copulaFitGaussian

copulaFitGaussian(X)

Fits a Gaussian Copula to data in the (0, 1) space and estimates the linear correlation matrix.

Gaussian Copula is suitable for modeling symmetric linear dependence and does not capture tail dependence.

X is a non-empty two-dimensional numeric matrix or table with dimensions n×p (p = 2), specifying the input data.

• All elements must be finite and strictly within the open interval (0, 1), and no column can be constant or nearly constant.
• n denotes the number of samples, and p denotes the number of variables. The input data should satisfy n > p.

Returns a dictionary with the following key-value pairs:

• rho: A matrix indicating the estimated p×p linear correlation matrix.
• logLikelihood: A DOUBLE scalar indicating the log-likelihood of the Gaussian Copula model.

Note: The outputs in the following examples are for illustration only. Actual results may vary.

Example 1. Generate two-dimensional positively correlated data with a true correlation of 0.7, fit a Gaussian Copula, and verify whether the estimated correlation is close to the true value.

n = 1000
x1 = norm(0, 1, n)
x2 = 0.7 * x1 + sqrt(1 - 0.49) * norm(0, 1, n)
u1 = (rank(x1) + 1) \ (n + 1.0)
u2 = (rank(x2) + 1) \ (n + 1.0)
X = matrix(u1, u2)

res = copulaFitGaussian(X)
res.logLikelihood
// Output: 326.382767
res.rho

Example 2. Generate negatively correlated data with a true correlation of −0.5 and fit a Gaussian Copula to verify its estimation of negative dependence.

n = 1000
x1 = norm(0, 1, n)
x2 = -0.5 * x1 + sqrt(1 - 0.25) * norm(0, 1, n)
u1 = (rank(x1) + 1) \ (n + 1.0)
u2 = (rank(x2) + 1) \ (n + 1.0)
Xneg = matrix(u1, u2)
resNeg = copulaFitGaussian(Xneg)
resNeg.logLikelihood
// Output: 171.85922
resNeg.rho

Example 3. Simulate daily returns of two stocks, convert them to pseudo-observations, and fit a Gaussian Copula using a table as input to serve as a baseline model for portfolio risk modeling.

// Simulate daily returns of two stocks and transform them into pseudo-observations in (0, 1)
n = 1000
retA = norm(0.0005, 0.02, n)
retB = 0.6 * retA + sqrt(1 - 0.36) * norm(0.0003, 0.015, n)
uA = (rank(retA) + 1) \ (n + 1.0)
uB = (rank(retB) + 1) \ (n + 1.0)

// Construct a table as input
stockReturns = table(uA as stockA, uB as stockB)
res = copulaFitGaussian(stockReturns)
res.logLikelihood
// Output: 295.109231
res.rho

Related functions: copulaFitStudent, copulaFitClayton, copulaFitFrank, copulaFitGumbel, copulaRandGaussian, copulaPdfGaussian, copulaCdfGaussian