internal override double[] CentralMoments(int rMax)
{
// This is just a recursive formulation of the direct formula.
double[] C = new double[rMax + 1];
C[0] = 1.0;
double t = 1.0; // Keep track of r! s^r
for (int r = 2; r <= rMax; r += 2)
{
t *= r * (r - 1) * (s * s);
C[r] = 2.0 * t * AdvancedMath.DirichletEta(r);
}
return(C);
}
/*
* public override double Variance {
* get {
* return (Global.HalfPI * (Math.PI / 6.0 - Global.LogTwo * Global.LogTwo) - Global.SqrtHalfPI * Global.LogTwo / 12.0 / Math.Sqrt(n));
* }
* }
*/
public override double Moment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException("r");
}
else if (r == 0)
{
return(1.0);
}
else
{
// we can get these expressions just by integrating Q_0' and Q_1' term by term
double M0 = AdvancedMath.DirichletEta(r) * AdvancedMath.Gamma(r / 2.0 + 1.0) / Math.Pow(2.0, r / 2.0 - 1.0);
double M1 = -2.0 / 3.0 * AdvancedMath.DirichletEta(r - 1) * AdvancedMath.Gamma((r + 1) / 2.0) / Math.Pow(2.0, (r + 1) / 2.0) * r;
return(M0 + M1 / Math.Sqrt(n));
}
}
/// <inheritdoc />
public override double CentralMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else if (r % 2 != 0)
{
return(0.0);
}
else
{
return(2.0 * AdvancedIntegerMath.Factorial(r) * AdvancedMath.DirichletEta(r) * MoreMath.Pow(s, r));
}
}
/// <inheritdoc />
public override double RawMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else if (r == 1)
{
return(Mean);
}
else
{
return(AdvancedMath.Gamma(r / 2.0 + 1.0) * AdvancedMath.DirichletEta(r) / Math.Pow(2.0, r / 2.0 - 1.0));
}
}
// The P-series expression for p(x) can be multiplied by x^r and integrated term-by term to give expressions
// for <x^r> involving powers of 1/k, i.e. zeta series.
public override double RawMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else
{
double q = Math.Pow(2.0, r / 2.0);
double d0 = AdvancedMath.DirichletEta(r);
double d1 = AdvancedMath.DirichletEta(r - 1);
double g0 = AdvancedMath.Gamma(r / 2.0 + 1.0);
double m0 = 2.0 / q * g0 * d0;
double m1 = -Global.SqrtTwo / 3.0 / q * AdvancedMath.Gamma(r / 2.0 + 1.0 / 2.0) * d1 * r;
// The O(1/N) expression has some canceling infinities for r = 1 so we form the limit by hand for that cases
double m2;
if (r == 1)
{
m2 = Global.SqrtHalfPI / 36.0 * (3.0 * Global.LogTwo - 1.0);
}
else
{
m2 = -1.0 / 18.0 / q * g0 * (
r * (r - 4) * d0 -
2 * (r - 1) * AdvancedMath.DirichletEta(r - 2) +
2 * (r - 1) * (1.0 - MoreMath.Pow(2, -r)) / (1.0 - MoreMath.Pow(2, 1 - r)) * d0
);
}
return(m0 + m1 / Math.Sqrt(n) + m2 / n);
}
}
