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/* File altered for inclusion in cephes module for Python:
* Main loop commented out.... */
/* Travis Oliphant Nov. 1998 */
/* Re Kolmogorov statistics, here is Birnbaum and Tingey's formula for the
* distribution of D+, the maximum of all positive deviations between a
* theoretical distribution function P(x) and an empirical one Sn(x)
* from n samples.
*
* +
* D = sup [P(x) - S (x)]
* n -inf < x < inf n
*
*
* [n(1-e)]
* + - v-1 n-v
* Pr{D > e} = > C e (e + v/n) (1 - e - v/n)
* n - n v
* v=0
*
* [n(1-e)] is the largest integer not exceeding n(1-e).
* nCv is the number of combinations of n things taken v at a time. */
#include "mconf.h"
extern double MAXLOG;
/* Exact Smirnov statistic, for one-sided test. */
double smirnov(int n, double e)
{
int v, nn;
double evn, omevn, p, t, c, lgamnp1;
/* This comparison should assure returning NaN whenever
* e is NaN itself. In original || form it would proceed */
if (!(n > 0 && e >= 0.0 && e <= 1.0))
return (NPY_NAN);
if (e == 0.0)
return 1.0;
nn = (int) (floor((double) n * (1.0 - e)));
p = 0.0;
if (n < 1013) {
c = 1.0;
for (v = 0; v <= nn; v++) {
evn = e + ((double) v) / n;
p += c * pow(evn, (double) (v - 1))
* pow(1.0 - evn, (double) (n - v));
/* Next combinatorial term; worst case error = 4e-15. */
c *= ((double) (n - v)) / (v + 1);
}
}
else {
lgamnp1 = lgam((double) (n + 1));
for (v = 0; v <= nn; v++) {
evn = e + ((double) v) / n;
omevn = 1.0 - evn;
if (fabs(omevn) > 0.0) {
t = lgamnp1 - lgam((double) (v + 1))
- lgam((double) (n - v + 1))
+ (v - 1) * log(evn)
+ (n - v) * log(omevn);
if (t > -MAXLOG)
p += exp(t);
}
}
}
return (p * e);
}
/* Kolmogorov's limiting distribution of two-sided test, returns
* probability that sqrt(n) * max deviation > y,
* or that max deviation > y/sqrt(n).
* The approximation is useful for the tail of the distribution
* when n is large. */
double kolmogorov(double y)
{
double p, t, r, sign, x;
if (y < 1.1e-16)
return 1.0;
x = -2.0 * y * y;
sign = 1.0;
p = 0.0;
r = 1.0;
do {
t = exp(x * r * r);
p += sign * t;
if (t == 0.0)
break;
r += 1.0;
sign = -sign;
}
while ((t / p) > 1.1e-16);
return (p + p);
}
/* Functional inverse of Smirnov distribution
* finds e such that smirnov(n,e) = p. */
double smirnovi(int n, double p)
{
double e, t, dpde;
int iterations;
if (!(p > 0.0 && p <= 1.0)) {
mtherr("smirnovi", DOMAIN);
return (NPY_NAN);
}
/* Start with approximation p = exp(-2 n e^2). */
e = sqrt(-log(p) / (2.0 * n));
iterations = 0;
do {
/* Use approximate derivative in Newton iteration. */
t = -2.0 * n * e;
dpde = 2.0 * t * exp(t * e);
if (fabs(dpde) > 0.0)
t = (p - smirnov(n, e)) / dpde;
else {
mtherr("smirnovi", UNDERFLOW);
return 0.0;
}
e = e + t;
if (e >= 1.0 || e <= 0.0) {
mtherr("smirnovi", OVERFLOW);
return 0.0;
}
if (++iterations > MAXITER) {
mtherr("smirnovi", TOOMANY);
return (e);
}
}
while (fabs(t / e) > 1e-10);
return (e);
}
/* Functional inverse of Kolmogorov statistic for two-sided test.
* Finds y such that kolmogorov(y) = p.
* If e = smirnovi (n,p), then kolmogi(2 * p) / sqrt(n) should
* be close to e. */
double kolmogi(double p)
{
double y, t, dpdy;
int iterations;
if (!(p > 0.0 && p <= 1.0)) {
mtherr("kolmogi", DOMAIN);
return (NPY_NAN);
}
if ((1.0 - p) < 1e-16)
return 0.0;
/* Start with approximation p = 2 exp(-2 y^2). */
y = sqrt(-0.5 * log(0.5 * p));
iterations = 0;
do {
/* Use approximate derivative in Newton iteration. */
t = -2.0 * y;
dpdy = 4.0 * t * exp(t * y);
if (fabs(dpdy) > 0.0)
t = (p - kolmogorov(y)) / dpdy;
else {
mtherr("kolmogi", UNDERFLOW);
return 0.0;
}
y = y + t;
if (++iterations > MAXITER) {
mtherr("kolmogi", TOOMANY);
return (y);
}
}
while (fabs(t / y) > 1.0e-10);
return (y);
}
/* Type in a number. */
/* void
* getnum (s, px)
* char *s;
* double *px;
* {
* char str[30];
*
* printf (" %s (%.15e) ? ", s, *px);
* gets (str);
* if (str[0] == '\0' || str[0] == '\n')
* return;
* sscanf (str, "%lf", px);
* printf ("%.15e\n", *px);
* }
*/
/* Type in values, get answers. */
/*
* void
* main ()
* {
* int n;
* double e, p, ps, pk, ek, y;
*
* n = 5;
* e = 0.0;
* p = 0.1;
* loop:
* ps = n;
* getnum ("n", &ps);
* n = ps;
* if (n <= 0)
* {
* printf ("? Operator error.\n");
* goto loop;
* }
*/
/*
* getnum ("e", &e);
* ps = smirnov (n, e);
* y = sqrt ((double) n) * e;
* printf ("y = %.4e\n", y);
* pk = kolmogorov (y);
* printf ("Smirnov = %.15e, Kolmogorov/2 = %.15e\n", ps, pk / 2.0);
*/
/*
* getnum ("p", &p);
* e = smirnovi (n, p);
* printf ("Smirnov e = %.15e\n", e);
* y = kolmogi (2.0 * p);
* ek = y / sqrt ((double) n);
* printf ("Kolmogorov e = %.15e\n", ek);
* goto loop;
* }
*/
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