RESAPLE: An Approximate One-Step Restricted Likelihood Estimator of Spatial Dependence for Exploratory Spatial Analysis

1. Problem Statement

In Exploratory Spatial Data Analysis (ESDA), researchers frequently need to assess the strength of spatial dependence (autocorrelation) in data, particularly after adjusting for large-scale trends and covariates.

• Current Limitations: The two most common tools are Moran's $I$ and the Approximate Profile Likelihood Estimator (APLE).
  • Moran's $I$: While excellent for detecting the presence of autocorrelation, it is a biased estimator of the spatial dependence parameter ($\rho$) when $\rho \neq 0$. It is often applied to regression residuals in practice, but its theoretical derivation does not explicitly account for the removal of fixed effects (covariates) in a statistically principled way.
  • APLE: Derived as a one-step approximation to the Maximum Likelihood (ML) profile score, APLE is computationally efficient and interpretable. However, standard APLE formulations are based on the full likelihood, which treats regression coefficients ($\beta$) as nuisance parameters to be estimated jointly with $\rho$. In applied workflows, practitioners typically estimate $\beta$ first (e.g., via OLS) and then analyze the resulting residuals. This creates a misalignment: existing APLE-type estimators do not strictly correspond to the "residual-based" workflow used in practice, leading to potential bias and sensitivity to the specification of the design matrix $X$.
• The Gap: There is a lack of a closed-form, computationally inexpensive estimator that is explicitly derived from the Restricted Maximum Likelihood (REML) framework. REML is the standard for estimating covariance parameters in linear mixed models because it eliminates fixed effects, reducing bias and ensuring invariance to the parameterization of covariates.

2. Methodology: The RESAPLE Estimator

The authors propose RESAPLE (REstricted Spatial Approximate Profile Likelihood Estimator), a new statistic designed to bridge the gap between theoretical optimality and practical residual-based workflows.

Core Concept

RESAPLE is constructed in the residual space obtained by projecting the data onto the orthogonal complement of the covariate space.

1. Model: The authors assume a Gaussian Spatial Error Model (SEM):
   $$Z = X\beta + U, \quad U = \rho W U + \epsilon$$
   where $Z$ is the response, $X$ are covariates, $W$ is the spatial weight matrix, and $\epsilon \sim N(0, \sigma^2 I)$.
2. Residual Space Construction:
   • Let $P = X(X^\top X)^{-1}X^\top$ be the projection onto the column space of $X$.
   • Let $M = I - P$ be the projection onto the residual space (dimension $r = n - p$).
   • Let $H$ be an orthonormal basis for the residual space ($H^\top H = I_r, HH^\top = M$).
   • Define residualized contrasts: $e = H^\top Z$. These contrasts contain all information in $Z$ orthogonal to $X$ and are independent of $\beta$.
3. Derivation:
   • RESAPLE is derived as a one-step approximation to the restricted profile score equation at $\rho = 0$.
   • It takes the form of a Rayleigh quotient (a ratio of quadratic forms):
     $$\hat{\rho}_{RESAPLE} = \frac{e^\top (K_r - \mu_r I_r) e}{e^\top (W_r^\top W_r + \nu_r I_r) e}$$
     Where $W_r = H^\top W H$, $K_r = (W_r + W_r^\top)/2$, and $\mu_r, \nu_r$ are trace-based normalization constants.
4. Key Properties:
   • Invariance: RESAPLE is invariant to reparameterizations of the design matrix $X$ and the choice of the orthonormal basis $H$.
   • Unbiasedness: It is locally approximately unbiased for $\rho$ around 0, with bias errors of order $O(|\rho|^2)$.
   • Generalization: When no covariates are present ($p=0$), RESAPLE reduces exactly to the standard APLE.

3. Key Contributions

1. Theoretical Alignment: RESAPLE provides a principled statistical justification for the common practice of analyzing residuals. It aligns the estimator with the REML framework, which is the gold standard for covariance estimation in the presence of fixed effects.
2. Diagnostic Tool for Weight Matrices: The authors derive an analytic expression for the Restricted Fisher Information at the null, $\mathcal{I}_r(0)$. This quantity serves as a diagnostic for selecting spatial weight matrices ($W$). It allows analysts to compare candidate weight matrices based on their "local informativeness" and power to detect spatial dependence after covariate adjustment.
3. ESDA Extensions:
   • Scatterplot: A RESAPLE scatterplot is defined analogous to the Moran scatterplot. Unlike the Moran plot, its axes are defined intrinsically in the restricted residual space, providing a decomposition of global dependence into unit-level contributions that account for covariates.
   • Local Statistics: Local contributions ($C_i$) are defined to identify hotspots and cold spots in the residual space.
4. Testing Framework: The paper outlines three testing procedures for RESAPLE:
   • Exact Test: Based on the distribution of quadratic forms (using Imhof's method).
   • Permutation Test: Using the Freedman-Lane procedure, robust to distributional assumptions.
   • Asymptotic Z-test: Based on the one-step construction, suitable for large samples.

4. Results

The authors validate RESAPLE through extensive simulations and a real-world case study.

Simulation Studies

• Estimation Accuracy: Across various sample sizes ($n$), covariate dimensions ($p$), and spatial topologies (regular Queen lattices and irregular B07 graphs), RESAPLE consistently achieves the lowest Root Mean Squared Error (RMSE) compared to Moran's $I$, APLE, MAPLE (covariate-adjusted APLE), and REML.
  • The improvement is driven primarily by reduced bias, particularly when $n$ is small or $p$ is large (leaving few residual degrees of freedom).
  • While RESAPLE sometimes exhibits slightly higher sampling variability than alternatives, the bias reduction results in a superior overall trade-off.
• Testing Power: RESAPLE-based tests demonstrate higher statistical power than Moran's $I$ and APLE/MAPLE, especially in scenarios with small residual degrees of freedom ($r = n-p$) and irregular spatial topologies.
• Weight Matrix Selection: The study confirms that the Restricted Fisher Information $\mathcal{I}_r(0)$ is a strong predictor of test power. Weight matrices with higher $\mathcal{I}_r(0)$ yield higher power.
• Calibration:
  • Permutation tests and Exact tests maintain correct Type I error rates (size) across all scenarios.
  • Asymptotic Z-tests can be overly conservative or inaccurate in small samples or with irregular topologies (e.g., B07), highlighting the need for exact or permutation-based calibration in these settings.

Real Data Case Study

• Data: Median household income in King County, Washington, USA (494 census tracts), adjusted for socioeconomic covariates (education, unemployment, etc.).
• Findings:
  • RESAPLE estimates of $\rho$ tracked the computationally intensive REML estimates more closely than APLE or MAPLE across a "ladder" of increasingly complex mean models.
  • The RESAPLE scatterplot and local contribution maps successfully identified high-income outliers (e.g., Seattle area) and provided interpretable diagnostics for residual spatial dependence.

5. Significance and Conclusion

• Practical Impact: RESAPLE resolves a critical misalignment in spatial analysis workflows. It offers a computationally cheap, closed-form estimator that is theoretically consistent with the "residual-first" approach used by practitioners.
• Robustness: It is robust to the choice of covariates and provides a more accurate summary of spatial dependence than existing indices, particularly in finite samples with many covariates.
• Guidance for Practice: The paper provides a concrete, analytic method ($\mathcal{I}_r(0)$) for resolving the ambiguity of spatial weight matrix selection, moving beyond heuristic choices to information-theoretic criteria.
• Future Directions: The authors suggest extending the framework to SAR-lag models and investigating robustness to heteroscedasticity and distributional misspecification.

In summary, RESAPLE represents a significant advancement in ESDA by unifying the computational convenience of quadratic-form statistics with the statistical rigor of restricted likelihood estimation, offering a superior tool for diagnosing and testing spatial dependence in the presence of covariates.