多变量空间自相关分析：Moran指数与二元变量应用

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"这篇文章探讨了如何在地理信息系统（GIS）中实现二元变量的空间自相关系数计算，特别是通过使用Moran指数。它介绍了一种替代方法，即利用专用函数库构建功能，而不是依赖现有的完整软件套件。这种方法是模块化的，并且独立于其他系统。在动态链接窗口的框架下，它结合了地图上的数据 cartographic 表示、传统的统计图形，如直方图、箱线图和散点图，特别扩展到了多变量的空间自相关性可视化，引入了Moran散点图矩阵和多元LISA地图。" 在地理数据分析中，空间自相关是一种重要的概念，它涉及到地理位置相邻的观测值之间的相似性。Moran指数是衡量这种空间自相关性的常用统计量，由PatrickJ. Moran在1950年提出。这个指数的值介于-1和1之间，其中1表示完全正相关，-1表示完全负相关，0表示没有空间自相关。如果Moran指数接近1，那么意味着空间数据在空间上呈现出聚集特征；如果接近-1，则表明数据呈现分散或反向聚集。文章提出的动态链接窗口框架允许用户在多个视图之间交互，使得分析更为直观。例如，Moran散点图矩阵是一种展示各变量间空间关系的工具，每一行和列代表一个变量，点的位置反映了变量对之间的空间自相关性。如果点集中在对角线上，表明变量之间存在空间一致性；若远离对角线，可能揭示了空间异质性或负相关。此外，多元LISA（Local Indicators of Spatial Association，局部空间关联指标）地图则用于识别空间中的热点和冷点，这些热点和冷点是指那些具有显著高或低值的区域，且这些值与周围邻居的值相比异常。多元LISA可以识别出在多变量环境中哪些特定区域的变量组合表现出强烈的空间聚集。这种集成GIS和统计分析的新方法对理解和解释复杂的地理现象非常有价值，尤其在环境科学、城市规划、农业经济学等领域，帮助研究人员发现潜在的空间模式和趋势，从而做出更准确的决策。通过定制化的小型函数库，这种方法提供了更大的灵活性，可以根据具体研究需求进行调整和扩展。

Visualizing Multivariate Spatial Correlation

“map” is but one of several linked views.

Similar visions underlie several other recent

efforts to develop open and modular software frameworks for the visualization of high

dimensional (spatial) data.

In addition to being freestanding, DynESDA2 also includes a number of other advances

over its predecessors, such as the capability to handle both point and polygon coverages,

“true” brushing of maps, simultaneous linking of multiple maps with multiple statistical

graphics, and interactive LISA maps. It also extends the visualization of spatial correlation

to a multivariate setting. We turn to this ﬁrst.

3 Multivariate Spatial Correlation

The visualization and exploration of multivariate association is a core functionality of cur-

rent exploratory data analysis (EDA), knowledge discovery and data mining tools (Buja

et al. 1996, Han and Kamber 2001, Gahegan et al. 2002). The incorporation of “spatial”

association in this framework is still in its infancy, however. Most suggested approaches

pertain to geostatistical analysis, where data are represented as points and the measure of

spatial correlation is derived from the variogram (see, e.g. Cook et al. 1996, Majure and

Cressie 1997). Similar progress has not been made for the analysis of multivariate spatial

correlation for lattice data, i.e., spatial objects represented as discrete points or polygons.

We develop a visualization device for multivariate spatial correlation in lattice data by

building on some of the ideas originally advanced in Wartenberg (1985). There, a multi-

variate coefﬁcient of spatial autocorrelation between two standardized random variables z

and z

is deﬁned as:

= z

, (1)

where z

= [x

− ¯x

]/σ

and z

= [x

− ¯x

]/σ

have been standardized such that the mean

is zero and standard deviation equals one, and W

is a doubly standardized (or, stochastic)

spatial weights matrix. The weights matrix deﬁnes the “neighbor set” for each observation

(with non-zero elements for neighbors, zero for others) and has zero on the diagonal by

convention.

This concept of multivariate spatial correlation thus centers on the extent to which val-

ues for one variable (z

) observed at a given location show a systematic (more than likely

under spatial randomness) assocation with another variable (z

) observed at the “neighbor-

ing” locations. Note that this multivariate spatial correlation can be considered in addition

to or instead of the usual (non-spatial) correlation between the two variables at the same

location. Wartenberg (1985) used this statistic to develop a notion of spatial principal

components, for which the double standardization of the weights matrix (and the implied

symmetry) was necessary.

For the purposes of visualization, our focus is on the linear association between a vari-

able z

at a location i, z

and the corresponding “spatial lag” for the other variable, [Wz

]

In this context, the usual singly-standardized (row-standardized) form of the spatial weights

matrix can be used, which yields an interpretation of the spatial lag as an “average” of

neighboring values. Also, the cross-product statistic can be re-scaled by dividing by the

See Unwin (1996) and Wilhelm and Steck (1998) for recent examples. Similar ideas are behind the Tcl/Tk

based cdv toolkit of Dykes (Dykes 1997, 1998) as well as Brundson’s exploration of local spatial association

using a dynamically linked “map” constructed with tools available in Xlispstat (Brundson 1998).

See, for example, MacEachren et al. (1999), Sutherland et al. (2000) and Gahegan et al. (2002).

Note that the points used in geostatistical analysis are sample points from a continuous surface. In contrast,

for lattice data the points are not a “sample,” but ﬁxed locations at which a spatial pattern for a random variable

can be observed.

The notation indicates that the spatial lag for location i is the i-th element of the vector Wz

. See Anselin

(1988), for an extensive treatment of the notion of a spatial lag.

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