Covariance set

Cokriging of variable Z₁ accounting for variables Z₂,..., Z_M requires the covariances between all the variables: C_{i, j}($\bf h$), i, j = 1,..., M. These covariances are specified through a Covariance Set. Different models can actually be used to determine these covariances. The full cokriging approach (LMC approach) is very demanding because it requires the complete knowledge of all covariance functions C_{i, j}($\bf h$). Modeling cross-covariances from data is a difficult task, because all the covariances can not be modeled independently from one another. The Markov models MM1 and MM2 alleviate the difficulties of full cokriging by using only the collocated secondary variables: the cokriging of location u depends on the neighboring values of Z₁($\bf u$ + $\bf h_j$) and only the collocated variables Z_i($\bf u$), i = 2,..., M (instead of many neighborhoods of variables: Z_i($\bf u$ + $\bf h^i_j$)). Hence the covariances C_{i, j}($\bf h$), i, j = 2,..., M need only be modeled for distance $\bf h$ = 0. Moreover, the covariances C_{1, j}, j = 2,..., M are (approximately) proportional to C_{1, 1} or C_{j, j}, depending on the Markov approximation used (MM1 or MM2).

Associated Types

• Return Type

  A::return_type

  The type of C_{i, j}($\bf u_{1}^{}$,$\bf u_{2}^{}$) (see definition above).

• Argument Type

  A::argument_type

  The type of location u (see definition above).

Refinement of

Assignable

Notations

a		an object of a type that models Covariance set
U		a type that models Location
u1,u2		objects of type U
i, j		objects of type convertible to unsigned int

Valid Expressions

• Set-up the system

  a(i,j, u1,u2)

  Return type:		a type convertible to double
  Semantics:		returns the covariance value Ci, j($\bf u_{1}^{}$,$\bf u_{2}^{}$)

Models

• LMC_covariance
• MM1_covariance
• MM2_covariance