/* Natural logarithm of gamma function for values > 0. */
double ln_gamma(double alpha) {
/* Magic coefficients for calculating the gamma function. */
static double coef[6] = {76.18009173, -86.50532033, 24.01409822,
-1.231739516, 0.120858003e-2, -0.536382e-5};
double ln_factor;
double series;
double a;
int i;
/* Use the magic coefficients to calculate ln(GAMMA). */
if (alpha > 1.0)
{
ln_factor = alpha + 4.5;
ln_factor = (alpha - 0.5) * log(ln_factor) - (ln_factor);
for (i = 0, a = alpha, series = 1.0; i < 6; ++i, a = a + 1.0)
series = series + coef[i] / a;
return(ln_factor + log(2.50662827465 * series));
}
/* Use the relationship of GAMMA(z) = GAMMA(z+1) / z since
* the magic coefficients work less well between 0 and 1. */
else if (alpha > 0.0) return(ln_gamma(alpha + 1.0) - log(alpha));
/* Magic coefficients do not work with values <= 0. */
else
{
fprintf(stderr,
"The function \"ln_gamma()\" only works with values > 0\n");
return -1;
}
}
/* Determine the incomplete gamma function using the series approximation. */
double gamma_ser(double alpha, double ln_likelihood) {
double sum; /* The sum of the series. */
double alpha_n; /* "alpha" + "n". */
double term; /* Current term in the series. */
for (alpha_n = alpha + 1.0, sum = term = 1.0/alpha;
term >= sum * L_ERR; alpha_n = alpha_n + 1.0)
{
term = term * ln_likelihood / alpha_n;
sum = sum + term;
}
return(exp(-ln_likelihood + alpha * log(ln_likelihood)
+ log(sum) - ln_gamma(alpha)));
}
/* Determine the natural logarithm of the incomplete gamma function's
* complement using the continued fraction approximation. */
double ln_gamma_cf(double alpha, double ln_likelihood) {
double frac_old = 0.0; /* Old continued fraction. */
/* Variables A and B for calculating the continued fraction. */
double a_0 = 0.0, a_1 = 1.0;
double b_0 = 1.0, b_1 = ln_likelihood;
/* Variables for calculating A and B. */
double i;
double i_alpha; /* i - alpha. */
/* Do 2 cycles of the reiteration for each index. */
for (i = 1.0; ; i = i + 1.0)
{
/* Cycle one of reiteration. */
i_alpha = i - alpha;
a_0 = a_1 + i_alpha * a_0;
b_0 = b_1 + i_alpha * b_0;
/* Cycle two of the reiteration. */
a_1 = ln_likelihood * a_0 + i * a_1;
b_1 = ln_likelihood * b_0 + i * b_1;
/* Determine the continued fraction if the denominator is not 0. */
if (b_1)
{
a_0 = a_0 / b_1;
a_1 = a_1 / b_1;
b_0 = b_0 / b_1;
b_1 = 1.0;
if (fabs((a_1 - frac_old)/a_1) < L_ERR)
return(-ln_likelihood + alpha*log(ln_likelihood) + log(a_1)
- ln_gamma(alpha));
frac_old = a_1;
}
}
}
double log_chi_inv_cdf(double alpha, double ln_like) {
return ln_gamma_prob(alpha/2.0, ln_like/2.0)-ln_gamma(alpha/2.0);
}
/*!
\brief
Evaluate the log-density for a Gamma distribution.
The Gamma distribution has the following density function
with respect to \f$ x \f$:
\f[
{\bf p} ( x | a, b) = \frac{b^a}{ \Gamma (a) } x^{a-1} \exp ( - b x )
\f]
where \f$ a \f$ and \f$ b \f$ are parameters and
\f$ \Gamma (a) \f$ is the \ref ln_gamma "gamma function".
This distribution has mean \f$ a / b \f$ and variance \f$ a / b^2 \f$.
\param[in] x
is the value of the Gamma distributed random variate.
\param[in] a
is the shape parameter in the Gamma distribution
\param[in] b
is the reciprical of the rate parameter in the Gamma distribution
(\f$ 1 / b \f$ is called the scale parameter).
\return
The return value is \f$ \log [ {\bf p} (x | a, b) ] \f$.
*/
double ln_gamma_density(double x, double a, double b)
{ return a*log(b) - ln_gamma(a) + (a-1.0)*log(x) - b*x; }
/*!
\brief
Evaluate the log-density for a Possion distribution.
The Possion distribution has the following density function
with respect to \f$ y \f$:
\f[
{\bf p} (y | \lambda ) = \frac{ \lambda^y } { y ! } \exp( - \lambda )
\f]
where \f$ y \f$ is the number of occurnaces,
and \f$ \lambda \f$ is a parameter that is equal
to the expected number of occurances in the time interval
(during which the occurances are counted).
This distribution has mean
\f$ \lambda \f$ and variance \f$ \lambda \f$.
\param[in] y
is the number of occurances that occured in the time interval
(must be non-negative).
\param[in] \lambda
is the the expected number of occurances in the time interval.
\return
The return value is \f$ \log [ {\bf p} ( y | \lambda ) ] \f$.
*/
double ln_poisson_density(int y, double lambda )
{ return y * log(lambda ) - ln_gamma(y + 1.0) - lambda ; }
