code for computing quantile of chi-squared distribution
Varun Agrawal
parent
9eb9770a43
commit
d0b3f1dd25
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@ -0,0 +1,516 @@
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/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file GncHelpers.h
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* @brief Helper functions for use with the GncOptimizer
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* @author Varun Agrawal
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*/
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#pragma once
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#include <algorithm>
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#include <cmath>
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namespace gtsam {
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/// Template type for numeric limits
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template <class T>
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using LIM = std::numeric_limits<T>;
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template <typename T>
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using return_t =
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typename std::conditional<std::is_integral<T>::value, double, T>::type;
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template <typename... T>
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using common_t = typename std::common_type<T...>::type;
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template <typename... T>
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using common_return_t = return_t<common_t<T...>>;
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/// Check if integer is odd
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constexpr bool is_odd(const long long int x) noexcept { return (x & 1U) != 0; }
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/// Templated check for NaN
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template <typename T>
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constexpr bool is_nan(const T x) noexcept {
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return x != x;
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}
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/// @brief Gauss-Legendre quadrature: 50 points
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static const long double gauss_legendre_50_points[50] = {
|
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-0.03109833832718887611232898966595L, 0.03109833832718887611232898966595L,
|
||||
-0.09317470156008614085445037763960L, 0.09317470156008614085445037763960L,
|
||||
-0.15489058999814590207162862094111L, 0.15489058999814590207162862094111L,
|
||||
-0.21600723687604175684728453261710L, 0.21600723687604175684728453261710L,
|
||||
-0.27628819377953199032764527852113L, 0.27628819377953199032764527852113L,
|
||||
-0.33550024541943735683698825729107L, 0.33550024541943735683698825729107L,
|
||||
-0.39341431189756512739422925382382L, 0.39341431189756512739422925382382L,
|
||||
-0.44980633497403878914713146777838L, 0.44980633497403878914713146777838L,
|
||||
-0.50445814490746420165145913184914L, 0.50445814490746420165145913184914L,
|
||||
-0.55715830451465005431552290962580L, 0.55715830451465005431552290962580L,
|
||||
-0.60770292718495023918038179639183L, 0.60770292718495023918038179639183L,
|
||||
-0.65589646568543936078162486400368L, 0.65589646568543936078162486400368L,
|
||||
-0.70155246870682225108954625788366L, 0.70155246870682225108954625788366L,
|
||||
-0.74449430222606853826053625268219L, 0.74449430222606853826053625268219L,
|
||||
-0.78455583290039926390530519634099L, 0.78455583290039926390530519634099L,
|
||||
-0.82158207085933594835625411087394L, 0.82158207085933594835625411087394L,
|
||||
-0.85542976942994608461136264393476L, 0.85542976942994608461136264393476L,
|
||||
-0.88596797952361304863754098246675L, 0.88596797952361304863754098246675L,
|
||||
-0.91307855665579189308973564277166L, 0.91307855665579189308973564277166L,
|
||||
-0.93665661894487793378087494727250L, 0.93665661894487793378087494727250L,
|
||||
-0.95661095524280794299774564415662L, 0.95661095524280794299774564415662L,
|
||||
-0.97286438510669207371334410460625L, 0.97286438510669207371334410460625L,
|
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-0.98535408404800588230900962563249L, 0.98535408404800588230900962563249L,
|
||||
-0.99403196943209071258510820042069L, 0.99403196943209071258510820042069L,
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-0.99886640442007105018545944497422L, 0.99886640442007105018545944497422L};
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/// @brief Gauss-Legendre quadrature: 50 weights
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static const long double gauss_legendre_50_weights[50] = {
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0.06217661665534726232103310736061L, 0.06217661665534726232103310736061L,
|
||||
0.06193606742068324338408750978083L, 0.06193606742068324338408750978083L,
|
||||
0.06145589959031666375640678608392L, 0.06145589959031666375640678608392L,
|
||||
0.06073797084177021603175001538481L, 0.06073797084177021603175001538481L,
|
||||
0.05978505870426545750957640531259L, 0.05978505870426545750957640531259L,
|
||||
0.05860084981322244583512243663085L, 0.05860084981322244583512243663085L,
|
||||
0.05718992564772838372302931506599L, 0.05718992564772838372302931506599L,
|
||||
0.05555774480621251762356742561227L, 0.05555774480621251762356742561227L,
|
||||
0.05371062188899624652345879725566L, 0.05371062188899624652345879725566L,
|
||||
0.05165570306958113848990529584010L, 0.05165570306958113848990529584010L,
|
||||
0.04940093844946631492124358075143L, 0.04940093844946631492124358075143L,
|
||||
0.04695505130394843296563301363499L, 0.04695505130394843296563301363499L,
|
||||
0.04432750433880327549202228683039L, 0.04432750433880327549202228683039L,
|
||||
0.04152846309014769742241197896407L, 0.04152846309014769742241197896407L,
|
||||
0.03856875661258767524477015023639L, 0.03856875661258767524477015023639L,
|
||||
0.03545983561514615416073461100098L, 0.03545983561514615416073461100098L,
|
||||
0.03221372822357801664816582732300L, 0.03221372822357801664816582732300L,
|
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0.02884299358053519802990637311323L, 0.02884299358053519802990637311323L,
|
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0.02536067357001239044019487838544L, 0.02536067357001239044019487838544L,
|
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0.02178024317012479298159206906269L, 0.02178024317012479298159206906269L,
|
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0.01811556071348939035125994342235L, 0.01811556071348939035125994342235L,
|
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0.01438082276148557441937890892732L, 0.01438082276148557441937890892732L,
|
||||
0.01059054838365096926356968149924L, 0.01059054838365096926356968149924L,
|
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0.00675979919574540150277887817799L, 0.00675979919574540150277887817799L,
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0.00290862255315514095840072434286L, 0.00290862255315514095840072434286L};
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namespace internal {
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/// 50 point Gauss-Legendre quadrature
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template <typename T>
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constexpr T incomplete_gamma_quad_inp_vals(const T lb, const T ub,
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const int counter) noexcept {
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return (ub - lb) * gauss_legendre_50_points[counter] / T(2) +
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(ub + lb) / T(2);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_weight_vals(const T lb, const T ub,
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const int counter) noexcept {
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return (ub - lb) * gauss_legendre_50_weights[counter] / T(2);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_fn(const T x, const T a,
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const T lg_term) noexcept {
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return exp(-x + (a - T(1)) * log(x) - lg_term);
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_recur(const T lb, const T ub, const T a,
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const T lg_term,
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const int counter) noexcept {
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return (counter < 49 ? // if
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incomplete_gamma_quad_fn(
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incomplete_gamma_quad_inp_vals(lb, ub, counter), a, lg_term) *
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incomplete_gamma_quad_weight_vals(lb, ub, counter) +
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incomplete_gamma_quad_recur(lb, ub, a, lg_term, counter + 1)
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:
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// else
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incomplete_gamma_quad_fn(
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incomplete_gamma_quad_inp_vals(lb, ub, counter), a, lg_term) *
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incomplete_gamma_quad_weight_vals(lb, ub, counter));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_lb(const T a, const T z) noexcept {
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// break integration into ranges
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return (a > T(1000) ? std::max(T(0), std::min(z, a) - 11 * sqrt(a))
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: a > T(800) ? std::max(T(0), std::min(z, a) - 11 * sqrt(a))
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: a > T(500) ? std::max(T(0), std::min(z, a) - 10 * sqrt(a))
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: a > T(300) ? std::max(T(0), std::min(z, a) - 10 * sqrt(a))
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: a > T(100) ? std::max(T(0), std::min(z, a) - 9 * sqrt(a))
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: a > T(90) ? std::max(T(0), std::min(z, a) - 9 * sqrt(a))
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: a > T(70) ? std::max(T(0), std::min(z, a) - 8 * sqrt(a))
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: a > T(50) ? std::max(T(0), std::min(z, a) - 7 * sqrt(a))
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: a > T(40) ? std::max(T(0), std::min(z, a) - 6 * sqrt(a))
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: a > T(30) ? std::max(T(0), std::min(z, a) - 5 * sqrt(a))
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: std::max(T(0), std::min(z, a) - 4 * sqrt(a)));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad_ub(const T a, const T z) noexcept {
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return (a > T(1000) ? std::min(z, a + 10 * sqrt(a))
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: a > T(800) ? std::min(z, a + 10 * sqrt(a))
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: a > T(500) ? std::min(z, a + 9 * sqrt(a))
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: a > T(300) ? std::min(z, a + 9 * sqrt(a))
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: a > T(100) ? std::min(z, a + 8 * sqrt(a))
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: a > T(90) ? std::min(z, a + 8 * sqrt(a))
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: a > T(70) ? std::min(z, a + 7 * sqrt(a))
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: a > T(50) ? std::min(z, a + 6 * sqrt(a))
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: std::min(z, a + 5 * sqrt(a)));
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}
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template <typename T>
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constexpr T incomplete_gamma_quad(const T a, const T z) noexcept {
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return incomplete_gamma_quad_recur(incomplete_gamma_quad_lb(a, z),
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incomplete_gamma_quad_ub(a, z), a,
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lgamma(a), 0);
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}
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// reverse cf expansion
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// see: https://functions.wolfram.com/GammaBetaErf/Gamma2/10/0003/
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template <typename T>
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constexpr T incomplete_gamma_cf_2_recur(const T a, const T z,
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const int depth) noexcept {
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return (depth < 100 ? (1 + (depth - 1) * 2 - a + z) +
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depth * (a - depth) /
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incomplete_gamma_cf_2_recur(a, z, depth + 1)
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: (1 + (depth - 1) * 2 - a + z));
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_2(
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const T a,
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const T z) noexcept { // lower (regularized) incomplete gamma function
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return (T(1.0) - exp(a * log(z) - z - lgamma(a)) /
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incomplete_gamma_cf_2_recur(a, z, 1));
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}
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// cf expansion
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// see: http://functions.wolfram.com/GammaBetaErf/Gamma2/10/0009/
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template <typename T>
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constexpr T incomplete_gamma_cf_1_coef(const T a, const T z,
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const int depth) noexcept {
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return (is_odd(depth) ? -(a - 1 + T(depth + 1) / T(2)) * z
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: T(depth) / T(2) * z);
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_1_recur(const T a, const T z,
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const int depth) noexcept {
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return (depth < 55 ? // if
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(a + depth - 1) + incomplete_gamma_cf_1_coef(a, z, depth) /
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incomplete_gamma_cf_1_recur(a, z, depth + 1)
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:
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// else
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(a + depth - 1));
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}
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template <typename T>
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constexpr T incomplete_gamma_cf_1(
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const T a,
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const T z) noexcept { // lower (regularized) incomplete gamma function
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return (exp(a * log(z) - z - lgamma(a)) /
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incomplete_gamma_cf_1_recur(a, z, 1));
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}
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//
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template <typename T>
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constexpr T incomplete_gamma_check(const T a, const T z) noexcept {
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return ( // NaN check
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(is_nan(a) || is_nan(z)) ? LIM<T>::quiet_NaN() :
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//
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a < T(0) ? LIM<T>::quiet_NaN()
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:
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//
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LIM<T>::min() > z ? T(0)
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:
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//
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LIM<T>::min() > a ? T(1)
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:
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// cf or quadrature
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(a < T(10)) && (z - a < T(10)) ? incomplete_gamma_cf_1(a, z)
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: (a < T(10)) || (z / a > T(3)) ? incomplete_gamma_cf_2(a, z)
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:
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// else
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incomplete_gamma_quad(a, z));
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}
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template <typename T1, typename T2, typename TC = common_return_t<T1, T2>>
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constexpr TC incomplete_gamma_type_check(const T1 a, const T2 p) noexcept {
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return incomplete_gamma_check(static_cast<TC>(a), static_cast<TC>(p));
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}
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} // namespace internal
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/**
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* Compile-time regularized lower incomplete gamma function
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*
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* @param a a real-valued, non-negative input.
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* @param x a real-valued, non-negative input.
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*
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* @return the regularized lower incomplete gamma function evaluated at (\c a,
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* \c x), \f[ \frac{\gamma(a,x)}{\Gamma(a)} = \frac{1}{\Gamma(a)} \int_0^x
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* t^{a-1} \exp(-t) dt \f] When \c a is not too large, the value is computed
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* using the continued fraction representation of the upper incomplete gamma
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* function, \f$ \Gamma(a,x) \f$, using \f[ \Gamma(a,x) = \Gamma(a) -
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* \dfrac{x^a\exp(-x)}{a - \dfrac{ax}{a + 1 + \dfrac{x}{a + 2 - \dfrac{(a+1)x}{a
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* + 3 + \dfrac{2x}{a + 4 - \ddots}}}}} \f] where \f$ \gamma(a,x) \f$ and \f$
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* \Gamma(a,x) \f$ are connected via \f[ \frac{\gamma(a,x)}{\Gamma(a)} +
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* \frac{\Gamma(a,x)}{\Gamma(a)} = 1 \f] When \f$ a > 10 \f$, a 50-point
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* Gauss-Legendre quadrature scheme is employed.
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*/
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template <typename T1, typename T2>
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constexpr common_return_t<T1, T2> incomplete_gamma(const T1 a,
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const T2 x) noexcept {
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return internal::incomplete_gamma_type_check(a, x);
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}
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namespace internal {
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template <typename T>
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class IncompleteGammaInverse {
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/**
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* @brief Compute an initial value for the inverse gamma function which is
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* then iteratively updated.
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*
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* @param a
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* @param p
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* @return constexpr T
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*/
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static constexpr T initial_val(const T a, const T p) noexcept {
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if (a > T(1)) {
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// Inverse gamma function initial value when a > 1.0
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const T t_val = p > T(0.5) ? sqrt(-2 * log(T(1) - p)) : sqrt(-2 * log(p));
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const T sgn_term = p > T(0.5) ? T(-1) : T(1);
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const T initial_val_1 =
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t_val -
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(T(2.515517L) + T(0.802853L) * t_val + T(0.010328L) * t_val * t_val) /
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(T(1) + T(1.432788L) * t_val + T(0.189269L) * t_val * t_val +
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T(0.001308L) * t_val * t_val * t_val);
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const T signed_initial_val_1 = sgn_term * initial_val_1;
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return std::max(
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T(1e-04),
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a * pow(T(1) - T(1) / (9 * a) - signed_initial_val_1 / (3 * sqrt(a)),
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3));
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} else {
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// Inverse gamma function initial value when a <= 1.0
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T t_val = T(1) - T(0.253) * a - T(0.12) * a * a;
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if (p < t_val) {
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return pow(p / t_val, T(1) / a);
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} else {
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return T(1) - log(T(1) - (p - t_val) / (T(1) - t_val));
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}
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}
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}
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/**
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* @brief Compute the error value `f(x)` which we can use for root-finding.
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*
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* @param value
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* @param a
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* @param p
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* @return constexpr T
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*/
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static constexpr T err_val(const T value, const T a, const T p) noexcept {
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return (incomplete_gamma(a, value) - p);
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}
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/**
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* @brief Derivative of the incomplete gamma function w.r.t. value
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*
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* @param value
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* @param a
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* @param log_val
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* @return constexpr T
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*/
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static constexpr T derivative(const T value, const T a,
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const T lg_val) noexcept {
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return (exp(-value + (a - T(1)) * log(value) - lg_val));
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}
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/**
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* @brief Second derivative of the incomplete gamma function w.r.t. value
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*
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* @param value
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* @param a
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* @param derivative
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* @return constexpr T
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*/
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static constexpr T second_derivative(const T value, const T a,
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const T derivative) noexcept {
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return (derivative * ((a - T(1)) / value - T(1)));
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}
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|
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/**
|
||||
* @brief Compute \f[ \frac{f(x_n)}{f'(x_n)} \f] as part
|
||||
* of the update denominator.
|
||||
*
|
||||
* @param value
|
||||
* @param a
|
||||
* @param p
|
||||
* @param derivative
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T ratio_val_1(const T value, const T a, const T p,
|
||||
const T derivative) noexcept {
|
||||
return (err_val(value, a, p) / derivative);
|
||||
}
|
||||
|
||||
/**
|
||||
* @brief Compute \f[ \frac{f''(x_n)}{f'(x_n)} \f] as part
|
||||
* of the update denominator.
|
||||
*
|
||||
* @param value
|
||||
* @param a
|
||||
* @param derivative
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T ratio_val_2(const T value, const T a,
|
||||
const T derivative) noexcept {
|
||||
return (second_derivative(value, a, derivative) / derivative);
|
||||
}
|
||||
|
||||
/**
|
||||
* @brief Halley's method update step
|
||||
*
|
||||
* @param ratio_val_1
|
||||
* @param ratio_val_2
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T halley(const T ratio_val_1, const T ratio_val_2) noexcept {
|
||||
return (ratio_val_1 /
|
||||
std::max(T(0.8), std::min(T(1.2), T(1) - T(0.5) * ratio_val_1 *
|
||||
ratio_val_2)));
|
||||
}
|
||||
/**
|
||||
* @brief Recursive method for computing the iterative solution for the
|
||||
* incomplete inverse gamma function.
|
||||
*
|
||||
* @param value
|
||||
* @param a
|
||||
* @param p
|
||||
* @param derivative
|
||||
* @param lg_val
|
||||
* @param iter_count
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T recurse(const T value, const T a, const T p,
|
||||
const T derivative, const T lg_val,
|
||||
const int iter_count) noexcept {
|
||||
return decision(value, a, p,
|
||||
halley(ratio_val_1(value, a, p, derivative),
|
||||
ratio_val_2(value, a, derivative)),
|
||||
lg_val, iter_count);
|
||||
}
|
||||
|
||||
static constexpr T decision(const T value, const T a, const T p,
|
||||
const T direc, const T lg_val,
|
||||
const int iter_count) noexcept {
|
||||
const int GAMMA_INV_MAX_ITER = 35;
|
||||
if (iter_count <= GAMMA_INV_MAX_ITER) {
|
||||
return recurse(value - direc, a, p, derivative(value, a, lg_val), lg_val,
|
||||
iter_count + 1);
|
||||
} else {
|
||||
return value - direc;
|
||||
}
|
||||
}
|
||||
|
||||
/**
|
||||
* @brief Start point for numerical computation of the incomplete gamma
|
||||
* inverse funtion.
|
||||
*
|
||||
* @param initial_val Initial value guess
|
||||
* @param a
|
||||
* @param p
|
||||
* @param lg_val
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T begin(const T initial_val, const T a, const T p,
|
||||
const T lg_val) noexcept {
|
||||
return recurse(initial_val, a, p, derivative(initial_val, a, lg_val),
|
||||
lg_val, 1);
|
||||
}
|
||||
|
||||
public:
|
||||
/**
|
||||
* @brief Compute the percent point function for the Gamma distribution.
|
||||
*
|
||||
* @param a
|
||||
* @param p
|
||||
* @return constexpr T
|
||||
*/
|
||||
static constexpr T compute(const T a, const T p) noexcept {
|
||||
// Perform checks on the input and return the corresponding best answer
|
||||
if (isnan(a) || isnan(p)) { // NaN check
|
||||
return LIM<T>::quiet_NaN();
|
||||
} else if (LIM<T>::min() > p) { // Check lower bound
|
||||
return T(0);
|
||||
} else if (p > T(1)) { // Check upper bound
|
||||
return LIM<T>::quiet_NaN();
|
||||
} else if (LIM<T>::min() > abs(T(1) - p)) {
|
||||
return LIM<T>::infinity();
|
||||
} else if (LIM<T>::min() > a) { // Check lower bound for degrees of freedom
|
||||
return T(0);
|
||||
} else {
|
||||
return begin(initial_val(a, p), a, p, lgamma(a));
|
||||
}
|
||||
}
|
||||
};
|
||||
|
||||
} // namespace internal
|
||||
|
||||
/**
|
||||
* Compile-time inverse incomplete gamma function
|
||||
*
|
||||
* Compute the value \f$ x \f$
|
||||
* such that \f[ f(x) := \frac{\gamma(a,x)}{\Gamma(a)} - p \f] equal to zero,
|
||||
* for a given \c p.
|
||||
*
|
||||
* We find this root using Halley's method:
|
||||
* \f[ x_{n+1} = x_n - \frac{f(x_n)/f'(x_n)}{1 - 0.5 \frac{f(x_n)}{f'(x_n)}
|
||||
* \frac{f''(x_n)}{f'(x_n)} } \f] where
|
||||
* \f[ \frac{\partial}{\partial x} \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) =
|
||||
* \frac{1}{\Gamma(a)} x^{a-1} \exp(-x) \f] \f[ \frac{\partial^2}{\partial x^2}
|
||||
* \left(\frac{\gamma(a,x)}{\Gamma(a)}\right) = \frac{1}{\Gamma(a)} x^{a-1}
|
||||
* \exp(-x) \left( \frac{a-1}{x} - 1 \right) \f]
|
||||
*
|
||||
* @param a The degrees of freedom for the gamma distribution.
|
||||
* @param p The quantile value for computing the percent point function.
|
||||
*
|
||||
* @return Computes the inverse incomplete gamma function.
|
||||
*/
|
||||
template <typename T1, typename T2>
|
||||
constexpr common_return_t<T1, T2> incomplete_gamma_inv(const T1 a,
|
||||
const T2 p) noexcept {
|
||||
using TC = common_return_t<T1, T2>;
|
||||
return internal::IncompleteGammaInverse<TC>::compute(static_cast<TC>(a),
|
||||
static_cast<TC>(p));
|
||||
}
|
||||
|
||||
/**
|
||||
* @brief Compute the quantile function of the Chi squared distribution.
|
||||
*
|
||||
* @param dofs Degrees of freedom
|
||||
* @param alpha Quantile value
|
||||
* @return constexpr double
|
||||
*/
|
||||
constexpr double chi_squared_quantile(const size_t dofs, const double alpha) {
|
||||
// The quantile function of the Chi-squared distribution is the quantile of
|
||||
// the specific (inverse) incomplete Gamma distribution
|
||||
return 2 * incomplete_gamma_inv(dofs * 0.5, alpha);
|
||||
}
|
||||
|
||||
} // namespace gtsam
|
||||
Loading…
Reference in New Issue