/// <summary>
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
/// </summary>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the density at <paramref name="x"/>.</returns>
public double Density(double x)
{
// TODO JVG we can probably do a better job for Cauchy special case
if (_freedom >= 1e+8d)
{
return(Normal.PDF(_location, _scale, x));
}
var d = (x - _location) / _scale;
return(Math.Exp(SpecialFunctions.GammaLn((_freedom + 1.0) / 2.0) - SpecialFunctions.GammaLn(_freedom / 2.0))
* Math.Pow(1.0 + (d * d / _freedom), -0.5 * (_freedom + 1.0))
/ Math.Sqrt(_freedom * Math.PI)
/ _scale);
}
/// <summary>
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
/// </summary>
/// <param name="location">The location (μ) of the distribution.</param>
/// <param name="scale">The scale (σ) of the distribution. Range: σ > 0.</param>
/// <param name="freedom">The degrees of freedom (ν) for the distribution. Range: ν > 0.</param>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the density at <paramref name="x"/>.</returns>
/// <seealso cref="Density"/>
public static double PDF(double location, double scale, double freedom, double x)
{
if (scale <= 0.0 || freedom <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
// TODO JVG we can probably do a better job for Cauchy special case
if (freedom >= 1e+8d)
{
return(Normal.PDF(location, scale, x));
}
var d = (x - location) / scale;
return(Math.Exp(SpecialFunctions.GammaLn((freedom + 1.0) / 2.0) - SpecialFunctions.GammaLn(freedom / 2.0))
* Math.Pow(1.0 + (d * d / freedom), -0.5 * (freedom + 1.0))
/ Math.Sqrt(freedom * Math.PI)
/ scale);
}
/// <summary>
/// Computes the probability density of the distribution (PDF) at x, i.e. ∂P(X ≤ x)/∂x.
/// </summary>
/// <param name="alpha">The stability (α) of the distribution. Range: 2 ≥ α > 0.</param>
/// <param name="beta">The skewness (β) of the distribution. Range: 1 ≥ β ≥ -1.</param>
/// <param name="scale">The scale (c) of the distribution. Range: c > 0.</param>
/// <param name="location">The location (μ) of the distribution.</param>
/// <param name="x">The location at which to compute the density.</param>
/// <returns>the density at <paramref name="x"/>.</returns>
/// <seealso cref="Density"/>
public static double PDF(double alpha, double beta, double scale, double location, double x)
{
if (alpha <= 0.0 || alpha > 2.0 || beta < -1.0 || beta > 1.0 || scale <= 0.0)
{
throw new ArgumentException(Resources.InvalidDistributionParameters);
}
if (alpha == 2d)
{
return(Normal.PDF(location, Constants.Sqrt2 * scale, x));
}
if (alpha == 1d && beta == 0d)
{
return(Cauchy.PDF(location, scale, x));
}
if (alpha == 0.5d && beta == 1d && x >= location)
{
return((Math.Sqrt(scale / Constants.Pi2) * Math.Exp(-scale / (2 * (x - location)))) / Math.Pow(x - location, 1.5));
}
throw new NotSupportedException();
}