Lecture 6 - Analysis of Pattern Measures at the...

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Unformatted text preview: Analysis of Pattern Measures at the Local/Regional Scale 9. Local Moran’s I 10. The Score Statistic 11. Tango’s C F 12. Cumulative χ 2 test 13. Maximum χ 2 test 14. Local Quadrat Test 15. Fuch’s & Kennett’s M Test(s) 16. Diggle & Rowlingson’s Maximum Likelihood Approach 17. Kernel Density Estimators/Tests 18. Besag & Newell’s Test 19. Kulldorff’s Likelihood Ratio Test 20. Rogerson’s Method(s) Also Getis G-statistic (which we will review later) Local Statistics for analyzing spatial patterns and clusters 9. Local Moran – used to determine if there is evidence of “local spatial autocorrelation” for a given variable (y: y i , i=1,…, n sample observations) around a specific sub-region, neighborhood, or locality. Local Moran’s I may be defined (for a given i-th sub-region) as n n I i = [n (y i – y ) / ∑ (y j – y ) 2 ] ∙ ∑ w ij (y j – y ) , j=1 j=1 where w ij are the typical elements of the spatial weights matrix W (nxn). Note that the sum of the local Moran values obtained for all sub-regions in the study region is equal to the “global Moran” coefficient multiplied by the sum of the spatial weights (w ij ). Anselin (1995) Data Requirements (a) Valued Point data for n points/locations; or (b) Raster/Grid cell or Polygon data i. Measured at interval or ratio scale ii. Count data or rate data (raw, smoothed, or adjusted) … for n cells/polygons; and (c) Specification of Spatial Weights/Connectivity Matrix Binary or Standardized Inverse Distance Weighting Shared Boundary (Rook’s, Queen’s case joins) Thiessen polygons, Delaunay Tessellation Gabriel graph k-th Nearest Neighbor Note: Large Sample size preferred (N>100) Output: Local Moran’s i for each polygon, Z-score, and p-value Global Moran’s I is an aggregate coefficient composed of the scaled sum of the Local Moran statistics. More formally, n n n I = ∑ ∑ w ij ∙ ∑ I i . i=1 j=1 i=1 Global Moran’s I a general test for global spatial autocorrelation (SA) Local Moran’s I i i=1,…, n tests for local SA …for a given data set and pre-specified W (nxn) spatial weights/connectivity matrix. As defined by Anselin (1995), the expected value of the local Moran coefficient is n E[I i ] = – w i / (n – 1), where w i = ∑ w ij , j=1 and the variance of I i under the “randomization” hypothesis may be expressed as V[I i ] = A + B, where A = [w i(2) (n – b 2 )]/(n – 1); and B = [2w i(kh) (2b 2 – n)/(n – 1)(n – 2)] – w i 2 /(n – 1) 2 , where n n n n n w i(2) = ∑ w ij 2 , w i(kh) = ∑ ∑ w ik ∙ w ih , b 2 = n ∙∑(y i – y) 4 /[∑(y i – y) 2 ] 2 j=1 k=1 h=1 i=1 i=1 j≠i k≠i h≠i Caveat. While the test of significance is typically carried out under the assumption that the test statistic has a normal distribution for the null hypothesis of no (zero) spatial autocorrelation (SA)....
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This note was uploaded on 02/15/2012 for the course GEO 6938 taught by Professor Staff during the Summer '08 term at University of Florida.

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Lecture 6 - Analysis of Pattern Measures at the...

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