An analytical process of spatial autocorrelation functions based on Moran's index

Abstract

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 the time lag in 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 using the 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.

Fig 1. A flow chart of data processing, parameter estimation, and spatial autocorrelation function analyses.

(Note: The analytical process is based on the improved mathematical expressions of Moran’s I, Geary’s C and Getis-Ord’s G. Compared with x, y represents the unitized size variable, and z, the standardized variable).

Fig 2. Spatial autocorrelation function and partial autocorrelation function of Chinese cities based on generalized Moran’s index and correlation cumulation (2000).

(Note: The blue lines in the histograms are termed “two-standard-error bands”, according to which we can know whether or not there is significant difference between ACF or PACF values and zero. The same below.).

Fig 3. Spatial autocorrelation function and partial autocorrelation function of Chinese cities based on generalized Moran’s index and correlation cumulation (2010).

Fig 4. Spatial autocorrelation function and partial autocorrelation function of Chinese cities based on conventional Moran’s index and correlation density (2000).

Fig 5. Spatial autocorrelation function and partial autocorrelation function of Chinese cities based on conventional Moran’s index and correlation density (2010).

Fig 6. The ratios of SACF to PSACF based on correlation density for the main cities of China.

(Note: Inside the scaling ranges of spatial correlation dimension, the ratios of spatial autocorrelation function to the partial spatial autocorrelation function are stable; In contrast, outside the scaling range, the ratio curves fluctuate significantly. From 2000 to 2010, the scaling range extended from about 2850 to 3350 km.).

Fig 7. The canonical Moran’s scatterplots of spatial autocorrelation based on correlation density function for the main cities of China (examples for 2010).

(Note: The scatter points are based on the inner product correlation, z^TzWz = Iz, and the relation is Eq (14). The trend line is based on the outer product correlation, zz^TWz = Iz, and the relation is Eq (16). The Moran’s index difference values are as follows. (a) For 450<r≤550, ΔI = -0.0043; (b) For 1050<r≤1150, ΔI = -0.0004; (c) For 1450<r≤1550, ΔI = 0.0199; (d) For 2550<r≤2650, ΔI = 0.0061.).

Fig 8. The scaling relations for the spatial autocorrelation function based on cumulative correlation and Geary’s coefficient.

Note: The solid dots represent all points of spatial autocorrelation functions, and the hollow blocks represent the points within the scaling range. The scaling range comes between 250 and 2750 km.

Fig 9. The scaling relations for the spatial autocorrelation function based on cumulative correlation and Getis-Ord’s index.

Note: The solid dots represent all points of spatial autocorrelation functions, and the hollow blocks represent the points within the scaling range. The scaling range comes between 150 and 2750 km.

Fig 10. The curves of Geary’s C based on correlation density for the main cities of China.

(Note: Inside the scaling ranges of spatial correlation dimension, the curves of Geary’s C fluctuate sharply. From 2000 to 2010, the curve shapes have no significant change.).

Fig 11. The curves of Getis-Ord’s G based on correlation density for the main cities of China.

(Note: Inside the scaling ranges of spatial correlation dimension, the curves of Getis-Ord’s G fluctuate significantly. From 2000 to 2010, the curve shapes have slight change.).