*2.4. Bivariate Spatial Association*

Lee (2001) developed a bivariate spatial association based on univariate spatial correlation analysis to examine the spatial association between bivariate observations [38]. The bivariate spatial association is defined as follows:

$$I\_{kl}^{\dot{i}} = Z\_k^{\dot{i}} \sum\_{j=l}^n W\_{\hat{i}j} Z\_l^{\dot{i}}$$

where *Wij* denotes the spatial weight matrix, *Z<sup>i</sup> <sup>k</sup>* = - *xi <sup>k</sup>* − *xk* /*σk*, *Z<sup>i</sup> <sup>k</sup>* = [*x<sup>j</sup> <sup>k</sup>* <sup>−</sup> *xl*]/*σl*, *<sup>x</sup><sup>i</sup> <sup>k</sup>* is the observation *k* at location *i*, *x<sup>j</sup> <sup>l</sup>* is the observation *l* at location *j*, and *σ<sup>k</sup>* and *σ<sup>l</sup>* denote the variance of *xk* and *xl*, respectively.

As with spatial correlation analysis, the bivariate spatial association visualizes the results using a Moran's *I* scatterplot and the bivariate LISA cluster map [39]. We used this method to examine the spatial association between newly increased construction land and newly increased polluting enterprises.
