Kriging with Trend Constraints

KT_constraints<forward_iterator>

KT_constraints adds constraints to account for the variations of the mean of the krigged variable Z. The mean is assumed to be of the form:

m($$\bf u$$) = $$\sum_{k=0}^{K}$$a_k($$\bf u$$)f_k($$\bf u$$)

where a_k are unknown but locally constant and f_k are known functions of u.

The kriging system at location u is then given by:

$$\left\{\vphantom{ \begin{array}{l} Var \Big( \sum_{\alpha=1}^{n}... ...alpha}) = f_k(\rm {\bf u}) \quad \forall k \in [1,K] \\ \end{array} }\right.$$$$\begin{array}{l} Var \Big( \sum_{\alpha=1}^{n} \lambda_{\alpha} ... ...\bf u}_{\alpha}) = f_k(\rm {\bf u}) \quad \forall k \in [1,K] \\ \end{array}$$

Kriging with trend is rarely used with secondary variables. Hence KT_constraints assumes no secondary variable is to be accounted for. KT_constraints computes the kriging system size, resizes the kriging matrix and the second member, computes the terms of the system that are associated with the constraints on the kriging weights and finally returns the system size.

Where Defined

In header file <kriging.h>

Template Parameters

forward_iterator

is a model of Forward Iterator.

Model of

Kriging Constraint

Type Requirements

See 2.4 for a thorough description of the requirements on the matrix library.

Members

• KT_constraints::KT_constraints(forward_iterator begin, forward_iterator end)
  
  Constructs a KT_constraint. The range [begin,end) contains the functions f_i, i = 1,..., K that define the mean of Z. These functions must be unary functions and associate to a location a value of type convertible to double. The prototype of each function f_i must be:

  double f(location u)

  where location is a model of Location.

• template<class InputIterator, class Location, class Matrix, class Vector>
  unsigned int
  KT_constraints::operator()(Matrix& A, Vector& b,
                             const Location& u,
                             InputIterator first_neigh, InputIterator last_neigh)
  
  Function call operator. first_neigh,last_neigh) is a range of Neighborhood, and Location is a model of Location. A is the kriging matrix and b the right hand side of the kriging system. u is the location being estimated. The function returns the total number of neighbors, i.e. the sum of the number of neighbors in each neighborhoods. The requirements on concepts Matrix and Vector are fully described in 2.4.