Fast Spatial Autocorrelation

TL;DR

This paper introduces a new spatial autocorrelation statistic, $S_A$, which can be computed in linear time and effectively detects spatial correlations in large datasets, outperforming traditional methods like Moran's I and Geary's C.

Contribution

The authors propose the $S_A$ statistic and a linear-time algorithm for its computation, enabling fast and scalable spatial autocorrelation analysis.

Findings

$S_A$ can be computed in 1 second for 63,000 points.

$S_A$ detects spatial correlations earlier than Moran's I and Geary's C.

$S_A$ has desirable theoretical properties, including being a true correlation and invariant under scaling.

Abstract

Physical or geographic location proves to be an important feature in many data science models, because many diverse natural and social phenomenon have a spatial component. Spatial autocorrelation measures the extent to which locally adjacent observations of the same phenomenon are correlated. Although statistics like Moran's $I$ and Geary's $C$ are widely used to measure spatial autocorrelation, they are slow: all popular methods run in $\Omega(n^2)$ time, rendering them unusable for large data sets, or long time-courses with moderate numbers of points. We propose a new $S_A$ statistic based on the notion that the variance observed when merging pairs of nearby clusters should increase slowly for spatially autocorrelated variables. We give a linear-time algorithm to calculate $S_A$ for a variable with an input agglomeration order (available at https://github.com/aamgalan/spatial_autocorrelation). For a typical dataset of $n \approx 63,000$ points, our $S_A$ autocorrelation measure can be computed in 1 second, versus 2 hours or more for Moran's $I$ and Geary's $C$. Through simulation studies, we demonstrate that $S_A$ identifies spatial correlations in variables generated with spatially-dependent model half an order of magnitude earlier than either Moran's $I$ or Geary's $C$. Finally, we prove several theoretical properties of $S_A$: namely that it behaves as a true correlation statistic, and is invariant under addition or multiplication by a constant.