/// <inheritdoc/>
public override double RawMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else
{
double[] central = CentralMoments(r);
return(MomentMath.CentralToRaw(a, central, r));
}
}
/// <inheritdoc/>
public override double CentralMoment(int r)
{
if (r < 0)
{
throw new ArgumentOutOfRangeException(nameof(r));
}
else if (r == 0)
{
return(1.0);
}
else
{
double[] K = Cumulants(r);
return(MomentMath.CumulantToCentral(K, 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
{
double[] C = CentralMoments(r);
return(MomentMath.CentralToRaw(Mean, C, r));
}
}
/// <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 == 1)
{
return(0.0);
}
else if (r == 2)
{
return(Variance);
}
else
{
double[] K = Cumulants(r);
return(MomentMath.CumulantToCentral(K, r));
// Begin with M_r = e^{r \mu + \frac{1}{2} r^2 \sigma^2} and use
// C_r = <(x-M_1)^r> = \sum_{k=0}^{r} {r \choose k} M_k (-M_1)^{r-k}
// = \sum_{k=0}^{r} (-1)^{r-k} {r \choose k} e^{k \mu + \frac{1}{2} k^2 \sigma^2} e^{(r-k)(\mu + \frac{1}{2} \sigma^2)}
// = e^{r (\mu + \frac{1}{2} \sigma^2)} \sum_{k=0}^{r} (-1)^{r-k} {r \choose k} e^{\frac{1}{2} k (k - 1) \sigma^2}
// So far, this is just the usual raw to central moment conversion, subject to the usual significant cancellation errors,
// particularly for small \sigma, in which case all exponentials will be nearly equal. Write
// e^{\frac{1}{2} k (k - 1) \sigma^2} = (e^{\frac{1}{2} k (k - 1) \sigma^2} - 1) + 1.
// When summed over all k, the contribution of the final +1 vanishes, so
// C_r = e^{r (\mu + \frac{1}{2} \sigma^2)} \sum_{k=0}^{r} (-1)^{r-k} {r \choose k} (e^{\frac{1}{2} k (k - 1) \sigma^2} - 1)
// and since ( ) is zero for k = 0 and k = 2, we need not start the sum until k = 2.
/*
* double s = 0.0;
* double x = sigma * sigma / 2.0;
* for (int k = 2; k <= r; k++) {
* double ds = AdvancedIntegerMath.BinomialCoefficient(r, k) * MoreMath.ExpMinusOne(k * (k - 1) * x);
* if ((r - k) % 2 != 0) ds = -ds;
* s += ds;
* }
* return (Math.Exp(r * (mu + x)) * s);
*/
/*
* // This follows from a straightforward expansion of (x-m)^n and substitution of expressions for M_k.
* // It eliminates some arithmetic but is still subject to loss of significance due to cancelation.
* double s2 = sigma * sigma / 2.0;
*
* double C = 0.0;
* for (int i = 0; i <= r; i++) {
* double dC = AdvancedIntegerMath.BinomialCoefficient(r, i) * Math.Exp((MoreMath.Sqr(r - i) + i) * s2);
* if (i % 2 != 0) dC = -dC;
* C += dC;
* }
* return (Math.Exp(r * mu) * C);
*/
// this isn't great, but it does the job
// expand in terms of moments about the origin
// there is likely to be some cancelation, but the distribution is wide enough that it may not matter
/*
* double m = -Mean;
* double C = 0.0;
* for (int k = 0; k <= n; k++) {
* C += AdvancedIntegerMath.BinomialCoefficient(n, k) * Moment(k) * Math.Pow(m, n - k);
* }
* return (C);
*/
}
}
/// <summary>
/// Computes a central moment of the distribution.
/// </summary>
/// <param name="r">The order of the moment to compute.</param>
/// <returns>The rth central moment of the distribution.</returns>
/// <seealso cref="RawMoment" />
public virtual double CentralMoment(int r)
{
// This is a terrible way to compute central moments, subject to significant cancelations, so replace it if at all possible.
double[] M = RawMoments(r);
return(MomentMath.RawToCentral(M, r));
}
/// <summary>
/// Computes a cumulant of the distribution.
/// </summary>
/// <param name="r">The index of the cumulant to compute.</param>
/// <returns>The rth cumulant of the distribution.</returns>
/// <seealso href="http://en.wikipedia.org/wiki/Cumulant"/>
public virtual double Cumulant(int r)
{
double[] C = CentralMoments(r);
double[] K = MomentMath.CentralToCumulant(Mean, C);
return(K[r]);
}