!!! note
This section describes the Kriging models used in the [`Interpolate`](@ref) transform.
Most users don't want to use models directly because they lack features such as
neighborhood search and change of support.
A Kriging model has the form:
with Z\colon \R^m \times \Omega \to \R
a random field.
This package implements the following Kriging variants:
- Simple Kriging
- Ordinary Kriging
- Universal Kriging
- External Drift Kriging
All these variants follow the same interface: an object is first created with a given set of parameters, it is then combined with the data to obtain predictions at new geometries.
The fit
function takes care of building the Kriging system and factorizing the LHS with
an appropriate decomposition (e.g. Cholesky, LU), and the predict
(or predictprob
)
function performs the estimation for a given variable and geometry.
All variants work with general Hilbert spaces, meaning that one can interpolate any data type that implements scalar multiplication, vector addition and inner product.
In Simple Kriging, the mean \mu
of the random field is assumed to be constant and known.
The resulting linear system is:
or in matricial form \C\l = \c
. We subtract the given mean from the observations
\boldsymbol{y} = \z - \mu \1
and compute the mean and variance at location \x_0
:
GeoStatsModels.SimpleKriging
In Ordinary Kriging the mean of the random field is assumed to be constant and unknown. The resulting linear system is:
with \nu
the Lagrange multiplier associated with the constraint \1^\top \l = 1
. The mean and variance at
location \x_0
are given by:
GeoStatsModels.OrdinaryKriging
In Universal Kriging, the mean of the random field is assumed to be a polynomial of the spatial coordinates:
with N_d
monomials f_k
of degree up to d
. For example, in 2D there are 6
monomials of degree up to 2
:
The choice of the degree d
determines the size of the polynomial matrix
and polynomial vector \f = \begin{bmatrix} f_1(\x_0) & f_2(\x_0) & \cdots & f_{N_d}(\x_0) \end{bmatrix}^\top
.
The variogram determines the variogram matrix:
and the variogram vector
\g = \begin{bmatrix} \gamma(\x_1,\x_0) & \gamma(\x_2,\x_0) & \cdots & \gamma(\x_n,\x_0) \end{bmatrix}^\top
.
The resulting linear system is:
with \boldsymbol{\nu}
the Lagrange multipliers associated with the universal constraints. The mean and
variance at location \x_0
are given by:
GeoStatsModels.UniversalKriging
In External Drift Kriging, the mean of the random field is assumed to be a combination of known smooth functions:
Differently than Universal Kriging, the functions m_k
are not necessarily polynomials of the spatial coordinates.
In practice, they represent a list of variables that is strongly correlated (and co-located) with the variable being
estimated.
External drifts are known to cause numerical instability. Give preference to other Kriging variants if possible.
GeoStatsModels.ExternalDriftKriging