1 | // boost\math\distributions\non_central_chi_squared.hpp |
2 | |
3 | // Copyright John Maddock 2008. |
4 | |
5 | // Use, modification and distribution are subject to the |
6 | // Boost Software License, Version 1.0. |
7 | // (See accompanying file LICENSE_1_0.txt |
8 | // or copy at http://www.boost.org/LICENSE_1_0.txt) |
9 | |
10 | #ifndef BOOST_MATH_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP |
11 | #define BOOST_MATH_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP |
12 | |
13 | #include <boost/math/distributions/fwd.hpp> |
14 | #include <boost/math/special_functions/gamma.hpp> // for incomplete gamma. gamma_q |
15 | #include <boost/math/special_functions/bessel.hpp> // for cyl_bessel_i |
16 | #include <boost/math/special_functions/round.hpp> // for iround |
17 | #include <boost/math/distributions/complement.hpp> // complements |
18 | #include <boost/math/distributions/chi_squared.hpp> // central distribution |
19 | #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks |
20 | #include <boost/math/special_functions/fpclassify.hpp> // isnan. |
21 | #include <boost/math/tools/roots.hpp> // for root finding. |
22 | #include <boost/math/distributions/detail/generic_mode.hpp> |
23 | #include <boost/math/distributions/detail/generic_quantile.hpp> |
24 | |
25 | namespace boost |
26 | { |
27 | namespace math |
28 | { |
29 | |
30 | template <class RealType, class Policy> |
31 | class non_central_chi_squared_distribution; |
32 | |
33 | namespace detail{ |
34 | |
35 | template <class T, class Policy> |
36 | T non_central_chi_square_q(T x, T f, T theta, const Policy& pol, T init_sum = 0) |
37 | { |
38 | // |
39 | // Computes the complement of the Non-Central Chi-Square |
40 | // Distribution CDF by summing a weighted sum of complements |
41 | // of the central-distributions. The weighting factor is |
42 | // a Poisson Distribution. |
43 | // |
44 | // This is an application of the technique described in: |
45 | // |
46 | // Computing discrete mixtures of continuous |
47 | // distributions: noncentral chisquare, noncentral t |
48 | // and the distribution of the square of the sample |
49 | // multiple correlation coefficient. |
50 | // D. Benton, K. Krishnamoorthy. |
51 | // Computational Statistics & Data Analysis 43 (2003) 249 - 267 |
52 | // |
53 | BOOST_MATH_STD_USING |
54 | |
55 | // Special case: |
56 | if(x == 0) |
57 | return 1; |
58 | |
59 | // |
60 | // Initialize the variables we'll be using: |
61 | // |
62 | T lambda = theta / 2; |
63 | T del = f / 2; |
64 | T y = x / 2; |
65 | boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); |
66 | T errtol = boost::math::policies::get_epsilon<T, Policy>(); |
67 | T sum = init_sum; |
68 | // |
69 | // k is the starting location for iteration, we'll |
70 | // move both forwards and backwards from this point. |
71 | // k is chosen as the peek of the Poisson weights, which |
72 | // will occur *before* the largest term. |
73 | // |
74 | int k = iround(lambda, pol); |
75 | // Forwards and backwards Poisson weights: |
76 | T poisf = boost::math::gamma_p_derivative(static_cast<T>(1 + k), lambda, pol); |
77 | T poisb = poisf * k / lambda; |
78 | // Initial forwards central chi squared term: |
79 | T gamf = boost::math::gamma_q(del + k, y, pol); |
80 | // Forwards and backwards recursion terms on the central chi squared: |
81 | T xtermf = boost::math::gamma_p_derivative(del + 1 + k, y, pol); |
82 | T xtermb = xtermf * (del + k) / y; |
83 | // Initial backwards central chi squared term: |
84 | T gamb = gamf - xtermb; |
85 | |
86 | // |
87 | // Forwards iteration first, this is the |
88 | // stable direction for the gamma function |
89 | // recurrences: |
90 | // |
91 | int i; |
92 | for(i = k; static_cast<boost::uintmax_t>(i-k) < max_iter; ++i) |
93 | { |
94 | T term = poisf * gamf; |
95 | sum += term; |
96 | poisf *= lambda / (i + 1); |
97 | gamf += xtermf; |
98 | xtermf *= y / (del + i + 1); |
99 | if(((sum == 0) || (fabs(term / sum) < errtol)) && (term >= poisf * gamf)) |
100 | break; |
101 | } |
102 | //Error check: |
103 | if(static_cast<boost::uintmax_t>(i-k) >= max_iter) |
104 | return policies::raise_evaluation_error( |
105 | "cdf(non_central_chi_squared_distribution<%1%>, %1%)" , |
106 | "Series did not converge, closest value was %1%" , sum, pol); |
107 | // |
108 | // Now backwards iteration: the gamma |
109 | // function recurrences are unstable in this |
110 | // direction, we rely on the terms diminishing in size |
111 | // faster than we introduce cancellation errors. |
112 | // For this reason it's very important that we start |
113 | // *before* the largest term so that backwards iteration |
114 | // is strictly converging. |
115 | // |
116 | for(i = k - 1; i >= 0; --i) |
117 | { |
118 | T term = poisb * gamb; |
119 | sum += term; |
120 | poisb *= i / lambda; |
121 | xtermb *= (del + i) / y; |
122 | gamb -= xtermb; |
123 | if((sum == 0) || (fabs(term / sum) < errtol)) |
124 | break; |
125 | } |
126 | |
127 | return sum; |
128 | } |
129 | |
130 | template <class T, class Policy> |
131 | T non_central_chi_square_p_ding(T x, T f, T theta, const Policy& pol, T init_sum = 0) |
132 | { |
133 | // |
134 | // This is an implementation of: |
135 | // |
136 | // Algorithm AS 275: |
137 | // Computing the Non-Central #2 Distribution Function |
138 | // Cherng G. Ding |
139 | // Applied Statistics, Vol. 41, No. 2. (1992), pp. 478-482. |
140 | // |
141 | // This uses a stable forward iteration to sum the |
142 | // CDF, unfortunately this can not be used for large |
143 | // values of the non-centrality parameter because: |
144 | // * The first term may underflow to zero. |
145 | // * We may need an extra-ordinary number of terms |
146 | // before we reach the first *significant* term. |
147 | // |
148 | BOOST_MATH_STD_USING |
149 | // Special case: |
150 | if(x == 0) |
151 | return 0; |
152 | T tk = boost::math::gamma_p_derivative(f/2 + 1, x/2, pol); |
153 | T lambda = theta / 2; |
154 | T vk = exp(-lambda); |
155 | T uk = vk; |
156 | T sum = init_sum + tk * vk; |
157 | if(sum == 0) |
158 | return sum; |
159 | |
160 | boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); |
161 | T errtol = boost::math::policies::get_epsilon<T, Policy>(); |
162 | |
163 | int i; |
164 | T lterm(0), term(0); |
165 | for(i = 1; static_cast<boost::uintmax_t>(i) < max_iter; ++i) |
166 | { |
167 | tk = tk * x / (f + 2 * i); |
168 | uk = uk * lambda / i; |
169 | vk = vk + uk; |
170 | lterm = term; |
171 | term = vk * tk; |
172 | sum += term; |
173 | if((fabs(term / sum) < errtol) && (term <= lterm)) |
174 | break; |
175 | } |
176 | //Error check: |
177 | if(static_cast<boost::uintmax_t>(i) >= max_iter) |
178 | return policies::raise_evaluation_error( |
179 | "cdf(non_central_chi_squared_distribution<%1%>, %1%)" , |
180 | "Series did not converge, closest value was %1%" , sum, pol); |
181 | return sum; |
182 | } |
183 | |
184 | |
185 | template <class T, class Policy> |
186 | T non_central_chi_square_p(T y, T n, T lambda, const Policy& pol, T init_sum) |
187 | { |
188 | // |
189 | // This is taken more or less directly from: |
190 | // |
191 | // Computing discrete mixtures of continuous |
192 | // distributions: noncentral chisquare, noncentral t |
193 | // and the distribution of the square of the sample |
194 | // multiple correlation coefficient. |
195 | // D. Benton, K. Krishnamoorthy. |
196 | // Computational Statistics & Data Analysis 43 (2003) 249 - 267 |
197 | // |
198 | // We're summing a Poisson weighting term multiplied by |
199 | // a central chi squared distribution. |
200 | // |
201 | BOOST_MATH_STD_USING |
202 | // Special case: |
203 | if(y == 0) |
204 | return 0; |
205 | boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); |
206 | T errtol = boost::math::policies::get_epsilon<T, Policy>(); |
207 | T errorf(0), errorb(0); |
208 | |
209 | T x = y / 2; |
210 | T del = lambda / 2; |
211 | // |
212 | // Starting location for the iteration, we'll iterate |
213 | // both forwards and backwards from this point. The |
214 | // location chosen is the maximum of the Poisson weight |
215 | // function, which ocurrs *after* the largest term in the |
216 | // sum. |
217 | // |
218 | int k = iround(del, pol); |
219 | T a = n / 2 + k; |
220 | // Central chi squared term for forward iteration: |
221 | T gamkf = boost::math::gamma_p(a, x, pol); |
222 | |
223 | if(lambda == 0) |
224 | return gamkf; |
225 | // Central chi squared term for backward iteration: |
226 | T gamkb = gamkf; |
227 | // Forwards Poisson weight: |
228 | T poiskf = gamma_p_derivative(static_cast<T>(k+1), del, pol); |
229 | // Backwards Poisson weight: |
230 | T poiskb = poiskf; |
231 | // Forwards gamma function recursion term: |
232 | T xtermf = boost::math::gamma_p_derivative(a, x, pol); |
233 | // Backwards gamma function recursion term: |
234 | T xtermb = xtermf * x / a; |
235 | T sum = init_sum + poiskf * gamkf; |
236 | if(sum == 0) |
237 | return sum; |
238 | int i = 1; |
239 | // |
240 | // Backwards recursion first, this is the stable |
241 | // direction for gamma function recurrences: |
242 | // |
243 | while(i <= k) |
244 | { |
245 | xtermb *= (a - i + 1) / x; |
246 | gamkb += xtermb; |
247 | poiskb = poiskb * (k - i + 1) / del; |
248 | errorf = errorb; |
249 | errorb = gamkb * poiskb; |
250 | sum += errorb; |
251 | if((fabs(errorb / sum) < errtol) && (errorb <= errorf)) |
252 | break; |
253 | ++i; |
254 | } |
255 | i = 1; |
256 | // |
257 | // Now forwards recursion, the gamma function |
258 | // recurrence relation is unstable in this direction, |
259 | // so we rely on the magnitude of successive terms |
260 | // decreasing faster than we introduce cancellation error. |
261 | // For this reason it's vital that k is chosen to be *after* |
262 | // the largest term, so that successive forward iterations |
263 | // are strictly (and rapidly) converging. |
264 | // |
265 | do |
266 | { |
267 | xtermf = xtermf * x / (a + i - 1); |
268 | gamkf = gamkf - xtermf; |
269 | poiskf = poiskf * del / (k + i); |
270 | errorf = poiskf * gamkf; |
271 | sum += errorf; |
272 | ++i; |
273 | }while((fabs(errorf / sum) > errtol) && (static_cast<boost::uintmax_t>(i) < max_iter)); |
274 | |
275 | //Error check: |
276 | if(static_cast<boost::uintmax_t>(i) >= max_iter) |
277 | return policies::raise_evaluation_error( |
278 | "cdf(non_central_chi_squared_distribution<%1%>, %1%)" , |
279 | "Series did not converge, closest value was %1%" , sum, pol); |
280 | |
281 | return sum; |
282 | } |
283 | |
284 | template <class T, class Policy> |
285 | T non_central_chi_square_pdf(T x, T n, T lambda, const Policy& pol) |
286 | { |
287 | // |
288 | // As above but for the PDF: |
289 | // |
290 | BOOST_MATH_STD_USING |
291 | boost::uintmax_t max_iter = policies::get_max_series_iterations<Policy>(); |
292 | T errtol = boost::math::policies::get_epsilon<T, Policy>(); |
293 | T x2 = x / 2; |
294 | T n2 = n / 2; |
295 | T l2 = lambda / 2; |
296 | T sum = 0; |
297 | int k = itrunc(l2); |
298 | T pois = gamma_p_derivative(static_cast<T>(k + 1), l2, pol) * gamma_p_derivative(static_cast<T>(n2 + k), x2); |
299 | if(pois == 0) |
300 | return 0; |
301 | T poisb = pois; |
302 | for(int i = k; ; ++i) |
303 | { |
304 | sum += pois; |
305 | if(pois / sum < errtol) |
306 | break; |
307 | if(static_cast<boost::uintmax_t>(i - k) >= max_iter) |
308 | return policies::raise_evaluation_error( |
309 | "pdf(non_central_chi_squared_distribution<%1%>, %1%)" , |
310 | "Series did not converge, closest value was %1%" , sum, pol); |
311 | pois *= l2 * x2 / ((i + 1) * (n2 + i)); |
312 | } |
313 | for(int i = k - 1; i >= 0; --i) |
314 | { |
315 | poisb *= (i + 1) * (n2 + i) / (l2 * x2); |
316 | sum += poisb; |
317 | if(poisb / sum < errtol) |
318 | break; |
319 | } |
320 | return sum / 2; |
321 | } |
322 | |
323 | template <class RealType, class Policy> |
324 | inline RealType non_central_chi_squared_cdf(RealType x, RealType k, RealType l, bool invert, const Policy&) |
325 | { |
326 | typedef typename policies::evaluation<RealType, Policy>::type value_type; |
327 | typedef typename policies::normalise< |
328 | Policy, |
329 | policies::promote_float<false>, |
330 | policies::promote_double<false>, |
331 | policies::discrete_quantile<>, |
332 | policies::assert_undefined<> >::type forwarding_policy; |
333 | |
334 | BOOST_MATH_STD_USING |
335 | value_type result; |
336 | if(l == 0) |
337 | return invert == false ? cdf(boost::math::chi_squared_distribution<RealType, Policy>(k), x) : cdf(complement(boost::math::chi_squared_distribution<RealType, Policy>(k), x)); |
338 | else if(x > k + l) |
339 | { |
340 | // Complement is the smaller of the two: |
341 | result = detail::non_central_chi_square_q( |
342 | static_cast<value_type>(x), |
343 | static_cast<value_type>(k), |
344 | static_cast<value_type>(l), |
345 | forwarding_policy(), |
346 | static_cast<value_type>(invert ? 0 : -1)); |
347 | invert = !invert; |
348 | } |
349 | else if(l < 200) |
350 | { |
351 | // For small values of the non-centrality parameter |
352 | // we can use Ding's method: |
353 | result = detail::non_central_chi_square_p_ding( |
354 | static_cast<value_type>(x), |
355 | static_cast<value_type>(k), |
356 | static_cast<value_type>(l), |
357 | forwarding_policy(), |
358 | static_cast<value_type>(invert ? -1 : 0)); |
359 | } |
360 | else |
361 | { |
362 | // For largers values of the non-centrality |
363 | // parameter Ding's method will consume an |
364 | // extra-ordinary number of terms, and worse |
365 | // may return zero when the result is in fact |
366 | // finite, use Krishnamoorthy's method instead: |
367 | result = detail::non_central_chi_square_p( |
368 | static_cast<value_type>(x), |
369 | static_cast<value_type>(k), |
370 | static_cast<value_type>(l), |
371 | forwarding_policy(), |
372 | static_cast<value_type>(invert ? -1 : 0)); |
373 | } |
374 | if(invert) |
375 | result = -result; |
376 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
377 | result, |
378 | "boost::math::non_central_chi_squared_cdf<%1%>(%1%, %1%, %1%)" ); |
379 | } |
380 | |
381 | template <class T, class Policy> |
382 | struct nccs_quantile_functor |
383 | { |
384 | nccs_quantile_functor(const non_central_chi_squared_distribution<T,Policy>& d, T t, bool c) |
385 | : dist(d), target(t), comp(c) {} |
386 | |
387 | T operator()(const T& x) |
388 | { |
389 | return comp ? |
390 | target - cdf(complement(dist, x)) |
391 | : cdf(dist, x) - target; |
392 | } |
393 | |
394 | private: |
395 | non_central_chi_squared_distribution<T,Policy> dist; |
396 | T target; |
397 | bool comp; |
398 | }; |
399 | |
400 | template <class RealType, class Policy> |
401 | RealType nccs_quantile(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& p, bool comp) |
402 | { |
403 | BOOST_MATH_STD_USING |
404 | static const char* function = "quantile(non_central_chi_squared_distribution<%1%>, %1%)" ; |
405 | typedef typename policies::evaluation<RealType, Policy>::type value_type; |
406 | typedef typename policies::normalise< |
407 | Policy, |
408 | policies::promote_float<false>, |
409 | policies::promote_double<false>, |
410 | policies::discrete_quantile<>, |
411 | policies::assert_undefined<> >::type forwarding_policy; |
412 | |
413 | value_type k = dist.degrees_of_freedom(); |
414 | value_type l = dist.non_centrality(); |
415 | value_type r; |
416 | if(!detail::check_df( |
417 | function, |
418 | k, &r, Policy()) |
419 | || |
420 | !detail::check_non_centrality( |
421 | function, |
422 | l, |
423 | &r, |
424 | Policy()) |
425 | || |
426 | !detail::check_probability( |
427 | function, |
428 | static_cast<value_type>(p), |
429 | &r, |
430 | Policy())) |
431 | return (RealType)r; |
432 | // |
433 | // Special cases get short-circuited first: |
434 | // |
435 | if(p == 0) |
436 | return comp ? policies::raise_overflow_error<RealType>(function, 0, Policy()) : 0; |
437 | if(p == 1) |
438 | return comp ? 0 : policies::raise_overflow_error<RealType>(function, 0, Policy()); |
439 | // |
440 | // This is Pearson's approximation to the quantile, see |
441 | // Pearson, E. S. (1959) "Note on an approximation to the distribution of |
442 | // noncentral chi squared", Biometrika 46: 364. |
443 | // See also: |
444 | // "A comparison of approximations to percentiles of the noncentral chi2-distribution", |
445 | // Hardeo Sahai and Mario Miguel Ojeda, Revista de Matematica: Teoria y Aplicaciones 2003 10(1-2) : 57-76. |
446 | // Note that the latter reference refers to an approximation of the CDF, when they really mean the quantile. |
447 | // |
448 | value_type b = -(l * l) / (k + 3 * l); |
449 | value_type c = (k + 3 * l) / (k + 2 * l); |
450 | value_type ff = (k + 2 * l) / (c * c); |
451 | value_type guess; |
452 | if(comp) |
453 | { |
454 | guess = b + c * quantile(complement(chi_squared_distribution<value_type, forwarding_policy>(ff), p)); |
455 | } |
456 | else |
457 | { |
458 | guess = b + c * quantile(chi_squared_distribution<value_type, forwarding_policy>(ff), p); |
459 | } |
460 | // |
461 | // Sometimes guess goes very small or negative, in that case we have |
462 | // to do something else for the initial guess, this approximation |
463 | // was provided in a private communication from Thomas Luu, PhD candidate, |
464 | // University College London. It's an asymptotic expansion for the |
465 | // quantile which usually gets us within an order of magnitude of the |
466 | // correct answer. |
467 | // Fast and accurate parallel computation of quantile functions for random number generation, |
468 | // Thomas LuuDoctorial Thesis 2016 |
469 | // http://discovery.ucl.ac.uk/1482128/ |
470 | // |
471 | if(guess < 0.005) |
472 | { |
473 | value_type pp = comp ? 1 - p : p; |
474 | //guess = pow(pow(value_type(2), (k / 2 - 1)) * exp(l / 2) * pp * k, 2 / k); |
475 | guess = pow(pow(value_type(2), (k / 2 - 1)) * exp(l / 2) * pp * k * boost::math::tgamma(k / 2, forwarding_policy()), (2 / k)); |
476 | if(guess == 0) |
477 | guess = tools::min_value<value_type>(); |
478 | } |
479 | value_type result = detail::generic_quantile( |
480 | non_central_chi_squared_distribution<value_type, forwarding_policy>(k, l), |
481 | p, |
482 | guess, |
483 | comp, |
484 | function); |
485 | |
486 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
487 | result, |
488 | function); |
489 | } |
490 | |
491 | template <class RealType, class Policy> |
492 | RealType nccs_pdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x) |
493 | { |
494 | BOOST_MATH_STD_USING |
495 | static const char* function = "pdf(non_central_chi_squared_distribution<%1%>, %1%)" ; |
496 | typedef typename policies::evaluation<RealType, Policy>::type value_type; |
497 | typedef typename policies::normalise< |
498 | Policy, |
499 | policies::promote_float<false>, |
500 | policies::promote_double<false>, |
501 | policies::discrete_quantile<>, |
502 | policies::assert_undefined<> >::type forwarding_policy; |
503 | |
504 | value_type k = dist.degrees_of_freedom(); |
505 | value_type l = dist.non_centrality(); |
506 | value_type r; |
507 | if(!detail::check_df( |
508 | function, |
509 | k, &r, Policy()) |
510 | || |
511 | !detail::check_non_centrality( |
512 | function, |
513 | l, |
514 | &r, |
515 | Policy()) |
516 | || |
517 | !detail::check_positive_x( |
518 | function, |
519 | (value_type)x, |
520 | &r, |
521 | Policy())) |
522 | return (RealType)r; |
523 | |
524 | if(l == 0) |
525 | return pdf(boost::math::chi_squared_distribution<RealType, forwarding_policy>(dist.degrees_of_freedom()), x); |
526 | |
527 | // Special case: |
528 | if(x == 0) |
529 | return 0; |
530 | if(l > 50) |
531 | { |
532 | r = non_central_chi_square_pdf(static_cast<value_type>(x), k, l, forwarding_policy()); |
533 | } |
534 | else |
535 | { |
536 | r = log(x / l) * (k / 4 - 0.5f) - (x + l) / 2; |
537 | if(fabs(r) >= tools::log_max_value<RealType>() / 4) |
538 | { |
539 | r = non_central_chi_square_pdf(static_cast<value_type>(x), k, l, forwarding_policy()); |
540 | } |
541 | else |
542 | { |
543 | r = exp(r); |
544 | r = 0.5f * r |
545 | * boost::math::cyl_bessel_i(k/2 - 1, sqrt(l * x), forwarding_policy()); |
546 | } |
547 | } |
548 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
549 | r, |
550 | function); |
551 | } |
552 | |
553 | template <class RealType, class Policy> |
554 | struct degrees_of_freedom_finder |
555 | { |
556 | degrees_of_freedom_finder( |
557 | RealType lam_, RealType x_, RealType p_, bool c) |
558 | : lam(lam_), x(x_), p(p_), comp(c) {} |
559 | |
560 | RealType operator()(const RealType& v) |
561 | { |
562 | non_central_chi_squared_distribution<RealType, Policy> d(v, lam); |
563 | return comp ? |
564 | RealType(p - cdf(complement(d, x))) |
565 | : RealType(cdf(d, x) - p); |
566 | } |
567 | private: |
568 | RealType lam; |
569 | RealType x; |
570 | RealType p; |
571 | bool comp; |
572 | }; |
573 | |
574 | template <class RealType, class Policy> |
575 | inline RealType find_degrees_of_freedom( |
576 | RealType lam, RealType x, RealType p, RealType q, const Policy& pol) |
577 | { |
578 | const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom" ; |
579 | if((p == 0) || (q == 0)) |
580 | { |
581 | // |
582 | // Can't a thing if one of p and q is zero: |
583 | // |
584 | return policies::raise_evaluation_error<RealType>(function, |
585 | "Can't find degrees of freedom when the probability is 0 or 1, only possible answer is %1%" , |
586 | RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy()); |
587 | } |
588 | degrees_of_freedom_finder<RealType, Policy> f(lam, x, p < q ? p : q, p < q ? false : true); |
589 | tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>()); |
590 | boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); |
591 | // |
592 | // Pick an initial guess that we know will give us a probability |
593 | // right around 0.5. |
594 | // |
595 | RealType guess = x - lam; |
596 | if(guess < 1) |
597 | guess = 1; |
598 | std::pair<RealType, RealType> ir = tools::bracket_and_solve_root( |
599 | f, guess, RealType(2), false, tol, max_iter, pol); |
600 | RealType result = ir.first + (ir.second - ir.first) / 2; |
601 | if(max_iter >= policies::get_max_root_iterations<Policy>()) |
602 | { |
603 | return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:" |
604 | " or there is no answer to problem. Current best guess is %1%" , result, Policy()); |
605 | } |
606 | return result; |
607 | } |
608 | |
609 | template <class RealType, class Policy> |
610 | struct non_centrality_finder |
611 | { |
612 | non_centrality_finder( |
613 | RealType v_, RealType x_, RealType p_, bool c) |
614 | : v(v_), x(x_), p(p_), comp(c) {} |
615 | |
616 | RealType operator()(const RealType& lam) |
617 | { |
618 | non_central_chi_squared_distribution<RealType, Policy> d(v, lam); |
619 | return comp ? |
620 | RealType(p - cdf(complement(d, x))) |
621 | : RealType(cdf(d, x) - p); |
622 | } |
623 | private: |
624 | RealType v; |
625 | RealType x; |
626 | RealType p; |
627 | bool comp; |
628 | }; |
629 | |
630 | template <class RealType, class Policy> |
631 | inline RealType find_non_centrality( |
632 | RealType v, RealType x, RealType p, RealType q, const Policy& pol) |
633 | { |
634 | const char* function = "non_central_chi_squared<%1%>::find_non_centrality" ; |
635 | if((p == 0) || (q == 0)) |
636 | { |
637 | // |
638 | // Can't do a thing if one of p and q is zero: |
639 | // |
640 | return policies::raise_evaluation_error<RealType>(function, |
641 | "Can't find non centrality parameter when the probability is 0 or 1, only possible answer is %1%" , |
642 | RealType(std::numeric_limits<RealType>::quiet_NaN()), Policy()); |
643 | } |
644 | non_centrality_finder<RealType, Policy> f(v, x, p < q ? p : q, p < q ? false : true); |
645 | tools::eps_tolerance<RealType> tol(policies::digits<RealType, Policy>()); |
646 | boost::uintmax_t max_iter = policies::get_max_root_iterations<Policy>(); |
647 | // |
648 | // Pick an initial guess that we know will give us a probability |
649 | // right around 0.5. |
650 | // |
651 | RealType guess = x - v; |
652 | if(guess < 1) |
653 | guess = 1; |
654 | std::pair<RealType, RealType> ir = tools::bracket_and_solve_root( |
655 | f, guess, RealType(2), false, tol, max_iter, pol); |
656 | RealType result = ir.first + (ir.second - ir.first) / 2; |
657 | if(max_iter >= policies::get_max_root_iterations<Policy>()) |
658 | { |
659 | return policies::raise_evaluation_error<RealType>(function, "Unable to locate solution in a reasonable time:" |
660 | " or there is no answer to problem. Current best guess is %1%" , result, Policy()); |
661 | } |
662 | return result; |
663 | } |
664 | |
665 | } |
666 | |
667 | template <class RealType = double, class Policy = policies::policy<> > |
668 | class non_central_chi_squared_distribution |
669 | { |
670 | public: |
671 | typedef RealType value_type; |
672 | typedef Policy policy_type; |
673 | |
674 | non_central_chi_squared_distribution(RealType df_, RealType lambda) : df(df_), ncp(lambda) |
675 | { |
676 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::non_central_chi_squared_distribution(%1%,%1%)" ; |
677 | RealType r; |
678 | detail::check_df( |
679 | function, |
680 | df, &r, Policy()); |
681 | detail::check_non_centrality( |
682 | function, |
683 | ncp, |
684 | &r, |
685 | Policy()); |
686 | } // non_central_chi_squared_distribution constructor. |
687 | |
688 | RealType degrees_of_freedom() const |
689 | { // Private data getter function. |
690 | return df; |
691 | } |
692 | RealType non_centrality() const |
693 | { // Private data getter function. |
694 | return ncp; |
695 | } |
696 | static RealType find_degrees_of_freedom(RealType lam, RealType x, RealType p) |
697 | { |
698 | const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom" ; |
699 | typedef typename policies::evaluation<RealType, Policy>::type eval_type; |
700 | typedef typename policies::normalise< |
701 | Policy, |
702 | policies::promote_float<false>, |
703 | policies::promote_double<false>, |
704 | policies::discrete_quantile<>, |
705 | policies::assert_undefined<> >::type forwarding_policy; |
706 | eval_type result = detail::find_degrees_of_freedom( |
707 | static_cast<eval_type>(lam), |
708 | static_cast<eval_type>(x), |
709 | static_cast<eval_type>(p), |
710 | static_cast<eval_type>(1-p), |
711 | forwarding_policy()); |
712 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
713 | result, |
714 | function); |
715 | } |
716 | template <class A, class B, class C> |
717 | static RealType find_degrees_of_freedom(const complemented3_type<A,B,C>& c) |
718 | { |
719 | const char* function = "non_central_chi_squared<%1%>::find_degrees_of_freedom" ; |
720 | typedef typename policies::evaluation<RealType, Policy>::type eval_type; |
721 | typedef typename policies::normalise< |
722 | Policy, |
723 | policies::promote_float<false>, |
724 | policies::promote_double<false>, |
725 | policies::discrete_quantile<>, |
726 | policies::assert_undefined<> >::type forwarding_policy; |
727 | eval_type result = detail::find_degrees_of_freedom( |
728 | static_cast<eval_type>(c.dist), |
729 | static_cast<eval_type>(c.param1), |
730 | static_cast<eval_type>(1-c.param2), |
731 | static_cast<eval_type>(c.param2), |
732 | forwarding_policy()); |
733 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
734 | result, |
735 | function); |
736 | } |
737 | static RealType find_non_centrality(RealType v, RealType x, RealType p) |
738 | { |
739 | const char* function = "non_central_chi_squared<%1%>::find_non_centrality" ; |
740 | typedef typename policies::evaluation<RealType, Policy>::type eval_type; |
741 | typedef typename policies::normalise< |
742 | Policy, |
743 | policies::promote_float<false>, |
744 | policies::promote_double<false>, |
745 | policies::discrete_quantile<>, |
746 | policies::assert_undefined<> >::type forwarding_policy; |
747 | eval_type result = detail::find_non_centrality( |
748 | static_cast<eval_type>(v), |
749 | static_cast<eval_type>(x), |
750 | static_cast<eval_type>(p), |
751 | static_cast<eval_type>(1-p), |
752 | forwarding_policy()); |
753 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
754 | result, |
755 | function); |
756 | } |
757 | template <class A, class B, class C> |
758 | static RealType find_non_centrality(const complemented3_type<A,B,C>& c) |
759 | { |
760 | const char* function = "non_central_chi_squared<%1%>::find_non_centrality" ; |
761 | typedef typename policies::evaluation<RealType, Policy>::type eval_type; |
762 | typedef typename policies::normalise< |
763 | Policy, |
764 | policies::promote_float<false>, |
765 | policies::promote_double<false>, |
766 | policies::discrete_quantile<>, |
767 | policies::assert_undefined<> >::type forwarding_policy; |
768 | eval_type result = detail::find_non_centrality( |
769 | static_cast<eval_type>(c.dist), |
770 | static_cast<eval_type>(c.param1), |
771 | static_cast<eval_type>(1-c.param2), |
772 | static_cast<eval_type>(c.param2), |
773 | forwarding_policy()); |
774 | return policies::checked_narrowing_cast<RealType, forwarding_policy>( |
775 | result, |
776 | function); |
777 | } |
778 | private: |
779 | // Data member, initialized by constructor. |
780 | RealType df; // degrees of freedom. |
781 | RealType ncp; // non-centrality parameter |
782 | }; // template <class RealType, class Policy> class non_central_chi_squared_distribution |
783 | |
784 | typedef non_central_chi_squared_distribution<double> non_central_chi_squared; // Reserved name of type double. |
785 | |
786 | // Non-member functions to give properties of the distribution. |
787 | |
788 | template <class RealType, class Policy> |
789 | inline const std::pair<RealType, RealType> range(const non_central_chi_squared_distribution<RealType, Policy>& /* dist */) |
790 | { // Range of permissible values for random variable k. |
791 | using boost::math::tools::max_value; |
792 | return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); // Max integer? |
793 | } |
794 | |
795 | template <class RealType, class Policy> |
796 | inline const std::pair<RealType, RealType> support(const non_central_chi_squared_distribution<RealType, Policy>& /* dist */) |
797 | { // Range of supported values for random variable k. |
798 | // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero. |
799 | using boost::math::tools::max_value; |
800 | return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>()); |
801 | } |
802 | |
803 | template <class RealType, class Policy> |
804 | inline RealType mean(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
805 | { // Mean of poisson distribution = lambda. |
806 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::mean()" ; |
807 | RealType k = dist.degrees_of_freedom(); |
808 | RealType l = dist.non_centrality(); |
809 | RealType r; |
810 | if(!detail::check_df( |
811 | function, |
812 | k, &r, Policy()) |
813 | || |
814 | !detail::check_non_centrality( |
815 | function, |
816 | l, |
817 | &r, |
818 | Policy())) |
819 | return r; |
820 | return k + l; |
821 | } // mean |
822 | |
823 | template <class RealType, class Policy> |
824 | inline RealType mode(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
825 | { // mode. |
826 | static const char* function = "mode(non_central_chi_squared_distribution<%1%> const&)" ; |
827 | |
828 | RealType k = dist.degrees_of_freedom(); |
829 | RealType l = dist.non_centrality(); |
830 | RealType r; |
831 | if(!detail::check_df( |
832 | function, |
833 | k, &r, Policy()) |
834 | || |
835 | !detail::check_non_centrality( |
836 | function, |
837 | l, |
838 | &r, |
839 | Policy())) |
840 | return (RealType)r; |
841 | return detail::generic_find_mode(dist, 1 + k, function); |
842 | } |
843 | |
844 | template <class RealType, class Policy> |
845 | inline RealType variance(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
846 | { // variance. |
847 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::variance()" ; |
848 | RealType k = dist.degrees_of_freedom(); |
849 | RealType l = dist.non_centrality(); |
850 | RealType r; |
851 | if(!detail::check_df( |
852 | function, |
853 | k, &r, Policy()) |
854 | || |
855 | !detail::check_non_centrality( |
856 | function, |
857 | l, |
858 | &r, |
859 | Policy())) |
860 | return r; |
861 | return 2 * (2 * l + k); |
862 | } |
863 | |
864 | // RealType standard_deviation(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
865 | // standard_deviation provided by derived accessors. |
866 | |
867 | template <class RealType, class Policy> |
868 | inline RealType skewness(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
869 | { // skewness = sqrt(l). |
870 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::skewness()" ; |
871 | RealType k = dist.degrees_of_freedom(); |
872 | RealType l = dist.non_centrality(); |
873 | RealType r; |
874 | if(!detail::check_df( |
875 | function, |
876 | k, &r, Policy()) |
877 | || |
878 | !detail::check_non_centrality( |
879 | function, |
880 | l, |
881 | &r, |
882 | Policy())) |
883 | return r; |
884 | BOOST_MATH_STD_USING |
885 | return pow(2 / (k + 2 * l), RealType(3)/2) * (k + 3 * l); |
886 | } |
887 | |
888 | template <class RealType, class Policy> |
889 | inline RealType kurtosis_excess(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
890 | { |
891 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::kurtosis_excess()" ; |
892 | RealType k = dist.degrees_of_freedom(); |
893 | RealType l = dist.non_centrality(); |
894 | RealType r; |
895 | if(!detail::check_df( |
896 | function, |
897 | k, &r, Policy()) |
898 | || |
899 | !detail::check_non_centrality( |
900 | function, |
901 | l, |
902 | &r, |
903 | Policy())) |
904 | return r; |
905 | return 12 * (k + 4 * l) / ((k + 2 * l) * (k + 2 * l)); |
906 | } // kurtosis_excess |
907 | |
908 | template <class RealType, class Policy> |
909 | inline RealType kurtosis(const non_central_chi_squared_distribution<RealType, Policy>& dist) |
910 | { |
911 | return kurtosis_excess(dist) + 3; |
912 | } |
913 | |
914 | template <class RealType, class Policy> |
915 | inline RealType pdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x) |
916 | { // Probability Density/Mass Function. |
917 | return detail::nccs_pdf(dist, x); |
918 | } // pdf |
919 | |
920 | template <class RealType, class Policy> |
921 | RealType cdf(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& x) |
922 | { |
923 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::cdf(%1%)" ; |
924 | RealType k = dist.degrees_of_freedom(); |
925 | RealType l = dist.non_centrality(); |
926 | RealType r; |
927 | if(!detail::check_df( |
928 | function, |
929 | k, &r, Policy()) |
930 | || |
931 | !detail::check_non_centrality( |
932 | function, |
933 | l, |
934 | &r, |
935 | Policy()) |
936 | || |
937 | !detail::check_positive_x( |
938 | function, |
939 | x, |
940 | &r, |
941 | Policy())) |
942 | return r; |
943 | |
944 | return detail::non_central_chi_squared_cdf(x, k, l, false, Policy()); |
945 | } // cdf |
946 | |
947 | template <class RealType, class Policy> |
948 | RealType cdf(const complemented2_type<non_central_chi_squared_distribution<RealType, Policy>, RealType>& c) |
949 | { // Complemented Cumulative Distribution Function |
950 | const char* function = "boost::math::non_central_chi_squared_distribution<%1%>::cdf(%1%)" ; |
951 | non_central_chi_squared_distribution<RealType, Policy> const& dist = c.dist; |
952 | RealType x = c.param; |
953 | RealType k = dist.degrees_of_freedom(); |
954 | RealType l = dist.non_centrality(); |
955 | RealType r; |
956 | if(!detail::check_df( |
957 | function, |
958 | k, &r, Policy()) |
959 | || |
960 | !detail::check_non_centrality( |
961 | function, |
962 | l, |
963 | &r, |
964 | Policy()) |
965 | || |
966 | !detail::check_positive_x( |
967 | function, |
968 | x, |
969 | &r, |
970 | Policy())) |
971 | return r; |
972 | |
973 | return detail::non_central_chi_squared_cdf(x, k, l, true, Policy()); |
974 | } // ccdf |
975 | |
976 | template <class RealType, class Policy> |
977 | inline RealType quantile(const non_central_chi_squared_distribution<RealType, Policy>& dist, const RealType& p) |
978 | { // Quantile (or Percent Point) function. |
979 | return detail::nccs_quantile(dist, p, false); |
980 | } // quantile |
981 | |
982 | template <class RealType, class Policy> |
983 | inline RealType quantile(const complemented2_type<non_central_chi_squared_distribution<RealType, Policy>, RealType>& c) |
984 | { // Quantile (or Percent Point) function. |
985 | return detail::nccs_quantile(c.dist, c.param, true); |
986 | } // quantile complement. |
987 | |
988 | } // namespace math |
989 | } // namespace boost |
990 | |
991 | // This include must be at the end, *after* the accessors |
992 | // for this distribution have been defined, in order to |
993 | // keep compilers that support two-phase lookup happy. |
994 | #include <boost/math/distributions/detail/derived_accessors.hpp> |
995 | |
996 | #endif // BOOST_MATH_SPECIAL_NON_CENTRAL_CHI_SQUARE_HPP |
997 | |
998 | |
999 | |
1000 | |