/// <summary>
/// Returns the population kurtosis of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The population kurtosis of the elements of v.</returns>
public static double PopulationKurtosis(IEnumerable <double> v)
{
double mean = Mean(v);
int count = 0;
double num = 0.0;
double sumsq = 0.0;
foreach (double item in v)
{
double c = item - mean;
double sq = c * c;
num += sq * sq;
sumsq += sq;
count++;
}
if (count < 4)
{
return(double.NaN);
}
double Stdev4 = ExMath.Pow(sumsq / count, 2);
return((num / count) / Stdev4 - 3);
}
/// <summary>
/// Returns the central moment of a specific order of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <param name="k">The order of the central moment.</param>
/// <returns>The central moment of k-order of the elements of v.</returns>
/// <exception cref="System.ArgumentException">The value of k less than one.</exception>
public static double PopulationCentralMoment(IEnumerable <double> v, int k)
{
if (k < 1)
{
throw new ArgumentException("The order of the central moment must be greater than zero.");
}
if (!v.Any())
{
return(double.NaN);
}
if (k == 1)
{
return(0.0);
}
int count = 0;
double mean = Mean(v);
double sum = 0.0;
foreach (double item in v)
{
sum += ExMath.Pow(item - mean, k);
count++;
}
return(sum / count);
}
/// <summary>
/// Returns the population skewness of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The population skewness of the elements of v.</returns>
public static double PopulationSkewness(IEnumerable <double> v)
{
double mean = Mean(v);
int count = 0;
double sumcube = 0.0;
double sumsq = 0.0;
foreach (double item in v)
{
double c = item - mean;
double sq = c * c;
sumcube += sq * c;
sumsq += sq;
count++;
}
if (count < 3)
{
return(double.NaN);
}
double stdev = Math.Sqrt(sumsq / count);
return((sumcube / count) / ExMath.Pow(stdev, 3));
}
/// <summary>
/// Returns the sample kurtosis of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The sample kurtosis of the elements of v.</returns>
public static double SampleKurtosis(IEnumerable <double> v)
{
double mean = Mean(v);
int count = 0;
double num = 0.0;
double sumsq = 0.0;
foreach (double item in v)
{
double c = item - mean;
double sq = c * c;
num += sq * sq;
sumsq += sq;
count++;
}
if (count < 4)
{
return(double.NaN);
}
double Stdev4 = ExMath.Pow(sumsq / (count - 1), 2);
return((count * (count + 1.0) / ((count - 1.0) * (count - 2.0) * (count - 3.0))) * (num / Stdev4) -
3 * ((count - 1.0) * (count - 1.0)) / ((count - 2.0) * (count - 3.0)));
}
/// <summary>
/// Returns the sample skewness of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The sample skewness of the elements of v.</returns>
public static double SampleSkewness(IEnumerable <double> v)
{
double mean = Mean(v);
int count = 0;
double sumcube = 0.0;
double sumsq = 0.0;
foreach (double item in v)
{
double c = item - mean;
double sq = c * c;
sumcube += sq * c;
sumsq += sq;
count++;
}
if (count < 3)
{
return(double.NaN);
}
double Stdev = Math.Sqrt(sumsq / (count - 1));
return((count * sumcube) / ((count - 1.0) * (count - 2.0) * ExMath.Pow(Stdev, 3)));
}
/// <summary>
/// Returns the population variance of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The population variance of the elements of v.</returns>
public static double PopulationVariance(IEnumerable <double> v)
{
int count = 0;
double sum = 0.0;
double mean = Mean(v);
foreach (double item in v)
{
sum += ExMath.Pow(item - mean, 2);
count++;
}
return(sum / count);
}
/// <summary>
/// Returns the sample variance of the elements of an array.
/// </summary>
/// <param name="v">An array of real numbers.</param>
/// <returns>The sample variance of the elements of v.</returns>
public static double SampleVariance(IEnumerable <double> v)
{
int count = 0;
double sum = 0.0;
double mean = Mean(v);
foreach (double item in v)
{
sum += ExMath.Pow(item - mean, 2);
count++;
}
if (count == 1)
{
return(0.0);
}
return(sum / (count - 1));
}
/// <summary>
/// Returns the value of the probability density function for the specified value.
/// </summary>
/// <param name="x">A real number within the domain of the distribution.</param>
/// <returns>The value of the probability density function for x.</returns>
public override double ProbabilityDensityFunction(double x)
{
double var = _stdev * _stdev;
return((1.0 / Math.Sqrt(ExMath.TwoPi * var)) * Math.Exp(-ExMath.Pow(x - _mean, 2) / (2 * var)));
}