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170 lines
6.9 KiB
C++
170 lines
6.9 KiB
C++
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// laplace_example.cpp
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// Copyright Paul A. Bristow 2008, 2010.
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// Use, modification and distribution are subject to the
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// Boost Software License, Version 1.0.
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// (See accompanying file LICENSE_1_0.txt
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// or copy at http://www.boost.org/LICENSE_1_0.txt)
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// Example of using laplace (& comparing with normal) distribution.
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// Note that this file contains Quickbook mark-up as well as code
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// and comments, don't change any of the special comment mark-ups!
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//[laplace_example1
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/*`
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First we need some includes to access the laplace & normal distributions
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(and some std output of course).
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*/
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#include <boost/math/distributions/laplace.hpp> // for laplace_distribution
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using boost::math::laplace; // typedef provides default type is double.
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#include <boost/math/distributions/normal.hpp> // for normal_distribution
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using boost::math::normal; // typedef provides default type is double.
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#include <iostream>
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using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
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#include <iomanip>
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using std::setw; using std::setprecision;
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#include <limits>
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using std::numeric_limits;
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int main()
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{
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cout << "Example: Laplace distribution." << endl;
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try
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{
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{ // Traditional tables and values.
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/*`Let's start by printing some traditional tables.
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*/
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double step = 1.; // in z
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double range = 4; // min and max z = -range to +range.
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//int precision = 17; // traditional tables are only computed to much lower precision.
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int precision = 4; // traditional table at much lower precision.
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int width = 10; // for use with setw.
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// Construct standard laplace & normal distributions l & s
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normal s; // (default location or mean = zero, and scale or standard deviation = unity)
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cout << "Standard normal distribution, mean or location = "<< s.location()
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<< ", standard deviation or scale = " << s.scale() << endl;
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laplace l; // (default mean = zero, and standard deviation = unity)
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cout << "Laplace normal distribution, location = "<< l.location()
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<< ", scale = " << l.scale() << endl;
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/*` First the probability distribution function (pdf).
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*/
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cout << "Probability distribution function values" << endl;
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cout << " z PDF normal laplace (difference)" << endl;
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cout.precision(5);
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for (double z = -range; z < range + step; z += step)
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{
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cout << left << setprecision(3) << setw(6) << z << " "
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<< setprecision(precision) << setw(width) << pdf(s, z) << " "
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<< setprecision(precision) << setw(width) << pdf(l, z)<< " ("
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<< setprecision(precision) << setw(width) << pdf(l, z) - pdf(s, z) // difference.
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<< ")" << endl;
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}
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cout.precision(6); // default
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/*`Notice how the laplace is less at z = 1 , but has 'fatter' tails at 2 and 3.
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And the area under the normal curve from -[infin] up to z,
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the cumulative distribution function (cdf).
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*/
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// For a standard distribution
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cout << "Standard location = "<< s.location()
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<< ", scale = " << s.scale() << endl;
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cout << "Integral (area under the curve) from - infinity up to z " << endl;
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cout << " z CDF normal laplace (difference)" << endl;
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for (double z = -range; z < range + step; z += step)
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{
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cout << left << setprecision(3) << setw(6) << z << " "
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<< setprecision(precision) << setw(width) << cdf(s, z) << " "
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<< setprecision(precision) << setw(width) << cdf(l, z) << " ("
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<< setprecision(precision) << setw(width) << cdf(l, z) - cdf(s, z) // difference.
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<< ")" << endl;
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}
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cout.precision(6); // default
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/*`
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Pretty-printing a traditional 2-dimensional table is left as an exercise for the student,
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but why bother now that the Boost Math Toolkit lets you write
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*/
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double z = 2.;
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cout << "Area for gaussian z = " << z << " is " << cdf(s, z) << endl; // to get the area for z.
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cout << "Area for laplace z = " << z << " is " << cdf(l, z) << endl; //
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/*`
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Correspondingly, we can obtain the traditional 'critical' values for significance levels.
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For the 95% confidence level, the significance level usually called alpha,
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is 0.05 = 1 - 0.95 (for a one-sided test), so we can write
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*/
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cout << "95% of gaussian area has a z below " << quantile(s, 0.95) << endl;
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cout << "95% of laplace area has a z below " << quantile(l, 0.95) << endl;
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// 95% of area has a z below 1.64485
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// 95% of laplace area has a z below 2.30259
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/*`and a two-sided test (a comparison between two levels, rather than a one-sided test)
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*/
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cout << "95% of gaussian area has a z between " << quantile(s, 0.975)
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<< " and " << -quantile(s, 0.975) << endl;
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cout << "95% of laplace area has a z between " << quantile(l, 0.975)
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<< " and " << -quantile(l, 0.975) << endl;
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// 95% of area has a z between 1.95996 and -1.95996
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// 95% of laplace area has a z between 2.99573 and -2.99573
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/*`Notice how much wider z has to be to enclose 95% of the area.
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*/
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}
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//] [/[laplace_example1]
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}
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catch(const std::exception& e)
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{ // Always useful to include try & catch blocks because default policies
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// are to throw exceptions on arguments that cause errors like underflow, overflow.
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// Lacking try & catch blocks, the program will abort without a message below,
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// which may give some helpful clues as to the cause of the exception.
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std::cout <<
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"\n""Message from thrown exception was:\n " << e.what() << std::endl;
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}
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return 0;
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} // int main()
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/*
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Output is:
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Example: Laplace distribution.
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Standard normal distribution, mean or location = 0, standard deviation or scale = 1
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Laplace normal distribution, location = 0, scale = 1
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Probability distribution function values
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z PDF normal laplace (difference)
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-4 0.0001338 0.009158 (0.009024 )
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-3 0.004432 0.02489 (0.02046 )
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-2 0.05399 0.06767 (0.01368 )
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-1 0.242 0.1839 (-0.05803 )
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0 0.3989 0.5 (0.1011 )
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1 0.242 0.1839 (-0.05803 )
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2 0.05399 0.06767 (0.01368 )
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3 0.004432 0.02489 (0.02046 )
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4 0.0001338 0.009158 (0.009024 )
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Standard location = 0, scale = 1
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Integral (area under the curve) from - infinity up to z
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z CDF normal laplace (difference)
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-4 3.167e-005 0.009158 (0.009126 )
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-3 0.00135 0.02489 (0.02354 )
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-2 0.02275 0.06767 (0.04492 )
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-1 0.1587 0.1839 (0.02528 )
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0 0.5 0.5 (0 )
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1 0.8413 0.8161 (-0.02528 )
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2 0.9772 0.9323 (-0.04492 )
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3 0.9987 0.9751 (-0.02354 )
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4 1 0.9908 (-0.009126 )
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Area for gaussian z = 2 is 0.97725
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Area for laplace z = 2 is 0.932332
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95% of gaussian area has a z below 1.64485
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95% of laplace area has a z below 2.30259
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95% of gaussian area has a z between 1.95996 and -1.95996
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95% of laplace area has a z between 2.99573 and -2.99573
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*/
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