Geographically Weighted Regression

AdaptiveBandwidth(k)

A nearest-neighbour bandwidth: each local regression uses its k closest observations, so the radius expands in sparse areas and contracts in dense ones.

The more robust default for irregularly spaced data, since every local fit gets the same number of observations. k must comfortably exceed the number of parameters.

Examples

fit_gwr(X, y, coords; bandwidth = AdaptiveBandwidth(30))

BandwidthType

How the GWR neighbourhood around each location is sized. Subtypes: FixedBandwidth and AdaptiveBandwidth.

The bandwidth is the single most consequential GWR choice: too small and the local coefficients chase noise, too large and they flatten towards the global regression.

BisquareKernel()

Bisquare weighting, (1 - (d/h)^2)^2 inside the bandwidth and exactly zero outside.

The default here and the usual GWR choice: the hard cut-off makes each local fit genuinely local and keeps the computation sparse.

See also GaussianKernel, ExponentialKernel.

ExponentialKernel()

Exponential weighting, exp(-d/h). Decays faster near the origin than GaussianKernel and, like it, has no hard cut-off.

FixedBandwidth(h)

A constant kernel bandwidth h, in the units of the coordinates, used at every location.

Appropriate when units are spread evenly. Where density varies, a fixed radius means sparse regions contribute few observations to their local fit — sometimes fewer than there are parameters — so AdaptiveBandwidth is usually safer with irregular data.

Examples

fit_gwr(X, y, coords; bandwidth = FixedBandwidth(2.5))

GWRModel

A fitted geographically weighted regression, returned by fit_gwr. Implements the StatsAPI.RegressionModel interface.

Rather than one coefficient vector, GWR estimates a separate regression at every location, so the headline output is the N x K surface coef_local.

Fields

• coef_local, se_local: N x K local coefficients and their standard errors, columns ordered as coefnames.
• coef_global: column means of coef_local, a summary only.
• y, X, coords, residuals, fitted: the data and the fit. X includes the intercept column that was added for you.
• local_R2: goodness of fit of each local regression.
• bandwidth, bandwidth_type, kernel: the neighbourhood actually used.
• AICc, R2, sigma2: global fit measures.
• hat_trace: tr(S), the effective number of parameters.
• coefnames: "(Intercept)" followed by the covariate names.

GaussianKernel()

Gaussian weighting, exp(-0.5 (d/h)^2).

Weights decline smoothly and never reach zero, so every observation contributes to every local fit. That makes the coefficient surface smooth but means the "local" regression is never strictly local.

See also BisquareKernel.

KernelFunction

Weighting applied to neighbours within a GWR bandwidth. Subtypes: GaussianKernel, BisquareKernel, ExponentialKernel.

MGWRModel

A fitted multiscale geographically weighted regression, returned by fit_mgwr.

Differs from GWRModel in carrying bandwidths, one per covariate, instead of a single bandwidth.

Fields

• coef_local, se_local: N x K local coefficients and standard errors.
• coef_global: column means of coef_local.
• y, X, coords, residuals, fitted: the data and the fit.
• bandwidths: the selected bandwidth per fitted coefficient, aligned with coefnames (intercept first). This is the substantive result — a large value means that relationship is close to constant in space, a small one that it varies locally.
• kernel, AICc, R2, sigma2, coefnames: as for GWRModel.
• iterations, converged: state of the back-fitting loop. Check converged before interpreting anything.

fit_gwr(X, y, coords; bandwidth = nothing, kernel = BisquareKernel(), varnames = nothing)

Fit a geographically weighted regression: a separate weighted least squares at every location.

Each local fit weights observations by their distance from the target location, so the result is a coefficient surface rather than a single vector. This describes how a relationship varies over space; it does not model spatial dependence in the error, which is what fit_spatial and fit_spatial_error are for.

Arguments

• X: N x K covariates, without an intercept — one is added.
• y: response of length N.
• coords: N x 2 matrix of locations.

Keywords

• bandwidth: a FixedBandwidth or AdaptiveBandwidth. nothing (the default) selects an adaptive bandwidth by minimizing corrected AIC, which costs an extra search but is the sensible default.
• kernel: a KernelFunction.
• varnames: covariate names; defaults to x1, x2, ....

Returns

A GWRModel.

Examples

m = fit_gwr(X, y, coords; varnames = ["income", "density"])
m.bandwidth
m.coef_local[:, 2]                        # the surface for "income"

fit_gwr(X, y, coords; bandwidth = AdaptiveBandwidth(30), kernel = GaussianKernel())
plot_gwr_coefficients(m, "income")        # needs a Makie backend

See also fit_mgwr, which lets each covariate have its own bandwidth.

References

Fotheringham, A. S., Brunsdon, C. & Charlton, M. (2002). Geographically Weighted Regression. Wiley.

fit_mgwr(X, y, coords; kernel = BisquareKernel(), varnames = nothing,
         max_iter = 50, tol = 1e-4)

Fit a multiscale GWR, giving every covariate its own bandwidth.

Standard fit_gwr forces one bandwidth on all relationships, which is a real restriction: some processes genuinely operate at a regional scale and others very locally. MGWR relaxes it by back-fitting, cycling through the covariates and re-selecting each bandwidth with the others held fixed until the coefficients stop moving.

The selected bandwidths are themselves the interesting output — a bandwidth near N says that relationship is effectively global.

Arguments

As fit_gwr: X without an intercept, y, and N x 2 coords.

Keywords

• kernel: a KernelFunction.
• varnames: covariate names.
• max_iter, tol: back-fitting iteration cap and convergence tolerance.

Returns

An MGWRModel.

Examples

m = fit_mgwr(X, y, coords; varnames = ["income", "density"])
m.converged || @warn "back-fitting did not converge"
m.bandwidths               # aligned with m.coefnames, intercept first

References

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