randomMvarTCopula

Given a Cholesky factorization of a correlation matrix, generates pseudorandom numbers from a Student’s t Copula distribution.

Synopsis

randomMvarTCopula (df, chol)

Required Arguments

float df (Input)

Degrees of freedom. df must be greater than 2.

float chol[[]] (Input)

An array of size n × n containing the upper-triangular Cholesky factorization of the correlation matrix of order n.

Return Value

An array of length n containing the pseudorandom numbers from a multivariate Student’s t Copula distribution.

Description

Function randomMvarTCopula generates pseudorandom numbers from a multivariate Student’s t Copula distribution which are uniformly distributed on the interval (0,1) representing the probabilities associated with Student’s t deviates with df degrees of freedom imprinted with correlation information from input upper-triangular Cholesky matrix chol. Cholesky matrix chol is defined as the “square root” of a user-defined correlation matrix. That is, chol is an upper triangular matrix such that the transpose of chol times chol is the correlation matrix. First, a length n array of independent random normal deviates with mean 0 and variance 1 is generated, and then this deviate array is post-multiplied by Cholesky matrix chol. Each of the n elements of the resulting vector of Cholesky-imprinted random deviates is then divided by

\[\sqrt{^s\!/_v},\]

where \(v\) = df and s is a random deviate taken from a chi-squared distribution with df degrees of freedom. Each element of the Cholesky-imprinted standard normal N(0,1) array is a linear combination of normally distributed random numbers and is therefore itself normal, and the division of each element by

\[\sqrt{^s\!/_v}\]

insures that each element of the resulting array is Student’s t distributed. Finally, each element of the Cholesky-imprinted Student’s t array is mapped to an output probability using the Student’s t cumulative distribution function (CDF) with df degrees of freedom.

Random deviates from arbitrary marginal distributions which are imprinted with the correlation information contained in Cholesky matrix chol can then be generated by inverting the output probabilities using user-specified inverse CDF functions.

Example: Using Student’s t Copulas to Imprint and Extract Correlation Information ———————————————————————————

This example uses function randomMvarTCopula to generate a multivariate sequence tcdevt whose marginal distributions are user-defined and imprinted with a user-specified input correlation matrix corrin and then uses function canonicalCorrelation to extract an output canonical correlation matrix corrout from this multivariate random sequence.

This example illustrates two useful copula related procedures. The first procedure generates a random multivariate sequence with arbitrary user-defined marginal deviates whose dependence is specified by a user-defined correlation matrix. The second procedure is the inverse of the first: an arbitrary multivariate deviate input sequence is first mapped to a corresponding sequence of empirically derived variates, i.e. cumulative distribution function values representing the probability that each random variable has a value less than or equal to the input deviate. The variates are then inverted, using the inverse standard normal CDF function, to N(0,1) deviates; and finally, a canonical covariance matrix is extracted from the multivariate N(0,1) sequence using the standard sum of products.

This example demonstrates that function randomMvarTCopula correctly imbeds the user-defined correlation information into an arbitrary marginal distribution sequence by extracting the canonical correlation from these sequences and showing that they differ from the original correlation matrix by a small relative error.

Recall that a Gaussian Copula array sequence, whose probabilities are mapped directly from Cholesky-imprinted N(0,1) deviates, has the property that the relative error between the input and output correlation matrices generally decreases as the number of multivariate sequence vectors increases. This is understandable because the correlation imprinting and extraction processes both act upon N(0,1) marginal distributions, and one would expect that a larger sample would therefore result in more accurate imprinting and extraction of correlation information.

In contrast, the imprinting of correlation information onto the Student’s t vector sequence is accomplished by imprinting onto an N(0,1) array and then dividing the array components by a scaled chi-squared random deviate, thereby introducing noise into the imprinting process. (An array of Student’s t deviates cannot be Cholesky-imprinted directly, because a linear combination of Student’s t deviates is not Student’s t distributed.) A larger sample would thus contain additional correlation information and additional noise, so the accuracy would be expected to plateau. This is illustrated in the example below, which should be compared with the Gaussian Copula example given for CNL function randomMvarGaussianCopula.

from __future__ import print_function
from numpy import *
from pyimsl.math.linSolPosdef import linSolPosdef
from pyimsl.stat.randomOption import randomOption
from pyimsl.stat.randomSeedSet import randomSeedSet
from pyimsl.stat.randomMvarTCopula import randomMvarTCopula
from pyimsl.stat.chiSquaredInverseCdf import chiSquaredInverseCdf
from pyimsl.stat.fInverseCdf import fInverseCdf
from pyimsl.stat.normalInverseCdf import normalInverseCdf
from pyimsl.stat.canonicalCorrelation import canonicalCorrelation

nvar = 3
lmax = 15000
df = 5.0
arg1 = 10.0
arg2 = 15.0
corrin = [[1.0, -0.9486832, 0.8164965],
          [-0.9486832, 1.0, -0.6454972],
          [0.8164965, -0.6454972, 1.0]]

print("Off-diagonal elements of Input Correlation Matrix:\n")
for i in range(nvar):
    for j in range(i):
        print(" CorrIn(%d,%d) = %10.6f" % (i, j, corrin[i][j]))
print("\n Degrees of freedom df = %6.2f" % df)
print("\n Imprinted random sequences distributions:")
print("\n 1: Chi, 2: F, 3: Normal;")
print("\nOff-diagonal elements of Output Correlation Matrices")
print("calculated from Student's t Copula imprinted")
print("multivariate sequence:")

#
# Compute the Cholesky factorization of corrin
#
# Use IMSL function linSolPosdef to generate
# the nvar by nvar upper triangular matrix chol from
# the Cholesky decomposition R*RT of input correlation
# matrix corrin:
#
chol = []
linSolPosdef(corrin, None, factor=chol, factorOnly=True)

kmax = lmax / 100
for kk in range(1, 4):
    tcdevt = zeros((int(kmax), nvar), dtype=double)
    print("\n# of vectors in multivariate sequence: %7d\n\n" % kmax)
    # use Congruential RN generator, with multiplier 16807
    randomOption(1)
    # set RN generator seed to be 123457
    randomSeedSet(123457)

    for k in range(int(kmax)):
        #
        # generate a NVAR-length random Student's t Copula
        # variate output vector tcvart which is uniformly
        # distributed on the interval [0,1] and imprinted
        # with correlation information from input Cholesky
        # matrix chol:
        tcvart = randomMvarTCopula(df, chol)
        for j in range(3):
            #
            # invert Student's t Copula probabilities to
            # deviates using variable-specific
            # inversions: j = 0: Chi Square; j = 1: F;
            # j = 2: Normal(0,1); will end up with deviate
            # sequences ready for mapping to canonical
            # correlation matrix:
            #
            if (j == 0):
                # convert probs into ChiSquare(df=10) deviates
                tcdevt[k, j] = chiSquaredInverseCdf(tcvart[j], arg1)
            elif (j == 1):
                # convert probs into F(dfn=15,dfd=10) deviates
                tcdevt[k, j] = fInverseCdf(tcvart[j], arg2, arg1)
            else:
                # convert probs into Normal(mean=0,variance=1) deviates:
                tcdevt[k, j] = normalInverseCdf(tcvart[j])
    #
    # extract Canonical Correlation matrix from arbitrarily
    # distributed deviate sequences tcdevt (k=1..kmax, j=1..NVAR)
    # which have been imprinted with corrin (i=1..NVAR, j=1..NVAR)
    # above:
    corrout = canonicalCorrelation(tcdevt)
    for i in range(nvar):
        for j in range(i):
            rs00 = corrin[i][j]
            rs = corrout[i][j]
            relerr = abs((rs - rs00) / rs00)
            print(" CorrOut(%d,%d) = %10.6f; relerr = %10.6f" %
                  (i, j, corrout[i][j], relerr))
    kmax *= 10

Output

Off-diagonal elements of Input Correlation Matrix:

 CorrIn(1,0) =  -0.948683
 CorrIn(2,0) =   0.816496
 CorrIn(2,1) =  -0.645497

 Degrees of freedom df =   5.00

 Imprinted random sequences distributions:

 1: Chi, 2: F, 3: Normal;

Off-diagonal elements of Output Correlation Matrices
calculated from Student's t Copula imprinted
multivariate sequence:

# of vectors in multivariate sequence:     150


 CorrOut(1,0) =  -0.953573; relerr =   0.005154
 CorrOut(2,0) =   0.774720; relerr =   0.051166
 CorrOut(2,1) =  -0.621419; relerr =   0.037302

# of vectors in multivariate sequence:    1500


 CorrOut(1,0) =  -0.944316; relerr =   0.004603
 CorrOut(2,0) =   0.810163; relerr =   0.007757
 CorrOut(2,1) =  -0.636348; relerr =   0.014174

# of vectors in multivariate sequence:   15000


 CorrOut(1,0) =  -0.946770; relerr =   0.002017
 CorrOut(2,0) =   0.808562; relerr =   0.009718
 CorrOut(2,1) =  -0.636322; relerr =   0.014215