New framework of Getis-Ord’s indexes associating spatial autocorrelation with interaction

Abstract

Spatial autocorrelation and spatial interaction are two important analytical processes for geographical analyses. However, the internal relations between the two types of models have not been brought to light. This paper is devoted to integrating spatial autocorrelation analysis and spatial interaction analysis into a logic framework by means of Getis-Ord's indexes. Based on mathematical derivation and transform, the spatial autocorrelation measurements of Getis-Ord's indexes are reconstructed in a new and simple form. A finding is that the local Getis-Ord's indexes of spatial autocorrelation are equivalent to the rescaled potential energy indexes of spatial interaction theory based on power-law distance decay. The normalized scatterplot is introduced into the spatial analysis based on Getis-Ord's indexes, and the potential energy indexes are proposed as a complementary measurement. The global Getis-Ord's index proved to be the weighted sum of the potential energy indexes and the direct sum of total potential energy. The empirical analysis of the system of Chinese cities are taken as an example to illustrate the effect of the improved methods and measurements. The mathematical framework newly derived from Getis-Ord's work is helpful for further developing the methodology of geographical spatial modeling and quantitative analysis.

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

“…Suppose that for the outer product x i x j , i = j can be kept, but for weight, w ij , i = j is rejected. Using unitized matrix and unitized vector, we can rewrite Eq ( 37 ) in the following simple form [ 34 ] where y = x / S = [ y 1 , y 2 , …, y n ] T represents the unitized vector of x . The elements of y is defined as below: where the sum of x is …”

“…In fact, many mathematical methods based on quadratic form, symmetric matrix, and reciprocal matrix can be outlined as a pair of equations based on inner product and outer product (Table 3). Based on normalized variables and normalized weight matrixes, Getis-Ord's index can also be expressed as two equations based on inner product and outer product (Chen, 2020).…”

“…The urban population is employed as a size measurement, while the distances by train between any two cities act as a spatial contiguity measurement. The census data of the urban population in 2000 and 2010 are available from the Chinese website, and the railroad distance matrix can be found in many Chinese traffic atlases [ 14 , 15 , 34 ]. The cities of Haikou and Lhasa are not taken into account in this study.…”

“…First of all, the staircase function was used to determine a spatial contiguity matrix based on a threshold distance r . Then, the three-step method was employed to calculate Moran’s index and Getis-Ord’s index [ 14 , 34 ]. The process is as follows.…”

Geospatial modelling to estimate the territory at risk of establishment of influenza type A in Mexico - An ecological study

“…A geospatial model was developed, by considering the similarities found when comparing different spatial patterns (Harding et al, 2020) that epidemiologically and probabilistically favour the establishment of EOITA. Spatial criteria as random, dispersed or grouped and a High/Low clustering tool have been spatially represented according to the quadrant methodology (Chen, 2020). These values represent spatial autocorrelation of the Getis-Ord (Gi*) indices expressed as High/High, Low/Low, High/Low, Low/High and non-significant values, which measure the concentration of values for a study area.…”