/// <summary>
/// Computes the circular kurtosis of the given circular angles.
/// </summary>
///
/// <param name="angles">A double array containing the angles in radians.</param>
///
/// <returns>The circular kurtosis for the given <paramref name="angles"/>.</returns>
///
public static double Kurtosis(double[] angles)
{
// Compute mean direction
double theta = Circular.Mean(angles);
// Compute central moments
double rho2 = CentralMoments(angles, 2).Magnitude;
double mu2 = NoncentralMoments(angles, 2).Phase;
// compute skewness
double k = 0;
for (int i = 0; i < angles.Length; i++) // Pewsey, Metrika, 2004
{
k += Math.Cos(2 * Circular.Distance(angles[i], theta));
}
k /= angles.Length;
/*
* double k0 = 0;
* double R4 = (R * R * R * R);
* double omR2 = (1 - R) * (1 - R);
* for (int i = 0; i < angles.Length; i++) // Fisher, Circular Statistics, p. 34
* k0 += (rho2 * Math.Cos(Circular.Distance(mu2, 2 * theta)) - R4) / omR2; // (formula 2.30)
*/
return(k);
}
/// <summary>
/// Computes the circular skewness of the given circular angles.
/// </summary>
///
/// <param name="angles">A double array containing the angles in radians.</param>
///
/// <returns>The circular skewness for the given <paramref name="angles"/>.</returns>
///
public static double Skewness(double[] angles)
{
// compute necessary values
double theta = Circular.Mean(angles);
System.Numerics.Complex m = CentralMoments(angles, 2);
double rho2 = m.Magnitude;
double mu2 = m.Phase;
// compute skewness
double b = 0; // Pewsey, Metrika, 2004
for (int i = 0; i < angles.Length; i++)
{
b += Math.Sin(2 * Distance(angles[i], theta));
}
b /= angles.Length;
/*
* // alternative skewness measure from Fisher
* // Statistical Analysis of Circular Data, p. 34
* double b0 = 0; // (formula 2.29)
* double omR = Math.Pow(1 - R, 3 / 2.0);
*
* for (int i = 0; i < angles.Length; i++)
* b0 += rho2 * Math.Sin(Distance(mu2, 2 * theta)) / omR;
*/
return(b);
}