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- // find_scale.cpp
- // Copyright Paul A. Bristow 2007, 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 finding scale (standard deviation) for normal (Gaussian).
- // Note that this file contains Quickbook mark-up as well as code
- // and comments, don't change any of the special comment mark-ups!
- //[find_scale1
- /*`
- First we need some includes to access the __normal_distrib,
- the algorithms to find scale (and some std output of course).
- */
- #include <boost/math/distributions/normal.hpp> // for normal_distribution
- using boost::math::normal; // typedef provides default type is double.
- #include <boost/math/distributions/find_scale.hpp>
- using boost::math::find_scale;
- using boost::math::complement; // Needed if you want to use the complement version.
- using boost::math::policies::policy; // Needed to specify the error handling policy.
- #include <iostream>
- using std::cout; using std::endl;
- #include <iomanip>
- using std::setw; using std::setprecision;
- #include <limits>
- using std::numeric_limits;
- //] [/find_scale1]
- int main()
- {
- cout << "Example: Find scale (standard deviation)." << endl;
- try
- {
- //[find_scale2
- /*`
- For this example, we will use the standard __normal_distrib,
- with location (mean) zero and standard deviation (scale) unity.
- Conveniently, this is also the default for this implementation's constructor.
- */
- normal N01; // Default 'standard' normal distribution with zero mean
- double sd = 1.; // and standard deviation is 1.
- /*`Suppose we want to find a different normal distribution with standard deviation
- so that only fraction p (here 0.001 or 0.1%) are below a certain chosen limit
- (here -2. standard deviations).
- */
- double z = -2.; // z to give prob p
- double p = 0.001; // only 0.1% below z = -2
- cout << "Normal distribution with mean = " << N01.location() // aka N01.mean()
- << ", standard deviation " << N01.scale() // aka N01.standard_deviation()
- << ", has " << "fraction <= " << z
- << ", p = " << cdf(N01, z) << endl;
- cout << "Normal distribution with mean = " << N01.location()
- << ", standard deviation " << N01.scale()
- << ", has " << "fraction > " << z
- << ", p = " << cdf(complement(N01, z)) << endl; // Note: uses complement.
- /*`
- [pre
- Normal distribution with mean = 0 has fraction <= -2, p = 0.0227501
- Normal distribution with mean = 0 has fraction > -2, p = 0.97725
- ]
- Noting that p = 0.02 instead of our target of 0.001,
- we can now use `find_scale` to give a new standard deviation.
- */
- double l = N01.location();
- double s = find_scale<normal>(z, p, l);
- cout << "scale (standard deviation) = " << s << endl;
- /*`
- that outputs:
- [pre
- scale (standard deviation) = 0.647201
- ]
- showing that we need to reduce the standard deviation from 1. to 0.65.
- Then we can check that we have achieved our objective
- by constructing a new distribution
- with the new standard deviation (but same zero mean):
- */
- normal np001pc(N01.location(), s);
- /*`
- And re-calculating the fraction below (and above) our chosen limit.
- */
- cout << "Normal distribution with mean = " << l
- << " has " << "fraction <= " << z
- << ", p = " << cdf(np001pc, z) << endl;
- cout << "Normal distribution with mean = " << l
- << " has " << "fraction > " << z
- << ", p = " << cdf(complement(np001pc, z)) << endl;
- /*`
- [pre
- Normal distribution with mean = 0 has fraction <= -2, p = 0.001
- Normal distribution with mean = 0 has fraction > -2, p = 0.999
- ]
- [h4 Controlling how Errors from find_scale are handled]
- We can also control the policy for handling various errors.
- For example, we can define a new (possibly unwise)
- policy to ignore domain errors ('bad' arguments).
- Unless we are using the boost::math namespace, we will need:
- */
- using boost::math::policies::policy;
- using boost::math::policies::domain_error;
- using boost::math::policies::ignore_error;
- /*`
- Using a typedef is convenient, especially if it is re-used,
- although it is not required, as the various examples below show.
- */
- typedef policy<domain_error<ignore_error> > ignore_domain_policy;
- // find_scale with new policy, using typedef.
- l = find_scale<normal>(z, p, l, ignore_domain_policy());
- // Default policy policy<>, needs using boost::math::policies::policy;
- l = find_scale<normal>(z, p, l, policy<>());
- // Default policy, fully specified.
- l = find_scale<normal>(z, p, l, boost::math::policies::policy<>());
- // New policy, without typedef.
- l = find_scale<normal>(z, p, l, policy<domain_error<ignore_error> >());
- /*`
- If we want to express a probability, say 0.999, that is a complement, `1 - p`
- we should not even think of writing `find_scale<normal>(z, 1 - p, l)`,
- but use the __complements version (see __why_complements).
- */
- z = -2.;
- double q = 0.999; // = 1 - p; // complement of 0.001.
- sd = find_scale<normal>(complement(z, q, l));
- normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(scale) shifted
- cout << "Normal distribution with mean = " << l << " has "
- << "fraction <= " << z << " = " << cdf(np95pc, z) << endl;
- cout << "Normal distribution with mean = " << l << " has "
- << "fraction > " << z << " = " << cdf(complement(np95pc, z)) << endl;
- /*`
- Sadly, it is all too easy to get probabilities the wrong way round,
- when you may get a warning like this:
- [pre
- Message from thrown exception was:
- Error in function boost::math::find_scale<Dist, Policy>(complement(double, double, double, Policy)):
- Computed scale (-0.48043523852179076) is <= 0! Was the complement intended?
- ]
- The default error handling policy is to throw an exception with this message,
- but if you chose a policy to ignore the error,
- the (impossible) negative scale is quietly returned.
- */
- //] [/find_scale2]
- }
- 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()
- //[find_scale_example_output
- /*`
- [pre
- Example: Find scale (standard deviation).
- Normal distribution with mean = 0, standard deviation 1, has fraction <= -2, p = 0.0227501
- Normal distribution with mean = 0, standard deviation 1, has fraction > -2, p = 0.97725
- scale (standard deviation) = 0.647201
- Normal distribution with mean = 0 has fraction <= -2, p = 0.001
- Normal distribution with mean = 0 has fraction > -2, p = 0.999
- Normal distribution with mean = 0.946339 has fraction <= -2 = 0.001
- Normal distribution with mean = 0.946339 has fraction > -2 = 0.999
- ]
- */
- //] [/find_scale_example_output]
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