  A number of spatial statistic measurements such as Moran's I and Geary's C
can be used for spatial autocorrelation analysis. Spatial autocorrelation
modeling proceeded from the 1-dimension autocorrelation of time series
analysis, with time lag replaced by spatial weights so that the autocorrelation
functions degenerated to autocorrelation coefficients. This paper develops
2-dimensional spatial autocorrelation functions based on the Moran index using
the relative staircase function as a weight function to yield a spatial weight
matrix with a displacement parameter. The displacement bears analogy with time
lag of time series analysis. Based on the spatial displacement parameter, two
types of spatial autocorrelation functions are constructed for 2-dimensional
spatial analysis. Then the partial spatial autocorrelation functions are
derived by Yule-Walker recursive equation. The spatial autocorrelation
functions are generalized to the autocorrelation functions based on Geary's
coefficient and Getis' index. As an example, the new analytical framework was
applied to the spatial autocorrelation modeling of Chinese cities. A conclusion
can be reached that it is an effective method to build an autocorrelation
function based on the relative step function. The spatial autocorrelation
functions can be employed to reveal deep geographical information and perform
spatial dynamic analysis, and lay the foundation for the scaling analysis of
spatial correlation.
