Multiscale Geographically Weighted Regression (GWRMultiscale)

Mathematical Foundation

Multiscale GWR (MGWR) extends basic GWR by allowing each independent variable to have its own bandwidth. Different spatial processes may operate at different spatial scales — the influence of some variables may be highly local (small bandwidth), while others may have regional or global scale (large bandwidth).

The MGWR model form is the same as basic GWR:

\[y_i = \beta_{i0}(b_0) + \sum_{k=1}^{p} \beta_{ik}(b_k) x_{ik} + \varepsilon_i\]

However, each coefficient \(\beta_{ik}\) is estimated using its own bandwidth \(b_k\). Bandwidth calibration is carried out via the backfitting algorithm [1]:

1. Initialisation: Compute initial coefficient estimates \(\boldsymbol{\beta}^{(0)}\) using standard GWR with an initial spatial weighting configuration.

2. Backfitting: For each iteration \(t = 1, 2, \dots\):

   1. For each variable \(k\):

      • Fix the coefficients of all other variables and compute the residual of the dependent variable for the current variable.

      • Select the optimal bandwidth \(b_k\) for variable \(k\) (golden section search with CV/AIC criterion).

      • Fit new coefficients for variable \(k\) using \(b_k\).

      • Update the residuals.

   2. Check the convergence criterion (CVR or dCVR); stop if satisfied.

3. Diagnostics: Compute RSS, AICc, ENP, EDF, R², and other diagnostic metrics.

Convergence Criteria

• CVR (Change in RSS):

  \[\text{CVR} = |RSS_t - RSS_{t-1}|\]

• dCVR (relative Change in RSS, default):

  \[\text{dCVR} = \sqrt{\frac{|RSS_t - RSS_{t-1}|}{RSS_t}}\]

Key Parameters

Parameter	Description	Default
weights	A list of bandwidth weights, one per variable (including intercept)	Required
distance	The distance metric shared by all variables	CRSDistance()
has_intercept	Whether to include an intercept term	True
bandwidth_initilize	Bandwidth initialisation type per variable	All Null (auto-select)
bandwidth_selection_approach	Bandwidth selection criterion per variable	All CV
preditor_centered	Whether to centre each predictor before fitting	All False
criterion_type	Backfitting convergence criterion type	dCVR
max_iteration	Maximum number of iterations	500
criterion_threshold	Convergence threshold	1e-6
has_hat_matrix	Whether to compute the hat matrix (for diagnostics)	True
adaptive_lower	Lower bound on neighbour count for adaptive bandwidth selection	10

Bandwidth Initialisation Types

Type	Meaning
BandwidthInitilizeType.Null	Not specified; auto-select via golden section search during backfitting
BandwidthInitilizeType.Specified	User-specified; skip bandwidth selection and use the fixed value from weights
BandwidthInitilizeType.Initial	Initially optimised; may still be adjusted in later backfitting iterations

Code Examples

Basic Usage (Auto-Select Bandwidths)

from pygwmodel import GWRMultiscale, BandwidthWeight, CRSDistance

n_var = 4  # intercept + 3 independent variables
weights = [BandwidthWeight(36.0, adaptive=True) for _ in range(n_var)]

algorithm = GWRMultiscale(
    data,
    depen_var="PURCHASE",
    indep_vars=["FLOORSZ", "UNEMPLOY", "PROF"],
    weights=weights,
    distance=CRSDistance()
).fit()

# Diagnostic information
print(algorithm.diagnostic)

# Optimised bandwidths per variable
for w in algorithm.weights:
    print(f"bandwidth={w.bandwidth}, adaptive={w.adaptive}")

# Result layer
result = algorithm.result_layer
print(result.columns)
# Intercept, FLOORSZ, UNEMPLOY, PROF,
# Intercept_SE, FLOORSZ_SE, ..., Intercept_TV, ..., fitted

Specified Bandwidths

from pygwmodel import GWRMultiscale

n_var = 4
spec = GWRMultiscale.BandwidthInitilizeType.Specified
cv = GWRMultiscale.BandwidthSelectionCriterionType.CV

algorithm = GWRMultiscale(
    data, y, x,
    weights=[BandwidthWeight(36.0, adaptive=True) for _ in range(n_var)],
    bandwidth_initilize=[spec] * n_var,
    bandwidth_selection_approach=[cv] * n_var
).fit()

Note: when bandwidth_initilize is set to Specified, the bandwidths remain fixed; otherwise they are iteratively optimised during backfitting.

Adjusting Convergence

algorithm = GWRMultiscale(data, y, x, weights)
algorithm.criterion_type = GWRMultiscale.BackFittingCriterionType.CVR
algorithm.criterion_threshold = 1e-4
algorithm.max_iteration = 100
algorithm.fit()

References

[1]

Fotheringham, A. S., Yang, W., & Kang, W. (2017). Multiscale geographically weighted regression (MGWR). Annals of the American Association of Geographers, 107(6), 1247-1265.

• Yu, H., Fotheringham, A. S., Li, Z., Oshan, T., Kang, W., & Wolf, L. J. (2020). Inference in multiscale geographically weighted regression. Geographical Analysis, 52(1), 87-106.