8  Unknown, varying mean

\[ \newcommand{\E}{{\rm E}} % E expectation operator \newcommand{\Var}{{\rm Var}} % Var variance operator \newcommand{\Cov}{{\rm Cov}} % Cov covariance operator \newcommand{\Cor}{{\rm Corr}} \]

For this, we need to know how the mean varies. Suppose we model this as a linear regression model in \(p\) known predictors: \[Z(s_i) = \sum_{j=0}^p \beta_j X_j(s_i) + e(s_i)\] \[Z(s) = \sum_{j=0}^p \beta_j X_j(s) + e(s) = X(s)\beta + e(s)\] with \(X(s)\) the matrix with predictors, and row \(i\) and column \(j\) containing \(X_j(s_i)\), and with \(\beta = (\beta_0,...\beta_p)\). Usually, the first column of \(X\) contains zeroes in which case \(\beta_0\) is an intercept.

Predictor: \[\hat{Z}(s_0) = x(s_0)\hat{\beta} + v'V^{-1} (Z-X\hat{\beta}) \] with \(x(s_0) = (X_0(s_0),...,X_p(s_0))\) and \(\hat{\beta} = (X'V^{-1}X)^{-1} X'V^{-1}Z\) it has prediction error variance \[\sigma^2(s_0) = \sigma^2_0 - v'V^{-1}v + Q\] with \(Q = (x(s_0) - X'V^{-1}v)'(X'V^{-1}X)^{-1}(x(s_0) - X'V^{-1}v)\)

This form is called external drift kriging, universal kriging or sometimes regression kriging.

Example in meuse data set: log(zinc) depending on sqrt(meuse)


    

  

  

  





Plotting them:



  
  

    

    

      

    

  

  

  







Linear trend:



  
  

    

    

      

    

  

  

  









With Gaussian covariance:



  
  

    

    

      

    

  

  

  









Estimating spatial correlation under the UK model


As opposed to the ordinary kriging model, the universal kriging model needs knowledge of the mean vector in order to estimate the semivariance (or covariance) from the residual vector: \[\hat{e}(s) = Z(s) - X\hat\beta\] but how to get \(\hat\beta\) without knowing \(V\)? This is a chicken-egg problem. The simplest, but not best, solution is to plug \(\hat{\beta}_{OLS}\) in, and from the \(e_{OLS}(s)\), estimate \(V\) (i.e., the variogram of \(Z\))






Spatial Prediction


… involves errors, uncertainties






Kriging varieties


    

  • Simple kriging: \(Z(s)=\mu+e(s)\), \(\mu\) known
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  • Ordinary kriging: \(Z(s)=m+e(s)\), \(m\) unknown
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  • Universal kriging: \(Z(s)=X\beta+e(s)\), \(\beta\) unknown
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  • SK: linear predictor \(\lambda'Z\) with \(\lambda\) such that \(\sigma^2(s_0) = \E(Z(s_0)-\lambda'Z)^2\) is minimized
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  • OK: linear predictor \(\lambda'Z\) with \(\lambda\) such that it

      

    • has minimum variance \(\sigma^2(s_0) = \E(Z(s_0)-\lambda'Z)^2\), and
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    • is unbiased \(\E(\lambda'Z) = m\)
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    • second constraint: \(\sum_{i=1}^n \lambda_i = 1\), weights sum to one.
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  • UK: \[\hat{Z}(s_0) = x(s_0)\hat{\beta} + v'V^{-1} (Z-X\hat{\beta}) \] with \(x(s_0) = (X_0(s_0),...,X_p(s_0))\) and \(\hat{\beta} = (X'V^{-1}X)^{-1} X'V^{-1}Z\) \[\sigma^2(s_0) = \sigma^2_0 - v'V^{-1}v + Q\] with \(Q = (x(s_0) - X'V^{-1}v)'(X'V^{-1}X)^{-1}(x(s_0) - X'V^{-1}v)\)
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  • OK: fill in a column vector with ones for \(X\): \(X=(1,1,...,1)'\) and \(X_0=1\)
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  • SK: take out the trend/unknown mean
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UK and linear regression


If \(Z\) has no spatial correlation, all covariances are zero and \(v=0\) and \(V=\mbox{diag}(\sigma^2)\). This implies that \[\hat{Z}(s_0) = x(s_0)\hat{\beta} + v'V^{-1} (Z-X\hat{\beta}) \] with \(\hat{\beta} = (X'V^{-1}X)^{-1} X'V^{-1}Z\) reduces to


\[\hat{Z}(s_0) = x(s_0)\hat{\beta}\] with \(\hat{\beta} = (X'X)^{-1} X'Z\), i.e., ordinary least squares regression.


Note that


    

  • under this model the residual does not carry information, as it is white noise
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  • in spatial prediction, UK can not be worse than linear regression, as linear regression is a limiting case of a more general model.
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