Spatial Statistics

Part of Statistical Methods — MORIE’s statistical-methods reference.

MORIE provides a comprehensive spatial statistics library covering global and local autocorrelation, geostatistical interpolation, geographically weighted regression, and point pattern analysis. The implementation achieves 100% chapter coverage of Schabenberger & Gotway (2005) across all 9 chapters.

All spatial functions live in morie/fn/ as individual files and accept arbitrary column names via keyword parameters.

Global Autocorrelation

Measures of spatial dependence across the entire study region.

  • morai — Moran’s \(I\). Global spatial autocorrelation index in \([-1, +1]\). Schabenberger & Gotway Ch. 7.

  • geary — Geary’s \(C\). Dissimilarity-based autocorrelation in \([0, 2]\). Ch. 7.

  • getgo — Getis-Ord general \(G\). Hot/cold-spot clustering for positive attributes. Ch. 7.

  • jncnt — Join count. Binary spatial autocorrelation on categorical data. Ch. 7.

from morie.fn import moran_permutation_test, gearyc
from morie.fn import _array_core as np   # numpy arrays work too

values = np.array([2.5, 3.1, 2.8, 4.0, 3.7])
W = np.array([[0, 1, 0, 0, 1],
              [1, 0, 1, 0, 0],
              [0, 1, 0, 1, 0],
              [0, 0, 1, 0, 1],
              [1, 0, 0, 1, 0]], dtype=float)

result = moran_permutation_test(values, W, nsim=199, seed=1)   # as spdep::moran.mc
print(f"Moran's I = {result['statistic']:.4f}, p = {result['p_value']:.3f}")
print(gearyc(values, W))                                        # Geary's C with its reading

Local Indicators of Spatial Association (LISA)

Local decompositions that identify clusters and spatial outliers.

  • lisa — local Moran’s \(I\). Per-location cluster / outlier classification (HH, HL, LH, LL).

  • getgi — Getis-Ord \(G_i^{*}\). Local hot/cold-spot z-scores.

  • lgeary — local Geary. Local dissimilarity decomposition.

Each function returns per-observation statistics, pseudo p-values from conditional permutation tests (default 999 permutations), and cluster classification labels.

Geostatistical Interpolation

Prediction of values at unsampled locations using spatial covariance structure.

Variogram modelling:

  • svari — semivariogram estimation. Parameters: lag_dist, n_lags, model (spherical / exponential / gaussian).

  • vfit — variogram fitting. Nugget, sill, range via weighted least squares.

Interpolation methods:

  • krige — ordinary kriging. BLUP with variogram model; returns predictions and variance.

  • ukrig — universal kriging. Kriging with external drift / trend surface.

  • idw — inverse distance weighting. Power parameter (default \(p = 2\)); no variance estimate.

  • cokriging / cokrg — co-kriging. Multivariate kriging using cross-variograms.

from morie.fn import empirical_variogram, krige
from morie.fn import _array_core as np

coords = np.array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [1.0, 1.0], [0.5, 0.5], [2.0, 1.0], [1.5, 0.2], [0.2, 1.7]])
z = np.array([1.0, 1.4, 0.9, 1.6, 1.2, 2.1, 1.5, 1.1])
gamma = empirical_variogram(coords, z, model="spherical")   # the sample variogram with a spherical fit
print(gamma["nugget"], gamma["psill"], gamma["range"])
# krige(z, coords, new_coords, model) predicts at new locations with that model (as gstat::krige)

Geographically Weighted Regression (GWR)

Spatially varying coefficient models that capture local heterogeneity in regression relationships.

  • gwr — basic GWR with Gaussian / bisquare kernel; bandwidth via AICc, cross-validation, or fixed.

  • gwpca — geographically weighted PCA; adaptive bandwidth.

  • stgwr — spatio-temporal GWR; joint space-time bandwidth.

Spatial Weight Matrices

Functions for constructing spatial connectivity structures:

  • wqueen — queen contiguity (shared edge or vertex).

  • wrook — rook contiguity (shared edge only).

  • wknn — k-nearest neighbours.

  • wdist — distance-based (threshold or inverse distance).

  • wrow — row-standardise any weight matrix.

Point Pattern Analysis

  • ripk — Ripley’s K-function and L-function for clustering / dispersion.

  • stkde — spatio-temporal kernel density estimation.

  • stscan — space-time scan statistic (Kulldorff).

Density-Based Clustering

  • stdbs — DBSCAN with spatio-temporal distance metric. Used in morie.tps_spatial_advanced to find connected hot clusters across the TPS incident feed and produce yearly small-multiples with a four-crime overlay (Assault / Robbery / Auto Theft / Break-and-Enter).

  • hdbsc — HDBSCAN hierarchical variant for density-varying data.

Kulldorff Space-Time Scan

The Kulldorff scan statistic detects significant spatio-temporal clusters by maximising a likelihood ratio over candidate cylindrical windows in space-time. Implementation in stscan returns the most likely cluster, its log-likelihood ratio, the Monte-Carlo p-value, and the included neighbourhoods + time window. The Hohl-style “panel d” visualisation is rendered by morie.tps_render.render_satscan_panel — it draws the significant cluster polygons over a base choropleth.

Reference: Kulldorff, M. (1997). A spatial scan statistic. Communications in Statistics 26(6):1481–1496.

Visualisation Helpers

The morie.tps_render and morie.tps_spatial_advanced modules expose publication-style plot helpers commonly paired with the spatial estimators above:

  • morie.tps_render.render_choropleth — choropleth (per 100k, 9 categories).

  • morie.tps_render.render_district_proportional — district proportional-symbol map (6 former-borough divisions).

  • morie.tps_render.render_quad — 4-panel quad (density / rate / LISA / \(G_i^{*}\)).

  • morie.tps_render.render_yearly_grid — yearly small-multiples (4-crime overlay + DBSCAN clusters).

  • morie.tps_render.render_dbscan — DBSCAN cluster overlay.

  • morie.tps_spatial_advanced.bivariate_moran — bivariate Moran scatter.

  • morie.tps_spatial_advanced.moran_sweep_heatmap — Moran sweep heatmap (categories × years).

  • morie.tps_render.render_satscan_panel — Kulldorff panel d (Hohl-style).

Style follows Hohl, A. Geographic visualisation of disease cluster detection results.

Spatio-Temporal Extensions

25 spatio-temporal functions cover panel data, trajectory analysis, and dynamic spatial processes: stacf, stscan, stkde, stvar, stgwr, trajd, stdbs, ripk, gwpca, stmrn, and 15 additional functions for spatio-temporal autocorrelation, clustering, and prediction.

References

[Schabenberger2005]

Schabenberger, O. & Gotway, C.A. (2005). Statistical Methods for Spatial Data Analysis. Chapman & Hall / CRC. 9 chapters, 100% coverage in MORIE fn/ files.

[Armstrong2000]

Armstrong, M.P. et al. (2000). Spatial voting analysis. 28/28 backends implemented.

[Anselin1995]

Anselin, L. (1995). Local Indicators of Spatial Association — LISA. Geographical Analysis 27(2):93–115.