Flow autocorrelation: a dyadic approach

Abstract

The paper proposes and investigates a new index of flow autocorrelation, based upon a generalization of Moran’s I, and made of two ingredients. The first one consists of a family of spatial weights matrix, the exchange matrix, possessing a freely adjustable parameter interpretable as the age of the network, and controlling for the distance decay range. The second one is a matrix of chi-square dissimilarities between outgoing or incoming flows. Flows have to be adjusted, that is their diagonal part must first be calibrated from their off-diagonal part, thanks to a new iterative procedure procedure aimed at making flows as independent as possible. Commuter flows in Western Switzerland as well as migration flows in Western US illustrate the statistical testing of flow autocorrelation, as well as the computation, mapping and interpretation of local indicators of flow autocorrelation. We prove the present dyadic formalism to be equivalent to the “origin-based” tetradic formalism found in alternative studies of flow autocorrelation.

Appendix: Computational details of diagonal flow adjustment

Define a family of independent flows by the model distribution $p^{ \text{ theo }}_{ij}=\beta _i\gamma _j$ where $\beta$ and $\gamma$ are unknown distributions. The free parameters are estimated so as the relative complete flows p, of the form (7) $p_{ij}=\sigma _i\, \delta _{ij}+\mu \, a_{ij}$, is as close as possible from independent relative flows $p^{ \text{ theo }}$, in the sense that the Kullback–Leibler divergence or relative entropy

$$\begin{aligned} K(p||p^{ \text{ theo }})=\sum _{ij}p_{ij}\ln \frac{p_{ij}}{\beta _i\gamma _j}=\sum _i \sigma _i\ln \frac{\sigma _i}{\beta _i\gamma _i}+ \sum _{ij}\mu \, a_{ij}\ln \frac{\mu \, a_{ij}}{\beta _i\gamma _j} \end{aligned}$$

must be minimum, under the constraints

$$\begin{aligned} \mu =1-\sigma _{\bullet }\ge 0\qquad \qquad \sigma _i,\beta _i,\gamma _j\ge 0 \qquad \qquad \sum _i \beta _i =\sum _j \gamma _j=1. \end{aligned}$$

Setting to 0 the derivative by $\sigma _i$ yields with $\mu =1-\sigma _{\bullet }$ :

$$\begin{aligned}&\ln \frac{\sigma _i}{\beta _i\gamma _i}+1-\sum _{ij}a_{ij}\ln \frac{\mu \, a_{ij}}{\beta _i\gamma _j} -\sum _{ij}a_{ij} =\ln \frac{\sigma _i}{\beta _i\gamma _i}-\ln \mu -K(a || \beta \gamma )=0 \end{aligned}$$

where $K(a || \beta \gamma ):=\sum _{ij}a_{ij}\ln \frac{a_{ij}}{\beta _i\gamma _j}$ is the “movers” relative entropy, which, incidentally, also appears in the likelihood ratio of the so-called quasi-independence models (e.g., Bishop et al. 1975). Derivating w.r.t. $\beta _i$ under the constraint $\sum _i \beta _i=1$ (and multiplier $\lambda$) yields

$$\begin{aligned} -\frac{\sigma _i}{\beta _i}-\mu \frac{a_{i\bullet }}{\beta _i}=-\lambda \qquad \qquad \beta _i\lambda =\sigma _i +\mu a_{i\bullet } \quad \Rightarrow \quad \lambda =1 \end{aligned}$$

in view of $\sigma _{\bullet }+\mu =1$. Hence $\beta _i=\sigma _i+\mu a_{i\bullet }$. Similarily, $\gamma _j=\sigma _j+\mu a_{\bullet j}$. Finally,

where $a_{ij}=m_{ij}/m_{\bullet \bullet }$ is the known proportion of movers from i to j Sect. 2.4, from which the unknown $\sigma ,\mu , \beta ,\gamma$ are to be estimated.

Equations (11) is iteratively solved from some initial solution such as $\beta ^0_i=a_{i\bullet }$, $\gamma ^0_j=a_{\bullet j}$ and $\sigma _i^0=a_{i\bullet }a_{\bullet i}$ until convergence, which occurs provided $a_{i\bullet }>0$ and $a_{\bullet j}>0$ for all origins i and destinations j. If needed, the latter conditions can be insured by considering augmented flows $t_{ij}+1$ instead of $t_{ij}$, a choice adopted for convenience in the case studies of Sect. 3, and justifiable in a Bayesian framework (Laplace rule of succession).

More generally, $a_{i\bullet }+a_{\bullet i}=0$ makes i irrelevant (a “non-region”). $a_{i\bullet }=0$ and $a_{\bullet i}>0$ makes $\beta _i=\sigma _i=0$ (absorbing destination). Similarly, $a_{i\bullet }>0$ and $a_{\bullet i}=0$ makes $\gamma _i=\sigma _i=0$ (transient origin). Finally, $a_{i\bullet }+a_{\bullet i}>0$ and $a_{i\bullet }a_{\bullet i}=0$ for all i splits the regions between two non-intersecting sets, namely transient origins and absorbing destinations, with $\sigma _{\bullet }=0$ and $\mu =1$, characterizing incommensurate, “rectangular” flows.