Construct Non-Parametric Cdf

1. template<class non_param_cdf, class random_iterator>
   void
   build_cdf(random_iterator first, random_iterator last,
             non_param_cdf& new_cdf, unsigned int nb_of_thresholds) 
   
   
2. template<class non_param_cdf, class random_iterator>
   void
   build_cdf(random_iterator first, random_iterator last,
             non_param_cdf& new_cdf) 
   
   

This function builds the cdf F of a random variable Z from a set of outcomes of Z, contained in range [first,last). Cdf F is a function defined by a set of thresholds z_i and the associated probabilities F(z_i), i = 1,..., n.

In version 1, function argument nb_of_thresholds indicates the number n of thresholds to be used to define the cdf. The n values retained are n equally-spaced quantiles of the distribution. For example, if n = 5, z₁ is the smallest value in range [first,last), z₂ is the first quartile, z₃ is the median, z₄ is the upper quartile, and z₅ is the highest value in range [first,last). With this version of the algorithm, F(z₁) = Prob(Z < z₁) = 0 and F(z_n) = Prob(Z < z_n) = 1.

Version 2 of the algorithm assumes that the cdf new_cdf is already initialized: it already contains the values z_i, i = 1,..., n, and function build_cdf computes the corresponding probabilities F(z_i). This version gives a complete flexibility in the choice of the thresholds values z_i.

Note that range [first,last) will be modified by build_cdf (it will be sorted). If the range must not be modified, a copy should be made first.

Where defined

In header file <univariate_stats.h>

Preconditions

• The range [first,last) is a valid range.

• The values in range [first,last) are of type cdf::value_type, or are convertible to this type.

Requirements on types

• random_iterator is a model of Random Access Iterator. Random Access Iterator is a refinement of Forward Iterator. If i is a model of Random Access Iterator pointing to the i-th element of a range, i-n is an iterator to the (i-n)-th element of the range, i[n] is equivalent to *(i+n), and i<j compares to iterators. For a thorough description of Random Access Iterator.
• non_param_cdf is a model of Non-Parametric Cdf.

Example

int main()
{
  gaussian_cdf normal_cdf(0,1);

  vector<double> gaussian_values;

  for(int i=1; i<=100; i++)
    gaussian_values.push_back( normal_cdf.inverse(drand48()) );  

  non_parametric_cdf new_cdf;

  build_cdf(gaussian_values.begin(), gaussian_values.end(),
            new_cdf);
}

new_cdf now contains a discrete representation of a standard normal cdf, and the elements of gaussian_values are sorted.