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