Applying%20Geostatistical%20Methods%20to%20Lattice%20Data:%20An%20Initial%20Examination%20of%20U.S.%20Presidential%20Elections%20in%20Iowa

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Applying Geostatistical Methods to Lattice Data
An Initial Examination of U.S. Presidential
Elections in Iowa

• A.C. Thomas
• Statistics 225
• December 14, 2004

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Sources/Guides

• Main source Hierarchical Models, chapters 2
  and 3 (geostatistical and spatial data)
• Data sources http//www.sos.state.ia.us/elections
  /results/ (1996/2000)
• http//www.cnn.com/ (2004)
• Special thanks Brad Carlin (UMN), Andy Gelman
  (Columbia), Paul Edlefsen (Harvard)
• GeoR P.J. Ribeiro and P.J. Diggle

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Motivation

• In this course, we have learned about three
  different methods of examining spatial data
  (depending on relevant conditions) with some
  interchangeabilities
• Often, we may not have the tools to examine data
  sets using one method (i.e. the shortcomings of R
  in manipulating lattice data)
• In this case, we will compare and contrast the
  effectiveness of a geostatistical method used on
  lattice data to a lattice method through self
  cross-validation

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Interrelationship

• Geostats and kriging using variograms and
  distance relationships to predict quantities
  across distances
• Lattices using neighbour relationships to
  predict quantities across distances
• Direct similarities some weighting schemes
  across distances directly resemble covariograms

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Why election data?

• Why not?
• Spatial organization is well understood and
  constant in time (county borders have not changed
  across data sets) and built into R (maps library)
• While specific challengers change over time,
  parties are relatively constant, as are other
  control variables
• Ramifications are germane to the functioning of
  society (and the insatiable appetite of news
  junkies and policy wonks)

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Questions

• For this data set, does a geostatistical
  approximation produce a result comparable in
  error to a lattice model?
• If so, can we use fitted information from one
  election to predict the complete results of the
  next one? (And how much are we off?)

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Chosen model Iowa
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Why Iowa?

• 99 counties which have roughly equal area,
  removing a possible nuisance (and are
  rectilinear, so easier to draw)
• Swing state, with a rough vote balance over time
• Not too big, not too small in either population
  or size

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Simplification No third parties

• For now, considering only the votes for Democrat
  and Republican candidates in presidential
  elections from 1996-2004
• Not so bad in 2000/2004, when independent vote
  was about 3 of total
• Worse in 1996 (Perots successful campaign drew a
  lot), up to 10 of total votes

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Iowa in 1996 (Dole, Clinton)
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Iowa in 2000 (Bush, Gore)
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Iowa in 2004 (Bush, Kerry)
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Initial impressions

• There seems to be a tendency to vote more
  Republican the further west we look
• (Observation, courtesy Matt Anthony as we go
  east, we hit Illinois, a Democratic core.)
• What is the population distribution by county
  over time?

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Iowas total voters, 1996
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Iowas total voters, 2000
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Iowas total voters, 2004
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Quick-and-dirty non-spatial analysis

• Question how does population size correlate with
  the Democratic vote?
• Correlation between blue vote and total vote
• 1996 r 0.18
• 2000 r 0.30
• 2004 r 0.29.
• So population would appear to be an important
  covariate.

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Geostatistical analysis

• Locations centroids of each county (obtained
  through centroid.polygon function in maps library
  of R)
• Data Republican percentage of vote (arbitrarily
  chosen, not necessarily personal political
  affiliation)

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Initial data plots Unaltered
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Initial fitting

• Semivariogram appears to increase without bound,
  suggesting nonstationarity
• Plan use Universal Kriging with this
  semivariogram
• Problem Trend appears to be power law, with
  power greater than 2 (impossible to fit with
  conventional definitions
• Possible solutions a) remove trend from data. b)
  dont care.

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Plan A Remove trend from data

• What it does lets us remove known spatial
  dependence, look at other trends
• Initial look
• major discrepancies.

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Plan B Dont care.

• The goodness of fit only tails off at the end
• Preliminary results show the other option to be
  extremely inaccurate due to noise levels in
  residual data

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Second trend removed, data centered
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Exploratory Kriging
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Meaningful Kriging

• Since we want to test the predictive power of
  this method, we should test it on our current
  data through cross-validation
• Key remove one point, use semivariogram with
  remaining points to interpolate the value at each
  centroid
• Then, return trend to data and compare with
  original values
• Use universal kriging with second-degree trend

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1996 Redux Predicted Values

• In total, Dole receives 9,726 more votes than
  predicted.
• Absolute error 43,526
• Total 2-party votes 1,112,902

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Fitting variograms between models

• For all, power model was appropriate choice g
  t2 s2 tj
• 1996 s2 9.24e-4, j1.98, t20.031
• 2000 s2 9.93e-4, j2.00, t20
• 2004 s2 1.16e-3, j2.00, t20.025
• All roughly identical, even with different total
  averages

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2000 Predicted

• Prediction Bush gets 26,000 more votes
• Absolute error 181,880
• Total Bush/Gore votes 1,272,890

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2004 Prediction

• Prediction Bush gets 32,094 more votes
• Absolute difference 74,458
• Total votes 1,479,702

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Naïve Neighbour

• For a baseline comparison, take the simplest
  (stupidest) lattice cross-validation test ask
  your neighbour, trivial SAR weights
• Predicted value at a square is simply the mean of
  border-sharing neighbours (data is Republican
  percentage of vote)

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NN 1996

• Dole 10,819 more predicted
• Total deviation 40,923

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NN 2000

• Bush gets 28,535 extra in prediction
• Total deviation 59,670

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NN 2004

• Bush gets 37,175 more
• Total deviation 76,926

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Cross-validation summary
Geostat error NN error Geostat total error NN total error Voting pop.
1996 9,726 10,819 43,526 40,923 1,112,902
2000 26,000 28,535 61,485 59,670 1,272,890
2004 32,094 37,175 74,458 76,926 1,479,702
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Conclusions

• Data is definitely not stationary, even after
  removing trends
• Good kriging is about as effective as naïve
  neighbour, both without covariates
• Prediction with these tools at this simple level
  is not yet accurate enough
• Each method overpredicts the Republican vote
• Fitting information for each year is very close

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Future Developments and Unanswered Questions
New!

• Ive since introduced universal co-kriging with
  population, past voting behavior and
  second-degree spatial dependences using the gstat
  package.
• Needed data from the last 4 elections,
  conveniently packaged. Other prediction using
  spatial methods.