10 Geographically Weighted Regression

10.1 Geographically Weighted Regression

Simple linear regression assumes the relationship between variables is constant everywhere. But in spatial data, relationships often change across space.

For example:

income may strongly affect housing prices in cities, but less in rural areas
rainfall may strongly influence vegetation in one region, but weakly in another
temperature may affect crop growth differently across climate zones

This means the relationship itself varies by location.

A Geographically Weighted Regression (GWR) models this by allowing regression coefficients to change across space.

In GWR, every location gets its own local regression equation.

GWR is useful when:

relationships vary across space
one global regression does not fit everywhere
regional differences are important
you want to understand where relationships are stronger or weaker

10.2 GWR Formula

The equation for Geographically Weighted Regression is:

\[ y_i = \beta_0(u_i,v_i) + \sum_k \beta_k(u_i,v_i)x_{ik} + \varepsilon_i \]

where:

(yi) = response at location (i)
(xik) = predictor variable(s) at location (i)
(beta(ui,vi)) = local coefficient estimated at coordinates ((ui,vi))
(ui,vi) = spatial location of observation (i)
(epsilon) = random error term

10.3 Step 1: Define a Neighborhood

10.4 Step 2: Create Spatial Weights

10.5 Step 3: Fit Local Regressions

Instead of fitting one regression for the entire dataset, GWR fits one regression at every location.

Each local model uses:

nearby observations
distance-based weights
local predictor-response relationships

This produces local coefficients for every point.

10.6 Fitting GWR

A GWR model can be fit using Python:

import numpy as np
from mgwr.gwr import GWR
from mgwr.sel_bw import Sel_BW

coords = list(zip(df["x_coord"], df["y_coord"]))
X = df[["x1", "x2"]].values
y = df["y"].values.reshape((-1,1))

# choose bandwidth
bw = Sel_BW(coords, y, X).search()

# fit model
gwr_model = GWR(coords, y, X, bw)
results = gwr_model.fit()

print(results.summary())

The output estimates:

local coefficients
local intercepts
local R squared values

10.7 Mapping Coefficients

One major strength of GWR is that coefficients can be mapped.

For example, mapping the coefficient for rainfall may show:

strong positive effect in one region weak effect in another negative effect elsewhere

This reveals:

where relationships are strongest/weakest

10.8 Strengths and Limitationss

Strengths

captures spatially varying relationships
reveals local patterns
produces mappable coefficients

Limitations

sensitive to bandwidth choice
can overfit data

--- title: "Geographically Weighted Regression" --- ## Geographically Weighted Regression Simple linear regression assumes the relationship between variables is constant everywhere. But in spatial data, relationships often change across space. For example: - income may strongly affect housing prices in cities, but less in rural areas - rainfall may strongly influence vegetation in one region, but weakly in another - temperature may affect crop growth differently across climate zones This means the relationship itself varies by location. A **Geographically Weighted Regression (GWR)** models this by allowing regression coefficients to change across space. In GWR, *every location gets its own local regression equation*. GWR is useful when: - relationships vary across space - one global regression does not fit everywhere - regional differences are important - you want to understand *where* relationships are stronger or weaker --- ## GWR Formula The equation for Geographically Weighted Regression is: $$ y_i = \beta_0(u_i,v_i) + \sum_k \beta_k(u_i,v_i)x_{ik} + \varepsilon_i $$ where: - $yi) = response at location \(i$ - $xik) = predictor variable(s) at location \(i$ - $beta(ui,vi)) = local coefficient estimated at coordinates \((ui,vi)$ - $ui,vi) = spatial location of observation \(i$ - (epsilon) = random error term --- ## Step 1: Define a Neighborhood --- ## Step 2: Create Spatial Weights --- ## Step 3: Fit Local Regressions Instead of fitting one regression for the entire dataset, GWR fits one regression at every location. Each local model uses: - nearby observations - distance-based weights - local predictor-response relationships This produces local coefficients for every point. --- ## Fitting GWR A GWR model can be fit using Python: ```python import numpy as np from mgwr.gwr import GWR from mgwr.sel_bw import Sel_BW coords = list(zip(df["x_coord"], df["y_coord"])) X = df[["x1", "x2"]].values y = df["y"].values.reshape((-1,1)) # choose bandwidth bw = Sel_BW(coords, y, X).search() # fit model gwr_model = GWR(coords, y, X, bw) results = gwr_model.fit() print(results.summary()) ``` The output estimates: - local coefficients - local intercepts - local R squared values --- ## Mapping Coefficients One major strength of GWR is that coefficients can be mapped. For example, mapping the coefficient for rainfall may show: strong positive effect in one region weak effect in another negative effect elsewhere This reveals: where relationships are strongest/weakest --- ## Strengths and Limitationss **Strengths** - captures spatially varying relationships - reveals local patterns - produces mappable coefficients **Limitations** - sensitive to bandwidth choice - can overfit data ---