1. University of Wisconsin-Milwaukee Geographic Information Science Geography 625 Intermediate Geographic Information Science Instructor: Changshan Wu Department of Geography The University of Wisconsin-Milwaukee Fall 2006 Week 10_11: Area objects and spatial autocorrelation

2. University of Wisconsin-Milwaukee Geographic Information Science Outline 1.Introduction 2.Geometric properties of areas 3.Spatial autocorrelation: joins count approach 4.Spatial autocorrelation: Moran’s I 5.Spatial autocorrelation: Geary’s C 6.Spatial autocorrelation: weight matrices 7.Local indicators of spatial association (LISA) 8.Spatial regression 9.Spatial expansion method 10.Geographical weighted regression

3. University of Wisconsin-Milwaukee Geographic Information Science 1. Introduction 1. Natural areas: self-defining, their boundaries are defined by the phenomenon itself (e.g. lake, land use) - Types of area object Lake map

4. University of Wisconsin-Milwaukee Geographic Information Science 1. Introduction - Types of area object 2. Imposed areas: area objects are imposed by human beings, such as countries, states, counties etc. Boundaries are defined independently of any phenomenon, and attribute values are enumerated by surveys or censuses. Potential Problems 1.may bear little relationship to underlying patterns 2.Arbitrary and modifiable 3.Danger of ecological fallacies (aggregated format)

5. University of Wisconsin-Milwaukee Geographic Information Science 1. Introduction - Types of area object 3. Raster: space is divided into small regular grid cells. In a raster, the area objects are identical and together cover the region of interest. Each cell can be considered an area object. Raster data are always used to represent continuous phenomenon. Squares Hexagons

6. University of Wisconsin-Milwaukee Geographic Information Science 1. Introduction - Types of area object Planar enforced: area objects mesh together neatly and exhaust the study region, so that there is no holes, and every location is inside just a single area. Not planar enforced: the areas do not fill or exhaust the space, the entities are isolated from one another, or perhaps overlapped

7. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Area x y (x 1, y 1 ) (x 2, y 2 ) (x 3, y 3 ) (x 4, y 4 ) Assume x 1 = x n+1

8. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Skeleton The skeleton of a polygon is a network of lines inside a polygon constructed so that each point on the network is equidistant from the nearest two edges in the polygon boundary.

9. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Skeleton center Arithmetic center Center derived by skeleton analysis

10. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Shape Shape: a set of relationships of relative position between points on their perimeters. In ecology, the shapes of patches of a specified habitat are thought to have significant effects on what happens and around them. In urban studies, urban shapes change from traditional monocentric to polycentric sprawl

11. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Shape Perimeter: P Area: a Longest axis: L 1 Second axis: L 2 The radius of the largest internal circle: R 1 The radius of the smallest enclosing circle: R 2

12. University of Wisconsin-Milwaukee Geographic Information Science Compactness ratio 2. Geometric Properties of Areas - Shape a is the area of the polygon a 2 is the area of the circle having the same perimeter (P) as the object What is the compactness ratio for a circle? What is the compactness ratio for a square?

13. University of Wisconsin-Milwaukee Geographic Information Science 2. Geometric Properties of Areas - Shape Other measurements Elongation ratio: L 1 /L 2 Form ratio:

14. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach Developed by Cliff and Ord (1973) in their book: Spatial Autocorrelation The joins count statistic is applied to a map of areal units where each unit is classified as either black (B) or white (W). The joins count is determined by counting the number of occurrences in the map of each of the possible joins (e.g. BB, WW, BW) between neighboring areal units.

15. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach Possible joins: J BB : the number of joins of BB J WW : the number of joins of WW J BW : the number of joins of BW or WB Neighbor definition Rook’s case: four neighbors (North-South-West-East) Queen’s case: eight neighbors (including diagonal neighbors)

16. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach

17. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach

18. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach

19. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach Statistical tests for spatial correlation Independent Random Process (IRP) Where k is the total number of joins on the map p B is the probability of an area being coded B p W is the probability of an area being coded W Mean

20. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach Independent Random Process (IRP) The expected standard deviations are as follows

21. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach k i is the number of joins to the ith area m = 0.5 [(4×2×1) + (16×3×2)+(16×4×3)] corners edges center = 148 Independent Random Process (IRP)

22. University of Wisconsin-Milwaukee Geographic Information Science Independent Random Process (IRP) 3. Spatial autocorrelation - Joins count approach

23. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach Independent Random Process (IRP)

24. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach A large negative Z-score on J BW indicates positive autocorrelation since it indicates that there are fewer BW joins than expected. A large positive Z-score on J BW is indicative of negative autocorrelation.

25. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach

26. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach (example) B: Bush W: Gore State-level results for the 2000 U.S. presidential election

27. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach (example) Adjacency (joins) matrix: if two states share a common boundary, they are adjacent.

28. University of Wisconsin-Milwaukee Geographic Information Science 3. Spatial autocorrelation - Joins count approach (example) Bush 48,021,500 p B = 0.49885 Gore 48,242,921 p w = 0.50115

29. University of Wisconsin-Milwaukee Geographic Information Science 4. Spatial autocorrelation - Moran’s I Limitations of Joins Count Statistics 1.It can only be applied on binary data 2.Although the approach provides an indication of the strength of autocorrelation present in terms of Z-scores, it is not readily interpreted, particularly if the results of different tests appear contradictory 3.The equations for the expected values of counts are fairly formidable.

30. University of Wisconsin-Milwaukee Geographic Information Science 4. Spatial autocorrelation - Moran’s I w ij = 1If zone i an zone j are adjacent 0 otherwise

31. University of Wisconsin-Milwaukee Geographic Information Science 4. Spatial autocorrelation - Moran’s I 20 20 1 2 3 4 A = 12341234 1 2 3 4

32. University of Wisconsin-Milwaukee Geographic Information Science 4. Spatial autocorrelation - Moran’s I 20 02 1 2 3 4 A = 12341234 1 2 3 4

33. University of Wisconsin-Milwaukee Geographic Information Science 4. Spatial autocorrelation - Moran’s I For Moran’s I, a positive value indicates a positive autocorrelation, and a negative value indicates a negative autocorrelation. Moran’s I is not strictly in the range of -1 to +1.

34. University of Wisconsin-Milwaukee Geographic Information Science 5. Spatial autocorrelation - Geary’s C w ij = 1If zone i an zone j are adjacent 0 otherwise Proposed by Geary’s contiguity ratio C

35. University of Wisconsin-Milwaukee Geographic Information Science 5. Spatial autocorrelation - Geary’s C 20 02 1 2 3 4 A = 12341234 1 2 3 4

36. University of Wisconsin-Milwaukee Geographic Information Science 5. Spatial autocorrelation - Geary’s C The value generally varies between 0 and 2. The theoretical value of C is 1 under independent random process. values less than 1 indicate positive spatial autocorrelation while values greater than 1 indicate negative autocorrelation.

37. University of Wisconsin-Milwaukee Geographic Information Science 6. Spatial autocorrelation - Other Weighting Matrices 1. Using distance Where d ij < D and z < 0 Where d ij > D

38. University of Wisconsin-Milwaukee Geographic Information Science 6. Spatial autocorrelation - Other Weighting Matrices 2. Using the length of shared boundary Where l i is the length of the boundary of zone i l ij is the length of boundary shared by area i and j

39. University of Wisconsin-Milwaukee Geographic Information Science 6. Spatial autocorrelation - Other Weighting Matrices 3. Using both distance and the length of shared boundary

40. University of Wisconsin-Milwaukee Geographic Information Science 7. Spatial autocorrelation - Local Indicators Global statistics tell us whether or not an overall configuration is autocorrelated, but not where the unusual interactions are. Local indicators of spatial association (LISA) were proposed in Getis and Ord (1992) and Anselin (1995). These are disaggregate measures of autocorrelation that describe the extent to which particular areal units are similar to, or different from, their neighbors.

41. University of Wisconsin-Milwaukee Geographic Information Science 7. Spatial autocorrelation - Local Indicators Local G i Used to detect possible nonstationarity in data, when clusters of similar values are found in specific subregions of the area studied.

42. University of Wisconsin-Milwaukee Geographic Information Science 20 02 1 2 3 4 7. Spatial autocorrelation - Local Indicators w ij = 1If zone i an zone j are adjacent 0 otherwise

43. University of Wisconsin-Milwaukee Geographic Information Science 7. Spatial autocorrelation - Local Indicators Local Moran’s I Where W matrix is a row-standardized (i.e. scaled so that each row sums to 1)

44. University of Wisconsin-Milwaukee Geographic Information Science 7. Spatial autocorrelation - Local Indicators = 20 02 1 2 3 4 12341234 1 2 3 4

45. University of Wisconsin-Milwaukee Geographic Information Science 7. Spatial autocorrelation - Local Indicators Local Geary’s C 20 02 1 2 3 4

46. University of Wisconsin-Milwaukee Geographic Information Science 8. Spatial regression Models Where U is a zero-mean vector of errors with variance- covariance matrix C If C = I, this is the ordinary least square (OLS) model

47. University of Wisconsin-Milwaukee Geographic Information Science 8. Spatial regression Models - Simultaneous autocorrelation model (SAR) and

48. University of Wisconsin-Milwaukee Geographic Information Science 8. Spatial regression Models - Simultaneous autocorrelation model (SAR) Software for SAR model ArcView 3.2 + S-Plus R programming language

49. University of Wisconsin-Milwaukee Geographic Information Science 9. Spatial Expansion Models Proposed by Casetti (1972) OLS model Expansion model

50. University of Wisconsin-Milwaukee Geographic Information Science 10. Geographic Weighted Regression OLS model GWR model The coefficients b vary with respect to the location i

51. University of Wisconsin-Milwaukee Geographic Information Science 10. Geographic Weighted Regression Software R program GWR package (free from Fotheringham)