# Kriging 

Let suppose that 

$Z(s_i) = X_i\beta + Y(s_i)$

where $Y$ is a second order stationary random field with mean 0 and covariance function $C$.

If $Z$ is a vector of observations, we denote 
$Z = X\beta + Y$ with $\Sigma$ the covariance of Y

## Simple Kriging 
If $\beta$ is known, we can obtain the simple kriging 

$Z_0^{SK} = X_0\beta + \Sigma_0^t\Sigma^{-1}(Z-X\beta) = X_0\beta + \lambda_{SK}^t(Z-X\beta)$

with:

- the simple kriging weights

$\lambda_{SK}=\Sigma^{-1}\Sigma_0$ 

- the variance of the estimator

$\textrm{Var}(Z_0^{SK})=\lambda_{SK}^t\Sigma\lambda_{SK}=\lambda_{SK}^t\Sigma_0$

- the estimation variance

$\sigma_{SK}^2 = \textrm{Var}(Z_0-Z_0^{SK}) = \sigma_0^2-\Sigma_0^t\Sigma^{-1}\Sigma_0=\sigma_0^2-\lambda_{SK}^t\Sigma_0$

### In matrix notation:

Simple Kriging System

$
      \begin{bmatrix}
	\Sigma
      \end{bmatrix}
      \times
      \begin{bmatrix}
	\lambda_{SK}
      \end{bmatrix}
      =
      \begin{bmatrix}
        \Sigma_0
      \end{bmatrix}
$

Estimation

$    Z_0^{SK} =
     \begin{bmatrix}
	Z
     \end{bmatrix}^t
     \times
     \begin{bmatrix}
	\lambda_{SK}
     \end{bmatrix}
     + m ({ 1 - \sum{\lambda_{SK}}} )
$

Variance of Estimation error

$
   \sigma_{SK}^2 = \sigma_0^2 -
   \begin{bmatrix}
     \lambda_{SK}
   \end{bmatrix}^t
   \times
   \begin{bmatrix}
     \Sigma_0
   \end{bmatrix}
$

Variance of Estimator

$
   \textrm{Var}(Z_0^{SK}) =
   \begin{bmatrix}
     \lambda_{SK}
   \end{bmatrix}^t
   \times
   \begin{bmatrix}
     \Sigma
   \end{bmatrix}
   \times
   \begin{bmatrix}
     \lambda_{SK}
   \end{bmatrix} =
   \begin{bmatrix}
     \lambda_{SK}
   \end{bmatrix}^t
   \times
   \begin{bmatrix}
     \Sigma_0
   \end{bmatrix}
$

## Universal kriging

If $\beta$ is unknown, we can estimate it by 

$\hat\beta =  \Sigma_c X^t\Sigma^{-1}Z$ 

Introducing the notation

$\Sigma_c =  (X^t\Sigma^{-1}X)^{-1} $

then

$\hat\beta = \Sigma_c X^t\Sigma^{-1}Z$ 

$\textrm{Var}(\hat\beta)=\Sigma_c$

The Universal kriging is obtained by first computing $\hat\beta$ and then pluging $\hat\beta$  in the simple kriging procedure.

$Z^{UK}_0 = X_0\hat\beta + \Sigma_0^t\Sigma^{-1}(Z-X\hat\beta)= \Sigma_0^t\Sigma^{-1}Z + (X_0 - \Sigma_0^t\Sigma^{-1}X)\hat\beta$

We can rewrite everything with respect to $Z$

$Z^{UK}_0 =  (\Sigma_0^t\Sigma^{-1} + (X_0 - \Sigma_0^t\Sigma^{-1}X)\Sigma_c X^t\Sigma^{-1})Z \\
=(\lambda_{SK}^t+(X_0-\lambda_{SK}^tX) \Sigma_c X^t\Sigma^{-1})Z\\
=\lambda_{UK}^tZ$ 

with

- the Universal Kriging Weights

$\lambda_{UK}=\lambda_{SK}+\Sigma^{-1}X \Sigma_c(X_0^t-X^t\lambda_{SK})$

- the variance of the estimator is

$\textrm{Var}(Z^{UK}_0) = \lambda_{UK}^t\Sigma\lambda_{UK} \\
=\textrm{Var}(Z^{SK}_0) +2\lambda_{SK}^tX \Sigma_c \Sigma_c (X_0^t-X^t\lambda_{SK})+(X_0-\lambda_{SK}^tX)\Sigma_c X^t\Sigma^{-1}X\Sigma_c (X_0^t-X^t\lambda_{SK})\\
=\textrm{Var}(Z^{SK}_0) +2\lambda_{SK}^tX\Sigma_c (X_0^t-X^t\lambda_{SK})+(X_0-\lambda_{SK}^tX)\Sigma_c (X_0^t-X^t\lambda_{SK})\\
=\textrm{Var}(Z^{SK}_0)+(\lambda_{SK}^tX+X_0)\Sigma_c (X_0^t-X^t\lambda_{SK})\\
=\textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX\Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t$

- the estimation variance

$\sigma_{UK}^2 = \sigma_0^2 - 2\textrm{Cov}(Z_0,Z^{UK}_0)+ \textrm{Var}(Z^{UK}_0)\\
= \sigma_0^2 -2\Sigma_0^t\lambda_{UK}+\textrm{Var}(Z^{UK}_0)\\
= \sigma_0^2 -2\Sigma_0^t(\lambda_{SK}+\Sigma^{-1}X \Sigma_c(X_0^t-X^t\lambda_{SK}))+\textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\
=  \sigma_0^2 -\Sigma_0^t\lambda_{SK} -2\Sigma_0^t\Sigma^{-1}X \Sigma_c (X_0^t-X^t\lambda_{SK})-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\
=\sigma_{SK}^2-2\lambda_{SK}^tX \Sigma_c (X_0^t-X^t\lambda_{SK})-\lambda_{SK}^tX \Sigma_c X^t\lambda_{SK}+X_0 \Sigma_c X_0^t\\
=\sigma_{SK}^2+(X_0-\lambda_{SK}^tX) \Sigma_c (X_0^t-X^t\lambda_{SK})
$

### In matrix notation:

Universal Kriging System

$
      \begin{bmatrix}
	\Sigma & X \\
         X^t   & 0
      \end{bmatrix}
      \times
      \begin{bmatrix}
	\lambda_{UK} \\
	-\mu
      \end{bmatrix}
      =
      \begin{bmatrix}
        \Sigma_0 \\
	X_0^t
      \end{bmatrix}
$

Estimation

$    Z_0^{UK} =
     \begin{bmatrix}
	Z \\
	0
     \end{bmatrix}^t
     \times
     \begin{bmatrix}
	\lambda_{UK} \\
	-\mu
     \end{bmatrix}
$

Variance of estimation error

$
   \sigma_{UK}^2 = \sigma_0^2 -
   \begin{bmatrix}
     \lambda_{UK} \\
     -\mu
   \end{bmatrix}^t
   \times
   \begin{bmatrix}
     \Sigma_0 \\
     X_0^t
   \end{bmatrix}
$

Variance of estimator

$
   \textrm{Var}(Z^{UK}_0) =
     \begin{bmatrix}
     \lambda_{UK}
   \end{bmatrix}^t
   \times
   \begin{bmatrix}
     \Sigma
   \end{bmatrix}
   \times
   \begin{bmatrix}
     \lambda_{UK}
   \end{bmatrix}
$


# Summary

## Simple Kriging

- the estimator

$Z_0^{SK} = X_0\beta + \lambda_{SK}^t(Z-X\beta) =  \lambda_{SK}^tZ +(X_0 -\lambda_{SK}^tX)\beta$

where 

$\lambda_{SK}=\Sigma^{-1}\Sigma_0$ 

- the variance of the estimator

$\textrm{Var}(Z_0^{SK})=\lambda_{SK}^t\Sigma\lambda_{SK}=\lambda_{SK}^t\Sigma_0$

- the variance of the estimation error

$\sigma_{SK}^2 = \textrm{Var}(Z_0-Z_0^{SK}) = \sigma_0^2-\Sigma_0^t\Sigma^{-1}\Sigma_0=\sigma_0^2-\lambda_{SK}^t\Sigma_0$

## Universal Kriging

$\hat\beta =  \Sigma_c X^t\Sigma^{-1}Z$ 

$\textrm{Var}(\hat\beta)= \Sigma_c $

- the estimator

$Z^{UK}_0 =\lambda^t_{SK}Z + (X_0 - \lambda_{SK}^tX)\hat\beta= \lambda^t_{UK}Z$

$\lambda_{UK}=\lambda_{SK}+\Sigma^{-1}X \Sigma_c (X_0^t-X^t\lambda_{SK})$

$\mu_{UK}=\Sigma_c (X_0 - \lambda_{SK}^tX)^t$

- the variance of the estimator

$\textrm{Var}(Z^{UK}_0) = \textrm{Var}(Z^{SK}_0)-\lambda_{SK}^tX\Sigma_c X^t\lambda_{SK}+X_0\Sigma_c X_0^t$

- the variance of the estimation error

$\sigma_{UK}^2 
=\sigma_{SK}^2+(X_0-\lambda_{SK}^tX) \Sigma_c (X_0^t-X^t\lambda_{SK})
$

# Collocated Option (Unique Neighborhood)

This is an interesting case for:

- estimating a Target
- in a multivariate case (say with N variables)
- based on the information in the input Db (note that all 'N' variable(s) do not have to be known in the 'heterotopic' case)
- when the Target contains information on some of the 'N' variables (these are the **collocated** variables) 

When working in **Unique** Neighborhood, the relevant information (from the input data base) do not change for any of the targets. But the Collocated information changes at each target.

Hence the interest of benefiting of the inversion of the covariance matrix (restricted to the information of the Input data base).