Elements of Spatial Regression

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Elements of Spatial Regression

• Patrick J. Sullivan
• Department of Natural Resources

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References

• Cressie (1973) Statistics for Spatial Data,
  Revised Edition, Wiley
• Kaluzny et al. (1998) S Spatial Stats, Springer
• Splus Statistics Software

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Birth Rates
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Elementary Linear Regression
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Elementary Linear Regression
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Wolfcamp Aquifer Data
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Mean Value Over Y
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Mean Value Over X
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Standard Linear Modelvs.Spatial Correlation
Model
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my.before.data lt- rnorm(20) my.after.data lt-
my.Lmatrix my.before.data
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Spatial Correlation Model
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So where does the problem lie?
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Independent Correlated
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Combined Linear and Correlative Model
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The Catch-22
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Possible Approaches

• If small-scale spatial correlation, then regress
  and assess remaining correlation
• REML (Restricted Maximum Likelihood)
• Through first differences
• Still a tricky business, so take care!
• Should try to assess priorities
• Trend or Variation
• Response to External Drivers or Self organization

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Universal Kriging Example

• For data over a continuous spatial metric
• Coal Ash Gomez and Hazen (1970)
• Pittsburgh coal seam
• Robena Mine, Greene County, PA

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Universal Kriging

• Exploratory Data Analysis
• Removal of Trend Via Median Polish
• Examination of Residual Variation
• Estimate Variogram Correlation Model
• Fit of Trend coal loc(x,y)xx2
• Prediction Using Trend and Variation

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Analyzing Lattice Data
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sids15,
county county name id county identification
number easting x coordinate northing relative
y coordinate sid number of sids deaths births
number of births nwbirths nonwhite
births group county groupings sid.ft
Freeman-Tukey transf. of of sids to of
births nwbirth.ft Freeman-Tukey transf. of
nonwhite births to of births
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Birth Rates
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Sids Rates
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1 2 3 4 5 6 7 8 2 1 3 4 7 8
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1 2 1 3 1 4 1 5 1 6 1 7 1 8 2 1 2 3 2 4 2 7 2 8
1 2 3 4 5 6 7 8 2 1 3 4 7 8
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Joint Probability Distributions

• Conditional probability
• Markov property
• Independence

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Distance-decay Correlation

• Cressie

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Application to SIDS Data

• Per birth SIDS Rate Nonwhite Births
• CAR Covariance Family
• Coefficients
• Value Std. Error t value
• (Intercept) 1.6456 0.2385 6.8990
• nwbirths.ft 0.0345 0.0066 5.2068
• rho 0.6454

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Conclusions

• Trade-off in Modeling Trend vs. Variation
• Identify Nature of Data and Objectives
• Scale
• Biophysical Understanding
• Use
• Visualize