Addressing Spatial Dependencies
(SEM)
Remote sensing, a field that deals with tons of spatial data extracted and processed from satellite images, aerial photos, and other sensor-based technologies, or any field using data with spatial features, presents a non-trivial challenge. When we analyze all this data, we have to deal with spatial dependencies (i.e., how things that are close together can influence each other). As Crawford (2009) aptly puts it:
Spatial dependence refers to the degree of spatial autocorrelation between independently measured values observed in geographical space.
These spatial dependencies can often lead to autocorrelated errors in statistical models, where observations near each other tend to exhibit similar error characteristics not captured by the explanatory variables alone.
To dig deep into spatial autocorrelation, check a nice YouTube video of a lecture by Luc Anselin at the University of Chicago (October 2016). I also invite you to briefly check my previous post, Spatial Cross-Validation in Geographic Data Analysis (March 22, 2024), where I expose the importance of modeling spatial relationships accounting for spatial correlation to improve the performance, reliability, and predictive power of a model.
One technique particularly useful in spatial analysis to address spatial dependencies in the error terms of a regression model is the Spatial Error Model (SEM).
What is the SEM?
This is a statistical technique that incorporates spatial autocorrelation into regression analyses. Unlike traditional regression models, which assume independence among observations, spatial regression models such as SEM consider the location-based interdependencies among data points.
Due to proximity, regressing spatial data often yields error terms correlated with each other. That means we can hypothesize that missing processes (not captured by the considered covariates) likely spillover into our outcomes. Therefore, this correlation can lead to biased and inefficient estimates if not properly addressed.
Well… SEM handles this by incorporating spatial weights or adding spatial autocorrelation structure into the error terms to provide more robust and reliable estimates. Feel free to read my post, Five Key Techniques in Spatial Analysis (April 11, 2024).
How Does SEM Work?
The key to SEM is including a spatially structured error component in the regression model. To be practical, let's try to make it easy to understand and follow (for details, check LeSage and Pace, 2009).
Assume a conventional linear form where y is the dependent variable, X is a set of covariates, and
