Indicator Cdf Estimator

indicator_cdf_estimator<covar_iterator,constraints_iterator,matrix_lib>

Indicator kriging estimates a non-parametric cdf $\Big($z_i, F(z_i)$\Big)$, i = 1,..., n of variable Z by kriging n indicator variables I($\bf u$, z_i):

i($$\bf u$$, z_i) = $$\left\{\vphantom{ \begin{array}{ll} 1 & \textrm{if } z(\rm {\bf u}) \le z_i \\ 0 & \textrm{otherwise} \end{array} }\right.$$$$\begin{array}{ll} 1 & \textrm{if } z(\rm {\bf u}) \le z_i \\ 0 & \textrm{otherwise} \end{array}$$

The kriging estimate of I($\bf u$, z_i) is indeed the least-squares estimate of
Prob(Z($\bf u$)$\le$z_i).

Each indicator variable can be estimated using a different kriging method, e.g. with different kriging constraints. This cdf estimator only allows to change the kriging constraints, and none of the kriging systems can account for multiple variables (no cokriging).

It must be stressed that an Indicator Cdf Estimator expects n indicator variables I($\bf u$, z_i), not the single variable Z($\bf u$) itself.

Where Defined

In header file <cdf_estimators.h>

Template Parameters

covar_iterator		is a model of Forward Iterator
constraints_iterator		is model of Forward Iterator
matrix_lib		defines the library of linear algebra to be used. The default value is tnt_lib, the TNT library.

Model of

Multiple Variable Cdf Estimator

Type Requirements

• covar_iterator iterates on a set of pointers to covariance functions, i.e. pointers to objects that are models Covariance.
• constraints_iterator iterates on a set of pointers to kriging constraints, i.e. pointers to objects that are models of Kriging Constraint.

Members

The leading indicator_cdf_estimator:: is omitted in the list of member functions.

• indicator_cdf_estimator(covar_iterator cov_begin, 
                          covar_iterator cov_end,
                          constraints_iterator begin, 
                          constraints_iterator end)
  
  Constructs an indicator_cdf_estimator. Ranges of iterators
  [cov_begin,cov_end) and [begin,end) need not be of the same size, nor do they need to be of size n, the number of discretizations of the non-parametric cdf. If they are of size lesser than n, the last element of the range is used for kriging the remaining indicators. For example, if n = 5 and both [cov_begin,cov_end) and [begin,end) contain only two elements, the first indicator variable I($\bf u$, z₁) will be kriged using covariance *cov_begin and kriging constraints *begin, while all four remaining indicator variables I($\bf u$, z_i), i = 1,..., 4 will be kriged using the same covariance and the kriging constraints *(cov_begin+1) and *(begin+1).

• template<class location, class neighborhood, class non_parametric_cdf> 
  void operator()(const location& u, neighborhood** neighbors,
                  unsigned int nb_of_neighborhoods,
                  non_parametric_cdf& ccdf)
  
  Function call operator. It estimates the non-parametric cdf parameters F(z_i) and modifies ccdf accordingly. location is a model of Location, and
  neighborhood is a model of Neighborhood.

  u is the location at which the non-parametric conditional cdf is estimated.

  neighbors is an array of pointers to neighborhoods of location u (*(neighbors+i) points to the i^th neighborhood of u). There are nb_of_neighborhoods neighborhoods in the array. Neighborhood i informs variable I($\bf u$, z_i).

  ccdf must contain the values z_i, i = 1,..., n.