A Spatial Statistical Approach to Residential Differentiation (II): Exploratory Spatial Data Analysis Using a Local Spatial Separation Measure* Sang-Il Lee** Abstract The main purpose of the research is to illustrate the value of the spatial statistical approach to residential differentiation by providing a framework for exploratory spatial data analysis (ESDA) using a local spatial separation measure. ESDA aims, by utilizing a variety of statistical and cartographic visualization techniques, at seeking to detect patterns, to formulate hypotheses, and to assess statistical models for spatial data. The research is driven by a realization that ESDA based on local statistics has a great potential for substantive research. The main results are as follows. First, a local spatial separation measure is correspondingly derived from its global counterpart. Second, a set of significance testing methods based on both total and conditional randomization assumptions is provided for the local measure. Third, two mapping techniques, a spatial separation scatterplot map and a spatial separation anomaly map, are devised for ESDA utilizing the local measure and the related significance tests. Fourth, a case study of residential differentiation between the highly educated and the least educated in major Korean metropolitan cities shows that the proposed ESDA techniques are beneficial in identifying bivariate spatial clusters and spatial outliers. : exploratory spatial data analysis (ESDA), spatial separation measure, local statistics, residential segregation, spatial dependence 2005 KRF 2005 003 B00389 Assistant Professor, Department of Geography Education, Seoul National University), si_lee@snu.ac.kr 134
2007 spatial separation measure global local exploratory spatial data analysis ESDA ESDA exploratory data analysis EDA Tukey 1977 EDA detective work Good 1983 intermediate statistics EDA Lee 2005 276 EDA ESDA ESDA Lee 2005 275 ESDA Geographic Information Systems GIS local statistics GIS spatial data manipulation ESDA ESDA generic research platform GIS Openshaw 1990 Fischer and Nijkamp 1992 Goodchild et al. 1992 Bailey 1994 Fotheringham and Charlton 1994 Openshaw and Clarke 1996 Unwin 1996 Anselin 1998 Wise et al. 1999 ESDA spatial heterogeneity local turn Fotheringham 1997 2000 ESDA spatial association measures SAM SAM local Moran s I local Geary s c Getis-Ord G G SAM local indicators of spatial association LISA Anselin 1995 Getis and Ord 1996 LISA LISA Lee 2001b Anselin et al. 2002 LISA ESDA LISA 135
π spatial variation spatial cluster spatial outliers spatial regimes Anselin 1995 SAM ESDA ESDA ESDA 7 40 59 r X i columnproportions r i X p i p i rá rá Y contiguity proximity row-standardized 1 n-1 SSM= R R w ( z -z ) ( zá -zá ) 2 2n 2007 623 Geary spatial separation measure; SSM n-1 SSM= 2R R v R R v [( r -p ) -(r -p ) ][( rá -p ) -(rá -p ) ] ør π( r -p ) π ør π( rá -p ) π 1 v spatial proximity matrix w z zá z = zá = r -p ø π π π R ( r -p ) /n rá -p ø π π π R ( rá -p ) /n 3 additivity requirement SSM n(n -1) SSM = 2R R v R v [( r -p ) -(r -p ) ][( rá -p ) -(rá -p ) ] ør π ( r -p ) π ør π ( rá -p ) 4 136
quadratic form n-1 ( z )ˇ [X -(V +Vˇ ) ]zá SSM = 5 2 1ˇ V1 5 Lee 2008 v 0 -v y 0 0 X -(V +Vˇ )= -v yr v -v y -v «0 0 6 -v «0 v «y y X {v y ( v +R v ) y v «} diagonal matrix V {v y v y v «} i 0 Vˇ V row-standardized 2 n-1 SSM = R w ( z -z ) ( zá -zá ) 7 2n cross-product over-represented SSM 0 2007 0 y normality assumption randomization assumption Cliff and Ord 1981 superpopulation n random permutation n! Lee 2001a n! Lee 2008 n n! 137
SAM Cliff and Ord 1981 Sokal et al. 1998 Leung et al. 2003 SAM Lee, 2001b 2008 SAM Cliff and Ord 1981 total conditional Anselin 1995 Sokal et al. 1998 Lee 2001b 2008 n! n! extended Mantel test Lee 2004c 5 P P[ ] 2007 16 17 n-1 [X -(V +Vˇ ) ] P[ ] 8 2 1ˇ V1 Q z ( zá )ˇ 9 Lee, 2004c n E(SSM ) = (v -v ) r Á 10 1ˇ V1 v =R v 0 10 E(SSM ) =r Á X Y Pearson X Y 2007 20 3 Lee 2008 n-1 Var(SSM ) = 4n ( 2n(v[ ]-1)(1-r Á) ( 2nr Á) ) - + { n-2 n-1 } ª (v[ ]+3) nr Á+(v[ ]+1) nb Á r Á+2n v[ ] º 11 v[ ] =R v b Á =m Á /(m Á ) m Á =R (x - x )(y -y )/n m Á =R (x -x ) (y -y ) /n SSM SSM 138
generalized vector randomization test Lee 2001b 2008 Hubert 1984 1987 1 n E( ) = R p R q 12 Var( ) = 1 R ( p -p ) R ( q -q ) 13 n-1 SSM 5 6 n-1 p[ ] Y[v y -(v -v ) y v «]ˇ 14 2 1ˇ V1 q[ ] z zá-(z zá +zá z ) 15 1215 SSM n E(SSM ) = (v -v ) (z zá +r Á) 2 1ˇ V1 16 0 16 E(SSM ) =(z zá +r Á)/2 Lee 2008 Var(SSM ) 1 = [(n-1)(v[ ]-1)] (2n) (n-2) (b Á r Á-2z b Á -2zÁ b Á + ( ) (n-1) 2z zá ) nr Á+(z +z Á ) n { -z z Á } ª -(nr Á+z zá ) º 17 normal approximation Mal2000 Wong 2002 2003 Feitosa et al. 2007 2004 Brown and Chung 2006 Chung and Brown 2007 2007 622 623 139
2007 z- 2007 3 0 0 2007 625 3 z- 2007 626 0 0 2007 625 626 ESDA z- z- z- z- location quotient Brown and Chung 2006 Chung and Brown 2007 140
a b a/b A 040 0 200 0 2 0 1 0 2 0 5 0 1 B 240 0 300 0 8 0 6 0 3 2 0 0 3 C 000 0 200 0 0 0 0 0 2 0 0 0 2 D 080 0 100 0 8 0 2 0 1 2 0 0 1 E 040 0 200 0 2 0 1 0 2 0 5 0 1 400 1 000 1 0 1 0 * (3). Oden 1995 Rogerson 1999 3 2007 621 622 3 1 A~E 1 000 400 1 B D 0 8 2 0 B D 300 100 3 ESDA spatial regimes spatial anomalies Anselin 1996 Moran Moran scatterplot map 141
spatial separation scatterplot map SSSM clusters outliers SSSM spatial separation anomalies map SSAM SSAM 142
2007 2000 CD 40 59 4 4 S-Plus S- SSSM SSAMESDA SSAM 0 10 Anselin 1995 0 05 0 01 ESDA Tiefelsdorf, 1998 Sokal et al. 1998 Lee 2004 2 0 33 23 3 1 10 2 5 59 4 81 1 4 65 4 49 6 4 19 4 00 4 3 97 5 3 46 3 3 29 1 3 26 2 0 5 SSSM 0 8647 522 451 389 62 SSSM SSAM 106 71 62 1 2 2 4 4 3 1 2 1 2 1 3 2 2 outlier 2 2 0 6 11 07 2 3 88 2 3 37 9 2 49 1 2 37 2 2 09 143
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20077 7 36 4 2 SSSM 0 8759 SSAM 216 14 10 9 2 2 10 2 1 3 2 1 1 2 2 3 10 8 2 2 0 6 2 3 75 1 3 64 2 3 50 1 3 09 4 2 90 1 2 75 10 6 SSSM 0 8652 35 28 SSAM 1 1 4 2 2 3 2 3 1 2 1 4 145
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2 0 6 3 4 69 4 35 1 3 27 4 2 77 3 2 76 2 10 2 0 1 2 SSSM 114 SSAM 19 9 7 1 3 2 1 3 4 1 3 2 4 5 5 10km 2 0 7 2 4 67 1 2 53 2 47 2 29 4 2 13 2 08 2 00 2 0 SSSM SSAM 13 3 9 1 2 1 1 2 6 2 0 6 4 18 3 42 1 2 91 2 89 2 2 48 2 2 01 2 0 147
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SSSM SSAM 10 7 3 2 1 3 2 7 2 0 4 12 3 98 2 2 1 0 SSSM SSAM 7 3 2 2 149
2 SSSM SSAM 7 Geary Geary local Lee s L Lee 2001a 2001b 2008 Lee 2004a 2004b ESDA ESDA Boots and Tiefelsdorf 2000 150
ESDA GIScience 2007 42 4 616 631 2004 Anselin, L., 1995, Local indicators of spatial association: LISA, Geographical Analysis, 27(2), 93-115. Anselin, L., 1996, The Moran scatterplot as an ESDA tools to assess local instability in spatial association, in Fischer, M., Scholten, H., and Unwin, D. (eds.), Spatial Analytical Perspectives on GIS, Taylor & Francis, London, 111-125. Anselin, L., 1998, Exploratory spatial data analysis in a geocomputational environment, in Longley, P. A., Brooks, S. M., McDonell, R., and MacMillan, B. (eds.), Geocomputation: A Primer, John Wiley & Sons, Chichester, West Sussex, 77-94. Anselin, L., Syabri, I., and Smirnov, O., 2002, Visualizing multivariate spatial correlation with dynamically linked windows, in Anselin, L. and Rey, S. (eds.), New Tools for Spatial Data Analysis: Proceedings of the Specialist Meeting, Center for Spatially Integrated Social Science (CSISS), University of California, Santa Barbara. Bailey, T. C., 1994, A review of statistical spatial analysis in geographical information systems, in Fotheringham, A. S. and Rogerson, P. (eds.), Spatial Analysis and GIS, Taylor & Francis, London, 13-44. Boots, B. and Tiefelsdorf, M., 2000, Global and local spatial autocorrelation in bounded regular tessellations, Journal of Geographical Systems, 2(3), 319-348. Brown, L. A. and Chung, S.-Y., 2006, Spatial segregation, segregation indices and the geographical perspective, Population, Space and Place, 12(2), 125-143. Cliff, A. D. and Ord, J. K., 1981, Spatial Processes: Models & Applications, Pion Limited, London. Chung, S.-Y. and Brown, L. A., 2007, Racial/ethnic residential sorting in spatial context: Testing the exploratory frameworks, Urban Geography, 28(4), 312-339. Feitosa, F. F., Camara, G., Monteiro, A. M. V., Koschitzki, T., and Silva, M. P. S., 2007, Global and local spatial indices of urban segregation, International Journal of Geographical Information Science, 21(3), 299 323 Fischer, M. M. and Nijkamp, P., 1992, Geographic information systems and spatial analysis, Annals of Regional Science, 26(1), 3-17. Fotheringham, A. S., 1997, Trends in quantitative methods I: stressing the local, Progress in Human Geography, 21(1), 88-96. Fotheringham, A. S., 2000, Context-dependent spatial analysis: a role for GIS?, Journal of Geographical Systems, 2(1), 71-76. Fotheringham, A. S. and Charlton, M., 1994, GIS and exploratory spatial data analysis, an overview of some research issues, Geographical Systems, 1(4), 315-327. Getis, A. and Ord, J. K., 1996, Local spatial statistics: an overview, in Longley, P. and Batty, M. (eds.), Spatial Analysis: Modelling in a GIS Environment, GeoInformation International, Cambridge, 261-277. Good, I. J., 1983, The philosophy of exploratory data analysis, Philosophy of Science, 50(2), 283-295. Goodchild, M. F., Haining, R., Wise, S., et al., 1992, Integrating GIS and spatial data analysis: problems and possibilities, International Journal of Geographical Information Systems, 151
6(5), 407-423. Hubert, L. J., 1984, Statistical applications of linear assignment, Psychometrika, 49(4), 449-473. Hubert, L. J., 1987, Assignment Methods in Combinatorial Data Analysis, Marcel Dekker, New York. Lee, S.-I., 2001a, Developing a bivariate spatial association measure: An integration of Pearson s r and Moran s I, Journal of Geographical Systems, 3(4), 369-385. Lee, S.-I., 2001b, Spatial Association Measures for an ESDA-GIS Framework: Developments, Significance Tests, and Applications to Spatiotemporal Income Dynamics of U.S. Labor Market Areas, 1969-1999, Ph.D. Dissertation, Department of Geography, The Ohio State University. Lee, S.-I., 2004a, Exploratory spatial data analysis of convergence in the U.S. regional income distribution, 1969-1999, Journal of the Korean Urban Geographical Society, 7(1), 79-95. Lee, S.-I., 2004b, Spatial data analysis for the U.S. regional income convergence, 1969-1999: A critical appraisal of -convergence, Journal of the Korean Geographical Society, 39(2), 212-228. Lee, S.-I., 2004c, A generalized significance testing method for global measures of spatial association: An extension of the Mantel test, Environment and Planning A, 36(9), 1687-1703. Lee, S.-I., 2005, Between the quantitative and GIS revolutions: Towards an SDA-centered GIScience, Journal of Geography Education, 49, 268-284. Lee, S.-I., 2008, A generalized randomization approach to local measures of spatial association, Geographical Analysis, under revision. Leung, Y., Mei, C.-L., and Zhang, W.-X., 2003, Statistical test for local patterns of spatial association, Environment and Planning A, 35(4), 725-744. Maly, M. T., 2000, The neighborhood diversity index: a complementary measure of racial residential settlement, Journal of Urban Affairs, 22(1), 37-47. Oden, 1995, Adjusting Moran s I for population density, Statistics in Medicine, 14(1), 17-26. Openshaw, S., 1990, Spatial analysis and geographical information systems: a review of progress and possibilities, in Scholten, H. and Stillwell, J. (eds.), Geographical Information Systems for Urban and Regional Planning, Kluwer, Dordrecht, 153-163. Openshaw, S. and Clarke, G., 1996, Developing spatial analysis functions relevant to GIS environments, in Fischer, M., Scholten, H., and Unwin, D. (eds.), Spatial Analytical Perspectives on GIS, Taylor & Francis, London, 21-37. Rogerson, P. A., 1999, The detection of clusters using a spatial version of the chi-square goodness-of-fit statistic, Geographical Analysis, 31(1), 130-147. Sokal, R. R., Oden, N. L., and Thomson, B. A., 1998, Local spatial autocorrelation in a biological model, Geographical Analysis, 30(4), 331-354. Tiefelsdorf, M., 1998, Some practical applications of Moran s I s exact conditional distribution, Papers in Regional Science, 77(2), 101-129. Tukey, J. W., 1977, Exploratory Data Analysis, Addison- Wesley, Reading, MA. Unwin, D. J., 1996, GIS, spatial analysis and spatial statistics, Progress in Human Geography, 20(4), 540-551. Wise, S., Haining, R., and Signoretta, P., 1999, Scientific visualization and the exploratory analysis of area data, Environment and Planning A, 31(10), 1825-1838. Wong, D. W. S., 2002, Modeling local segregation: a spatial interaction approach, Geographical & Environmental Modelling, 6(1), 81-97. Wong, D. W. S., 2003, Spatial decomposition of segregation indices: A framework toward measuring segregation at multiple levels, Geographical Analysis, 35(3), 179-194. :, 151-748, 56-1, (: 152
si_lee@snu.ac.kr, : 02-880-9028, : 02-882-9873) Correspondence: Sang-Il Lee, Department of Geography Education, College of Education, Seoul National University, San 56-1, Silim-dong, Gwanak-gu, Seoul 151-748, Korea (e-mail: si_lee@snu.ac.kr, phone: +82-2-880-9028, fax: +82-2-882-9873) 153