Decomposing the Pearson’s Correlation Coefficient Based on the Eigenvector Spatial Filtering Approach

doi:10.22776/kgs.2019.54.5.545

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2019 Vol.54, Issue 5 Preview Page Next Page

Research Article

Decomposing the Pearson’s Correlation Coefficient Based on the Eigenvector Spatial Filtering Approach 고유벡터공간필터링 접근법에 기반한 피어슨 상관계수의 요소분해: Sang-Il Lee
이상일

31 October 2019. pp. 545-560

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Abstract

본 논문의 주된 연구 목적은 고유벡터공간필터링 접근법에 기반한 피어슨 상관계수 요소분해 기법을 정련화하고, 이 기법이 공간적 자기상관이 전제된 상태에서의 이변량 상관관계 분석에 어떠한 공헌을 할 수 있을 지를 실험 연구를 통해 검토하는 것이다. 피어슨 상관계수 요소분해 기법은 공간적 패턴 요소분해, 하위 상관계수의 산출, 결정계수의 요소분해의 과정을 거쳐 이루어지며, 최종적으로 피어슨 상관계수를 네 가지 상관관계 요소, 즉 ‘잔차-잔차 상관관계 요소(EE)’, ‘공통-공통 상관관계 요소(CC)’, ‘특수-잔차 상관관계 요소(UE)’, ‘잔 차-특수 상관관계 요소(EU)’로 분해한다. 피어슨 상관계수 요소분해 기법의 유용성을 검토하기 위해 동일한 피 어슨 상관계수 값을 갖지만 서로 다른 수준의 이변량 공간적 자기상관을 보이는 가상의 8개 패턴쌍에 적용하였다. 실험 연구를 통해 밝혀진 주요 내용을 정리하면 다음과 같다. 첫째, 공간적 패턴 요소분해 결과, 공통 패턴 요 소와 특수 패턴 요소의 존재/부존재의 양상이 매우 다양하게 나타난다. 둘째, 공간적 패턴 요소의 존재/부존재의 양상과 개별 변수의 일변량 공간적 자기상관의 정도에 따라, 하위 상관계수의 상대적 크기, 그리고 공통 결정계 수와 특수 결정계수의 상대적 크기가 다양한 방식으로 나타난다. 셋째, 전체적인 이변량 공간적 자기상관의 수준뿐만 아니라 일변량 공간적 자기상관의 조합 양상에 따라 상관관계 요소분해의 결과는 달라진다. 본 연구는 공간데이터분석의 연구 관행에 새로운 제안을 하고자 하는데, 이변량 혹은 다변량 공간통계분석의 경우, 피어슨 상관계수와 그것의 유의성 검정 결과뿐만 아니라 피어슨 상관계수 요소분해 결과도 함께 병기한다면 공간적 자 기상관이 상관계수의 팽창/위축에 어떠한 방식과 강도로 작동하는지에 대한 새로운 통찰력을 제공할 수 있을 것으로 기대된다.

키워드

피어슨 상관계수

이변량 공간적 자기상관

고유벡터공간필터링

상관계수 요소분해

The main objective of this paper is to propose a refined version of the Pearson’s correlation coefficient decomposition technique and to examine how much contribution the technique can make to our understanding about what happens to the bivariate correlation when spatial autocorrelation is present. The technique employs sequential steps which are the spatial pattern decomposition, the sub-correlation coefficients calculation, and the determination coefficients decomposition, and it finally divides the Pearson’s correlation coefficient into four correlation components, the residual-residual correlation, the common-common correlation, the unique-residual correlation, and the residual-unique correlation components. The applicability and practicality of the technique is assessed on a hypothetical data set composed of 8 pairs which are identical in terms of the Pearson’s correlation coefficient but are different in terms of the level of bivariate spatial autocorrelation. Main findings from the experimental study are as follows. First, individual variables involved in the 8 pairs are diverse in terms of presence/absence of particular spatial pattern components. Second, the relative size and proportion of the sub-correlation coefficients and the sub-determination coefficients turn out to be substantially influenced by the presence/absence of particular spatial pattern components and the relative strength of spatial autocorrelation of two variables. Third, the correlation decomposition results are significantly subject to the relative strength of univariate spatial autocorrelation of each variable as well as the overall level of bivariate spatial autocorrelation. In conclusion, this paper proposes a new research practice that encourages researchers conducting a bivariate or multivariate spatial statistical analysis to report some of the results from the Pearson’s correlation coefficient decomposition analysis in addition to Pearson’s correlation coefficients and their associated p-values, which may lead to a new insight into our understanding about how spatial autocorrelation is involved in the process of inflating/deflating correlation coefficients.

Keywords

Pearson’s correlation coefficient

bivariate spatial autocorrelation

eigenvector spatial filtering

correlation coefficients decomposition

References

이상일, 2007, “거주지 분화에 대한 공간통계학적 접근 (I): 공간 분리성 측도의 개발,” 대한지리학회지, 42(4), 616-631.
이상일, 2008, “거주지 분화에 대한 공간통계학적 접근 (II): 국지적 공간 분리성 측도를 이용한 탐색적 공간데이터 분석,” 대한지리학회지, 43(1), 134-153.
이상일·조대헌·이민파, 2017, “공간적 자기상관 통계량 의 고유벡터 간 비교 연구: 공간근접성행렬의 효 과와 공간적 회귀분석에의 함의를 중심으로,” 대 한지리학회지, 52(5), 645-660.
이상일·조대헌·이민파, 2018, “피어슨 상관계수의 공간 화: 세 관련 기법 간의 비교 실험 연구,” 대한지리 학회지, 53(5), 761-776.
Bivand, R., 1980, A Monte Carlo study of correlation coefficient estimation with spatially correlate observations, Quaestiones Geographical, 6, 5-10.
Chun, Y. and Griffith, D. A., 2013, Spatial Statistics & Geostatistics, SAGE, Los Angeles.
Chun, Y., Griffith, D. A., Lee, M., and Sinha, P., 2016, Eigenvector selection with stepwise regression techniques to construct eigenvector spatial filters, Journal of Geographical Systems, 18(1), 67-85.
Clifford, P. and Richardson, S., 1985, Testing the association between two spatial processes. Statistics and Decisions, 2 (Supplementary issue), 155-160.
Clifford, P., Richardson, S., and Hémon, D., 1989. Assessing the significance of the correlation between two spatial processes, Biometrics, 45(1), 123-134.
Dutilleul, P., 1993, Modifying the t test for assessing the correlation between two spatial processes, Biometrics, 49(1), 305-314.
Getis, A. and Griffith, D. A., 2002, Comparative spatial filtering in regression analysis, Geographical Analysis, 34(2), 130-140.
Griffith, D. A., 1980, Towards a theory of spatial statistics, Geographical Analysis, 12(4), 325-339.
Griffith, D. A., 1993, Which spatial statistics techniques should be converted to GIS functions? in Fischer,
M. and Nijkamp, P. New Directions in Spatial Econometrics, Springer, Berlin, 101-114.
Griffith, D. A., 1996, Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data, The Canadian Geographer, 40(4), 351-367.
Griffith, D. A., 2000, A linear regression solution to the spatial autocorrelation problem, Journal of Geographical Systems, 2(2), 141-156.
Griffith, D. A., 2003, Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization, Springer, Berlin.
Griffith, D. A., 2010, Spatial filtering, in Fischer, M. M. and Getis, A. (eds.), Handbook of Applied Spatial Analysis: Software Tools, Methods and Applications, Springer, Berlin, 301-318.
Griffith, D. A., 2017, Spatial filtering, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 3, 2nd edition, Springer, Berlin, 2018-2031.
Griffith, D. A. and Chun, Y., 2014, Spatial autocorrelation and spatial filtering, in Fischer, M. M. and Nijkamp, P. (eds.), Handbook of Regional Science, Springer, Berlin, 1477-1507.
Griffith, D. A., Chun, Y., and Li, B., 2019, Spatial Regression Analysis Using Eigenvector Spatial Filtering, Cambridge, MA: Academic Press.
Griffith, D. A. and Paelinck, J. H. P., 2011, Non-standard Spatial Statistics and Spatial Econometrics, Springer, Berlin.
Haining, R. P., 1980, Spatial autocorrelation problems, in Herbert, D. T. and Johnston, R. J. (eds.), Geography and the Urban Environment, Wiley, New York, 1-44.
Haining, R. P., 1991, Bivariate correlation with spatial data, Geographical Analysis, 23(3), 210-227.
Lee, S.-I., 2001, 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., 2004, 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., 2009, A generalized randomization approach to local measures of spatial association, Geographical Analysis, 41(2), 221-248.
Lee, S.-I., 2017, Correlation and spatial autocorrelation, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 1, 2nd edition, Springer, New York, 360-368.
Reich, R. M., Czaplewski, R. L., and Bechtold, W. A., 1994, Spatial cross-correlation of undisturbed, natural shortleaf pine stands in northern Georgia, Environmental and Ecological Statistics, 1(3), 201-217.
Richardson, S. and Hémon, D., 1981, On the variance of the sample correlation between two independent lattice processes, Journal of Applied Probability, 18(4), 943-948.
Thayn, J. B., 2017, Eigenvector spatial filtering and spatial autocorrelation, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 1, 2nd edition, Springer, New York, 511-522.
Tiefelsdorf, M. and Griffith, D. A., 2007, Semiparametric filtering of spatial autocorrelation: The eigenvector approach, Environment and Planning A, 39(5), 1193 -1221.
Wartenberg, D., 1985, Multivariate spatial correlation: a method for exploratory geographical analysis. Geographical Analysis, 17(4), 263-283.

Information

Publisher :The Korean Geographical Society
Publisher(Ko) :대한지리학회
Journal Title :Journal of the Korean Geographical Society
Journal Title(Ko) :대한지리학회지
Volume : 54
No :5
Pages :545-560
DOI :https://doi.org/10.22776/kgs.2019.54.5.545

[1] 이상일, 2007, “거주지 분화에 대한 공간통계학적 접근 (I): 공간 분리성 측도의 개발,” 대한지리학회지, 42(4), 616-631.

[2] 이상일, 2008, “거주지 분화에 대한 공간통계학적 접근 (II): 국지적 공간 분리성 측도를 이용한 탐색적 공간데이터 분석,” 대한지리학회지, 43(1), 134-153.

[3] 이상일·조대헌·이민파, 2017, “공간적 자기상관 통계량 의 고유벡터 간 비교 연구: 공간근접성행렬의 효 과와 공간적 회귀분석에의 함의를 중심으로,” 대 한지리학회지, 52(5), 645-660.

[4] 이상일·조대헌·이민파, 2018, “피어슨 상관계수의 공간 화: 세 관련 기법 간의 비교 실험 연구,” 대한지리 학회지, 53(5), 761-776.

[5] Bivand, R., 1980, A Monte Carlo study of correlation coefficient estimation with spatially correlate observations, Quaestiones Geographical, 6, 5-10.

[6] Chun, Y. and Griffith, D. A., 2013, Spatial Statistics & Geostatistics, SAGE, Los Angeles.

[7] Chun, Y., Griffith, D. A., Lee, M., and Sinha, P., 2016, Eigenvector selection with stepwise regression techniques to construct eigenvector spatial filters, Journal of Geographical Systems, 18(1), 67-85.

[8] Clifford, P. and Richardson, S., 1985, Testing the association between two spatial processes. Statistics and Decisions, 2 (Supplementary issue), 155-160.

[9] Clifford, P., Richardson, S., and Hémon, D., 1989. Assessing the significance of the correlation between two spatial processes, Biometrics, 45(1), 123-134.

[10] Dutilleul, P., 1993, Modifying the t test for assessing the correlation between two spatial processes, Biometrics, 49(1), 305-314.

[11] Getis, A. and Griffith, D. A., 2002, Comparative spatial filtering in regression analysis, Geographical Analysis, 34(2), 130-140.

[12] Griffith, D. A., 1980, Towards a theory of spatial statistics, Geographical Analysis, 12(4), 325-339.

[13] Griffith, D. A., 1993, Which spatial statistics techniques should be converted to GIS functions? in Fischer,

[14] M. and Nijkamp, P. New Directions in Spatial Econometrics, Springer, Berlin, 101-114.

[15] Griffith, D. A., 1996, Spatial autocorrelation and eigenfunctions of the geographic weights matrix accompanying geo-referenced data, The Canadian Geographer, 40(4), 351-367.

[16] Griffith, D. A., 2000, A linear regression solution to the spatial autocorrelation problem, Journal of Geographical Systems, 2(2), 141-156.

[17] Griffith, D. A., 2003, Spatial Autocorrelation and Spatial Filtering: Gaining Understanding Through Theory and Scientific Visualization, Springer, Berlin.

[18] Griffith, D. A., 2010, Spatial filtering, in Fischer, M. M. and Getis, A. (eds.), Handbook of Applied Spatial Analysis: Software Tools, Methods and Applications, Springer, Berlin, 301-318.

[19] Griffith, D. A., 2017, Spatial filtering, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 3, 2nd edition, Springer, Berlin, 2018-2031.

[20] Griffith, D. A. and Chun, Y., 2014, Spatial autocorrelation and spatial filtering, in Fischer, M. M. and Nijkamp, P. (eds.), Handbook of Regional Science, Springer, Berlin, 1477-1507.

[21] Griffith, D. A., Chun, Y., and Li, B., 2019, Spatial Regression Analysis Using Eigenvector Spatial Filtering, Cambridge, MA: Academic Press.

[22] Griffith, D. A. and Paelinck, J. H. P., 2011, Non-standard Spatial Statistics and Spatial Econometrics, Springer, Berlin.

[23] Haining, R. P., 1980, Spatial autocorrelation problems, in Herbert, D. T. and Johnston, R. J. (eds.), Geography and the Urban Environment, Wiley, New York, 1-44.

[24] Haining, R. P., 1991, Bivariate correlation with spatial data, Geographical Analysis, 23(3), 210-227.

[25] Lee, S.-I., 2001, Developing a bivariate spatial association measure: an integration of Pearson’s r and Moran’s I, Journal of Geographical Systems, 3(4), 369-385.

[26] Lee, S.-I., 2004, 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.

[27] Lee, S.-I., 2009, A generalized randomization approach to local measures of spatial association, Geographical Analysis, 41(2), 221-248.

[28] Lee, S.-I., 2017, Correlation and spatial autocorrelation, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 1, 2nd edition, Springer, New York, 360-368.

[29] Reich, R. M., Czaplewski, R. L., and Bechtold, W. A., 1994, Spatial cross-correlation of undisturbed, natural shortleaf pine stands in northern Georgia, Environmental and Ecological Statistics, 1(3), 201-217.

[30] Richardson, S. and Hémon, D., 1981, On the variance of the sample correlation between two independent lattice processes, Journal of Applied Probability, 18(4), 943-948.

[31] Thayn, J. B., 2017, Eigenvector spatial filtering and spatial autocorrelation, in Shekhar, S., Xiona, H., and Zhou, X. (eds.), Encyclopedia of GIS, Volume 1, 2nd edition, Springer, New York, 511-522.

[32] Tiefelsdorf, M. and Griffith, D. A., 2007, Semiparametric filtering of spatial autocorrelation: The eigenvector approach, Environment and Planning A, 39(5), 1193 -1221.

[33] Wartenberg, D., 1985, Multivariate spatial correlation: a method for exploratory geographical analysis. Geographical Analysis, 17(4), 263-283.

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