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