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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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