double chi_squared(int ndf, double chi2) {
// Calculate the chi-squared distribution function
// ndf = number of degrees of freedom
// chi2 = chi-squared value
// prob = probability of a chi-squared value <= chi2 (i.e. the left-hand
// tail area)
return gammad(0.5*chi2, 0.5*ndf);
}
double ppchi2 ( double p, double v, double g, int *ifault )
/******************************************************************************/
/*
Purpose:
PPCHI2 evaluates the percentage points of the Chi-squared PDF.
Discussion
Incorporates the suggested changes in AS R85 (vol.40(1),
pages 233-5, 1991) which should eliminate the need for the limited
range for P, though these limits have not been removed
from the routine.
Licensing:
This code is distributed under the GNU LGPL license.
Modified:
03 November 2010
Author:
Original FORTRAN77 version by Donald Best, DE Roberts.
C version by John Burkardt.
Reference:
Donald Best, DE Roberts,
Algorithm AS 91:
The Percentage Points of the Chi-Squared Distribution,
Applied Statistics,
Volume 24, Number 3, 1975, pages 385-390.
Parameters:
Input, double P, value of the chi-squared cumulative
probability density function.
0.000002 <= P <= 0.999998.
Input, double V, the parameter of the chi-squared probability
density function.
0 < V.
Input, double G, the value of log ( Gamma ( V / 2 ) ).
Output, int *IFAULT, is nonzero if an error occurred.
0, no error.
1, P is outside the legal range.
2, V is not positive.
3, an error occurred in GAMMAD.
4, the result is probably as accurate as the machine will allow.
Output, double PPCHI2, the value of the chi-squared random
deviate with the property that the probability that a chi-squared random
deviate with parameter V is less than or equal to PPCHI2 is P.
*/
{
double a;
double aa = 0.6931471806;
double b;
double c;
double c1 = 0.01;
double c2 = 0.222222;
double c3 = 0.32;
double c4 = 0.4;
double c5 = 1.24;
double c6 = 2.2;
double c7 = 4.67;
double c8 = 6.66;
double c9 = 6.73;
double c10 = 13.32;
double c11 = 60.0;
double c12 = 70.0;
double c13 = 84.0;
double c14 = 105.0;
double c15 = 120.0;
double c16 = 127.0;
double c17 = 140.0;
double c18 = 175.0;
double c19 = 210.0;
double c20 = 252.0;
double c21 = 264.0;
double c22 = 294.0;
double c23 = 346.0;
double c24 = 420.0;
double c25 = 462.0;
double c26 = 606.0;
double c27 = 672.0;
double c28 = 707.0;
double c29 = 735.0;
double c30 = 889.0;
double c31 = 932.0;
double c32 = 966.0;
double c33 = 1141.0;
double c34 = 1182.0;
double c35 = 1278.0;
double c36 = 1740.0;
double c37 = 2520.0;
double c38 = 5040.0;
double ch;
double e = 0.5E-06;
int i;
int if1;
int maxit = 20;
double pmax = 0.999998;
double pmin = 0.000002;
double p1;
double p2;
double q;
double s1;
double s2;
double s3;
double s4;
double s5;
double s6;
double t;
double value;
double x;
double xx;
/*
Test arguments and initialize.
*/
value = - 1.0;
if ( p < pmin || pmax < p )
{
*ifault = 1;
return value;
}
if ( v <= 0.0 )
{
*ifault = 2;
return value;
}
*ifault = 0;
xx = 0.5 * v;
c = xx - 1.0;
/*
Starting approximation for small chi-squared
*/
if ( v < - c5 * log ( p ) )
{
ch = pow ( p * xx * exp ( g + xx * aa ), 1.0 / xx );
if ( ch < e )
{
value = ch;
return value;
}
}
/*
Starting approximation for V less than or equal to 0.32
*/
else if ( v <= c3 )
{
ch = c4;
a = log ( 1.0 - p );
for ( ; ; )
{
q = ch;
p1 = 1.0 + ch * ( c7 + ch );
p2 = ch * (c9 + ch * ( c8 + ch ) );
t = - 0.5 + (c7 + 2.0 * ch ) / p1 - ( c9 + ch * ( c10 +
3.0 * ch ) ) / p2;
ch = ch - ( 1.0 - exp ( a + g + 0.5 * ch + c * aa ) * p2 / p1) / t;
if ( r8_abs ( q / ch - 1.0 ) <= c1 )
{
break;
}
}
}
else
{
/*
Call to algorithm AS 111 - note that P has been tested above.
AS 241 could be used as an alternative.
*/
x = ppnd ( p, ifault );
/*
Starting approximation using Wilson and Hilferty estimate
*/
p1 = c2 / v;
ch = v * pow ( x * sqrt ( p1 ) + 1.0 - p1, 3 );
/*
Starting approximation for P tending to 1.
*/
if ( c6 * v + 6.0 < ch )
{
ch = - 2.0 * ( log ( 1.0 - p ) - c * log ( 0.5 * ch ) + g );
}
}
/*
Call to algorithm AS 239 and calculation of seven term
Taylor series
*/
for ( i = 1; i <= maxit; i++ )
{
q = ch;
p1 = 0.5 * ch;
p2 = p - gammad ( p1, xx, &if1 );
if ( if1 != 0 )
{
*ifault = 3;
return value;
}
t = p2 * exp ( xx * aa + g + p1 - c * log ( ch ) );
b = t / ch;
a = 0.5 * t - b * c;
s1 = ( c19 + a * ( c17 + a * ( c14 + a * ( c13 + a * ( c12 +
c11 * a ))))) / c24;
s2 = ( c24 + a * ( c29 + a * ( c32 + a * ( c33 + c35 * a )))) / c37;
s3 = ( c19 + a * ( c25 + a * ( c28 + c31 * a ))) / c37;
s4 = ( c20 + a * ( c27 + c34 * a) + c * ( c22 + a * ( c30 + c36 * a ))) / c38;
s5 = ( c13 + c21 * a + c * ( c18 + c26 * a )) / c37;
s6 = ( c15 + c * ( c23 + c16 * c )) / c38;
ch = ch + t * ( 1.0 + 0.5 * t * s1 - b * c * ( s1 - b *
( s2 - b * ( s3 - b * ( s4 - b * ( s5 - b * s6 ))))));
if ( e < abs ( q / ch - 1.0 ) )
{
value = ch;
return value;
}
}
*ifault = 4;
value = ch;
return value;
}