Global_Measures_SA - Global Measures of Spatial...

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Global Measures of Spatial Autocorrelation Briggs Henan University 2010 1 China
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Last Time The concept of spatial autocorrelation. “Near things are more similar than distant things” The use of the weights matrix W ij to measure “nearness” The difficulty of measuring “nearness” This was a surprise! This Time Measures of Spatial Autocorrelation Join Count Statistic Moran’s I Geary’s C Getis-Ord G statistic Briggs Henan University 2010 2
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Global Measures and Local Measures Briggs Henan University 2010 3 An equivalent local measure can be calculated for most global measures China Global Measures A single value which applies to the entire data set The same pattern or process occurs over the entire geographic area An average for the entire area Local Measures A value calculated for each observation unit Different patterns or processes may occur in different parts of the region A unique number for each location
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Briggs Henan University 2010 4 Join (or Joins or Joint) Count Statistic Polygons only binary (1,0) data only Polygon has or does not have a characteristic For example, a candidate won or lost an election Based on examining polygons which share a border Do they have the same characteristic or not? Border same on each side Border not the same on each side Requires a contiguity matrix for polygons
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Different numbers of BW, BB and WW joins Briggs Henan University 2010 5 Join (or Joint or Joins) Count Statistic Uses binary (1,0) data Shown here as B/W (black/white) Measures the number of borders (“joins”) of each type (1,1), (0,0), (1,0 or 0,1) relative to total number of borders For 6 x 6 matrix, border totals are: 60 for Rook Case 110 for Queen Cas e Small number of BW joins (6 only for rook) Large proportion of BB and WW joins Large number of BW joins Small number of BB and WW joins
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Briggs Henan University 2010 6 Join Count: Test Statistic Test Statistic given by: Z= Observed - Expected SD of Expected Expected given by: Standard Deviation of Expected (standard error) given by: Where: k is the total number of joins (neighbors) p B is the expected proportion Black, if random p W is the expected proportion White m is calculated from k according to: Note: the formulae given here are for free (normality) sampling. Those for non-free (randomization) sampling are substantially more complex. See Wong and Lee 1 st ed. p. 151 compared to p. 155. Se next slide for explanation. Expected = random pattern generated by tossing a coin in each cell.
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A Note on Sampling Assumptions: applies to most tests for spatial autocorrelation Test results depend on the assumption made regarding the type of sampling: Free (or normality) sampling Analogous to sampling with replacement After a polygon is selected for a sample, it is returned to the population set The same polygon can occur more than one time in a sample Non-free (or randomization) sampling Analogous to sampling without replacement
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