=== versions === python 3.12.0 | mgwr 2.2.1 | spreg 1.9.1 | libpysal 4.15.1 | esda 2.10.0 | geopandas 1.2.0 | numpy 2.5.4 === data === 301 MSOAs; y = standardised ln(median price); X = standardised level4, detached, no_car, ln_density SD of ln(price) = 0.333; SDs of X: level4 9.66, detached 12.13, no_car 12.02, ln_density 0.99 === global models on the standardised data (as in Part 1) === OLS R2 0.739 AICc 462.0 residual Moran's I +0.457 (p 0.0001) betas -0.000 +0.487 +0.149 -0.409 +0.008 Spatial lag (ML) R2 0.848 AICc 319.9 residual Moran's I +0.094 (p 0.0031) betas -0.016 +0.291 +0.157 -0.269 +0.038 Spatial error (ML) R2 0.696 AICc 301.7 residual Moran's I -0.074 (p 0.0164) betas +0.031 +0.227 +0.179 -0.381 -0.059 === GWR bandwidth selection (adaptive bisquare, AICc) === optimal bandwidth: 62 nearest neighbours (21% of the MSOAs); search took 5.0 s AICc profile: 30: 352.2, 60: 282.9, 90: 284.0, 120: 291.8, 150: 298.6, 180: 308.9, 210: 322.7, 240: 341.3, 270: 361.6, 300: 405.2 GWR fit in 0.12 s =========================================================================== Model type Gaussian Number of observations: 301 Number of covariates: 5 Global Regression Results --------------------------------------------------------------------------- Residual sum of squares: 78.527 Log-likelihood: -224.878 AIC: 459.756 AICc: 462.041 BIC: -1610.778 R2: 0.739 Adj. R2: 0.736 Variable Est. SE t(Est/SE) p-value ------------------------------- ---------- ---------- ---------- ---------- X0 -0.000 0.030 -0.000 1.000 X1 0.487 0.033 14.623 0.000 X2 0.149 0.052 2.874 0.004 X3 -0.409 0.049 -8.277 0.000 X4 0.008 0.041 0.195 0.846 Geographically Weighted Regression (GWR) Results --------------------------------------------------------------------------- Spatial kernel: Adaptive bisquare Bandwidth used: 62.000 Diagnostic information --------------------------------------------------------------------------- Residual sum of squares: 27.995 Effective number of parameters (trace(S)): 56.517 Degree of freedom (n - trace(S)): 244.483 Sigma estimate: 0.338 Log-likelihood: -69.650 AIC: 254.334 AICc: 282.094 BIC: 467.555 R2: 0.907 Adjusted R2: 0.885 Adj. alpha (95%): 0.004 Adj. critical t value (95%): 2.868 Summary Statistics For GWR Parameter Estimates --------------------------------------------------------------------------- Variable Mean STD Min Median Max -------------------- ---------- ---------- ---------- ---------- ---------- X0 0.012 0.381 -0.622 -0.069 0.979 X1 0.283 0.125 0.024 0.262 0.628 X2 0.080 0.169 -0.641 0.092 0.540 X3 -0.487 0.179 -0.794 -0.506 -0.116 X4 -0.127 0.149 -0.554 -0.115 0.143 =========================================================================== === GWR local significance with the corrected critical t-value === effective number of parameters ENP = 56.5; adjusted alpha = 0.05 * k / ENP = 0.00442; critical |t| = 2.87 (vs 1.97 uncorrected) Intercept beta min -0.622 median -0.069 max +0.979 | significant: 177 (73 +, 104 -) | uncorrected: 211 level4 beta min +0.024 median +0.262 max +0.628 | significant: 157 (157 +, 0 -) | uncorrected: 210 detached beta min -0.641 median +0.092 max +0.540 | significant: 1 (0 +, 1 -) | uncorrected: 42 no_car beta min -0.794 median -0.506 max -0.116 | significant: 215 (0 +, 215 -) | uncorrected: 247 ln_density beta min -0.554 median -0.115 max +0.143 | significant: 36 (0 +, 36 -) | uncorrected: 103 === MGWR bandwidth selection (back-fitting, AICc) === bandwidth search 110.7 s; search + fit 111.7 s (GWR search took 5.0 s) =========================================================================== Model type Gaussian Number of observations: 301 Number of covariates: 5 Global Regression Results --------------------------------------------------------------------------- Residual sum of squares: 78.527 Log-likelihood: -224.878 AIC: 459.756 AICc: 462.041 BIC: -1610.778 R2: 0.739 Adj. R2: 0.736 Variable Est. SE t(Est/SE) p-value ------------------------------- ---------- ---------- ---------- ---------- X0 -0.000 0.030 -0.000 1.000 X1 0.487 0.033 14.623 0.000 X2 0.149 0.052 2.874 0.004 X3 -0.409 0.049 -8.277 0.000 X4 0.008 0.041 0.195 0.846 Multi-Scale Geographically Weighted Regression (MGWR) Results --------------------------------------------------------------------------- Spatial kernel: Adaptive bisquare Criterion for optimal bandwidth: AICc Score of Change (SOC) type: Smoothing f Termination criterion for MGWR: 1e-05 MGWR bandwidths --------------------------------------------------------------------------- Variable Bandwidth ENP_j Adj t-val(95%) Adj alpha(95%) X0 54.000 13.675 2.929 0.004 X1 298.000 1.373 2.102 0.036 X2 300.000 1.340 2.092 0.037 X3 300.000 1.203 2.047 0.042 X4 43.000 20.053 3.050 0.002 Diagnostic information --------------------------------------------------------------------------- Residual sum of squares: 30.090 Effective number of parameters (trace(S)): 37.645 Degree of freedom (n - trace(S)): 263.355 Sigma estimate: 0.338 Log-likelihood: -80.514 AIC: 238.317 AICc: 250.042 BIC: 381.579 R2 0.900 Adjusted R2 0.886 Summary Statistics For MGWR Parameter Estimates --------------------------------------------------------------------------- Variable Mean STD Min Median Max -------------------- ---------- ---------- ---------- ---------- ---------- X0 0.044 0.341 -0.516 -0.047 0.773 X1 0.260 0.007 0.247 0.259 0.278 X2 0.104 0.007 0.090 0.103 0.119 X3 -0.506 0.007 -0.516 -0.506 -0.489 X4 -0.128 0.171 -0.445 -0.088 0.161 =========================================================================== Intercept bandwidth 54 (95% interval 51-65) ENP_j 13.7 critical |t| 2.93 | beta min -0.516 median -0.047 max +0.773 | significant 188 (92 +, 96 -) level4 bandwidth 298 (95% interval 202-298) ENP_j 1.4 critical |t| 2.10 | beta min +0.247 median +0.259 max +0.278 | significant 301 (301 +, 0 -) detached bandwidth 300 (95% interval 240-300) ENP_j 1.3 critical |t| 2.09 | beta min +0.090 median +0.103 max +0.119 | significant 301 (301 +, 0 -) no_car bandwidth 300 (95% interval 240-300) ENP_j 1.2 critical |t| 2.05 | beta min -0.516 median -0.506 max -0.489 | significant 301 (0 +, 301 -) ln_density bandwidth 43 (95% interval 43-57) ENP_j 20.1 critical |t| 3.05 | beta min -0.445 median -0.088 max +0.161 | significant 83 (0 +, 83 -) === local R2 and local collinearity === GWR local R2: min 0.69, median 0.86, max 0.95; global GWR R2 0.907, MGWR R2 0.900 lowest local R2: Farnley East (Leeds) 0.69; Holbeck (Leeds) 0.70; Farnley West & Gamble Hill (Leeds) 0.71 GWR local condition number: min 3.0, median 5.7, max 18.4; > 30 in 0 MSOAs GWR local VIF max per variable: level4 3.7, detached 13.0, no_car 7.1, ln_density 5.2; MSOAs with any VIF > 10: 4 strongest local correlation: detached vs no_car, from -0.92 to -0.65 (global -0.78) MGWR local condition number: median 3.4, max 5.8; > 30 in 0 MSOAs === comparison with the global models === (R2 for the lag model is spreg's pseudo R2; for the error model it is the R2 of the non-spatial part) model R2 AICc resid_I resid_p ENP OLS 0.739 462.041 0.457 0.000 5.000 Spatial lag (ML) 0.848 319.916 0.094 0.003 6.000 Spatial error (ML) 0.696 301.679 -0.074 0.016 6.000 GWR (bw 62) 0.907 282.094 -0.027 0.254 56.517 MGWR 0.900 250.042 -0.034 0.189 37.645 wrote figures, output/gwr_mgwr_results.gpkg and output/metrics.json