Testing spatial autocorrelation in weighted networks: the modes permutation test

Abstract

In a weighted spatial network, as specified by an exchange matrix, the variances of the spatial values are inversely proportional to the size of the regions. Spatial values are no more exchangeable under independence, thus weakening the rationale for ordinary permutation and bootstrap tests of spatial autocorrelation. We propose an alternative permutation test for spatial autocorrelation, based upon exchangeable spatial modes, constructed as linear orthogonal combinations of spatial values. The coefficients obtain as eigenvectors of the standardized exchange matrix appearing in spectral clustering and generalize to the weighted case the concept of spatial filtering for connectivity matrices. Also, two proposals aimed at transforming an accessibility matrix into an exchange matrix with a priori fixed margins are presented. Two examples (inter-regional migratory flows and binary adjacency networks) illustrate the formalism, rooted in the theory of spectral decomposition for reversible Markov chains.

Appendix

Proof of (7)

U being orthogonal, ∑ _i f _i c _{i α} c _{i β}  = ∑ _i u _{i α} u _{i β}  = δ _αβ and $\sum_i f_i c_{i\alpha}=\sum_i \sqrt{f_i}u_{i\alpha} =\sum_i u_{i0}u_{i\alpha}=\delta_{\alpha0}$.

Proof of (9)

independence implies the functional form σ _ij  = δ _ij g(f _i ) where g(f) expresses a possible size dependence. Consider the aggregation of regions j into super-region J, with aggregated field $X_J=\sum\nolimits_{j\in J}f_j X_j/f_J$, where $f_J:=\sum\nolimits_{j\in J}f_j$. By construction,

$$ g(f_J)=\hbox{Var}(X_J)=\frac{1}{f_J^2}\sum_{i,j\in J}f_if_j \sigma_{ij}= \frac{1}{f_J^2}\sum_{j\in J}f_j^2 g(f_j) $$

that is $f_J^2 g(f_J)=\sum\nolimits_{j\in J}f_j^2 g(f_j)$, with unique solution g(f _j ) = σ²/f _j (and g(f _J ) = σ²/f _J ), where $\sigma^2=\hbox{Var}(\bar{X})$.

Proof of (10)

$\hat{\sigma}_{\alpha\beta}:=\hbox{Cov}(\hat{X}_\alpha,\hat{X}_\beta)= \sum\nolimits_{ij} f_i f_j c_{i\alpha}c_{j\beta}\hbox{Cov}(X_i,X_j) =\sigma^2 \sum\nolimits_i f_i c_{i\alpha}c_{i\beta}=\sigma^2\sum\nolimits_i u_{i\alpha}u_{i\beta}=\sigma^2\delta_{\alpha\beta}$.

Proof of (11)

$\sum\nolimits_{\alpha\ge1}\hat{x}^2_\alpha= \sum\nolimits_{ij}\sqrt{f_if_j}x_ix_j\sum\nolimits_{\alpha\ge0}u_{i\alpha} u_{j\alpha}-\hat{x}^2_0=\sum\nolimits_i f_i x_i^2-\bar{x}^2=\hbox{var}(x)$. Also, $\hbox{var}_{{\rm loc}}(x)=\frac{1}{2}\sum\nolimits_{ij}e_{ij}(x_i-x_j)^2=\sum\nolimits_i f_i x_i^2-\sum\nolimits_{ij}e_{ij}x_ix_j= \sum\nolimits_i f_i x_i^2-\bar{x}^2-\sum\nolimits_{\alpha\ge1}\lambda_\alpha\sum\nolimits_i c_{i\alpha} x_i \sum\nolimits_j c_{j\alpha} x_j= \hbox{var}(x)-\sum\nolimits_{\alpha\ge1}\lambda_\alpha\hat{x}^2_\alpha$.

Proof of (12) and (13)

define

$$ a_\alpha:=\frac{\hat{x}^2_\alpha}{\sum_{\beta\ge1} \hat{x}^2_\beta} \quad \hbox{with}\quad \sum_{\alpha\ge1}a_\alpha=1 \quad\hbox{and}\quad I(\hat{x})=\sum_{\alpha\ge1} \lambda_\alpha a_\alpha . $$

Under H ₀, the distribution of the non-trivial modes is exchangeable, i.e. f(a) = f(π(a)). By symmetry, E _π(a _α ) = 1/(n − 1), E _π(a ² _α ) = s(x)/(n − 1)² where s(x) = ∑ _β ≥ 1 a ² _β /(n − 1) and E _π(a _α a _β ) = (1 − s(x)/(n − 1))/[(n − 1)(n − 2)] for α ≠ β. Further substitution proves the result.

Proof of the semi-negative definiteness of Q in (20)

for any vector h,

$$ 0\le \frac{1}{2}\sum_{ij}\varepsilon_{ij}(h_i-h_j)^2=\sum_i\sigma_i h_i^2-\sum_{ij}\epsilon_{ij}h_ih_j =- \sum_{ij}(\epsilon_{ij}-\delta_{ij}\sigma_j)h_ih_j . $$

Relation between the eigen-decompositions of E ^(s) (t) and Q in (20)

in matrix notation, $Q=\Uppi^{\frac{1}{2}} R\Uppi^{-\frac{1}{2}}$, and hence, $Q\sqrt{f}=0$ by (19), showing $u_0=\sqrt{f}$ with μ ₀ = 0. Consider another, non-trivial eigenvector u _α of Q, with eigenvalue μ _α , orthogonal to $\sqrt{f}$ by construction. Identity $E(t)=\Uppi \exp(t R)$ together with (5) yield

$$ E^s(t)=\sum_{k\ge0}\frac{t^k}{k!}Q^k-\sqrt{f}\sqrt{f}^{\prime} \quad E^s(t)u_\alpha=\sum_{k\ge0}\frac{t^k\mu_\alpha^k }{k!} u_\alpha=\exp(\mu_\alpha t)u_\alpha . $$