[转] Moran's I

Introduction

In statistics, Moran’s I is a measure of spatial autocorrelation developed by Patrick Alfred Pierce Moran. Spatial autocorrelation is characterized by a correlationin a signal among nearby locations in space. Spatial autocorrelation is more complex than one-dimensional autocorrelation because spatial correlation is multi-dimensional (i.e. 2 or 3 dimensions of space) and multi-directional.

Global Moran’s I is defined as

where w_ijis the weight between observation i and j, and S₀ is the sum of all w_ij‘s, n is equal to the total number of pixels.:

The z_I-score for the statistic is computed as:

Where:

the p-value for the null hypothesis is written as

this formula sources from Aki-Hiro Sato. Applied Data-Centric Social Sciences: Concepts, Data, Computation and Theory. Japan, Springer, 2014. pp. 137-138.

Negative/positive
values indicate negative/positive spatial autocorrelation. Values range
from -1 (indicating perfect dispersion) to 1 (perfect correlation). A
zero value indicates a random spatial pattern. For statistical
hypothesis testing (the null hypothesis of no autocorrelation is not equal to zero but given by I₀ = -1/(n-1).), 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.

栅格数据的权重规则：In its simplest form, these weights will take values 1 for close neighbours, and 0 otherwise，并规定栅格邻接为四至，如Fig. 1，中间绿色的像元的四至是四个橘红像元，绿色像元与他们之间存在着公共边，四个蓝色像元虽与绿色像元临近但不存在公共边，故而，绿色像元与橘红像元的权重是1，而与蓝色像元权重是0。（Rooks Case, Bishops Case, Oueen’s (Kings) Case）

Fig. 1

Example

四张NDVI影像测试本帖代码，计算有效区域（mask影像数值是1像元）的Global Moran’s I。代码运算结果如Fig. 2，MODIS_NDVI_200308的Global Moran’s I是0.95467222，Z-score是418.50598145，ArcGIS计算结果分别是0.954661和418.498382，两者计算结果非常接近，考虑到二者计算中间数值的精度不同，这很可能就是引起二者差异的主要原因。其他三张数据情况类似，代码计算结果与ArcGIS结果非常接近。