Moran's I Statistics

空间分析中常见的两种统计算法，分别是Moran‘s I 和Geary’s C Statistics。

下面主要讨论前者的思想。

Moran's I  (Moran 1950 ) is a weighted correlation coefficient used to detect departures from spatial randomness. Moran's I  is used to determine whether neighboring areas are more similar than would be expected under the null hypothesis. 

在维基百科中的解释如下：

In statistics, Moran's I is a measure of spatial autocorrelation developed by Patrick A.P. Moran.^[1] Like autocorrelation, spatial autocorrelation means that adjacent observations of the same phenomenon are correlated. However, autocorrelation is about proximity in time. Spatial autocorrelation is about proximity in (two-dimensional) space. Spatial autocorrelation is more complex than autocorrelation because the correlation is two-dimensional and bi-directional.

Moran's I is defined as

where N is the number of spatial units indexed by i and j; X is the variable of interest;  is the mean of X; and w_ij is a matrix of spatial weights.

The expected value of Moran's I under hypothesis of no spatial autocorrelation is

Its variance equals

where

Negative (positive) values indicate negative (positive) spatial autocorrelation. Values range from −1 (indicating perfect dispersion) to +1 (perfect correlation). A zero values indicates a random spatial pattern. For statistical hypothesis testing, Moran's I values can be transformed to Z-scores in which values greater than 1.96 or smaller than −1.96 indicate spatial autocorrelation that is significant at the 5% level.

The significance of I can be judged by calculating the variance of I and then comparing the following statistic to the standard normal distribution

 

                                           .   （关于置信度的计算）

 

Moran's I is inversely related to Geary's C, but it is not identical. Moran's I is a measure of global spatial autocorrelation, while Geary's C is more sensitive to local spatial autocorrelation.

可以看出Moran’s I的相关特点。

转载于:https://www.cnblogs.com/w2william/archive/2009/10/11/1581028.html