Empirical Estimations for the Jacobian Term of an Auto-Normal Model

Authors: Nicholas Taliceo*, University of Texas At Dallas
Topics: Spatial Analysis & Modeling, Geographic Information Science and Systems, Quantitative Methods
Keywords: Jacobian, SAR Model, Empirical Surface Partitionings
Session Type: Paper
Day: 4/11/2018
Start / End Time: 1:20 PM / 3:00 PM
Room: Mid-City, Sheraton, 8th Floor
Presentation File: No File Uploaded

The normalizing factor for an auto-normal model specification can be written in terms of its associated geographic weights matrix. Eigenvalues of this normalizing factor, known as the Jacobian in calculus terminology, are not known analytically for most geographic weights matrices, and therefore, must be calculated. Calculating large spatial weights matrices for surface partitionings that extend into the tens of thousands of areal units is computationally heavy; a method in the literature to mitigate the computational limitation of massive spatial weights matrices is to estimate the percentage of the spectrum of eigenvalues. Griffith and Luhanga (2011) estimate the upper bound for the percentage of negative eigenvalues for a given number of areal units, n, to be 67%. However, Elphick and Wocjan (2016) prove this estimation to have an upper bound of 75%, citing the K_4 complete planar graph as a principle counterexample to the 67% upper bound. We build on Griffith and Luhanga’s work supporting their 67% conjecture for most empirical surface partitionings, with claims that this percentage is a tighter upper bound for most real world georeferenced data. This contention is argued through examples of empirical datasets of United States metropolitan statistical areas (MSAs). We focus on the Boston-Cambridge-Newton, MA MSA, and the Houston-The Woodlands-Sugarland, TX MSA, estimating spatial autoregressive models describing census tract population densities. These empirical results support the 67% upper bound, and are a practical representation of the relevant georeferenced data that are used in most practical estimations.

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