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Ordinal Patterns Based Testing of Spatial Independence in Irregular Spatial Structures

Giorgio Micali, David Garnés-Galindo, Mariano Matilla-García, Manuel Ruiz-Marín

Abstract

We propose a nonparametric test of spatial independence for data observed on irregular, non-lattice point clouds $\mathcal{V}_{n}\subset\mathbb{R}^{2}$. For each location $v\in\mathcal{V}_{n}$, we encode the local spatial configuration through the ordinal pattern of the $m$ nearest-neighbour observations, obtaining a symbolic representation that is invariant under strictly monotone transformations and robust to outliers. Under the null hypothesis of spatial independence, the local ordinal patterns are i.i.d.\ and uniformly distributed over the symmetric group $\mathcal{S}_{m}$, regardless of the unknown marginal distribution $F$. We exploit this characterisation to construct a test statistic $L_{n}$ based on the additive log-ratio (ALR) transformation of the empirical ordinal-pattern frequencies. Invoking a central limit theorem for graph-dependent processes under a graph-based $α$-mixing condition, we establish that $L_{n}$ converges in distribution to a $χ^{2}_{m!-1}$ random variable, yielding an asymptotically pivotal procedure with no nuisance parameters. An extensive Monte Carlo study confirms that the $χ^{2}_{m!-1}$ approximation is accurate already at moderate sample sizes, that the test controls size at the nominal level, and that power increases monotonically with the strength of spatial dependence. Notably, the test detects dependence in both linear and nonlinearly transformed spatial autoregressive models, illustrating the robustness that is characteristic of ordinal-pattern methods. Our framework extends the spatial ordinal-pattern testing paradigm from regular lattices to general spatial supports, opening the door to ordinal-pattern inference in the many applied settings where observations are irregularly located.

Ordinal Patterns Based Testing of Spatial Independence in Irregular Spatial Structures

Abstract

We propose a nonparametric test of spatial independence for data observed on irregular, non-lattice point clouds . For each location , we encode the local spatial configuration through the ordinal pattern of the nearest-neighbour observations, obtaining a symbolic representation that is invariant under strictly monotone transformations and robust to outliers. Under the null hypothesis of spatial independence, the local ordinal patterns are i.i.d.\ and uniformly distributed over the symmetric group , regardless of the unknown marginal distribution . We exploit this characterisation to construct a test statistic based on the additive log-ratio (ALR) transformation of the empirical ordinal-pattern frequencies. Invoking a central limit theorem for graph-dependent processes under a graph-based -mixing condition, we establish that converges in distribution to a random variable, yielding an asymptotically pivotal procedure with no nuisance parameters. An extensive Monte Carlo study confirms that the approximation is accurate already at moderate sample sizes, that the test controls size at the nominal level, and that power increases monotonically with the strength of spatial dependence. Notably, the test detects dependence in both linear and nonlinearly transformed spatial autoregressive models, illustrating the robustness that is characteristic of ordinal-pattern methods. Our framework extends the spatial ordinal-pattern testing paradigm from regular lattices to general spatial supports, opening the door to ordinal-pattern inference in the many applied settings where observations are irregularly located.
Paper Structure (9 sections, 4 theorems, 110 equations, 3 figures, 1 table)

This paper contains 9 sections, 4 theorems, 110 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical Framework
  3. central limit theorem for spatial ordinal patterns
  4. Covariance Estimation
  5. Test statistics
  6. Monte Carlo Simulations
  7. Conclusions
  8. Proof of Theorem 1
  9. Auxiliary Results

Key Result

Theorem 1

Suppose that the spatial process $(X_s)_{v_s\in\mathcal{V}_n}$ satisfies Assumption assumption1. Fix $m\ge 2$, and let $(\mathbf Y_s)_{v_s\in\mathcal{V}_n}$ denote the $(m!-1)$-dimensional random vectors defined in eq:Y_vector. For $r\ge 0$, define $U_n(h):=\#\{(v_s,v_t)\in\mathcal{V}_n \times \math Define the centered partial sum and its variance as respectively. Then $\Sigma_n$ is almost surely

Figures (3)

  • Figure 1: Ordinal patterns graph construction from initial process.
  • Figure 2: QQ plot for different sample sizes (left). Empirical density function versus theoretical density (right). Plots are obtained from $R=10000$ Monte Carlo replications.
  • Figure 3: Empirical power curves based on $R=10 000$ Monte Carlo repetitions for ordinal pattern dimensions $m\in\{3,4\}$ under three data-generating mechanisms: the linear SAR model $X=(I_n-\rho W)^{-1}\varepsilon$, the nonlinear model $X=\sin((I_n-\rho W)^{-1}\varepsilon)$, and the nonlinear model $X=\log |(I_n-\rho W)^{-1}\varepsilon|$.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['thm:cond_vector_clt']}
  • Definition 2
  • Definition 3
  • Theorem 2
  • Remark 2
  • ...and 2 more