Contents. Learning Outcomes 2012/2/26. Lecture 6: Area Pattern and Spatial Autocorrelation. Dr. Bo Wu

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1 01//6 LSGI 1: : GIS Spatial Applications Analysis Lecture 6: Lecture Area Pattern 1: Introduction and Spatial to GIS Autocorrelation Applications Contents Lecture 6: Area Pattern and Spatial Autocorrelation Dr. Bo Wu lsou@polyu.edu.hk Department of Land Surveying & Geo-Informatics The Hong Kong Polytechnic University Learning outcomes Characteristics of area oject Geometric properties of area ojects Area Skeleton Shape 1D Spatial autocorrelation D Spatial autocorrelation Joints count Moran s I and Geary s C 01//6 Learning Outcomes By the end of this lecture you should e ale to: Outline the types of area oject of interest Sho ho geometric properties of area ojects can e calculated Develop a simple measure for autocorrelation in one dimension Extend the measure for autocorrelation to to dimension using the joins count statistic Use alternatives to the joints count, hich can e applied to interval and ratio data such as Moran s I and Geary s C 01//6 Types of Area Oject Natural areas Self-defining Crisp and homogenous inside (e.g., Lake) Suject to interpretation Uncertain and suject oundary or content is fuzzy (e.g., soil coverage) Human imposed Common regions Provinces, states, census Involves data aout human eings Sampling of underlying social reality May e misleading Aritrary or ias (MAUP) Aggregations (ecological fallacies) Raster grid (Cell) Area ojects are identical Together cover the region of interest Polygonal Voronoi/Thiessen regions Forest urns over natural areas Imposed areas Natural ojects Imposed raster areas 01//6 Characteristics of Area Ojects in GIS Have oth geometric and topologic characteristics Isolated or overlapping Have holes or inside Planar enforced is required in GIS Fundamental assumptions of data models Area ojects all mesh together neatly and exhaust the study region Planar Enforced A polygon set is said to e planar enforced if every point in the set lies in exactly one polygon, or on the oundary eteen to or more polygons. Planar enforcement is used to uild ojects out of digitized lines (hence the phrase "uilding topology") It is a consistent and precise approach to the prolem of making meaningful ojects out of groups of lines 01//6 5 01//6 6 1

2 01//6 Trapezium Area Finding the area of a polygon from the coordinates of its vertices The skeleton of a polygon is a netork of lines inside a polygon constructed so that each point on the netork is equidistant from the nearest to edges in the polygon oundary. The process can e thought of as a process of peeling aay layers of the polygon. Transform eteen area and point representation Skeleton 01//6 7 Medial axis transformation 01//6 8 Shape Shape Parameters for descriing a shape: Perimeter P Area a Longest axis L 1 Second axis L The radius of the largest internal circle R 1 The radius of the smallest enclosing circle R Compactness P Compactness = a Most compact shape is a circle: π Another definition of compactness here a is the area of the circle having the same perimeter P Elongation ratio L 1 /L Form ratio a/l 1 01//6 9 01//6 10 Spatial Autocorrelation Spatial autocorrelation refers to the fact that spatial data from near locations are more likely to e similar than data from distant location. First la of geography (Toler 1970) everything is related to everything else, ut near things are more related than distant things. The orld is not random, so autocorrelation is extremely important to the discipline and to GIS analysis. The uiquity of spatial autocorrelation is a reason hy spatial is special. When you touch lack, you ecome lack, hen you touch red, you ecome red. 01//6 1 1D Spatial Autocorrelation &^&&^&^&^^^&^&&&^&^^^^&^&& Is this a random sequence? That is to say, could it have een generated y a random process? &&&&&&&&&&&&&^^^^^^^^^^^^^ Intuitively, e ould say this one is not random. Nor is, &^&^&^&^&^&^&^&^&^&^&^&^&^ Let s look at statistics of runs: & ^ && ^ & ^ & ^^^ & ^ &&& ^ & ^^^^ & ^ && 17 runs &&&&&&&&&&&&& ^^^^^^^^^^^^^ runs 01//6 1

3 01//6 Statistics of Runs If a sequence is randomly distriuted: Numer of runs is inomially distriuted, the expected numer of runs is given y Runs Analysis &^&&^&^&^^^&^&&&^&^^^^&^&& &&&&&&&&&&&&&^^^^^^^^^^^^^ 1 & (n 1 =1) 1 ^ (n =1) For the example ith 17 runs, e have its z-score: The expected standard deviation: For the example ith runs, a z-score given y Please calculate the z-score for the elo one and try to compare the three patterns! &^&^&^&^&^&^&^&^&^&^&^&^&^ 01//6 1 01//6 15 D Spatial Autocorrelation: The Joins Count Example: Joins Counts of Rook s Case The joins count is determined y counting the numer of occurrences in the map of each of the possile joins eteen neighoring areal units. E.g., the possile joins: BB, WW, and BW/WB. Different join patterns: Rook s case (<5,><5,6><5,8><5,>) Queen s case (four more <5,1><5,><5,9><5,7>) (A) (B) (C) 01// //6 17 Positive vs Negative Spatial Autocorrelation Positive Spatial Autocorrelation: cells are usually found next to similarly cells. Negative Spatial Autocorrelation: elements next to one another are usually different. Hoever, most of time, e have situations in eteen +SA and -SA Computing the Proailities If e have n Black cells and n = n n hite cells, the proaility of a lack cell is: p = n /n p = n /n Start ith first cell. The proaility it is lack is p and the proaility of hite is p The proaility of BB in to adjacent cells is: p * p or p Proaility of BW is p * p + p * p or p p 01// //6 0

4 01//6 Computing the Proailities Therefore, if there are L joins on a map, the expected numer of cells of each type: E(BB) = μ(bb) = p L E(WW) = μ(ww) = p L E(BW) = μ(bw) = p p L The expected standard deviation: σ ( BB) = σ ( WW ) = σ ( BW ) = p L + p K p ( L + K) p L + p K p ( L + K) p p L + p p K p p K = = L ( i 1) ( L + K) n i 1 i L Assumption: Each cell is assigned either Black or White Each pair of cells are defined as adjacent or nonadjacent in a consistent manner Hypotheses: Z-score: Statistic Test for Join Count (area pattern is random) (area pattern is not random) (area pattern is more dispersed than random) (area pattern is more clustered than random) 01//6 1 01//6 Example from 000 US Election Did the lue and red map really say something significant aout the locations here people voted y County? Example from 000 US Election 01//6 5 01//6 Example from 000 US Election The Limitations of the Joins Count The most ovious is that it can only e used on inary lack/hite, high/lo, on/off. Although the approach provides an indication of the strength of autocorrelation present in terms of z-scores, hich can e thought of in terms of proailities, it is NOT readily interpreted, particular if the results of different tests appear contradictory Spatial Autocorrelation is scale dependent Negative spatial autocorrelation is more sensitive to changes in scale 01//6 01//6 8

5 01//6 Moran s I One of the oldest indicators of spatial autocorrelation (Moran, 1950). Applied to zones or points ith continuous variales associated ith them. Compares the value of the variale at any one location ith the value at all other locations. n: numer of cases y i, y j : variale values at particular location i and j. W ij : eight applied to the comparison eteen location i and location j It varies eteen 1.0 and When autocorrelation is high, the coefficient is high. A high I value indicates positive autocorrelation. 01//6 9 Example of Moran s I Per Capita Income in Monroe County Moran s I = 0.66 Moran s I = //6 0 Geary s C Similar to Moran s I (Geary, 195). Interaction is not the cross-product of the deviations from the mean, ut the deviations in intensities of each oservation location ith one another. Revie Further readings Roger Bivand, 010, The Prolem of Spatial Autocorrelation: forty years on, ( Moran I and Geary C ( in Michael John De Smith, Michael F. Goodchild, Paul Longley, Geospatial Analysis - a comprehensive guide, rd edition, 006. Summarization of the main ideas presented in this lecture: Value typically range eteen 0 and If value of any one zone are spatially unrelated to any other zone, the expected value of C ill e 1 Values less than 1 (eteen 1 and ) indicate negative (positive) spatial autocorrelation Questions? 01//6 1 01//6 5