적 자기상관은 어디에나 존재합니다. 효과(spatial spillovers)와 관련이 있습니다. 공 간적 자기상관은 또한 무엇인가가 잘못되었다는 (Eigenvector Spatial Filtering) 방법론의 기반 을 이룹니다.
Matheron(geostatistics), Cliff와 Ord(공간적 자기상관), Besag(베이지언 지도 분석)의 연 구, 미국 지리정보분석센터(NCGIA), 캘리포 니아주립 산타바바라대학교의 공간통합사회과 학센터(CSISS), Luc Anselin의 SpaceStat 보 급, 그리고 미국 환경보호국의 GEOEAS 배포 수추정(parameter estimates)에 편향(bias)을 발 생시킬 수 있고, 분산 추정값과 그에 따른 추정량 (estimator)의 통계적 효율(efficiency)에 영향 을 끼치고, 점근 수렴(asymptotic convergence) 을 변화시켜 이로 인하여 일치성(consistency)에 영향을 끼칠 수 있다는 지각으로 이어집니다. 공 간적 자기상관은 표본조사 네트워크(sampling network)의 디자인을 복잡하게 만들고, 표본 분
This preceding discussion highlights that, at least in part, spatial autocorrelation means map pattern. The concept literally means self (auto) relationships (correlation) due to relative position in a geographic landscape (spatial). Expanding upon this self-correlation notion, spatial autocorrelation can take on an information content meaning: as spatial autocorrelation latent in data increases, information redundancy increases, and new information content decreases with the acquisition of additional coterminous locations.
Spatial autocorrelation also can refer to spatial spillovers, such as a river flooding its banks, or the price of a house being impacted upon by the price of its nearby houses. Spatial autocorrelation also can mean something is wrong. It can underscore gerrymandering in a surface partitioning. It can indicate that a variable is missing from a regression equation, or that the functional form of a regression equation is wrong. In other words, spatial autocorrelation has many faces, all of which link back to map pattern. This specific linkage constitutes much of the foundation of eigenvector spatial filtering methodology.
▶ Chun: We see scientists accounting for spatial autocorrelation in their research more frequently today than in years past. Why do you think this is happening?
▶▶ Griffith: The success of time series analysis set a precedent. Work by Matheron (geostatistics), by Cliff and Ord (spatial autoregression) and by Besag (Bayesian map analysis), the US NCGIA, the UC/Santa Barbara CSISS, and Luc Anselin’s promotion of SpaceStat and the US EPA’s distribution of GEOEAS were major contributors to raising the profile of spatial autocorrelation. Statisticians and econometricians originally argued that spatial autocorrelation was irrelevant, and tended to treat it as a nuisance. But the need to properly analyze cancer clusters in space, and the need to predict data values for all locations from data collected for a relatively few sampled locations in a region inspired many scientists to acknowledge spatial autocorrelation. This acknowledgement resulted in an awareness that spatial autocorrelation can introduce bias into some parameter estimates, impacts upon variance estimates and hence the statistical efficiency of estimators, and can alter asymptotic convergence and hence impact upon consistency. Spatial autocorrelation can complicate the design of sampling networks, and distort Type I and Type II errors in sampling distributions. Accounting for spatial autocorrelation allows scientists analyzing geospatial data to draw sound statistical inferences. Nevertheless, as several reviews of the literature to date show, many analyses of such data dismiss or ignore spatial effects. My 1988 paper co-authored with Luc Anselin argues against this latter viewpoint, demonstrating that spatial
포에서 제1종 오류(Type I error)와 제2종 오류 Planning A와 Applied Statistics와 같은 저널 들이 이러한 알고리즘들을 출판하기 시작하였습
제목-effects really do matter.
▶ Chun: I understand that your academic career overlapped with much of the development of spatial autocorrelation theory and methodology. How did research about spatial autocorrelation begin and then develop?
▶▶ Griffith: When I was a doctoral student at the University of Toronto during 1972-1978, few scholars were studying spatial autocorrelation. The literature already contained the classic papers by Moran and by Geary, as well as papers by Whittle and by Meade concerning spatial autocorrelation. Cliff and Ord were beginning to publish their pioneering work (their seminal paper appeared in 1969, and the first edition of their book appeared in 1973). These works quickly began to eclipse the single spatial statistics topic of the time, namely point pattern analysis. Most of the geostatistics literature was being published in French, whereas most of the spatial autoregression literature was being published in English; little interaction was taking place between these two bodies of literature because of this language difference.
Besag published his classic paper in 1974.
One of the major drawbacks then was an absence of software to support spatial statistical analyses. But in the late 1960s and early 1970s, little software existed for any type of statistical analysis (first version dates include 1961 for BMDP, 1966 for SAS, 1968 for SPSS, and 1972 for Minitab). This computer-software-poor environment meant that researchers had to write their own computer code, much of it in Fortran, to do data analyses. Libraries such as IMSL and NAG furnished the modules upon which computer code was built. This situation was very similar to the R project. Fortran was analogous to R, and the subroutine libraries were analogous to R packages. Journals such as E&PA and Applied Statistics began publishing these algorithms.
The study of spatial autocorrelation was scorned by many. When my 1978 dissertation defense was announced by the University of Toronto, The Toronto Star, a local newspaper, ridiculed its title, The Impact of Configuration and Spatial Autocorrelation on the Investigation and Interpretation of Geographical Models, especially questioning whether or not tax payers’ money should be wasted on such academic endeavors. Matheron had a similar experience, with his work not being very well accepted in the statistical community for a period of time. Cliff and Ord published in geography and regional science journals (although in 1975 Ord was successful in published a short paper in JASA). Meade, an
지리학 모델의 조사와 해석에 대한 상대적 배열 Paelinck와 Klaassen은 경제학으로 전파하였습 니다.
Goodchild의 Spatial Autocorrelation(1986), Odland의 Spatial Autocorrelation(1988), 그
리고 제가 집필한 Spatial Autocorrelation: A Primer(1988)의 출판은 많은 관심을 불러일으 켰습니다. 1988년 출판된 Anselin의 책과 저의 책(Advanced Spatial Statistics)은 Cressie가 1991년 출판한 책이 받았던 관심만큼이나 큰 호 Stephen(1934), Wolpert(1964)가 보여준 공 간적 자기상관의 중요성에 대한 인식이 있습니
ecologist, was criticized publically for not having a PhD. Statisticians such as Ripley, Martin and Besag became very hostile toward geographers, viewing them as unqualified to study spatial autocorrelation, in part because these very statisticians were struggling for acceptance by the statistics community. The research environment was not very hospitable. Nevertheless, during this period, Dorien transported spatial autocorrelation analysis to sociology, and Paelinck and Klaassen transported it to economics.
After a rough start, spatial autocorrelation research began taking off. Cliff and Ord revised their earlier book and published it as Spatial Processes (1981); the milestone signified here is that their book merited a second edition. In the early 1980s, Sokal and Legendre began popularized spatial autocorrelation in ecology. Journal’s arrival in the United States was a major contributor to much closer interaction between the geostatistics and spatial autoregression communities. Interest was piqued by the publication of Goodchild’s Spatial Autocorrelation (1986), of Odland’s Spatial Autocorrelation (1988), and of my Spatial Autocorrelation: A Primer (1988). 1988 books by Luc Anselin and by me (Advanced Spatial Statistics) were extremely well received, as was Cressie’s 1991 book. Anselin’s book marked a bifurcation point: spatial econometrics branched off from spatial statistics.
The overall trend in spatial autocorrelation research across this time horizon follows a rather standard trajectory: measurement, hypothesis testing, and then modeling. Paralleling this trajectory was one evolving from an application to a mathematical statistics orientation. When I studied at Penn State in 1985, Ord was teaching one of the only spatial statistics courses taught in the United States. In 1993, I was invited by Cornell University to give a guest lecture in its inaugural spatial statistics course, at that time one of only a handful of spatial statistics courses taught in the United States. Today, many universities in the United States offer spatial statistics courses.
▶ Chun: What do you consider to be the milestones during this development?
▶▶ Griffith: Important but low profile milestones were the pioneering papers by Whittle, work by Bartlett, and acknowledgement of the importance of spatial autocorrelation by Student (1914), by Stephen (1934), and by Wolpert (1964). One important milestone was Krige’s development of kriging in the 1950s. The critical milestone was Cliff and Ord’s seminal paper (1969) coupled with their 1973 book, which snatched from oblivion the Moran and Geary papers that had languished for many years.
The next milestone was the collection of short books by Goodchild, by Odland, and by me. The following
다. 하나의 중요한 사건은 1950년대에 Krige가 Analysis on the PC: Spatial Statistics Using SAS)과 이 책에 포함된 SAS 구현물을 보급하
milestone was the pair of 1988 books by Anselin and by me, followed by the 1991 book by Cressie.
One milestone that is difficult to classify is Matérn’s 1960 book (Spatial Variation), which did not make much of an impact until the mid-1980s, resulting in its second edition being published in 1986.
Software milestones include Anselin’s SpaceStat and the workshops he convened to disseminate it, my 1993 book (Spatial Regression Analysis on the PC: Spatial Statistics Using SAS) and the workshops I convened to disseminate the SAS implementations presented in it, the S+ Spatial Statistics module, the spatial statistics now available in ArcGIS, and GeoDa.
▶ Chun: Needless to say, spatial autocorrelation is an important component in spatial data analysis.
Besides geography, which disciplines acknowledge spatial autocorrelation in their research, and which disciplines contributed to the development of spatial autocorrelation theory and methodology?
▶▶ Griffith: Volume 41, Number 4 of Geographical Analysis (October 2009) furnishes a historical reference for this question. The principal contributors to the development of spatial autocorrelation theory and methodology are geographers, regional scientists, geoscientists, physicists (e.g., the Ising model), economists, statisticians, and ecologists. Naturally, scholars in these disciplines also acknowledge spatial autocorrelation in their research. Disciplines that tend to only acknowledge spatial autocorrelation in research include: archeology, criminology, demography, environmental science, epidemiology, forestry, political science, real estate, sociology, and urban/regional planning.
▶ Chun: People in the academic community tend to be interested in accounting for spatial autocorrelation in their data analyses. However, people who work with data concerning practical matters, such as policy makers and decision makers, may view this type of effort as being unnecessary. What do you think spatial autocorrelation means to these latter people?
▶▶ Griffith: Most practitioners view concepts like spatial autocorrelation as esoteric subjects that can be ignored. They have to convey information to laypeople, who tend to be uneducated about sophisticated quantitative analyses, and hence seek to utilize the simplest techniques possible. Even if they employ regression analysis, practitioners rarely engage in undertaking model diagnostics, let alone in spatial autoregression analysis. Many times, they substitute qualitative analyses for quantitative analyses.
That being said, eigenvector spatial filtering was employed by two Oesterreichische Nationalbank
식을 갖고 있지 않은 일반 사람들에게 정보를 전 행(Oesterreichische Nationalbank)의 두 정책 결정자들은 고유벡터 공간 필터링을 사용하였습 자기지수(auto-exponential) 모델을 소개합니 다. 그러나 연구자들은 또 다른 분포들에 대한 자 기확률모델을 필요로 합니다. 이러한 분포들 중 몇 가지를 언급하자면, Beta, Weibull, 그리고 Gumbel 등이 있습니다. Besag은 조건부 자기회 귀모델(conditional autoregressive model) 사양 을 사용하여, 양의 공간적 자기상관의 대부분의 경우, 심지어 포아송과 음이항의 경우도 고려할 수 있는 계층적 베이지언 모델 사양을 수립하였 습니다. 이러한 베이지언 환경은 연립자기회귀
policy makers, suggesting that sometimes more contemporary practitioners recognize the importance of accounting for spatial autocorrelation.
▶ Chun: What do you think are the principal future spatial autocorrelation research directions?
▶▶ Griffith: The following are my predictions of at least some of the future research directions:
1. Three-dimensional data analyses, especially with space-time data;
2. Handling massively large spatial datasets in an efficient and effective manner;
3. More fully developing eigenvector spatial filtering;
4. Expanding the family of auto- probability models;
5. Advancing Bayesian spatial statistics;
6. Improving spatial interpolation;
7. Multiple testing in the presence of spatial autocorrelation;
8. More sophisticated spatial weights matrix specifications;
9. Testing for spatial autocorrelation in non-normal regression residuals; and,
10. Expansion of computer software for spatial statistics (e.g., more R packages, more tools in ArcMap, and more procedures in SAS).
Because considerably more space-time data are available today, and because more and more studies are focusing on data tagged with both surface and elevation coordinates, three-dimensional data analysis is increasingly more popular and common. Extending spatial autocorrelation analysis to remotely sensed data involving millions or billions of pixels, and the increasing availability of higher resolution vector data involving thousands or hundreds of thousands of polygons demand improved methodology that handles massively large spatial datasets in an efficient and effective manner. As spatial autocorrelation becomes better understood, it should be able to be linked more directly to conventional statistics; this is one appealing feature of eigenvector spatial filtering, a feature that encourages it to be more fully developed. The normal, binomial, negative binomial and Poisson probability models have auto-forms.
Besag also presents the auto-exponential model. But researchers need auto-versions of other probability models, such as the Beta, the Weibull, and the Gumbel, to name a few. Besag established the hierarchical Bayesian model specification that accounts for the most common case of positive spatial autocorrelation, even in a Poisson or a negative binomial case, but with a conditional autoregressive model specification.
모델(simultaneous autoregressive model)-고 차의 공간적 자기상관의 포착을 위해서-과 고유 벡터 공간 필터링을 지원할 필요가 있습니다.
Cressie는 co-kriging을 다변량의 맥락으로 확장 할 수 있는 이론을 제공합니다. 저는 공간 자기회 귀와 고유벡터 공간 필터링을 결손 데이터 (missing data) 대치(imputation)를 위한 다변량 의 맥락으로 확장할 수 있는 이론을 제공합니다. 이루어집니다. Journal of Statistical Planning
& Inference에 실린 저의 최근 논문은 어떻게 Moran 계수가 비정규 데이터에 적용되는지에 대한 골자를 제공합니다. 그러나 우리는 여전히 일반화선형모델(Generalized Linear Model)의
회귀 잔차의 공간적 자기상관에 대한 통계적 분 Statistics and Spatial Econometrics. Springer.
Griffith, D. A. 2005. “Effective Geographic Sample Size in the Presence of Spatial Autocorrelation”. Annals of the Association of American Geographers vol. 95. pp740-760.
Griffith, D. A. 2003. Spatial Autocorrelation and Spatial Filtering:
Gaining Understanding through Theory and Scientific Visualization. Berlin: Springer-Verlag.
Griffith, D. A. 2004. “Eigen Function Properties and Approximations of Selected Incidence Matrices Employed in Spatial Analyses”.
Linear Algebra & Its Applications vol. 321. pp95-112.
Griffith, D. A. 2000. “A Linear Regression Solution to the Spatial Autocorrelation Problem”. J. of Geographical Systems vol. 2.
pp141-156.
Griffith, D. A. 1999. “Statistical and Mathematical Sources of Regional Science Theory: Map Pattern Analysis as an Example”.
Papers in Regional Science vol. 78. pp21-45.
Griffith, D. A. 1993. Spatial Regression Analysis on the PC: Spatial Statistics Using SAS. Washington, DC: Association of American Geographers “Resource Publications in Geography Series”.
Griffith, D. A. 1992. “What is Spatial Autocorrelation? Reflections on the Past 25 Years of Spatial Statistics”. l’Espace Géographique vol. 21. pp265-280.
Griffith, D. A. 1988. Advanced Spatial Statistics, Dordrecht: Martinus Nijhoff.
Anselin, L. and Griffith, D. A. 1988. “Do Spatial Effects Really Matter in Regression Analysis?”. Papers in Regional Science vol. 65.
pp11-34.
Griffith, D. A. 1987. Spatial Autocorrelation: A Primer, Washington, DC:
Association of American Geographers Resource Publication.
Griffith, D. A. 1980. “Towards a Theory of Spatial Statistics”.
Geographical Analysis vol. 12. pp325-339.
This Bayesian environment also needs to embrace the simultaneous autoregressive model, to capture higher levels of spatial autocorrelation, and eigenvector spatial filtering. Cressie furnishes theory for extending co-kriging to a multivariate context. I furnish theory for extending spatial autoregressive and eigenvector spatial filtering to a multivariate context for missing values imputation. These spatial interpolation situations need further development and wider dissemination. Perhaps local spatial statistics is the specific instance of multiple testing in spatial statistics that needs the most attention today. This problem must be solved for proper geographic cluster identification. To date, most analyses employ simple spatial weights matrices involving a single matrix. Several pixel-type data structures allow two matrices, one for, say, north-south dependencies, and the other for east-west dependencies. Geostatistics allows a wide range of anisotropy specifications. This technology needs to be transferred to spatial autoregression analysis. Also, matrix formulations are needed for hierarchical spatial structures. Today, considerably more spatial data analyses are undertaken with non-normal probability models. My recent paper in J. of Statistical Planning & Inference outlines how the Moran Coefficient extends to non-normal data. But we still need the statistical distribution theory for regression residual spatial autocorrelation for generalized linear models. Finally, a pressing need exists for more software development. Eigenvector spatial filtering
This Bayesian environment also needs to embrace the simultaneous autoregressive model, to capture higher levels of spatial autocorrelation, and eigenvector spatial filtering. Cressie furnishes theory for extending co-kriging to a multivariate context. I furnish theory for extending spatial autoregressive and eigenvector spatial filtering to a multivariate context for missing values imputation. These spatial interpolation situations need further development and wider dissemination. Perhaps local spatial statistics is the specific instance of multiple testing in spatial statistics that needs the most attention today. This problem must be solved for proper geographic cluster identification. To date, most analyses employ simple spatial weights matrices involving a single matrix. Several pixel-type data structures allow two matrices, one for, say, north-south dependencies, and the other for east-west dependencies. Geostatistics allows a wide range of anisotropy specifications. This technology needs to be transferred to spatial autoregression analysis. Also, matrix formulations are needed for hierarchical spatial structures. Today, considerably more spatial data analyses are undertaken with non-normal probability models. My recent paper in J. of Statistical Planning & Inference outlines how the Moran Coefficient extends to non-normal data. But we still need the statistical distribution theory for regression residual spatial autocorrelation for generalized linear models. Finally, a pressing need exists for more software development. Eigenvector spatial filtering