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/* -*- mode: c++; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*- */
/*
Copyright (C) 2014 Jose Aparicio
This file is part of QuantLib, a free-software/open-source library
for financial quantitative analysts and developers - http://quantlib.org/
QuantLib is free software: you can redistribute it and/or modify it
under the terms of the QuantLib license. You should have received a
copy of the license along with this program; if not, please email
<quantlib-dev@lists.sf.net>. The license is also available online at
<https://www.quantlib.org/license.shtml>.
This program is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
FOR A PARTICULAR PURPOSE. See the license for more details.
*/
#include <ql/experimental/math/convolvedstudentt.hpp>
#include <ql/errors.hpp>
#include <ql/math/factorial.hpp>
#include <ql/math/distributions/normaldistribution.hpp>
#include <ql/math/solvers1d/brent.hpp>
#include <ql/math/functional.hpp>
#include <boost/math/distributions/students_t.hpp>
namespace QuantLib {
CumulativeBehrensFisher::CumulativeBehrensFisher(const std::vector<Integer>& degreesFreedom,
const std::vector<Real>& factors)
: degreesFreedom_(degreesFreedom), factors_(factors), polyConvolved_(std::vector<Real>(1, 1.))
{
QL_REQUIRE(degreesFreedom.size() == factors.size(),
"Incompatible sizes in convolution.");
for (int i : degreesFreedom) {
QL_REQUIRE(i % 2 != 0, "Even degree of freedom not allowed");
QL_REQUIRE(i >= 0, "Negative degree of freedom not allowed");
}
for(Size i=0; i<degreesFreedom_.size(); i++)
polynCharFnc_.push_back(polynCharactT((degreesFreedom[i]-1)/2));
// adjust the polynomial coefficients by the factors in the linear
// combination:
for(Size i=0; i<degreesFreedom_.size(); i++) {
Real multiplier = 1.;
for(Size k=1; k<polynCharFnc_[i].size(); k++) {
multiplier *= std::abs(factors_[i]);
polynCharFnc_[i][k] *= multiplier;
}
}
//convolution, here it is a product of polynomials and exponentials
for (auto& i : polynCharFnc_)
polyConvolved_ = convolveVectorPolynomials(polyConvolved_, i);
// trim possible zeros that might have arised:
auto it = polyConvolved_.rbegin();
while (it != polyConvolved_.rend()) {
if (*it == 0.) {
polyConvolved_.pop_back();
it = polyConvolved_.rbegin();
}else{
break;
}
}
// cache 'a' value (the exponent)
for(Size i=0; i<degreesFreedom_.size(); i++)
a_ += std::sqrt(static_cast<Real>(degreesFreedom_[i]))
* std::abs(factors_[i]);
a2_ = a_ * a_;
}
std::vector<Real> CumulativeBehrensFisher::polynCharactT(Natural n) const {
Natural nu = 2 * n +1;
std::vector<Real> low(1,1.), high(1,1.);
high.push_back(std::sqrt(static_cast<Real>(nu)));
if(n==0) return low;
if(n==1) return high;
for(Size k=1; k<n; k++) {
std::vector<Real> recursionFactor(1,0.); // 0 coef
recursionFactor.push_back(0.); // 1 coef
recursionFactor.push_back(nu/((2.*k+1.)*(2.*k-1.))); // 2 coef
std::vector<Real> lowUp =
convolveVectorPolynomials(recursionFactor, low);
//add them up:
for(Size i=0; i<high.size(); i++)
lowUp[i] += high[i];
low = high;
high = lowUp;
}
return high;
}
std::vector<Real> CumulativeBehrensFisher::convolveVectorPolynomials(
const std::vector<Real>& v1,
const std::vector<Real>& v2) const {
#if defined(QL_EXTRA_SAFETY_CHECKS)
QL_REQUIRE(!v1.empty() && !v2.empty(),
"Incorrect vectors in polynomial.");
#endif
const std::vector<Real>& shorter = v1.size() < v2.size() ? v1 : v2;
const std::vector<Real>& longer = (v1 == shorter) ? v2 : v1;
Size newDegree = v1.size()+v2.size()-2;
std::vector<Real> resultB(newDegree+1, 0.);
for(Size polyOrdr=0; polyOrdr<resultB.size(); polyOrdr++) {
for(Size i=std::max<Integer>(0, polyOrdr-longer.size()+1);
i<=std::min(polyOrdr, shorter.size()-1); i++)
resultB[polyOrdr] += shorter[i]*longer[polyOrdr-i];
}
return resultB;
}
Probability CumulativeBehrensFisher::operator()(const Real x) const {
// 1st & 0th terms with the table integration
Real integral = polyConvolved_[0] * std::atan(x/a_);
Real squared = a2_ + x*x;
Real rootsqr = std::sqrt(squared);
Real atan2xa = std::atan2(-x,a_);
if(polyConvolved_.size()>1)
integral += polyConvolved_[1] * x/squared;
for(Size exponent = 2; exponent <polyConvolved_.size(); exponent++) {
integral -= polyConvolved_[exponent] *
Factorial::get(exponent-1) * std::sin((exponent)*atan2xa)
/std::pow(rootsqr, static_cast<Real>(exponent));
}
return .5 + integral / M_PI;
}
Probability
CumulativeBehrensFisher::density(const Real x) const {
Real squared = a2_ + x*x;
Real integral = polyConvolved_[0] * a_ / squared;
Real rootsqr = std::sqrt(squared);
Real atan2xa = std::atan2(-x,a_);
for(Size exponent=1; exponent <polyConvolved_.size(); exponent++) {
integral += polyConvolved_[exponent] *
Factorial::get(exponent) * std::cos((exponent+1)*atan2xa)
/std::pow(rootsqr, static_cast<Real>(exponent+1) );
}
return integral / M_PI;
}
InverseCumulativeBehrensFisher::InverseCumulativeBehrensFisher(
const std::vector<Integer>& degreesFreedom,
const std::vector<Real>& factors,
Real accuracy)
: normSqr_(std::inner_product(factors.begin(), factors.end(),
factors.begin(), Real(0.))),
accuracy_(accuracy), distrib_(degreesFreedom, factors) { }
Real InverseCumulativeBehrensFisher::operator()(const Probability q) const {
Probability effectiveq;
Real sign;
// since the distrib is symmetric solve only on the right side:
if(q==0.5) {
return 0.;
}else if(q < 0.5) {
sign = -1.;
effectiveq = 1.-q;
}else{
sign = 1.;
effectiveq = q;
}
Real xMin =
InverseCumulativeNormal::standard_value(effectiveq) * normSqr_;
// inversion will fail at the Brent's bounds-check if this is not enough
// (q is very close to 1.), in a bad combination fails around 1.-1.e-7
Real xMax = 1.e6;
return sign *
Brent().solve([&](Real x) -> Real { return distrib_(x) - effectiveq; },
accuracy_, (xMin+xMax)/2., xMin, xMax);
}
}
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