Local Measures of Spatial Autocorrelation Briggs Henan University 2010

Global Measures and Local Measures 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 China An equivalent local measure can be calculated for most global measures Briggs Henan University 2010

Local Indicators of Spatial Association (LISA) We will look at local versions of Moran’s I, Geary’s C, and the Getis-Ord G statistic Moran’s I is most commonly used, and the local version is often called Anselin’s LISA, or just LISA See: Luc Anselin 1995 Local Indicators of Spatial Association-LISA Geographical Analysis 27: 93-115 Briggs Henan University 2010

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 Briggs Henan University 2010

Local Indicators of Spatial Association (LISA) Raw data 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 LISA Briggs Henan University 2010

Calculating Anselin’s LISA The local Moran statistic for areal unit i is: where zi is the original variable xi in “standardized form” or it can be in “deviation form” and wij is the spatial weight The summation is across each row i of the spatial weights matrix. An example follows Briggs Henan University 2010

Briggs Henan University 2010

Briggs Henan University 2010 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

Briggs Henan University 2010 Contiguity Matrix 1 2 3 4 5 6 7 Code Anhui Zhejiang Jiangxi Jiangsu Henan Hubei Shanghai Sum Neighbors Illiteracy 6 5 4 3 2 14.49 7 4 3 1 9.36 6 2 1 6.49 7 2 1 8.05 6 1 7.36 1 3 5 7.69 2 4 3.97 1 5 4 3 6 7 2 Each row in the contiguity matrix describes the neighborhood for that location. Briggs Henan University 2010

Contiguity Matrix and Row Standardized Spatial Weights Matrix 1 2 3 4 5 6 7 Code Anhui Zhejiang Jiangxi Jiangsu Henan Hubei Shanghai Sum Row Standardized Spatial Weights Matrix 0.00 0.20 0.25 0.33 0.50 1/3 Briggs Henan University 2010

Calculating standardized (z) scores Deviations from Mean and z scores. X X-Xmean X-Mean2 z Anhui 14.49 6.29 39.55 2.101 Zhejiang 9.36 1.16 1.34 0.387 Jiangxi 6.49 (1.71) 2.93 (0.572) Jiangsu 8.05 (0.15) 0.02 (0.051) Henan 7.36 (0.84) 0.71 (0.281) Hubei 7.69 (0.51) 0.26 (0.171) Shanghai 3.97 (4.23) 17.90 (1.414) Mean and Standard Deviation Sum 57.41 0.00 62.71 Mean / 7 = 8.20 Variance / 7 = 8.96 SD √ 8.96 = 2.99 Briggs Henan University 2010

Calculating LISA wij zj wijzj Anhui Zhejiang Jiangxi Jiangsu Henan Row Standardized Spatial Weights Matrix Code Anhui Zhejiang Jiangxi Jiangsu Henan Hubei Shanghai 1 0.00 0.20 2 0.25 3 0.33 4 5 0.50 6 7 wij Z-Scores for row Province and its potential neighbors Anhui Zhejiang Jiangxi Jiangsu Henan Hubei Shanghai Zi 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) SumWijZj LISA Lisa from 0.000 GeoDA - 0.077 (0.114) (0.010) (0.056) (0.034) (0.137) -0.289 -0.248 0.525 (0.143) (0.013) (0.353) 0.016 0.006 0.005 0.700 0.129 (0.057) 0.772 -0.442 -0.379 (0.471) 0.358 -0.018 -0.016 1.050 (0.085) 0.965 -0.271 -0.233 (0.191) (0.094) 0.416 -0.071 -0.061 0.194 (0.025) 0.168 -0.238 -0.204 zj wijzj

Results Raw Data Moran’s I = -.01889 I expected Anhui to be High-Low! (high illiteracy surrounded by low) Low Significance levels are calculated by simulations. They may differ each time software is run. High Province Literacy % LISA Significance Anhui 14.49 -0.25 0.12 Zhejiang 9.36 0.01 0.46 Jiangxi 6.49 -0.38 0.04 Jiangsu 8.05 -0.02 0.32 Henan 7.36 -0.23 0.14 Hubei 7.69 -0.06 0.28 Shanghai 3.97 -0.20 0.37 Low-High

LISA for Illiteracy for all China Provinces Low High Illiteracy Rates LISA Moran’s I = 0.2047 Briggs Henan University 2010

Briggs Henan University 2010 Moran Scatter Plot 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. Low/High negative SA High/High positive SA Low/Low positive SA High/Low negative SA Briggs Henan University 2010

Quadrants of Moran Scatterplot Each quadrant corresponds to one of the four different types of spatial association (SA) Low/High negative SA High/High positive SA Locations of positive spatial association (“I’m similar to my neighbors”). WX Q2 Q1 Q1 (values [+], nearby values [+]): H-H Q3 (values [-], nearby values [-]): L-L Locations of negative spatial association (“I’m different from my neighbors”). Q3 Q4 X Low/Low positive SA High/Low negative SA Q2 (values [-], nearby values [+]): L-H Q4 (values [+], nearby values [-]): H-L GISC 7361 Spatial Statistics

Why is Moran’s I low for China provinces? For illiteracy = .2047 Are provinces really “local” Briggs Henan University 2010

Briggs Henan University 2010 LISA for Median Income, 2000 in D/FW Source: Eric Hajek, 2008 Moran’s I = .59 Briggs Henan University 2010

Examples of LISA for 7 Ohio counties: median income Lake Ashtabula Geauga Cuyahoga Trumbull Summit Portage Ashtabula has a statistically significant Negative spatial autocorrelation ‘cos it is a poor county surrounded by rich ones (Geauga and Lake in particular) Source: Lee and Wong Median Income (p< 0.10) (p< 0.05) Briggs Henan University 2010

Local Getis-Ord G Statistic It is the proportion of all x values in the study area accounted for by the neighbors of location i Local Moran’s I Global Getis-Ord G For comparison G will be high where high values cluster G will be low where low values cluster Interpreted relative to expected value if randomly distributed. Briggs Henan University 2010

LISA and Getis G with Different Distance Weights Getis with 0.5 distance Data for crime in Columbus, Ohio from Anselin, 1995 LISA with 0,1 contiguity Results for Getis G vary depending on distance band used. Getis with 1.0 distance Getis with 2.0 distance

Briggs Henan University 2010 Running in ArcGIS Local Getis G* with Fixed Distance Bands at 2, 1, and 0.5 LISA using Contiguity Weights Briggs Henan University 2010

Briggs Henan University 2010 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. Briggs Henan University 2010

Briggs Henan University 2010 Bivariate LISA Moran Scatter Plot for Crime v. Income Moran’s I is the correlation between X and Lag-X--the same variable but in nearby areas Univariate Moran’s I Bivariate Moran’s I is a correlation between X and a different variable in nearby areas. Moran Significance Map for Crime v. Income Moran Cluster Map for Crime v. Income Briggs Henan University 2010

Bivariate LISA and the Correlation Coefficient Scatter Diagram for relationship between income and crime Correlation coefficient r = 0.696 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. Bivariate Moran’s I: --less strong relationship --greater scatter --lower slope Moran’s I = -.45 Briggs Henan University 2010

Bivariate LISA: a local version of the correlation coefficient Can view Bivariate LISA as a “local” version of the correlation coefficient It shows how the nature & strength of the association between two variables varies over the study region For example, how home values are associated with crime in surrounding areas Classic Inner City: Low value/ High crime Gentrification? High value/ High crime Unique: Low value/ Low crime Classic suburb: high value/ low crime Briggs Henan University 2010

What have we learned today? Local Indicators of Spatial Autocorrelation Anselin’s LISA Local Getis Ord G Spatial autocorrelation can be calculated for each areal unit Spatial autocorrelation can vary across the region in strength and in type Next time (Friday) Using GeoDA software to explore spatial autocorrelation Next week Spatial regression and modeling Briggs Henan University 2010

Briggs Henan University 2010 References Getis, A. and Ord, J.K. (1992) The analysis of spatial association by use of distance statistics Geographical Analysis, 24(3) 189-206 Ord, J.K. and Getis A. (1995) Local Spatial Autocorrelation Statistics: distributional issues and an application Geographical Analysis, 27(4) 286-306 Anselin, L. (1995) Local Indicators of Spatial Association-LISA Geographical Analysis 27: 93-115 Briggs Henan University 2010

Briggs Henan University 2010