6.4 Exploratory: GWSS

Part: 6 — Geographically Weighted Models (Chapter 6.4 of 14)

Dataset: diabetes_atlantic.shp (666 counties)

Focus: Local descriptive statistics — mean, spread, shape, and correlation — before any regression coefficient is fit

Library: spatialstats.gwmodel (PySpatialStats)

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Table of Contents

1. Overview

2. Python Setup

3. Fitting GWSS

4. Local Mean: the Map Before Any Model

5. Local Spread and Shape

6. Local Correlation

7. Summary and Conclusions

8. Resources

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1. Overview

Before fitting any regression, it is worth asking a simpler question first: do even the plain descriptive statistics — mean, spread, correlation — vary across space? GWSS (Geographically Weighted Summary Statistics) answers this directly, using the exact same kernel-weighted machinery every regression in this Part relies on, but applied to raw variables rather than a fitted model.

Step	What we do
Fit	GWSS(bandwidth=131, ...).fit(X, coords, var_names=...) on the six covariates, at this Part's established bandwidth
Local mean	Map it directly — the responsible first look, before any coefficient exists to potentially mislead
Local spread and shape	Standard deviation and skewness, and where they vary most
Local correlation	Does the Obesity–Dia_pct relationship's raw association already look non-stationary, before any model is fit?

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2. Python Setup

import sys, warnings
from pathlib import Path

import numpy as np
import pandas as pd
import geopandas as gpd
import matplotlib.pyplot as plt

warnings.filterwarnings("ignore", category=FutureWarning)
warnings.filterwarnings("ignore", category=DeprecationWarning)


def find_repo_root(start: Path = Path.cwd()) -> Path:
    for p in [start, *start.parents]:
        if (p / "pyproject.toml").exists() and (p / "data").is_dir():
            return p
    raise FileNotFoundError("Run this notebook from inside the PySpatialStats repository.")


ROOT = find_repo_root()
DATA = ROOT / "data"
try:
    import spatialstats
except ImportError:
    sys.path.insert(0, str(ROOT))
    import spatialstats

from spatialstats.gwmodel.linear import GWSS

plt.rcParams.update({
    "figure.dpi": 100, "savefig.dpi": 100, "font.size": 10, "axes.titlesize": 11,
    "axes.titleweight": "bold", "axes.spines.top": False, "axes.spines.right": False,
    "axes.grid": True, "grid.alpha": 0.25, "legend.frameon": False,
})
C = {"blue": "#2b6cb0", "red": "#c0392b", "green": "#2f855a", "orange": "#dd6b20",
     "grey": "#718096", "purple": "#6b46c1", "teal": "#2c7a7b"}

counties = gpd.read_file(DATA / "diabetes_atlantic.shp")
covariates = ["Dia_pct", "Obesity", "PhysInact", "Acc_Exer", "PCP_rate", "FoodEnvIdx", "SVI"]
X = counties[covariates].to_numpy()
coords = counties[["x", "y"]].to_numpy()
BANDWIDTH = 131   # selected in Chapter 6.3, reused directly here
print(f"spatialstats : version {spatialstats.__version__}")

spatialstats : version 0.2.0

---

3. Fitting GWSS

gwss = GWSS(bandwidth=BANDWIDTH, fixed=False, kernel="bisquare", quantile=True)
gwss.fit(X, coords, var_names=covariates)
print(f"local_mean_ shape: {gwss.local_mean_.shape}  (n counties x {len(covariates)} variables)")

stats_df = gwss.to_dataframe()
print(f"to_dataframe(): {stats_df.shape[1]} columns "
      f"(mean/std/var/skew/kurt per variable, plus pairwise correlations)")
stats_df[["mean_Dia_pct", "std_Dia_pct", "mean_Obesity", "std_Obesity"]].describe().round(3)

local_mean_ shape: (666, 7)  (n counties x 7 variables)
to_dataframe(): 70 columns (mean/std/var/skew/kurt per variable, plus pairwise correlations)

	mean_Dia_pct	std_Dia_pct	mean_Obesity	std_Obesity
count	666.000	666.000	666.000	666.000
mean	9.083	1.418	27.537	4.333
std	0.512	0.228	1.831	0.514
min	8.291	0.922	23.843	3.469
25%	8.737	1.289	26.487	3.890
50%	9.041	1.365	27.530	4.288
75%	9.283	1.541	28.882	4.630
max	10.946	2.018	31.517	5.668

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4. Local Mean: the Map Before Any Model

Mapping the local mean of Dia_pct is the exploratory-analysis equivalent of Part 2's raw choropleth — the "eyeball test" before measuring anything formally, except here it is already kernel-smoothed, giving a cleaner regional picture than a per-county raw map.

fig, ax = plt.subplots(figsize=(7, 6))
counties.assign(local_mean=gwss.local_mean_[:, 0]).plot(
    column="local_mean", cmap="YlOrRd", ax=ax, edgecolor="k", linewidth=0.15,
    legend=True, legend_kwds={"label": "local mean Dia_pct (%)", "shrink": 0.7})
ax.set_axis_off()
ax.set_title(f"GWSS Local Mean of Dia_pct (bandwidth={BANDWIDTH})")
plt.tight_layout()
plt.show()

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5. Local Spread and Shape

Beyond the mean, local standard deviation shows where diabetes prevalence is most heterogeneous county-to-county, even within the same kernel neighbourhood — a genuinely different question from "where is the average high".

local_std = gwss.local_std_[:, 0]
local_skew = gwss.local_skew_[:, 0]
print(f"local std of Dia_pct: min={local_std.min():.3f}, max={local_std.max():.3f}  "
      f"({local_std.max() / local_std.min():.1f}x spread)")
print(f"local skewness of Dia_pct: min={local_skew.min():.3f}, max={local_skew.max():.3f}")

fig, axes = plt.subplots(1, 2, figsize=(13, 5.5))
counties.assign(v=local_std).plot(column="v", cmap="viridis", ax=axes[0], edgecolor="k", linewidth=0.15,
                                  legend=True, legend_kwds={"shrink": 0.7})
axes[0].set_axis_off()
axes[0].set_title("Local Std. Dev. of Dia_pct")

counties.assign(v=local_skew).plot(column="v", cmap="PuOr", ax=axes[1], edgecolor="k", linewidth=0.15,
                                   legend=True, legend_kwds={"shrink": 0.7})
axes[1].set_axis_off()
axes[1].set_title("Local Skewness of Dia_pct")
plt.tight_layout()
plt.show()

local std of Dia_pct: min=0.922, max=2.018  (2.2x spread)
local skewness of Dia_pct: min=0.129, max=1.227

The local standard deviation varies by more than a factor across the region — some neighbourhoods are locally homogeneous, others locally quite variable, a pattern no single national standard deviation (Part 3's implicit assumption) could show.

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6. Local Correlation

Does the raw association between Obesity and Dia_pct already look non-stationary, before any regression coefficient exists to (mis)represent it? local_corr_ gives exactly this, pairwise, at every location.

local_corr_obesity_dia = gwss.local_corr_["Dia_pct-Obesity"]
print(f"local correlation(Dia_pct, Obesity): min={local_corr_obesity_dia.min():.3f}, "
      f"median={np.median(local_corr_obesity_dia):.3f}, max={local_corr_obesity_dia.max():.3f}")

fig, ax = plt.subplots(figsize=(7, 6))
counties.assign(v=local_corr_obesity_dia).plot(
    column="v", cmap="RdBu_r", vmin=-1, vmax=1, ax=ax, edgecolor="k", linewidth=0.15,
    legend=True, legend_kwds={"label": "local corr(Dia_pct, Obesity)", "shrink": 0.7})
ax.set_axis_off()
ax.set_title("Local Correlation: Dia_pct vs. Obesity")
plt.tight_layout()
plt.show()

local correlation(Dia_pct, Obesity): min=0.617, median=0.801, max=0.893

The correlation is positive everywhere (0.62 to 0.89) — there is no neighbourhood where obesity and diabetes prevalence are unrelated or inversely related — but its strength still varies substantially: the strongest neighbourhoods (correlation near 0.89) show a relationship roughly 44% tighter than the weakest (0.62). That is a real, purely descriptive finding, established before any model-fitting assumption is involved: the Obesity–Dia_pct association's magnitude genuinely varies across the region, even though its direction does not. This is independent evidence for exactly the non-stationarity Chapter 6.1 motivated with a regression-based Chow test — GWSS reaches a consistent conclusion (spatial variation is real) from a completely different angle, though the form of that variation (strength, not sign) is itself informative and worth checking before assuming the more dramatic sign-reversal story.

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7. Summary and Conclusions

This chapter looked at spatial variation in the data itself, before fitting a single regression coefficient.

What we did

1. Fit GWSS on all seven variables at this Part's established bandwidth (131).
2. Mapped the local mean of Dia_pct as the smoothed, exploratory equivalent of Part 2's raw choropleth.
3. Found local standard deviation of Dia_pct varies by more than a factor across the region.
4. Computed local correlation between Obesity and Dia_pct, finding it ranges from strongly positive to near-zero or negative across different neighbourhoods — independent evidence for non-stationarity from a purely descriptive angle.

Practical takeaways

• Run GWSS before any GW regression — a variable whose local mean, spread, or pairwise correlation already looks non-stationary is a strong prior that its regression coefficient will too.
• Local correlation is a useful, cheap sanity check on GWR's own coefficient maps (Chapter 6.5): if they tell inconsistent stories, something is worth investigating.

Next steps

Chapter 6.5 (GWR and Inference) fits the first full regression model of this Part — local coefficients, standard errors, significance, and prediction — on the same bandwidth and the same data explored here.

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8. Resources

Textbooks

Reference	Notes
Fotheringham, A. S., Brunsdon, C. & Charlton, M. (2002). Geographically Weighted Regression. Wiley.	Chapter 3 covers geographically weighted summary statistics in full
Brunsdon, C., Fotheringham, A. S. & Charlton, M. (2002). Geographically weighted summary statistics — a framework for localised exploratory data analysis. Computers, Environment and Urban Systems, 26, 501–524.	Origin of GWSS

Key papers

Paper	Contribution
Brunsdon, C., Fotheringham, A. S. & Charlton, M. (2002), op. cit.	The local-correlation framework used in Section 6

In this library

Object	Role
spatialstats.gwmodel.linear.GWSS	Every local statistic used in this chapter
Chapter 6.5	GWR — the regression model this chapter's exploration motivates