The Earth’s surface is very dynamic, particularly in those aspects that fall in the social domain. People move, houses are constructed, areas are devastated by war, and severe storms modify coastlines. Patterns of phenomena on the Earth’s surface reflect these processes, and each process tends to leave a distinctive footprint. Interpreting those footprints in order to gain insight into process is one of the most important motivations for spatial analysis.
The pervasiveness of Tobler’s First Law suggests that processes that leave a smooth pattern on the Earth’s surface are generally more prevalent than those that cause sharp discontinuities, but spatial analysts have developed many ideas in both categories.
Many processes involve some form of convolution, when the outcome at a point is determined by the conditions in some immediate neighborhood. For example, many social outcomes are responses to neighborhood conditions — individual obesity may result in part from a physical design of the neighborhood that discourages walking, alcoholism may be due in part to the availability of alcohol in the neighborhood, and asthma incidence rates may be raised due to pollution caused by high local density of traffic. Mathematically, a convolution is a weighted average of a point’s neighborhood, the weights decreasing with distance from the point, and bears a strong technical relationship to density estimation. As long as the weights are positive, the resulting pattern will be smoother than the inputs. The blurring of an out-of-focus image is a form of convolution.