Spatial Statistics

Spatial Statistics

Purpose: answer "is it clustered, where, and why" with defensible inference. The core discipline: spatial data violates independence assumptions, so standard statistics silently overstate significance — every analysis here starts with weights design and ends with residual diagnostics.

Spatial weights (W) — the analysis IS the weights

Every result downstream depends on W; choose it for substantive reasons and run a sensitivity check with one alternative:

from libpysal.weights import Queen

w = Queen.from_dataframe(gdf, use_index=True)
print(f"islands: {w.islands}")   # unconnected units break stats — fix or document
w.transform = "r"                # row-standardize (default for Moran/lag models)

Always report: weights type, parameters, number of islands, and whether results survive an alternative W.

Global → local workflow

1. Global Moran's I (esda.Moran, permutation inference ≥999) — answers "any clustering at all?" Report I, p_sim, and the permutation distribution, not the analytical p.
2. LISA / local Moran (esda.Moran_Local) — maps WHERE: High-High, Low-Low clusters, High-Low/Low-High outliers. Correct for multiple testing (FDR at minimum) before coloring a map — uncorrected LISA maps overstate clusters and this is the field's most common abuse.
3. Getis-Ord Gi* (esda.G_Local, star=True) — hot/cold spots of intensity (a distinct question from Moran clusters — Gi* finds concentrations of high values, LISA finds similarity structure).
4. Rates, not counts, for population-based phenomena; use Empirical Bayes smoothing (esda.smoothing) for small-population units before any of the above — raw rates in sparse units are noise.

Point patterns

• Separate first-order intensity (density varies) from second-order interaction (points attract/repel) — KDE describes the former, Ripley's K/L (pointpats) tests the latter.
• Always test against an inhomogeneous null when the study area has obvious density gradients (population, roads); CSR against a city is a strawman.
• KDE bandwidth drives the story: report it, justify it (Silverman/CV), and show one alternative.

Spatial regression decision path

Run OLS first, then diagnose — never start with a spatial model:

from spreg import OLS
ols = OLS(y, X, w=w, spat_diag=True, moran=True, name_y="price", name_x=xnames)

Decision (Anselin's rule via LM tests): LM-Lag significant & LM-Error not → spatial lag (SAR); reverse → spatial error (SEM); both → compare robust LM versions; neither → OLS stands (report that as a finding). Interpretation caveats: in SAR, coefficients are NOT marginal effects — report direct/indirect (spillover) effects. In SEM, spatial structure is nuisance correlation, no spillover story allowed.

GWR/MGWR (mgwr): when relationships plausibly vary over space. Bandwidth by AICc search; map local coefficients WITH local t-values masked for insignificance; MGWR when predictors operate at different scales. GWR is exploratory — resist causal language on local coefficients.

Inference honesty

• Permutation p-values over analytical ones wherever available.
• Multiple testing: n local tests = n units; FDR-correct.
• MAUP (modifiable areal unit problem): results can flip with unit aggregation — if the aggregation level is a choice, test one alternative and disclose.
• Spatial autocorrelation in residuals after modeling = model still wrong; report residual Moran's I for every final model.
• Correlation ≠ causation applies doubly here: spatially confounded variables (everything correlates with "distance to coast") demand explicit identification strategies before causal claims.

Reporting template

## Spatial analysis: <question>
- Units & n, variable(s), rate smoothing: <...>
- W: <type/params>, islands: <n>, sensitivity W: <type>
- Global: Moran's I = <> (p_perm = <>)
- Local: <k> significant clusters after FDR; map attached
- Model: <OLS/SAR/SEM/GWR> chosen because <LM diagnostics>
- Residual Moran's I: <> — <interpretation>
- Caveats: MAUP, W-sensitivity, causal limits

Execution contract

• Workflow: define inferential question and unit; inspect distributions and rates; construct and justify spatial weights; run global before local tests; fit models if needed; diagnose residual dependence; report uncertainty.
• Decision rules: use spatial statistics for dependence and inference, geostatistics for interpolating sampled continuous surfaces, and predictive ML when out-of-sample prediction is the primary goal.
• Verification protocol: test alternative weights and aggregation, use valid permutation or model inference, correct local multiplicity, inspect residual Moran's I, and distinguish association from causation.
• Failure modes: withhold inferential claims for arbitrary weights, islands ignored, unstable MAUP results, uncorrected multiple tests, residual autocorrelation, or unsupported causal language.
• Deliverables: analysis-ready variables, weights specification, global and local results, corrected significance, diagnostic maps, model and residual checks, sensitivity analysis, and caveats.
• Source freshness: consult the authoritative source registry before applying version-sensitive statistical APIs or defaults.