Modeling spatial autocorrelation in spatial interaction data: empirical evidence from 2002 Germany journey-to-work flows

Abstract

Since before the inception of work by Okabe, the intermingling of spatial autocorrelation (i.e., local distance and configuration) and distance decay (i.e., global distance) effects has been suspected in spatial interaction data. This convolution was first treated conceptually because technology and methodology did not exist at the time to easily or fully address spatial autocorrelation effects within spatial interaction model specifications. Today, however, sufficient computer power coupled with eigenfunction-based spatial filtering offers a means for accommodating spatial autocorrelation effects within a spatial interaction model for modest-sized problems. In keeping with Okabe’s more recent efforts to dissemination spatial analysis tools, this paper summarizes how to implement the methodology utilized to analyze a particular empirical flows dataset.

Appendices

Appendix A

Percentage of variance accounted for (i.e., redundant information) by spatial autocorrelation

$$\begin{aligned}{\text{MC}} &= \frac{n}{{{\mathbf{1}}^\prime {\mathbf{C1}}}}\frac{{\mathbf{z}^\prime {\mathbf{Cz}}}}{n - 1},\quad{\text{where}}\,\mathbf{z}\,{\text{is}}\,{\text{a}}\,z {\text{-score}} \\\mathbf{z} &= a + b{\mathbf{Cz}} + e \\b &= \frac{{\mathbf{z}^\prime {\mathbf{Cz}}}}{{\mathbf{z}^\prime{\mathbf{C}}^{2} \mathbf{z}}} = \frac{{s_{z} }}{{s_{\text{Cz}} }}r =\frac{\sqrt 1 }{{s_{\text{Cz}} }}r \\\frac{\text{MC}}{{{\text{MC}}_{ \max } }}\frac{{n\lambda_{1}}}{{{\mathbf{1}}^\prime {\mathbf{C1}}}}\frac{{{\mathbf{1}}^\prime{\mathbf{C1}}}}{n}\frac{n - 1}{{{\mathbf{z}}^\prime {\mathbf{C}}^{2}\mathbf{z}}} &= \frac{1}{{\sqrt {\frac{{{\mathbf{z}}^\prime{\mathbf{C}}({\mathbf{I}} - {\mathbf{11}}^\prime/n){\mathbf{Cz}}}}{n - 1}} }}r \\\left( {\frac{\text{MC}}{{{\text{MC}}_{ \max } }}} \right)^{2}\lambda_{1}^{2} \frac{{(n - 1)^{2} }}{{\left( {{\mathbf{z}}^\prime{\mathbf{C}}^{2} \mathbf{z}} \right)^{2} }} &= \frac{n -1}{{{\mathbf{z}}^\prime {\mathbf{C}}({\mathbf{I}} -{\mathbf{11}}^\prime /n){\mathbf{Cz}}}}r^{2} \\r^{2} &= \left( {\frac{\text{MC}}{{{\text{MC}}_{ \max } }}}\right)^{2} \lambda_{1}^{2} (n - 1)\frac{{{\mathbf{z}}^\prime{\mathbf{C}}({\mathbf{I}} - {\mathbf{11}}^\prime/n){\mathbf{Cz}}}}{{\left( {{\mathbf{z}}^\prime {\mathbf{C}}^{2}\mathbf{z}} \right)^{2} }} \\ \end{aligned}$$

1.1 A.1 The absence of spatial autocorrelation

z iid:

$$\begin{aligned}{\mathbf{z}}^\prime {\mathbf{C}}^{2} {\mathbf{z}} &={\mathbf{1}}^\prime {\mathbf{C1}} \hfill \\{\mathbf{z}}^\prime {\mathbf{Cz}} &\approx 0 \hfill \\r^{2} &\approx \frac{{({\mathbf{1}}^\prime {\mathbf{C1}})^{2}}}{{n^{2} (n - 1){\mathbf{1}}^\prime {\mathbf{C}}^{2} {\mathbf{1}}}}\approx 0\end{aligned}$$

1.2 A.2 Some limiting cases: regular square (rook, queen) and hexagonal tessellations

$$ \begin{aligned} &{\mathbf{C1}} \to k{\mathbf{1}} \\&{\mathbf{Cz}} \to k{\mathbf{z}}\,({\text{but}}\,{\text{with}}\,{\text{some}}\,{\text{permuting}})\\ &{\text{MC}}_{ \max } \to 1 \\ &r^{2} = \left( {\frac{\text{MC}}{1}} \right)^{2} k^{2} (n -1)\frac{{k^{2} (n - 1)- \tfrac{{0^{2} }}{n}}}{{[k^{2} (n - 1)^{2} ]^{2} }} \\ &\quad= {\text{MC}}^{2} \\ \end{aligned} $$

Appendix B

Table 3.

Table 3 Frequencies of eigenvectors appearing in the Kronecker products