20  Spatial correlation in RF


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20.1 Machine learning for spatial prediction

Advantages

  • Machine learning (ML) handles today’s requirements well

    • Large dataset sizes (n observations, n covariates)

    • Lowering computational demands

    • Automatic model selection and building of model structure

    • Non-linear models, relaxed model assumptions (non-Gaussian)

    • Classification tasks

    • Uncertainty for point locations

Disadvantages

  • Spatial auto-correlation is only integrated in ad-hoc manner

  • Standard errors for spatial averages only via workaround (e.g. to report quantities per municipality or arable land parcel)

  • Solved problems within classical geostatistics reoccur for ML

Overview of “fixes” for random forest

Random forest is widespread for spatial mapping. Many more or less ad-hoc implementations have been proposed.

  • Kriging of residuals
  • Spatial coordinate axis as predictors
  • Relative distances
  • Neighboring points as predictors

20.2 Apply geostatistics on residuals

Often termed “regression kriging”, even if combined with machine learning.


Two step approach:

  1. Fit non-spatial model, compute its residuals.

  2. Fit ordinary kriging to residuals.

Predictions are a sum of model outputs from (1) and (2).

Non-spatial model can also be a linear regression.

(see e.g. analysis of Wolfcamp data in Geostatistics section).


Difficulty:

  • non-spatial method often ignores spatial correlation in model fit.
  • quantification of prediction uncertainty.
  • some challenges with kriging are re-introduced.

20.3 Spatial coordinates as covariates

Add x- and y-coordinate axis as additional covariates.

Allows random forest to partition the study area into smaller sections and fit local models.


Model trend not only North-South and East-West, rotate coordinate axis by \(\alpha\)

(in figure: 30° and 60°)

20.4 Relative spatial distances as covariates

An additional set of covariates is defined based on the relative distance to each observation:

\[ X_G = \{d_{p_1}, d_{p_1}, ..., d_{p_n}\} \]

where \(d_{p_i}\) is the euclidean distance (or any other more complex proximity distance) to the observed location \(p_i\) and \(n\) is the total number of training observations.

For each \(n\) observations one covariate is added (R packages GSIF, spatialRF).

Example covariate layer for one observation by euclidean distance (left) or travel time distance (right).

Hengl et al., 2018. PeerJ.

20.5 Spatial neighbors as covariates

Include observed values from \(n\) nearest locations and distances to these locations.

For each \(n\) nearest neighbor 2 colums are added as covariates:

  • value at \(n\)th nearest observation
  • distance corresponding to the location contributing the observed value

\(n\) is a tuning parameter.


R package RFSI

20.6 Spatial random forest – summary


  • Currently no satisfactory solution.

  • Proposed methods are merely ad-hoc.

  • I suggest to keep it simple: Use roated coordiante axis.