Spatial autocorrelation equation based on Moran's index

Area of Science:

• Spatial Statistics
• Geographic Information Science
• Econometrics

Background:

• Moran's index is a key spatial statistic for detecting spatial autocorrelation.
• Existing limitations: Moran's index is primarily a statistical measure, not a mathematical model.
• Need for a model-based approach to spatial autocorrelation analysis.

Purpose of the Study:

• To establish spatial autocorrelation models using linear regression analysis.
• To represent Moran's index as a parameter within a mathematical framework.
• To explore the mathematical structure and implications of spatial autocorrelation models.

Main Methods:

• Linear regression analysis utilizing standardized vectors as independent variables and spatial weighted vectors as dependent variables.
• Derivation of normalized linear autocorrelation equations through quadratic form and vector inner product.
• Mathematical analysis to reveal the inherent structure of model parameters.

Main Results:

• The slope of the derived linear equation directly corresponds to Moran's index.
• The intercept of the equation represents the average value of the standardized spatial weight variable.
• A negative correlation was found between the square of the intercept and the square of Moran's index.

Conclusions:

• The proposed inner product equation for spatial autocorrelation based on Moran's index is effective.
• These models extend the capabilities of spatial analysis.
• The study aids in understanding the boundary values and behavior of Moran's index.