Spatially Assessing Model Error Using Geographically Weighted Regression

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Spatially Assessing Model Error Using
Geographically Weighted Regression
Shawn Laffan Geography Dept ANU
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• Non-spatial methods are increasingly used to
  model and map continuous spatial properties
• Artificial Neural Networks, Decision Trees,
  Expert Systems...
• These can use more ancillary variables than
  explicitly spatial methods
• Usually assessed using non-spatial global error
  measures
• Summarise many data points
• Cannot easily identify where model is correct

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• Error residuals may be mapped
• But usually points
• Difficult to visually identify spatial clustering
• Large point symbols
• no multi-scale
• no quantification
• Can use spatial error analysis to detect clusters
  of similar prediction
• Use these areas with confidence
• Areas with unacceptable error indicate need for
  different variables or approach

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• To spatially assess model error a method should
• Locally calculate omission, commission total
  error in original data units in one assessment
• one dataset each
• Assess error for unsampled locations
• generate spatially continuous surfaces for easier
  interpretation
• Provide confidence information about the
  assessment
• uncertainty estimate

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• Possible approaches
• Mean, StdDev, Range for spatial window
• three attributes to interpret for each of
  omission, commission and total error
• mean will often not equal zero
• Co-variograms
• global assessment
• work only for sampled locations
• Local Spatial Autocorrelation
• Geographically Weighted Regression

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• Local Spatial Autocorrelation
• indices of spatial association
• easy to interpret
• multi-scale
• calculate residuals and assess spatial clustering
• some indices calculable for unsampled locations
• Getis-Ord Gi, Openshaws GAM

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• Local Spatial Autocorrelation
• Give difference from expected (global mean)
• mean will not normally be zero
• Must analyse omission commission separately
• partly cancel out
• leads to numeric and sample density problems
• confidence information

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• Geographically Weighted Regression
• multivariate spatial analysis in the presence of
  non-stationarity
• perform regression within a moving spatial window
• multi-scaled
• can directly assess residual error without prior
  calculation
• simultaneous omission, commission and total error
  assessment
• estimates for unsampled locations
• r2 parameter gives confidence information

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• The approach
• Ordinary Least Squares
• Y a bX
• calculated for circles of increasing radius
  across the entire dataset
• minimum 5 sample points
• no spatial weight decay with distance
• does not force an assumed distribution on the
  data
• optimal spatial scale when r2 is maximum

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• Interpreting regression parameters for error
• error is the square root of the area between the
  fitted and the optimal lines
• this is bounded by the min and max of the
  predicted distribution
• as b approaches 1 the intercept approaches /-
  infinity causing extremely large error values
• use the intersection of the fitted line with the
  optimal line (11, YX) to determine omission
  commission

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• The r2 parameter
• high r2 means reliable b parameters and therefore
  reliable error measures
• low values indicate low confidence caused by
  dispersed data values
• these areas cannot be used as b is meaningless

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• Example application
• feed-forward ANN to infer aluminium oxide
• used topographic and vegetation indices
• 1100 km2 area at Weipa, Far North Queensland,
  Australia
• 16000 drill cores
• 30.4 accurate within /- 1 original unit
• 48.7 accurate within /- 2 original units

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Subset of study area
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Total error 4, 7 10 cell radius
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Total
Omission
Commission
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Optimal spatial lag
Max r2
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Visualising error distribution with confidence
information
r2 red omission green commission blue
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• Limitations
• sample density distribution
• outliers
• data spatial
• cause low r2
• landscape does not operate in circles

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• Extended Utility
• can use the regression parameters to correct the
  ANN prediction
• similar to universal kriging but ANN allows for
  the inclusion of more ancillary variables
• have not taken into account r2 values

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Comparison with universal kriging
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• Conclusions
• GWR allows the spatial investigation of
  non-spatial model error
• calculates total, omission and commission error
  in one assessment, with confidence information
• identified locations of good and poor model
  prediction in a densely sampled dataset
• not immediately obvious without GWR
• currently exploratory
• significance tests would be useful