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Find Location (Mean) Example

First we need some includes to access the normal distribution, the algorithms to find location (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/cauchy.hpp> // for cauchy_distribution
  using boost::math::cauchy; // typedef provides default type is double.
#include <boost/math/distributions/find_location.hpp>
  using boost::math::find_location; // for mean
#include <boost/math/distributions/find_scale.hpp>
  using boost::math::find_scale; // for standard devation
  using boost::math::complement; // Needed if you want to use the complement version.
  using boost::math::policies::policy;

#include <iostream>
  using std::cout; using std::endl;
#include <iomanip>
  using std::setw; using std::setprecision;
#include <limits>
  using std::numeric_limits;

For this example, we will use the standard normal distribution, with mean (location) zero and standard deviation (scale) unity. This is also the default for this implementation.

normal N01;  // Default 'standard' normal distribution with zero mean and 
double sd = 1.; // normal default standard deviation is 1.

Suppose we want to find a different normal distribution whose mean is shifted so that only fraction p (here 0.001 or 0.1%) are below a certain chosen limit (here -2, two standard deviations).

double z = -2.; // z to give prob p
double p = 0.001; // only 0.1% below z

cout << "Normal distribution with mean = " << N01.location()
  << ", standard deviation " << N01.scale()
  << ", 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.

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

We can now use ''find_location'' to give a new offset mean.

double l = find_location<normal>(z, p, sd);
cout << "offset location (mean) = " << l << endl;

that outputs:

offset location (mean) = 1.09023

showing that we need to shift the mean just over one standard deviation from its previous value of zero.

Then we can check that we have achieved our objective by constructing a new distribution with the offset mean (but same standard deviation):

normal np001pc(l, sd); // Same standard_deviation (scale) but with mean (location) shifted.

And re-calculating the fraction below 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;

Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001
Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999
Controlling Error Handling from find_location

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 often 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_location with new policy, using typedef.
l = find_location<normal>(z, p, sd, ignore_domain_policy());
// Default policy policy<>, needs "using boost::math::policies::policy;"
l = find_location<normal>(z, p, sd, policy<>());
// Default policy, fully specified.
l = find_location<normal>(z, p, sd, boost::math::policies::policy<>());
// A new policy, ignoring domain errors, without using a typedef.
l = find_location<normal>(z, p, sd, policy<domain_error<ignore_error> >());

If we want to use a probability that is the complement of our probability, we should not even think of writing find_location<normal>(z, 1 - p, sd), but, to avoid loss of accuracy, use the complement version.

z = 2.;
double q = 0.95; // = 1 - p; // complement.
l = find_location<normal>(complement(z, q, sd));

normal np95pc(l, sd); // Same standard_deviation (scale) but with mean(location) 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;

See find_location_example.cpp for full source code: the program output looks like this:

Example: Find location (mean).
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
offset location (mean) = 1.09023
Normal distribution with mean = 1.09023 has fraction <= -2, p = 0.001
Normal distribution with mean = 1.09023 has fraction > -2, p = 0.999
Normal distribution with mean = 0.355146 has fraction <= 2 = 0.95
Normal distribution with mean = 0.355146 has fraction > 2 = 0.05

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