# Boost C++ Libraries

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#### Distribution Construction Examples

The structure of distributions is rather different from some other statistical libraries, for example, those written in less object-oriented language like FORTRAN and C: these provide a few arguments to each free function.

Boost.Math library provides each distribution as a template C++ class. A distribution is constructed with a few arguments, and then member and non-member functions are used to find values of the distribution, often a function of a random variate.

For this demonstration, first we need some includes to access the negative binomial distribution (and the binomial, beta and gamma distributions too).

To demonstrate the use with a high precision User-defined floating-point type cpp_dec_float we also need an include from Boost.Multiprecision.

#include <boost/math/distributions/negative_binomial.hpp> // for negative_binomial_distribution
using boost::math::negative_binomial_distribution; // default type is double.
using boost::math::negative_binomial; // typedef provides default type is double.
#include <boost/math/distributions/binomial.hpp> // for binomial_distribution.
#include <boost/math/distributions/beta.hpp> // for beta_distribution.
#include <boost/math/distributions/gamma.hpp> // for gamma_distribution.
#include <boost/math/distributions/normal.hpp> // for normal_distribution.

#include <boost/multiprecision/cpp_dec_float.hpp> // for cpp_dec_float_100

Several examples of constructing distributions follow:

First, a negative binomial distribution with 8 successes and a success fraction 0.25, 25% or 1 in 4, is constructed like this:

boost::math::negative_binomial_distribution<double> mydist0(8., 0.25);

But this is inconveniently long, so we might be tempted to write

using namespace boost::math;

but this might risk ambiguity with names in std random so much better is explicit using boost::math:: statements, for example:

using boost::math::negative_binomial_distribution;

and we can still reduce typing.

Since the vast majority of applications use will be using double precision, the template argument to the distribution (RealType) defaults to type double, so we can also write:

negative_binomial_distribution<> mydist9(8., 0.25); // Uses default `RealType = double`.

But the name negative_binomial_distribution is still inconveniently long, so, for most distributions, a convenience typedef is provided, for example:

typedef negative_binomial_distribution<double> negative_binomial; // Reserved name of type double.
Caution This convenience typedef is not provided if a clash would occur with the name of a function: currently only beta and gamma fall into this category.

So, after a using statement,

using boost::math::negative_binomial;

we have a convenient typedef to negative_binomial_distribution<double>:

negative_binomial mydist(8., 0.25);

Some more examples using the convenience typedef:

negative_binomial mydist10(5., 0.4); // Both arguments double.

And automatic conversion takes place, so you can use integers and floats:

negative_binomial mydist11(5, 0.4); // Using provided typedef double, int and double arguments.

This is probably the most common usage.

negative_binomial mydist12(5., 0.4F); // Double and float arguments.
negative_binomial mydist13(5, 1); // Both arguments integer.

Similarly for most other distributions like the binomial.

binomial mybinomial(1, 0.5); // is more concise than
binomial_distribution<> mybinomd1(1, 0.5);

For cases when the typdef distribution name would clash with a math special function (currently only beta and gamma) the typedef is deliberately not provided, and the longer version of the name must be used. For example do not use:

using boost::math::beta;
beta mybetad0(1, 0.5); // Error beta is a math FUNCTION!

Which produces the error messages:

error C2146: syntax error : missing ';' before identifier 'mybetad0'
warning C4551: function call missing argument list

Instead you should use:

using boost::math::beta_distribution;

or for the gamma distribution:

We can, of course, still provide the type explicitly thus:

// Explicit double precision:  both arguments are double:
negative_binomial_distribution<double>        mydist1(8., 0.25);

// Explicit float precision, double arguments are truncated to float:
negative_binomial_distribution<float>         mydist2(8., 0.25);

// Explicit float precision, integer & double arguments converted to float:
negative_binomial_distribution<float>         mydist3(8, 0.25);

// Explicit float precision, float arguments, so no conversion:
negative_binomial_distribution<float>         mydist4(8.F, 0.25F);

// Explicit float precision, integer arguments promoted to float.
negative_binomial_distribution<float>         mydist5(8, 1);

// Explicit double precision:
negative_binomial_distribution<double>        mydist6(8., 0.25);

// Explicit long double precision:
negative_binomial_distribution<long double>   mydist7(8., 0.25);

And you can use your own RealType, for example, boost::math::cpp_dec_float_50 (an arbitrary 50 decimal digits precision type), then we can write:

using namespace boost::multiprecision;

negative_binomial_distribution<cpp_dec_float_50>  mydist8(8, 0.25);
// `integer` arguments are promoted to your RealType exactly, but
// `double` argument are converted to RealType,
// possibly losing precision, so don't write:

negative_binomial_distribution<cpp_dec_float_50>  mydist20(8, 0.23456789012345678901234567890);
// to avoid truncation of second parameter to `0.2345678901234567`.

negative_binomial_distribution<cpp_dec_float_50>  mydist21(8, cpp_dec_float_50("0.23456789012345678901234567890") );

// Ensure that all potentially significant digits are shown.
std::cout.precision(std::numeric_limits<cpp_dec_float_50>::digits10);
cpp_dec_float_50 x("1.23456789012345678901234567890");
std::cout << pdf(mydist8, x) << std::endl;
showing  0.00012630010495970320103876754721976419438231705359935
Warning When using multiprecision, it is all too easy to get accidental truncation!

For example, if you write

std::cout << pdf(mydist8, 1.23456789012345678901234567890) << std::endl;

showing 0.00012630010495970318465064569310967179576805651692929, which is wrong at about the 17th decimal digit!

This is because the value provided is truncated to a double, effectively double x = 1.23456789012345678901234567890;

Then the now double x is passed to function pdf, and this truncated double value is finally promoted to cpp_dec_float_50.

Another way of quietly getting the wrong answer is to write:

std::cout << pdf(mydist8, cpp_dec_float_50(1.23456789012345678901234567890)) << std::endl;

A correct way from a multi-digit string value is

std::cout << pdf(mydist8, cpp_dec_float_50("1.23456789012345678901234567890")) << std::endl;
Tip Getting about 17 decimal digits followed by many zeros is often a sign of accidental truncation.
##### Default arguments to distribution constructors.

Note that default constructor arguments are only provided for some distributions. So if you wrongly assume a default argument, you will get an error message, for example:

negative_binomial_distribution<> mydist8;
error C2512 no appropriate default constructor available.

No default constructors are provided for the negative binomial distribution, because it is difficult to chose any sensible default values for this distribution.

For other distributions, like the normal distribution, it is obviously very useful to provide 'standard' defaults for the mean (zero) and standard deviation (unity) thus:

normal_distribution(RealType mean = 0, RealType sd = 1);

So in this case we can write:

using boost::math::normal;

normal norm1;       // Standard normal distribution.
normal norm2(2);    // Mean = 2, std deviation = 1.
normal norm3(2, 3); // Mean = 2, std deviation = 3.

}
catch(std::exception &ex)
{
std::cout << ex.what() << std::endl;
}

return 0;
}  // int main()

There is no useful output from this demonstration program, of course.

See distribution_construction.cpp for full source code.

 Copyright © 2006-2010, 2012, 2013 Paul A. Bristow, Christopher Kormanyos, Hubert Holin, Bruno Lalande, John Maddock, Johan Råde, Gautam Sewani, Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang Distributed under 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)