[ceph.git] / ceph / src / boost / libs / math / example / laplace_example.cpp

// 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

*/
Commit	Line	Data
7c673cae FG	1	// laplace_example.cpp
	2
	3	// Copyright Paul A. Bristow 2008, 2010.
	4
	5	// Use, modification and distribution are subject to the
	6	// Boost Software License, Version 1.0.
	7	// (See accompanying file LICENSE_1_0.txt
	8	// or copy at http://www.boost.org/LICENSE_1_0.txt)
	9
	10	// Example of using laplace (& comparing with normal) distribution.
	11
	12	// Note that this file contains Quickbook mark-up as well as code
	13	// and comments, don't change any of the special comment mark-ups!
	14
	15	//[laplace_example1
	16	/*`
	17	First we need some includes to access the laplace & normal distributions
	18	(and some std output of course).
	19	*/
	20
	21	#include <boost/math/distributions/laplace.hpp> // for laplace_distribution
	22	using boost::math::laplace; // typedef provides default type is double.
	23	#include <boost/math/distributions/normal.hpp> // for normal_distribution
	24	using boost::math::normal; // typedef provides default type is double.
	25
	26	#include <iostream>
	27	using std::cout; using std::endl; using std::left; using std::showpoint; using std::noshowpoint;
	28	#include <iomanip>
	29	using std::setw; using std::setprecision;
	30	#include <limits>
	31	using std::numeric_limits;
	32
	33	int main()
	34	{
	35	cout << "Example: Laplace distribution." << endl;
	36
	37	try
	38	{
	39	{ // Traditional tables and values.
	40	/*`Let's start by printing some traditional tables.
	41	*/
	42	double step = 1.; // in z
	43	double range = 4; // min and max z = -range to +range.
	44	//int precision = 17; // traditional tables are only computed to much lower precision.
	45	int precision = 4; // traditional table at much lower precision.
	46	int width = 10; // for use with setw.
	47
	48	// Construct standard laplace & normal distributions l & s
	49	normal s; // (default location or mean = zero, and scale or standard deviation = unity)
	50	cout << "Standard normal distribution, mean or location = "<< s.location()
	51	<< ", standard deviation or scale = " << s.scale() << endl;
	52	laplace l; // (default mean = zero, and standard deviation = unity)
	53	cout << "Laplace normal distribution, location = "<< l.location()
	54	<< ", scale = " << l.scale() << endl;
	55
	56	/*` First the probability distribution function (pdf).
	57	*/
	58	cout << "Probability distribution function values" << endl;
	59	cout << " z PDF normal laplace (difference)" << endl;
	60	cout.precision(5);
	61	for (double z = -range; z < range + step; z += step)
	62	{
	63	cout << left << setprecision(3) << setw(6) << z << " "
	64	<< setprecision(precision) << setw(width) << pdf(s, z) << " "
65	<< setprecision(precision) << setw(width) << pdf(l, z)<< " ("
66	<< setprecision(precision) << setw(width) << pdf(l, z) - pdf(s, z) // difference.
67	<< ")" << endl;
68	}
69	cout.precision(6); // default
70	/*`Notice how the laplace is less at z = 1 , but has 'fatter' tails at 2 and 3.
71
72	And the area under the normal curve from -[infin] up to z,
73	the cumulative distribution function (cdf).
74	*/
75	// For a standard distribution
76	cout << "Standard location = "<< s.location()
77	<< ", scale = " << s.scale() << endl;
78	cout << "Integral (area under the curve) from - infinity up to z " << endl;
79	cout << " z CDF normal laplace (difference)" << endl;
80	for (double z = -range; z < range + step; z += step)
81	{
82	cout << left << setprecision(3) << setw(6) << z << " "
83	<< setprecision(precision) << setw(width) << cdf(s, z) << " "
84	<< setprecision(precision) << setw(width) << cdf(l, z) << " ("
85	<< setprecision(precision) << setw(width) << cdf(l, z) - cdf(s, z) // difference.
86	<< ")" << endl;
87	}
88	cout.precision(6); // default
89
90	/*`
91	Pretty-printing a traditional 2-dimensional table is left as an exercise for the student,
92	but why bother now that the Boost Math Toolkit lets you write
93	*/
94	double z = 2.;
95	cout << "Area for gaussian z = " << z << " is " << cdf(s, z) << endl; // to get the area for z.
96	cout << "Area for laplace z = " << z << " is " << cdf(l, z) << endl; //
97	/*`
98	Correspondingly, we can obtain the traditional 'critical' values for significance levels.
99	For the 95% confidence level, the significance level usually called alpha,
100	is 0.05 = 1 - 0.95 (for a one-sided test), so we can write
101	*/
102	cout << "95% of gaussian area has a z below " << quantile(s, 0.95) << endl;
103	cout << "95% of laplace area has a z below " << quantile(l, 0.95) << endl;
104	// 95% of area has a z below 1.64485
105	// 95% of laplace area has a z below 2.30259
106	/*`and a two-sided test (a comparison between two levels, rather than a one-sided test)
107
108	*/
109	cout << "95% of gaussian area has a z between " << quantile(s, 0.975)
110	<< " and " << -quantile(s, 0.975) << endl;
111	cout << "95% of laplace area has a z between " << quantile(l, 0.975)
112	<< " and " << -quantile(l, 0.975) << endl;
113	// 95% of area has a z between 1.95996 and -1.95996
114	// 95% of laplace area has a z between 2.99573 and -2.99573
115	/*`Notice how much wider z has to be to enclose 95% of the area.
116	*/
117	}
118	//] [/[laplace_example1]
119	}
120	catch(const std::exception& e)
121	{ // Always useful to include try & catch blocks because default policies
122	// are to throw exceptions on arguments that cause errors like underflow, overflow.
123	// Lacking try & catch blocks, the program will abort without a message below,
124	// which may give some helpful clues as to the cause of the exception.
125	std::cout <<
126	"\n""Message from thrown exception was:\n " << e.what() << std::endl;
127	}
128	return 0;
129	} // int main()
130
131	/*
132
133	Output is:
134
135	Example: Laplace distribution.
136	Standard normal distribution, mean or location = 0, standard deviation or scale = 1
137	Laplace normal distribution, location = 0, scale = 1
138	Probability distribution function values
139	z PDF normal laplace (difference)
140	-4 0.0001338 0.009158 (0.009024 )
141	-3 0.004432 0.02489 (0.02046 )
142	-2 0.05399 0.06767 (0.01368 )
143	-1 0.242 0.1839 (-0.05803 )
144	0 0.3989 0.5 (0.1011 )
145	1 0.242 0.1839 (-0.05803 )
146	2 0.05399 0.06767 (0.01368 )
147	3 0.004432 0.02489 (0.02046 )
148	4 0.0001338 0.009158 (0.009024 )
149	Standard location = 0, scale = 1
150	Integral (area under the curve) from - infinity up to z
151	z CDF normal laplace (difference)
152	-4 3.167e-005 0.009158 (0.009126 )
153	-3 0.00135 0.02489 (0.02354 )
154	-2 0.02275 0.06767 (0.04492 )
155	-1 0.1587 0.1839 (0.02528 )
156	0 0.5 0.5 (0 )
157	1 0.8413 0.8161 (-0.02528 )
158	2 0.9772 0.9323 (-0.04492 )
159	3 0.9987 0.9751 (-0.02354 )
160	4 1 0.9908 (-0.009126 )
161	Area for gaussian z = 2 is 0.97725
162	Area for laplace z = 2 is 0.932332
163	95% of gaussian area has a z below 1.64485
164	95% of laplace area has a z below 2.30259
165	95% of gaussian area has a z between 1.95996 and -1.95996
166	95% of laplace area has a z between 2.99573 and -2.99573
167
168	*/
169