Review of Two Papers on Spatial Interpolation Methods

1
Review of Two Papers on Spatial Interpolation
Methods

• M. Savoie and M.J. Brodzik
• CVEN 6833
• December 19, 2006

2
Brief description of the domain of application

• We compare the methods and results of 2 papers,
  submitted to a spatial interpolation competition
  for precipitation data whose results were
  published in the Journal of Geographic
  Information and Decision Analysis, vol. 2, no. 2
• Atkinson, Peter M. and Chris D. Lloyd. 1998.
  Mapping Precipitation in Switzerland with
  Ordinary and Indicator Kriging. pp. 65-76.
• Rajagopalan, Balaji and Upmanu Lall. 1998.
  Locally Weighted Polynomial Estimation of Spatial
  Precipitation. pp. 44-51.

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Brief description of the domain of application

• Given measurements of a field like precipitation
  that are only available
• at irregularly spaced locations
• Why do people want to perform spatial
  interpolatation?
• to estimate values of a variable at unsampled
  locations
• to evaluate the spatial distribution of data
  values (e.g. for agriculture or natural resource
  management)
• to correlate with a related field (e.g. the
  contest was motivated by high correlation of
  precipitation with radioactive exposure from
  Chernobyl fallout)
• to describe the data field on a regularly-spaced
  grid for modellers to use
• to describe the data field at consistent
  locations in space, (e.g. satellite observations
  are not at fixed locations.)

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Brief description of the domain of application

• Spatial interpolation methods require assigning a
  series of weights to
• neighboring points to be used to compute an
  interpolated value.
• Methods range from simple to complex, and
  include
• nearest-neighbor
• equal-weights (drop-in-the-bucket)
• inverse-distance weighting
• "optimal" interpolation ("greater use of the
  information provided by spatial arrangement of
  neighbors"), including kriging and LOCFIT

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Questions/Hypothesis proposed by these papers

• Object of the contest was to estimate values of
  precipitation at locations
• from which measurements had not been taken, and
  compare the results to measurements that were
  held back from the original data, using a common
  set of required statistics
• minimum, maximum, mean, median and standard
  deviation of estimated values
• RMSE
• bias of the errors
• mean relative and absolute errors
• correlation between the errors
• correlation plot of the estimated vs. observed
  values
• map with isolines showing the levels of the
  estimated values

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Description of the Data Set

• 467 daily rainfall measurements (mm) made in
  Switzerland on 8-May-1986 by Environment
  Institute of the Joint Research Centre
• 100 random locations were selected and submitted
  to the contestants for their spatial
  interpolation analysis

• After each group finished their analysis, the
  entire data set was release for comparison.
  Here's how those data look

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Description of Methodology

• ordinary kriging (OK) vs. LOCFIT
• kriging (OK) make a variogram from data,
  relating correlation as a function of distance.
  Fit a curve to it. Use this curve to weight some
  number of neighbors to each interpolation point

LOCFIT like we did in class, locally fitted
polynomial, chose best fit number of neighbors
k32, and polynomial order p2, using minimum GCV
criterion
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Results and conclusion
Full Data
kriging (OK)
LOCFIT
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Results and conclusion
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Results and conclusion

• Based on the comparison statistics table and
  spatial patterns of the estimates, we conclude
  that both methods perform about the same on the
  limited contest data set.
• We expect that kriging would perform less well on
  a larger region, because variograms will only
  make sense when they describe correlations that
  hold true for the entire region of interest. We
  also expect that LOCFIT will perform better when
  the right combination of additional predictor
  variables (e.g. elevation) are included.
• Which leads into our project...