  Based on standardized vector and globally normalized weight matrix, Moran's
index of spatial autocorrelation analysis has been expressed as a formula of
quadratic form. Further, based on this formula, an inner product equation and
outer product equation of the standardized vector can be constructed for
Moran's index. However, the theoretical foundations and application direction
of these equations are not yet clear. This paper is devoted to exploring the
inner and outer product equations of Moran's index. The methods include
mathematical derivation and empirical analysis. The results are as follows.
First, based on the inner product equation, two spatial autocorrelation models
can be constructed. One bears constant terms, and the other bear no constant
term. The spatial autocorrelation models can be employed to calculate Moran's
index by regression analysis. Second, the inner and outer product equations can
be used to improve Moran's scatterplot. The normalized Moran's scatterplot can
show more geospatial information than the conventional Moran's scatterplot. A
conclusion can be reached that the spatial autocorrelation models are useful
spatial analysis tools, complementing the uses of spatial autocorrelation
coefficient and spatial autoregressive models. These models are helpful for
understanding the boundary values of Moran's index and spatial autoregressive
modeling process.
