Interpolation sometimes enhances and sometimes impedes spatial correlation: simple pedagogical examples

A natural way to check whether there is a dependence between two quantities is to estimate their correlation. For spatial quantities, such an estimation is complicated by the fact that, in general, we measure the values of the two quantities of interest in somewhat different locations. In this case, one possibility is to correlate each value of the first quantity with the value of the second quantity measured at a nearby point. An alternative idea is to first apply an appropriate interpolation to each of the quantities, and then look for the correlation between the resulting spatial maps. Empirical results show that sometimes one of these techniques leads to a larger correlation, and sometimes the other one. In this paper, we provide simple pedagogical examples explaining why sometimes interpolation enhances spatial correlation and sometimes interpolation impedes correlation. 1 Two Approaches to Spatial Correlation: Empirical Evidence and Need for Simple Pedagogical Examples Formulation of the problem. In many practical situations, we want to find the correlation between spatially distributed quantities a(x, y) and a′(x, y). For example: • we have pollution measurements at different spatial points, • we have frequencies of allergies in different geographic locations, and • we want to see if some of these allergy cases are caused by pollution.