Chapter 2 Spatial Autocorrelation
Much of classical statistical theory assumes that observations are iid (independent and identically distributed).
Correlated samples theory: repeated measures —-> multivariate statistical theory: observations are paired with the distribution of pairs being iid while pairs themselves are correlated.
（也就是一组观测值与另一组观测值认为无关，但一组观测值内的各个变量认为是相关的，所以称为autocorrelation）
—-> time series analysis —-> spatial series analysis (spatial autocorrelation)
2.1 Indices measuring spatial dependency
Moran coefficient (MC)
- a covariation measurement that relates directly to the Pearson product moment coeffient（也就是皮尔逊相关系数，即Pearson correlation coefficient）.
- is capable of indexing data across all four scales (nominal, ordinal, interval and ratio).
- PPMCC:
\[\rho=\frac{\sum_{i=1}^{n}{1(x_i-\overline{x})(y_i-\overline{y})}/n}{\sqrt{\sum_{i=1}^{n}{(x_i-\overline{x})^2}/n}\sqrt{\sum_{i=1}^{n}{(y_i-\overline{y})^2}/n}}=\frac{\sum_{i=1}^{n}{(x_i-\overline{x})(y_i-\overline{y})}}{\sqrt{\sum_{i=1}^{n}{(x_i-\overline{x})^2}}\sqrt{\sum_{i=1}^{n}{(y_i-\overline{y})^2}}}\]
- where
- 1 is the frequency of observation \(i\);
- \(x_i\) and \(y_i\) are paired values of two variables for observation \(i\);
- \(\overline{x}\) and \(\overline{y}\) are the respective mean of these two variables;
- the numerator is a covariation term and the denominator is the product of the standard deviations of the two variables;
- division by \(n\) can be replaced by \(n-1\) for unbiased estimates.
- MC for variable Y (similar to PPMCC):
\[MC=\frac{\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}(y_i-\overline{y})(y_j-\overline{y})/\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}}}}{\sqrt{\sum_{i=1}^{n}{(y_i-\overline{y})^2}/n}\sqrt{\sum_{j=1}^{n}{(y_i-\overline{y})^2}/n}}=\frac{n}{\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}}\frac{\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}(y_i-\overline{y})(y_j-\overline{y})}}}{\sum_{i=1}^{n}{(y_i-\overline{y})^2}}\]
- where
- \(c_{ij}\) is the corresponding cell value in the distance matrix that replaces the frequence 1 in PPMCC.
- autocorrelation refers to correlation with in a single variable.
- 计算时空间权重阵一般需要进行行标准化
Important properties of MC
- the range of MC is no longer [-1, 1] – this interval maybe shrink, but usually expands – and midpoint is \(\frac{1}{n-1}\) rather than 0 with denoting 0-spatial autocorrelation.（对于正方形剖分的空间而言，MC的值域仍然为[-1,1]，正值越大表明空间上出现了属性相似的聚集（高-高，低-低），负值的绝对值越大表明出现了属性相反的聚集（高-低，低-高），越接近0表明在空间上的关联性越小）。
- \(H_0\): \(n\)个观测单元的属性值不存在空间自相关的关系，标准化MC \(Z_\alpha=\frac{I-E(I)}{\sqrt{Var(I)}}\)，其中I就是MC的值，\(E(I)\)是MC的期望，\(Var(I)\)是MC的方差，具体值比较复杂，可以参照MORAN P A P. Notes on continuous stochastic phenomena.[J]. Biometrika,1950,37(1-2). 还可以证明，在零假设成立的情况下，\(Z_\alpha\)近似服从正态分布，可以根据 \(Z_\alpha\) 判断是否存在着空间自相关现象。
- MC方差的渐进值为\(\frac{2}{\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}}\)，对于非正态分布，观测区域数至少为25，最好至少100。
Geary Ratio (GR)
- a paired comparisons measure that related directly to the semi-variogram plot utilized in geostatistics.
- GR=\(\frac{(n-1)\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}(x_i-x_j)^2}{2\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}\sum_{k=1}^{n}{(x_k-\overline{x})^2}}\)
- GR的值一般在[0,2]，大于1表示负相关，等于1表示不相关，小于1表示正相关，GR的标准化方式与MC的类似，也服从渐近正态分布，正的Z(GR)表示高值聚集，负的Z(GR)表示低值聚集。
- 计算GR时，空间权重阵一般不进行行标准化。
Relationships between MC and GR
\[GR=\frac{n-1}{2\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}}}}\frac{\sum_{i=1}^{n}{(x_i-\overline{x})^2}(\sum_{j=1}^{n}{c_{ij}})}{\sum_{i=1}^{n}{(x_i-\overline{x})^2}}-\frac{n-1}{n}MC\]
这也与上述两个系数之间的属性相符（inverse relationship）,GR也更容易受离群值影响，如果GR+MC≈1，那么说明数据质量较好（没有极端离群值），因为GR变化的范围更大（更不稳定），且MC在任意量表下都适用，所以MC是更优的度量空间自相关的指数。
小结——二者的比较
| Method |
MC |
GR |
| What does it measure? |
Clustering data across the space |
Dissimmilarities between data points of juxtaposition |
| What’s the reference? |
Global mean of data points |
Differences between neighboring data points |
| Similar to |
PPMCC |
Variogram |
| Weakness |
Have to know the mean |
Not influenced by sample size and spatial weight:less robust statistics |
补充：Getis-Ord G
- Indicates that if high or low values are clusterred (only global status), but not for both.
-
\[G=\frac{\sum_{i=1}^{n}{\sum_{j=1}^{n}{c_{ij}x_ix_j}}}{\sum_{i=1}^{n}{\sum_{j=1}^{n}{x_ix_j}}}\]
- 若是热点区域，则G值将较大；反之，若是冷点区域，则G值将较小。
- 变体：\(G_i^*\)检验，用于寻找局部聚类，\(G_i^*=\frac{\sum_{j=1}^{n}c_{ij}x_i}{\sum_{j=1}^{n}{x_j}}\)，再标准化即可进行判别。
2.2 Graphic portrayals: the Moran scatterplot and semi-variogram plot
The Moran scatterplot is a two-dimensional diagram using Cartesian coordinates to display pairs of values in a manner that summarizes the relationship between the observations comprising a univariate georeferenced dataset.
Methods:
- Step1: Convert each \(y_i\) to a z-score.
- Step2: Plot each z-score (horizontal axis) against its corresponding sum of surrounding z-scores (vertical axis).
注意：这里的莫兰散点图也就是所谓的LISA（Local Indicators of Spatial Association）图，只不过表述方式有区别（《空间数据分析》教材上的表述是“描述观测变量\(x\)和其空间滞后变量\(W_x\)（即该空间单元周围单元的观测变量的值得加权平均值）”），LISA图的斜率就是未标准化的MC（需要除以距离矩阵权重之和），右上、左下、右下、左上四个区域分别代表高-高、低-低、高-低、低-高四种聚类形式。
这里再来补充一下局部莫兰指数的定义。
探究是否存在观测值的局部聚集、哪个空间单元对全局空间自相关的有更大的贡献时，需要使用局部空间自相关分析。局部莫兰指数就是一种常用的度量指数。
\[MC_i=\frac{(x_i-\overline{x})}{S^2}\sum_{j=1}^{n}{c_{ij}(x_j-\overline{x})}=\frac{n(x_i-\overline{x})\sum_{j=1}^{n}{c_{ij}(x_j-\overline{x})}}{\sum_{i=1}^{n}{(x_i-\overline{x})^2}}=\frac{nz_i\sum_{j=1}^{n}{c_{ij}z_j}}{Z^TZ}\]
其中 \(S^2=\frac{\sum_{i=1}^{n}{(x_i-\overline{x})^2}}{n}\), \(\overline{x}=\frac{1}{n}\sum_{i=1}^{n}{x_i}\), \(z_i\) 和 \(z_j\)是经过标准差标准化后的观测值。
有\(\sum_{i=1}^{n}{MC_i}=S_0MC\)
（此外，Join Count统计量也是一种衡量空间自相关的方法（主要用于衡量名义量表变量的空间自相关，也有简单的介绍，这里跳过了）
The semi-variogram scatterplot is a two-dimensional diagram using Cartesian coordinates of the first quadrant (i.e., all values are non-negative) to display pairs of values in a manner that summarizes the relationship between the variation for a univariate georeferenced variable and distance separating the georeferenced observations.
也就是变差函数图，横轴为标准化之后的距离，纵轴为半方差值，与横轴为Topological Lag（空间滞后），纵轴为GR值的散点图趋势一致（因为二者都与方差有关），由三个主要部分组成：
- 变程（Range）: the distance at which spatial autocorrelation becomes zero.
- 块金常数（Nugget）: a discontinuity at the origin, such that the trend line doesn’t go to zero when the distance is zero.
- 基台值（Sill）: the limit of the trend line implied by the scatter of points as distance goes to infinity.
(后面应该还会见到)
2.3 Impacts of spatial autocorrelation
Variance Inflation
也就是方差膨胀，变量之间存在着相关性（多重共线性）
2.4 Testing for spatial autocorrelation in regression residuals
回归残差的空间自相关检验，对于不同的回归方程Cliff和Ord给出了不同的检验量。（跳过）
2.5 R Code for concept implementations
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