/// <summary>
/// Returns the value of the density function of a copula, given two input cumulative distribution fucntions, u and v.
/// </summary>
public static double CopulaDensityFunction(CopulaType type, double u, double v, double alpha)
{
CheckCopulaAlpha(type, alpha);
switch (type)
{
case CopulaType.Clayton:
return (1.0 + alpha) * Math.Pow(u * v, -alpha - 1.0) * Math.Pow(Math.Pow(u, -alpha) + Math.Pow(v, -alpha) - 1.0, -1.0 / alpha - 2.0);
case CopulaType.FGM:
return 1.0 + alpha * (1.0 - 2.0 * u) * (1.0 - 2.0 * v);
case CopulaType.Frank:
double d1 = FrankPart(u, alpha) * FrankPart(v, alpha) + FrankPart(1, alpha);
return -alpha * FrankPart(1, alpha) * Math.Exp(-alpha * (u + v)) / (d1 * d1);
case CopulaType.Gumbel:
double lnu = Math.Log(u);
double lnv = Math.Log(v);
double g1 = Math.Pow(lnu * lnv, alpha - 1.0) / (u * v);
double g2 = CopulaCumulativeDistributionFunction(type, u, v, alpha);
double g3 = Math.Pow(-lnu + -lnv, 1.0 / alpha - 2.0);
double g4 = alpha - 1.0 + Math.Pow(-lnu + -lnv, 1.0 / alpha);
return g1 * g2 * g3 * g4;
case CopulaType.Independent:
return 1.0;
case CopulaType.Joe:
double ju = Math.Pow(1.0 - u, alpha);
double jv = Math.Pow(1.0 - v, alpha);
double d3 = ju + jv - ju * jv;
return Math.Pow(1.0 - u, alpha - 1.0) * Math.Pow(1.0 - v, alpha) * Math.Pow(d3, (1.0 / alpha) - 2.0) * (1.0 - alpha - d3);
default:
throw new ArgumentException("Copula type not expected.");
}
}
/// <summary>
/// Returns the value of a copula, given two input cumulative distribution fucntions, u and v.
/// </summary>
public static double CopulaCumulativeDistributionFunction(CopulaType type, double u, double v, double alpha)
{
CheckCopulaAlpha(type, alpha);
switch (type)
{
case CopulaType.Clayton:
return Math.Pow(Math.Pow(u, -alpha) + Math.Pow(v, -alpha) - 1.0, -1.0 / alpha);
case CopulaType.FGM:
return u * v * (1.0 + alpha * (1.0 - u) * (1.0 - v));
case CopulaType.Frank:
double d1 = 1 + (FrankPart(u, alpha) * FrankPart(v, alpha)) / FrankPart(1.0, alpha);
return -(1.0/alpha) * Math.Log(d1);
case CopulaType.Gumbel:
double d2 = -Math.Pow(Math.Pow(-Math.Log(u), alpha) + Math.Pow(-Math.Log(v), alpha), 1.0 / alpha);
return Math.Exp(d2);
case CopulaType.Independent:
return u * v;
case CopulaType.Joe:
double ju = Math.Pow(1.0 - u, alpha);
double jv = Math.Pow(1.0 - v, alpha);
double d3 = ju + jv - ju * jv;
return 1 - Math.Pow(d3, 1.0 / alpha);
default:
throw new ArgumentException("Copula type not expected.");
}
}
private static void CheckCopulaAlpha(CopulaType type, double alpha)
{
switch (type)
{
case CopulaType.Clayton:
if (alpha <= 0.0) throw new ArgumentException("Invalid alpha. Alpha should be greater than zero.");
return;
case CopulaType.FGM:
if (alpha < -1.0 || alpha > 1.0) throw new ArgumentException("Invalid alpha. Should be: -1 <= alpha <= +1.");
return;
case CopulaType.Frank:
if (alpha == 0.0) throw new ArgumentException("Invalid alpha. Alpha cannot equal zero.");
return;
case CopulaType.Gumbel:
if (alpha < 1.0) throw new ArgumentException("Invalid alpha. Should be: alpha >= 1.");
return;
case CopulaType.Independent:
return; //no alpha for independent
case CopulaType.Joe:
if (alpha < 0.0) throw new ArgumentException("Invalid alpha. Should be: alpha >= 0.");
return;
default:
throw new ArgumentException("Copula type not expected.");
}
}
/// <summary>
/// Returns the inverse of the first marginal CDF of a copula, for two cumulative distribution fucntions, u and v.
/// Used in Monte Carlo simulation.
/// </summary>
/// <param name="type"></param>
/// <param name="u">random number [0-1]</param>
/// <param name="C1">random number [0-1], independent of u</param>
/// <param name="alpha"></param>
/// <returns>v, random number [0-1]</returns>
public static double CopulaFirstMarginalInverse(CopulaType type, double u, double C1, double alpha)
{
CheckCopulaAlpha(type, alpha);
switch (type)
{
case CopulaType.Clayton:
double d = Math.Pow(C1, -alpha / (1.0 + alpha)) + Math.Pow(u, alpha) - 1.0;
return u * Math.Pow(d, -1.0 / alpha);
case CopulaType.Frank:
double f1 = C1 * (Math.Exp(-alpha) - 1.0);
double f2 = 1.0 + (Math.Exp(-alpha * u) - 1.0) * (1.0 - C1);
return -(1.0 / alpha) * Math.Log(1.0 + f1 / f2);
case CopulaType.Independent:
return C1;
default:
throw new ArgumentException("Copula type not expected.");
}
}