local measures of spatial autocorrelation briggs henan university 2010 1

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  • Slide 1
  • Local Measures of Spatial Autocorrelation Briggs Henan University 2010 1
  • Slide 2
  • Global Measures and Local Measures Briggs Henan University 2010 2 An equivalent local measure can be calculated for most global measures China Global Measures (last time) 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 (this time) 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
  • Slide 3
  • Local Indicators of Spatial Association (LISA) We will look at local versions of Morans I, Gearys C, and the Getis-Ord G statistic Morans I is most commonly used, and the local version is often called Anselins LISA, or just LISA 3 Briggs Henan University 2010 See: Luc Anselin 1995 Local Indicators of Spatial Association-LISA Geographical Analysis 27: 93-115
  • Slide 4
  • Local Indicators of Spatial Association (LISA) The statistic is calculated for each areal unit in the data For each polygon, the index is calculated based on neighboring polygons with which it shares a border 4 Briggs Henan University 2010
  • Slide 5
  • Local Indicators of Spatial Association (LISA) Since a measure is available for each polygon, these can be mapped to indicate how spatial autocorrelation varies over the study region Since each index has an associated test statistic, we can also map which of the polygons has a statistically significant relationship with its neighbors, and show type of relationship 5 Briggs Henan University 2010 Raw data LISA
  • Slide 6
  • Calculating Anselins LISA The local Moran statistic for areal unit i is: where z i is the original variable x i in standardized form or it can be in deviation form and w ij is the spatial weight The summation is across each row i of the spatial weights matrix. An example follows Briggs Henan University 2010 6
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  • Slide 8
  • Example using seven China provinces --caution: edge effects will strongly influences the results because we have a very small number of observations Briggs Henan University 2010 8
  • Slide 9
  • 9 1 5 4 3 6 7 2 Contiguity Matrix1234567 Code AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai Sum Neighbors Illiteracy Anhui 1011111056 5 4 3 2 14.49 Zhejiang 2101100147 4 3 1 9.36 Jiangxi 3110001036 2 1 6.49 Jiangsu 4110000137 2 1 8.05 Henan 5100001026 1 7.36 Hubei 6101010031 3 5 7.69 Shanghai 7010100022 4 3.97 Each row in the contiguity matrix describes the neighborhood for that location.
  • Slide 10
  • Briggs Henan University 2010 10 Contiguity Matrix1234567 Code AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai Sum Anhui 101111105 Zhejiang 210110014 Jiangxi 311000103 Jiangsu 411000013 Henan 510000102 Hubei 610101003 Shanghai 701010002 Row Standardized Spatial Weights Matrix Code AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai Sum Anhui 10.000.20 0.001 Zhejiang 20.250.000.25 0.00 0.251 Jiangxi 30.33 0.00 0.330.001 Jiangsu 40.33 0.00 0.331 Henan 50.500.00 0.500.001 Hubei 60.330.000.330.000.330.00 1 Shanghai 70.000.500.000.500.00 1 Contiguity Matrix and Row Standardized Spatial Weights Matrix 1/3
  • Slide 11
  • Calculating standardized (z) scores Briggs Henan University 2010 11 Deviations from Mean and z scores. XX-XmeanX-Mean2z Anhui14.49 6.29 39.55 2.101 Zhejiang9.36 1.16 1.34 0.387 Jiangxi6.49 (1.71) 2.93 (0.572) Jiangsu8.05 (0.15) 0.02 (0.051) Henan7.36 (0.84) 0.71 (0.281) Hubei7.69 (0.51) 0.26 (0.171) Shanghai3.97 (4.23) 17.90 (1.414) Mean and Standard Deviation Sum 57.41 0.00 62.71 Mean 57.41/ 7 = 8.20 Variance 62.71 / 7 = 8.96 SD 8.96 = 2.99
  • Slide 12
  • Row Standardized Spatial Weights Matrix Code AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai Anhui 10.000.20 0.00 Zhejiang 20.250.000.25 0.00 0.25 Jiangxi 30.33 0.00 0.330.00 Jiangsu 40.33 0.00 0.33 Henan 50.500.00 0.500.00 Hubei 60.330.000.330.000.330.00 Shanghai 70.000.500.000.500.00 Z-Scores for row Province and its potential neighbors AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai Zi Anhui 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Zhejiang 0.387 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Jiangxi (0.572) 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Jiangsu (0.051) 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Henan (0.281) 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Hubei (0.171) 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Shanghai (1.414) 2.101 0.387 (0.572) (0.051) (0.281) (0.171) (1.414) Spatial Weight Matrix multiplied by Z-Score Matrix (cell by cell multiplication) AnhuiZhejiangJiangxiJiangsuHenanHubeiShanghai SumWijZjLISA Lisa from Zi 0.000GeoDA Anhui 2.101 - 0.077 (0.114) (0.010) (0.056) (0.034) - (0.137) -0.289 -0.248 Zhejiang 0.387 0.525 - (0.143) (0.013) - - (0.353) 0.016 0.006 0.005 Jiangxi (0.572) 0.700 0.129 - - - (0.057) - 0.772 -0.442 -0.379 Jiangsu (0.051) 0.700 0.129 - - - - (0.471) 0.358 -0.018 -0.016 Henan (0.281) 1.050 - - - - (0.085) - 0.965 -0.271 -0.233 Hubei (0.171) 0.700 - (0.191) - (0.094) - - 0.416 -0.071 -0.061 Shanghai (1.414) - 0.194 - (0.025) - - - 0.168 -0.238 -0.204 Calculating LISA w ij zjzj w ij z j
  • Slide 13
  • Significance levels are calculated by simulations. They may differ each time software is run. I expected Anhui to be High-Low! (high illiteracy surrounded by low) High Low Low-High Morans I = -.01889 Results Raw Data ProvinceLiteracy %LISASignificance Anhui14.49 -0.250.12 Zhejiang9.36 0.010.46 Jiangxi6.49 -0.380.04 Jiangsu8.05 -0.020.32 Henan7.36 -0.230.14 Hubei7.69 -0.060.28 Shanghai3.97 -0.200.37
  • Slide 14
  • LISA for Illiteracy for all China Provinces 14 Briggs Henan University 2010 Illiteracy Rates LISA High Low Morans I = 0.2047
  • Slide 15
  • Moran Scatter Plot Briggs Henan University 2010 15 Low/High negative SA High/High positive SA Low/Low positive SA Scatter Diagram between X and Lag-X, the spatial lag of X formed by averaging all the values of X for the neighboring polygons Identifies which type of spatial autocorrelation exists. High/Low negative SA
  • Slide 16
  • Quadrants of Moran Scatterplot GISC 7361 Spatial Statistics 16 0 0 X WX Q 1 (values [+], nearby values [+]): H-H Q 3 (values [-], nearby values [-]): L-L Q 2 (values [-], nearby values [+]): L-H Q 4 (values [+], nearby values [-]): H-L Q2Q2 Q1Q1 Q4Q4 Q3Q3 Locations of positive spatial association (Im similar to my neighbors). Locations of negative spatial association (Im different from my neighbors). Low/High negative SA High/High positive SA Low/Low positive SA High/Low negative SA Each quadrant corresponds to one of the four different types of spatial association (SA)
  • Slide 17
  • Why is Morans I low for China provinces? For illiteracy =.2047 Are provinces really local Briggs Henan University 2010 17
  • Slide 18
  • Briggs Henan University 2010 18 LISA for Median Income, 2000 in D/FW Source: Eric Hajek, 2008 Morans I =.59
  • Slide 19
  • Briggs Henan University 2010 19 Examples of LISA for 7 Ohio counties: median income (p< 0.05) (p< 0.10) Source: Lee and Wong Ashtabula Geauga Lake Trumbull PortageSummit Cuyahoga Median Income Ashtabula has a statistically significant Negative spatial autocorrelation cos it is a poor county surrounded by rich ones (Geauga and Lake in particular)
  • Slide 20
  • Local Getis-Ord G Statistic Briggs Henan University 2010 20 Local Getis-Ord It is the proportion of all x values in the study area accounted for by the neighbors of location i G will be high where high values cluster G will be low where low values cluster Interpreted relative to expected value if randomly distributed. Local Morans I Global Getis-Ord G For comparison
  • Slide 21
  • Getis with 0.5 distance Getis with 1.0 distance LISA and Getis G with Different Distance Weights Getis with 2.0 distance Results for Getis G vary depending on distance band used. Data for crime in Columbus, Ohio from Anselin, 1995 LISA with 0,1 contiguity
  • Slide 22
  • Briggs Henan University 2010 22 LISA using Contiguity Weights Local Getis G* with Fixed Distance Bands at 2, 1, and 0.5 Running in ArcGIS
  • Slide 23
  • Briggs Henan University 2010 23 Can map the statistical significance level and use it as a measure of the strength of the spatial autocorrelation --note how the significance level is higher at the center of each cluster.
  • Slide 24
  • Briggs Henan University 2010 24 Bivariate LISA Morans I is the correlation between X and Lag-X--the same variable but in nearby areas Univariate Morans I Bivariate Morans I is a correlation between X and a different variable in nearby areas. Moran Scatter Plot for Crime v. Income Moran Cluster Map for Crime v. Income Moran Significance Map for Crime v. Income
  • Slide 25
  • Bivariate LISA and the Correlation Coefficient Correlation Coefficient is the relationship between two different variables in the same area Bivariate LISA is a correlation between two different variables in an area and in nearby areas. Briggs Henan University 2010 25 Scatter Diagram for relationship between income and crime Correlation coefficient r = 0.696 Bivariate Morans I: --less strong relationship --greater scatter --lower slope Morans I = -.45
  • Slide 26
  • Briggs Henan University 2010 26 Bivar

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