/// <summary>
/// Computes the natural logarithm of the cumulative bivariate normal distribution.
/// </summary>
/// <param name="x">First upper limit.</param>
/// <param name="y">Second upper limit.</param>
/// <param name="r">Correlation coefficient.</param>
/// <returns><c>ln(phi(x,y,r))</c></returns>
public static double NormalCdfLn(double x, double y, double r)
{
if (Double.IsNegativeInfinity(x) || Double.IsNegativeInfinity(y))
{
return(Double.NegativeInfinity);
}
else if (Double.IsPositiveInfinity(x))
{
return(NormalCdfLn(y));
}
else if (Double.IsPositiveInfinity(y))
{
return(NormalCdfLn(x));
}
else if (r == 0)
{
return(NormalCdfLn(x) + NormalCdfLn(y));
}
else if (r == 1)
{
return(NormalCdfLn(Math.Min(x, y)));
}
else if (r == -1)
{
if (x > 0)
{
if (y > 0)
{
// 1-NormalCdf(-x) + 1-NormalCdf(-y)-1 = 1 - NormalCdf(-x) - NormalCdf(-y)
return(Log1MinusExp(LogSumExp(NormalCdfLn(-x), NormalCdfLn(-y))));
}
else
{
// 1-NormalCdf(-x) + NormalCdf(y) - 1 = NormalCdf(y) - NormalCdf(-x)
double nclx = NormalCdfLn(-x);
double diff = NormalCdfLn(y) - nclx;
if (diff < 0)
{
return(double.NegativeInfinity);
}
return(LogExpMinus1(diff) + nclx);
}
}
else
{
if (y > 0)
{
// NormalCdf(x) - NormalCdf(-y)
if (x < -y)
{
return(double.NegativeInfinity);
}
double ncly = NormalCdfLn(-y);
double diff = NormalCdfLn(x) - ncly;
if (diff < 0)
{
return(double.NegativeInfinity);
}
return(LogExpMinus1(diff) + ncly);
}
else
{
// x < 0 and y < 0
return(double.NegativeInfinity);
}
}
}
// at this point, both x and y are finite.
// swap to ensure |x| > |y|
if (Math.Abs(y) > Math.Abs(x))
{
double t = x;
x = y;
y = t;
}
double logOffset = double.NegativeInfinity;
double scale = 1;
// ensure x <= 0
if (x > 0)
{
// phi(x,y,r) = phi(inf,y,r) - phi(-x,y,-r)
logOffset = MMath.NormalCdfLn(y);
scale = -1;
x = -x;
r = -r;
}
// ensure r <= 0
if (r > 0)
{
// phi(x,y,r) = phi(x,inf,r) - phi(x,-y,-r)
double logOffset2 = MMath.NormalCdfLn(x);
if (scale == 1)
{
logOffset = logOffset2;
}
else
{
// the difference here must always be positive since y > -x
// offset -= offset2;
// logOffset = log(exp(logOffset) - exp(logOffset2))
logOffset = MMath.LogDifferenceOfExp(logOffset, logOffset2);
}
scale *= -1;
y = -y;
r = -r;
}
double omr2 = (1 - r) * (1 + r); // more accurate than 1-r*r
double ymrx = (y - r * x) / Math.Sqrt(omr2);
double exponent;
double result = NormalCdf_Helper(x, y, r, omr2, ymrx, out exponent);
double logResult = exponent + Math.Log(result);
if (scale == -1)
{
return(MMath.LogDifferenceOfExp(logOffset, logResult));
}
else
{
return(MMath.LogSumExp(logOffset, logResult));
}
}
