# Trivariate t-copula

## Coding of multivariate t-copula¶

THis code shows how to fit a trivariate t-copula and simulate it

In [1]:
import math
import numpy as np
import scipy as sp
import bokeh as bk
import scipy.optimize as opt
import matplotlib.pyplot as plt
import scipy.stats as stats
import seaborn as sns
import xlwt
from bokeh.plotting import figure, output_file, show


The loglikelihood of a 3D time-varying t-copula

In [2]:
def tCopula_3D_TVP_LL(theta,U,exog12,exog13,exog23,exogNU):
'''Returns -ve LL of the 3-dimension Student R copula'''
T,N = U.shape

k12 = exog12.shape[1]
k13 = exog13.shape[1]
k23 = exog23.shape[1]
kNU = exogNU.shape[1]

beta12 = theta[:k12]
beta13 = theta[k12:k12+k13]
beta23 = theta[k12+k13:k12+k13+k23]
betaNU = theta[k12+k13+k23:]

NUt = 2+np.exp(np.dot(exogNU,betaNU))
rho12 = -1+2/(1+np.exp(np.dot(-exog12,beta12)))
rho13 = -1+2/(1+np.exp(np.dot(-exog13,beta13)))
rho23 = -1+2/(1+np.exp(np.dot(-exog23,beta23)))

RHOt = np.empty((3,3,T))
counter = 0
for tt in np.arange(T):
#        allRHOt = np.concatenate((rho12[:tt],rho13[:tt],rho23[:tt]),axis=0)
allRHOt = np.vstack((rho12[tt],rho13[tt],rho23[tt]))
RHOt[:,:,tt] = theta2rho(allRHOt)
if is_pos_def(RHOt[:,:,tt])== 0:
counter = counter+1;

if counter > 0:
LL=1e7
else:
trmU = np.empty((T,N))
LLa = np.empty((T,1))

for tt in np.arange(T):
NU = NUt[tt]

if NU < 100:
trmU[tt,:] = stats.t.ppf(U[tt,:],NU)
else:
trmU[tt,:] = stats.norm.ppf(U[tt,:])

allRHOt = np.vstack((rho12[tt],rho13[tt],rho23[tt]))
RHO = theta2rho(allRHOt)

LLa[tt] = math.lgamma((NU+N)/2) + (N-1)*math.lgamma(NU/2)-N*math.lgamma((NU+1)/2)-0.5*math.log(np.linalg.det(RHO))
t1 = trmU[tt,:].dot(np.linalg.inv(RHO)).dot(trmU[tt,:])
#np.reduce(np.dot,[trmU[tt,:],np.linalg.inv(RHO),trmU[tt,:]])
# Equiv. to t0 =
LLa[tt] = LLa[tt] - ((NU+N)/2)*np.log(1+t1/NU)
LLa[tt] = LLa[tt] + ((NU+1)/2)*sum(np.log(1+(trmU[tt,:]**2/NU)))
LL = -sum(LLa)
print(LL)
return LL
#return LL,rho12,rho13,rho23,NUt


Produce M samples of d-dimensional multivariate t distribution Input: mu = mean (d dimensional numpy array or scalar) Sigma = scale matrix (dxd numpy array) df = degrees of freedom N = # of samples to produce

In [3]:
def multivariatet(mu,Sigma,df,N):

dim = len(Sigma)
g = np.tile(np.random.gamma(df/2.,2./df,M),(dim,1)).T
Z = np.random.multivariate_normal(np.zeros(dim),Sigma,N)
return mu + Z/np.sqrt(g)


Multivariate t-student density: output: the density of the given element input: x = parameter (d dimensional numpy array or scalar) mu = mean (d dimensional numpy array or scalar) Sigma = scale matrix (dxd numpy array) df = degrees of freedom d: dimension

In [4]:
def multivariate_t_distribution(x,mu,Sigma,df):
d = len(Sigma)
Num = gamma(1. * (d+df)/2)
Denom = ( gamma(1.*df/2) * pow(df*pi,1.*d/2) * pow(np.linalg.det(Sigma),1./2) * pow(1 + (1./df)*np.dot(np.dot((x - mu),np.linalg.inv(Sigma)), (x - mu)),1.* (d+df)/2))
d = 1. * Num / Denom
return d


Check if a matrix is positive and definite output: 1 (or 0) if the matrix is positive and definite input: x = the matrix (d dimensional numpy array)

In [5]:
def is_pos_def(x):

return np.all(np.linalg.eigvals(x) > 0)


The code below converds code from Kendall's tau to Pearson's correlation

In [6]:
def theta2rho(theta):
m = len(theta)
k = int((1+np.sqrt(1+8*m))/2)

out1 = np.empty((k,k))
counter=0
for ii in np.arange(k):
for jj in np.arange(ii,k):
if ii==jj:
out1[ii,jj]=1
else:
out1[ii,jj]=theta[counter]
out1[jj,ii]=theta[counter]
counter = counter+1
return out1


The code is to generate random variables from a multivaraite t-distribution

In [7]:
def mvtrnd(C,df,cases):
(m,n) = C.shape
s = np.diag(C)
s_not_1 = s != 1
if any(s_not_1):
C = C/np.sqrt(s*s.T)
T = np.linalg.cholesky(C)

r = np.dot(np.random.randn(cases,len(T)),T)
x = np.sqrt(np.random.gamma(df/2,2,cases)/df)
r = r/x    # No need the ./x(:,ones(n,1)) as Python does elem by elem operations
return r



The code is to generate random variables based on a t-copula with the degrees of freedom and the VCV Generates random vectors froma t-copula output: u = random vectors (NxP matrix). Each column of u is a sample from a Uniiform(0,1) marginal distribution input: Rho = PxP llinear correlation matrix nu = degrees of freedom n = N random vectors

In [8]:
def tCop_rnd(Rho,nu,n):
mvtVar = mvtrnd(Rho,nu,n)
u = stats.t.cdf(mvtVar,nu)
return u


The below is the script to generate the tri-variate copula, and then estimate the parameters of the model

In [9]:
if  __name__=="__main__":
#%% Time-varying trivariate t-Copula (Simulation)
np.random.seed(10)
T = 1000
x1 = np.ones([T,1])
x2 = np.reshape(0.7*sp.sin((np.arange(1,T+1)/100)),(T,1))
x3 = np.reshape(0.7*sp.cos((np.arange(1,T+1)/200)),(T,1))
x4 = np.reshape(0.7*sp.sin((np.arange(1,T+1)/50)),(T,1))
x5 = np.reshape(0.1*sp.cos((np.arange(1,T+1)/50)),(T,1))
x = np.arange(1,T+1)

fig1, ax1 = plt.subplots(1,1)
ax1.plot(x,x2,x,x3,x,x4,x,x5)
ax1.set_title('Co-variates')
ax1.set_xlabel('x Numbers')
ax1.set_ylabel('y Numbers')

beta12 = np.array([[0.5],[-1]])
beta13 = np.array([[1],[-0.5]])
beta23 = np.array([[1],[0.5]])
betaNU = np.array([[2],[5]])

exog12 = np.concatenate((x1,x2),axis=1)
exog13 = np.concatenate((x1,x3),axis=1)
exog23 = np.concatenate((x1,x4),axis=1)
exogNU = np.concatenate((x1,x5),axis=1)

rho12 = -1+2/(1+np.exp(np.dot(-exog12,beta12)))
rho13 = -1+2/(1+np.exp(np.dot(-exog13,beta13)))
rho23 = -1+2/(1+np.exp(np.dot(-exog23,beta23)))
NUt = 2+np.exp(np.dot(exogNU,betaNU))

fig2 = plt.figure()
ax1.plot(x,rho12,label='rho12')
ax1.plot(x,rho13,label='rho13')
ax1.plot(x,rho23,label='rho23')
ax1.plot(x,NUt,label='NUt')
plt.title('Information Variables')
plt.xlabel('x Numbers')
plt.ylabel('y Numbers')
plt.legend()

RHOt = np.empty((3,3,T))
for tt in np.arange(T):
allRHOt = np.vstack((rho12[tt],rho13[tt],rho23[tt]))
RHOt[:,:,tt] = theta2rho(allRHOt)
if is_pos_def(RHOt[:,:,tt])==0:
print(tt)

dataU = np.zeros((T,3))
data = np.zeros((T,3))

for tt in np.arange(T):
dataU[tt,:] = tCop_rnd(RHOt[:,:,tt],NUt[tt],1)
data[tt,:] = stats.norm.ppf(dataU[tt,:])

fig3,axes = plt.subplots(3,1,sharex=True,sharey=True)
for i in np.arange(3):
axes[i].plot(data[:,i])
title = 'Random variable generation from student tCopula Asset '
axes[i].set_title(title+str(i))

fig4,axes = plt.subplots(2,2,sharex=True,sharey=True)
axes[0,0].scatter(data[:,0],data[:,1], marker = ">"),axes[0,0].set_xlabel('x1'),axes[0,0].set_ylabel('x2')
axes[0,1].scatter(data[:,0],data[:,2], marker = "."),axes[0,1].set_xlabel('x1'),axes[0,1].set_ylabel('x3')
axes[1,0].scatter(data[:,1],data[:,2]),axes[1,0].set_xlabel('x2'),axes[1,0].set_ylabel('x3')

tCopData = dataU

# Time-varying trivariate t-copula (Estimation)
theta0 = np.zeros(8)
Udata = tCopData
thetahat_powell = opt.fmin_powell(tCopula_3D_TVP_LL,theta0,args=(Udata,exog12,exog13,exog23,exogNU,))


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Optimization terminated successfully.
Current function value: -174.422014
Iterations: 4
Function evaluations: 344

In [11]:
LL= tCopula_3D_TVP_LL(thetahat_powell,Udata,exog12,exog13,exog23,exogNU)

[-174.42201363]

In [12]:
thetahat_simplex = opt.fmin(tCopula_3D_TVP_LL,theta0,args=(Udata,exog12,exog13,exog23,exogNU))

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Warning: Maximum number of function evaluations has been exceeded.

In [ ]:
thetahat_DE = opt.differential_evolution(tCopula_3D_TVP_LL,theta0,args=(Udata,exog12,exog13,exog23,exogNU))


Write to Excel

In [ ]:
from tempfile import TemporaryFile
book = xlwt.Workbook()

for row, array in enumerate(tCopData):
for col, value in enumerate(array):
sheet1.write(row,col,value)

name = "tCop.xls"
book.save(name)
book.save(TemporaryFile())

ax1.plot(data[:,0])
ax2.plot(data[:,1])
ax3.plot(data[:,2])
plt.setp([a.get_xticklabels() for a in f.axes[:-1]], visible=False)
ax1.set_title('Random variable generation from Student t Copula')
plt.autoscale(tight='x')
plt.tight_layout()

In [ ]:
Output graphics to static html file

In [ ]:
output_file("covariates.html", title="Covariates")

# create a new plot
p = figure(
tools = "pan,box_zoom,reset,save",
title = "Covariates",
x_axis_label='x', y_axis_label='y'
)