Gamerman-Moreira-Rue (2003) Space-Varying Regression Models

Summary

Gamerman, Moreira, and Rue (2003) extend the standard linear regression model by allowing regression coefficients to vary across spatial units, specifying spatial smoothness via a multivariate pairwise difference prior — a Markov random field (MRF) prior that penalises differences between neighbouring units. Bayesian inference is via Markov chain Monte Carlo (MCMC); the paper's central contribution is a systematic comparison of three sampling schemes that differ in how aggressively they block the high-dimensional coefficient vector. The approach is illustrated on simulated Brazilian microregion data and applied to a space-varying re-estimation of the Andersen-Granger-Reis (1997) vector autoregression (VAR) model of Amazon land use, showing clear North-to-South spatial heterogeneity in cropland-to-pasture conversion dynamics.

Key Claims

• Space-Varying Regression Model (SVRM). Replace the global coefficient $\beta$ with unit-specific vectors $\beta_i$, so that $\mu_i = x_i'\beta_i$. The full parameter matrix $B$ (rows $\beta_1', \ldots, \beta_n'$) is given the pairwise difference prior: $$f(\beta|\kappa,\eta) \propto \kappa^{nr/2}|\eta|^{n/2}\exp\!\Bigl\{-\tfrac{\kappa}{2}\sum_{i\sim j}(\beta_i-\beta_j)'\eta(\beta_i-\beta_j)\Bigr\}$$ where $\eta$ is an $r\times r$ precision hyperparameter and neighbours $i\sim j$ are defined by shared boundaries. This is an improper prior (rows of the graph Laplacian $W$ sum to zero) but yields proper posteriors under proper priors on $\kappa$ and $\eta$.

• Full conditional of $\beta_i$ (Scheme A — single-site Gibbs). Given neighbouring values $\beta_{\partial i}$: $$({\beta_i}|\beta_{\partial i},\kappa,\eta,y) \sim N(a_i,\, \kappa^{-1}R_i)$$ $$R_i = (x_ix_i' + w_{i+}\eta)^{-1}, \qquad a_i = R_i(x_i y_i + w_{i+}\eta\bar\beta_{\partial i})$$ where $\bar\beta_{\partial i}=(1/w_{i+})\sum_{j\sim i}w_{ij}\beta_j$ is the weighted-neighbour mean. Fast per iteration ($r\times r$ inversions only) but slow mixing due to strong neighbour correlation.

• Scheme B — block Normal-Gamma update. Jointly sample $(\beta,\kappa)$ from the Normal-Gamma marginal $({\beta,\kappa}|\eta,y)\sim\text{NG}(a,R,\hat\nu_\kappa/2,\hat S_\kappa/2)$ where $R=(X'X+W\otimes\eta)^{-1}$ and $a=RX'y$. The $nr\times nr$ matrix $R$ is sparse (MRF structure); Rue (2001) Cholesky enables efficient computation.

• Scheme C — marginalise out $(\beta,\kappa)$, sample $\eta$ only. Analytically integrate over $(\beta,\kappa)$ to obtain the marginal $\pi(\eta|y)$, then draw $\eta$ via Metropolis-Hastings or sampling importance resampling (SIR). Achieves the best chain mixing; requires no large matrix storage per iteration.

• MCMC efficiency ranking. C > B > A in terms of effective sample size per unit time, confirmed on the simulated dataset. The gap is smaller than in time-series state-space comparisons (Frühwirth-Schnatter 1994): strong likelihood information for $\beta_i$ makes single-site updating competitive. Scheme A is approximately 20× faster per iteration.

• Proper prior extension. Adding a pinning term $\delta\sum_i(\beta_i-b_i)'\eta(\beta_i-b_i)$ makes the prior proper; the full conditional becomes a weighted average of the spatial neighbour mean and anchor $b_i$ with weight $w_i = w_{i+}/(w_{i+}+\delta)$.

• Amazon land use application. Space-varying version of the Andersen-Granger-Reis (1997) VAR for $n=228$ Brazilian counties, $K=4$ land usages (forest, cropland, pasture, fallow), $T=4$ time points (1970–1985). Spatially varying model beats fixed-coefficient model by the Gelfand-Ghosh (1998) predictive loss criterion ($D=0.016$ vs. $D=0.035$). Main finding: the effect of lagged cropland area on pasture conversion is substantially stronger in the South (the main human access zone) than in the North — a clear spatial gradient consistent with low-technology slash-and-burn agriculture.

Concepts Introduced or Extended

• Space-Varying Regression Model
• Conditional Autoregressive Model
• Bayesian Hierarchical Model

Entities Mentioned

• Dani Gamerman
• Ajax R.B. Moreira
• Håvard Rue

Quotes

"Results show that the scheme that integrates out regression coefficients and observation precision from the model provides better mixing properties and should be preferred whenever computation is feasible."

"The application reveals important spatial heterogeneity: the conversion coefficient from cropland to pasture is strongest in the South of the Amazon where human access is greatest, suggesting low-technology land clearing predominates near the agricultural frontier."

My Take

A clean methodological paper with a single useful idea — the multivariate pairwise difference prior for spatially varying regression — and a thorough MCMC comparison. The core result (scheme C best, but A not catastrophic because the likelihood is informative) is practically useful for applied work. The Amazon application is illustrative but thin: four time points and a single-equation summary obscure the VAR dynamics. The paper's lasting contribution is showing that the improper intrinsic conditional autoregressive (CAR) prior extends naturally to multivariate coefficient vectors, and that analytical marginalisation of the conjugate Normal-Gamma layer eliminates mixing problems at modest computational cost.