueError:
continue
lst = []
for jj in range(0, N):
U_conditioned = U[:, jj]
# if there are any 1's, condition it
U_conditioned[U_conditioned == 1] = 0.99
if (jj % 2 == 0):
lst.append(norm.ppf(U_conditioned))
else:
lst.append(expon.ppf(U_conditioned))
# combine X and Y into the joint distribution w/ the copula
X = np.vstack(lst)
X = X.T
ret = optimalCopulaFamily(X, family_search=copulaFamiliesToTest)
ret_family = ret[0].lower()
# aggregate results
results[ret_family] = results[ret_family] + 1.0
# display some progress
sys.stdout.write("\rComputing " + str(familyToTest) +
" Copula (DIM=%d) (tau=%f)-- %d%%" % (N, tau, ii + 1))
sys.stdout.flush()
sys.stdout.write("\r")
# convert results to percentage
for fam in copulaFamiliesToTest:
results[fam.lower()] = results[fam.lower()] / float(numMCSims) * 100
return results