I plan to include coordinates as covariates in the regression equation in order to adjust for the spatial trend that exists in the data. After that, I want to test residuals on spatial autocorrelation in random variation. I have several questions:
Should I perform linear regression in which only independent variables are $x$ and $y$ coordinates and then tests residuals on spatial autocorrelation, or should I rather include not only coordinates as covariates but also other variables and then test residuals.
If I expect to have quadratic trend, and then include not only $x,y$, but also , $xy$, $x^2$ and $y^2$, but then some of them ($xy$ and $y^2$) have the $p$-value higher than the threshold --should I exclude those variables with higher $p$-value as being nonsignificant? How should I then interpret the trend, it is certainly not quadratic anymore?
I guess I should treat $x$ and $y$ coordinates as any other covariates, and test them on having linear relationship with dependent variable by constructing partial residual plots ... but then once I transform them (if they show they need transformation), that will not be that kind of trend any more (especially if I include $xy$, $x^2$ and $y^2$ for quadratic trend). It may show that $x^2$, for example, needs transformation, while $x$ does not or so? How should I react in these situations?