Geography 625

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

2. Outline • Introduction • Geometric properties of areas • Spatial autocorrelation: joins count approach • Spatial autocorrelation: Moran’s I • Spatial autocorrelation: Geary’s C • Spatial autocorrelation: weight matrices • Local indicators of spatial association (LISA) • Spatial regression • Spatial expansion method • Geographical weighted regression

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

4. 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 • may bear little relationship to underlying patterns • Arbitrary and modifiable • Danger of ecological fallacies (aggregated format)

5. 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. 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. 2. Geometric Properties of Areas - Area y (x3, y3) (x4, y4) (x2, y2) Assume x1 = xn+1 (x1, y1) x

8. 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. 2. Geometric Properties of Areas - Skeleton Center derived by skeleton analysis center Arithmetic center

10. 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. 2. Geometric Properties of Areas - Shape Perimeter: P Area: a Longest axis: L1 Second axis: L2 The radius of the largest internal circle: R1 The radius of the smallest enclosing circle: R2

12. 2. Geometric Properties of Areas - Shape Compactness ratio a is the area of the polygon a2 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. 2. Geometric Properties of Areas - Shape Other measurements Elongation ratio: L1/L2 Form ratio:

14. 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. 3. Spatial autocorrelation - Joins count approach Possible joins: JBB: the number of joins of BB JWW: the number of joins of WW JBW: 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. 3. Spatial autocorrelation - Joins count approach

17. 3. Spatial autocorrelation - Joins count approach

18. 3. Spatial autocorrelation - Joins count approach

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

20. 3. Spatial autocorrelation - Joins count approach Independent Random Process (IRP) The expected standard deviations are as follows

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

22. 3. Spatial autocorrelation - Joins count approach Independent Random Process (IRP)

23. 3. Spatial autocorrelation - Joins count approach Independent Random Process (IRP)

24. 3. Spatial autocorrelation - Joins count approach A large negativeZ-score on JBW indicates positive autocorrelation since it indicates that there are fewer BW joins than expected. A large positiveZ-score on JBW is indicative of negative autocorrelation.

25. 3. Spatial autocorrelation - Joins count approach

26. 3. Spatial autocorrelation - Joins count approach (example) B: Bush W: Gore State-level results for the 2000 U.S. presidential election

27. 3. Spatial autocorrelation - Joins count approach (example) Adjacency (joins) matrix: if two states share a common boundary, they are adjacent.

28. 3. Spatial autocorrelation - Joins count approach (example) Gore 48,242,921 pw = 0.50115 Bush 48,021,500 pB = 0.49885

29. 4. Spatial autocorrelation - Moran’s I Limitations of Joins Count Statistics • It can only be applied on binary data • 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 • The equations for the expected values of counts are fairly formidable.

30. 4. Spatial autocorrelation - Moran’s I • If zone i an zone j are adjacent • 0 otherwise wij =

31. 4. Spatial autocorrelation - Moran’s I 1 2 2 0 2 0 3 4 1 2 3 4 1 2 3 4 A =

32. 4. Spatial autocorrelation - Moran’s I 1 2 2 0 0 2 3 4 1 2 3 4 1 2 3 4 A =

33. 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. 5. Spatial autocorrelation - Geary’s C Proposed by Geary’s contiguity ratio C • If zone i an zone j are adjacent • 0 otherwise wij =

35. 5. Spatial autocorrelation - Geary’s C 1 2 2 0 0 2 3 4 1 2 3 4 1 2 3 4 A =

36. 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. 6. Spatial autocorrelation - Other Weighting Matrices 1. Using distance Where dij < D and z < 0 Where dij > D

38. 6. Spatial autocorrelation - Other Weighting Matrices 2. Using the length of shared boundary Where li is the length of the boundary of zone i lij is the length of boundary shared by area i and j

39. 6. Spatial autocorrelation - Other Weighting Matrices 3. Using both distance and the length of shared boundary

40. 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. 7. Spatial autocorrelation - Local Indicators Local Gi Used to detect possible nonstationarity in data, when clusters of similar values are found in specific subregions of the area studied.

42. 7. Spatial autocorrelation - Local Indicators 1 2 2 0 0 2 3 4 • If zone i an zone j are adjacent • 0 otherwise wij =

43. 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. 7. Spatial autocorrelation - Local Indicators 1 2 2 0 0 2 3 4 1 2 3 4 1 2 3 4 =

45. 7. Spatial autocorrelation - Local Indicators Local Geary’s C 1 2 2 0 0 2 3 4

46. 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. 8. Spatial regression Models - Simultaneous autocorrelation model (SAR) and

48. 8. Spatial regression Models - Simultaneous autocorrelation model (SAR) Software for SAR model ArcView 3.2 + S-Plus R programming language

49. 9. Spatial Expansion Models Proposed by Casetti (1972) OLS model Expansion model

50. 10. Geographic Weighted Regression OLS model GWR model The coefficients b vary with respect to the location i