9  Local predictors, block kriging

Global vs. local predictors

In many cases, instead of using all data, the number of observations used for prediction are limited to a selection of nearest observations, based on

  • number of observations or
  • distance to prediction location \(s_0\)
  • possibly, in addition, directions

The reason for this is usually either

  • statistical, allowing for a more flexible mean/trend structure
  • practical, if \(n\) gets large

Statistical arguments for local prediction

  • estimating \(\beta\) locally instead of globally means that
    • \(\beta\) will adjust to local situations (less bias)
    • it will be harder to estimate \(\beta\) from less information, so (slightly?) larger prediction errors will result (larger variance)
    • \(X\) needs to be non-singular in every neighbourhood
  • some authors claim that local trends are so adaptive, that one can ignore spatial correlation of the residual
  • Using local linear regression with weights that decay with distance is called geographically weighted regression, GWR

Practical arguments for local prediction

  • The number of observations, \(n\) may become very large.
    • lidar data, 3D chemical, satellite sensors, geotechnical, seismic, …
  • Computing \(V^{-1}v\) is the expensive part; it is \(O(N^2)\) or \(O(N^3)\) as \(V\) is usually not of simple structure
  • there is a trade-off; for a global neighbourhood, the expensive part, factoring \(V\) needs only be done once, for a local neighbourhood for each unique neighbourhood (in practice: for each \(s_0\)).
  • selecting local neighbourhoods also costs time; naive selection \(O(n \log n)\) doesn’t scale well
  • gstat uses quadtrees/octtrees, inspired by http://donar.umiacs.umd.edu/quadtree/index.html (defunct)

Predicting block means

Instead of predicting \(Z(s_0)\) for a “point” location, one might be interested at predicting the average of \(Z(s)\) over a block, \(B_0\), i.e. \[Z(B_0) = \frac{1}{|B_0|}\int_{B_0} Z(u)du\]

  • This can (naively) be done by predicting \(Z\) over a large number of points \(s_0\) inside \(B_0\), and averaging
  • For the prediction error, of \(\hat{Z}(B_0)\), we then need the covariances between all point predictions
  • a more efficient way is to use block kriging, which does both at once

Reason why one wants block means

Examples

  • mining: we cannot mine point values
  • soil remediation: we cannot remediate points
  • RS: we can match satellite image pixels
  • disaster management: we cannot evacuate points
  • environment: legislation may be related to blocks
  • accuracy: block means can be estimated with smaller errors than points