/// <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);
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);
}
/// <summary>
/// Computes the circular quartiles of the given circular samples.
/// The minimum possible value for a sample must be zero and the maximum must
/// be indicated in the parameter <paramref name="length"/>.
/// </summary>
///
/// <param name="samples">A double array containing the circular samples.</param>
/// <param name="length">The maximum possible value of the samples.</param>
/// <param name="range">The sample quartiles, as an out parameter.</param>
/// <param name="median">The median value of the <paramref name="samples"/>, if already known.</param>
/// <param name="wrap">
/// Whether range values should be wrapped to be contained in the circle. If
/// set to false, range values could be returned outside the [+pi;-pi] range.
/// </param>
/// <param name="type">The quartile definition that should be used. See <see cref="QuantileMethod"/> for datails.</param>
///
/// <returns>The median of the given samples.</returns>
///
public static double Quartiles(double[] samples, double length, out DoubleRange range, double median, bool wrap = true, QuantileMethod type = QuantileMethod.Default)
{
double angleMedian = Circular.ToRadians(median, length);
double q2 = Quartiles(ToRadians(samples, length), out range, angleMedian, wrap, type: type);
range.Min = ToCircular(range.Min, length, wrap);
range.Max = ToCircular(range.Max, length, wrap);
return(ToCircular(q2, length));
}
/// <summary>
/// Computes the circular quartiles of the given circular samples.
/// The minimum possible value for a sample must be zero and the maximum must
/// be indicated in the parameter <paramref name="length"/>.
/// </summary>
///
/// <param name="samples">A double array containing the circular samples.</param>
/// <param name="length">The maximum possible value of the samples.</param>
/// <param name="q1">The first quartile, as an out parameter.</param>
/// <param name="q3">The third quartile, as an out parameter.</param>
/// <param name="median">The median value of the <paramref name="samples"/>, if already known.</param>
/// <param name="wrap">
/// Whether range values should be wrapped to be contained in the circle. If
/// set to false, range values could be returned outside the [+pi;-pi] range.
/// </param>
/// <param name="type">The quartile definition that should be used. See <see cref="QuantileMethod"/> for datails.</param>
///
/// <returns>The median of the given samples.</returns>
///
public static double Quartiles(double[] samples, double length, out double q1, out double q3, double median, bool wrap = true, QuantileMethod type = QuantileMethod.Default)
{
double angleMedian = Circular.ToRadians(median, length);
double q2 = Quartiles(ToRadians(samples, length), out q1, out q3, angleMedian, wrap, type: type);
q1 = ToCircular(q1, length, wrap);
q3 = ToCircular(q3, length, wrap);
return(ToCircular(q2, length));
}