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TopIntroduction
Spread of people, goods, information, or ideas can sometimes be more complex than what a simple wave-like diffusion model can describe. This is because the way such things are transported or transmitted is no longer confined to a localized geography or through a limited network of contacts. For example, a contagious influenza spreading in a place may become a complex diffusion process if virus carriers were dispersed over highway networks to faraway places. Such influenza may spread beyond a locality and even become international. Consequently, studies of spatial diffusion processes need to become more sophisticated and specialized than traditional approaches of fitting an infected population into a one-dimensional S-shape diffusion curve.
Rogers (1983) describes and discusses past efforts in modeling spatial diffusion processes that extend the one-dimensional S-shape diffusion curves by adding spatial terms into the diffusion models. Doing so required complex modifications of the S-shape models to approximate the diffusion processes. This makes the analysis of spatial diffusion processes difficult to understand. Furthermore, it also makes the resulting models difficult to apply to real world problems.
Given a geographic phenomenon that is spreading over space and time, let it be represented by a set of points where each point is defined by a pair of coordinates that gives its spatial attribute and a time stamp that gives its temporal attribute. First, we argue that the nearest neighbor distances at each stage of a spatial process over time are important in characterizing the spatial diffusion processes. Second, we suggest that the way nearest neighbor distances change at each diffusion stage is critical to the form of the spatial diffusion process.
Based on these two concepts, we describe our experiment that uses the nearest neighbor ratios and then regression curve estimation to model spatial diffusion processes so that main characteristics of diffusion processes can be detected and distinguished. We applied these methods to three spatio-temporal data sets. The results suggest that the methods discussed here provide a feasible way to quantitatively model and distinguish spatial diffusion processes. To validate the effectiveness of this approach, we tested it against randomly simulated spatial diffusion processes. The results of this experiment confirm our suggestion that nearest neighbor distances do play a critical role in the spatial structure of diffusion processes.
TopSpatial Diffusion Processes
The most notable early effort in quantitative modeling of spatial diffusion was Hägerstrand (1966). He developed stochastic models using statistical techniques to simulate how a spatial diffusion process progresses over time. Using a mean information field (MIF), which is a matrix with cell values representing the likelihoods that cells may receive diffusing phenomenon from source cells, Hägerstrand (1967) illustrates how the spread of a geographic phenomenon can be modeled quantitatively by moving an MIF through source cells in the diffusing pattern and how other cells would be affected in subsequent steps. Essentially, a spatial diffusion process is decomposed into matrix cells that represent locations and an MIF that represents how cells are related.
Efforts of quantitative modeling of spatial diffusion processes since Hägerstrand had taken many forms, such as attempts with spatial interaction models, expansion methods and others. These were first reviewed by Brown (1968). Gregory and Urry (1985) provide a critique to Hägerstrand’s models, noting that such models lack the ability to deal with spatial diffusion over intangible media such as social networks and the models’ inability to deal with conflicts and resistance that may exist in the networks. Following this, Morrill, Gaile, and Thrall (1988) reviewed and discussed quantitative models of spatial diffusion processes by categorizing them as stochastic models or deterministic models.