A relationship can be true in one place and weak somewhere else without either place being wrong.
More tree cover may strongly reduce local heat in one urban district and have a smaller effect in another because building form, wind, humidity and surface materials differ. Proximity to rail may raise housing prices sharply in one city and more modestly in another because car ownership, service quality, zoning and neighbourhood reputation differ. Rainfall may translate into flood risk differently across catchments because soils, slopes and drainage systems differ.
Geographers call this spatial nonstationarity: the possibility that the relationship between variables changes across space.
Spatial nonstationarity asks a difficult question: what if there is no single coefficient that belongs to the whole map?
Quick Read: The Nonstationarity Mechanism
SAME VARIABLES + DIFFERENT LOCAL CONTEXT → DIFFERENT STRENGTH / DIRECTION / FORM OF RELATIONSHIP → GLOBAL AVERAGE HIDES LOCAL STRUCTURE
1. A Global Relationship Is an Average
When one model estimates one relationship for an entire study area, it assumes the underlying relationship is reasonably stable across space. Sometimes that is useful. Sometimes the average hides important local differences.
2. Nonstationarity Is More Specific Than Heterogeneity
Spatial Heterogeneity owns the broad fact that places differ. Spatial nonstationarity asks something narrower: does the relationship between two or more variables itself vary from place to place?
3. Context Can Change Effect Size
The same intervention may have a strong effect where infrastructure is weak and a small effect where infrastructure is already excellent. The cause exists in both places, but the marginal effect differs.
4. Context Can Change Direction
A new road may raise nearby land values in one region by improving accessibility and lower them elsewhere because noise and traffic dominate. The sign of the relationship can reverse when different mechanisms outweigh one another.
5. Physical Geography Produces Local Relationships
Slope, soil, vegetation, rainfall and geology vary. A rainfall–runoff relationship estimated in one catchment may perform badly in another because the landscape processes differ.
6. Human Geography Does the Same
Income, household structure, land markets, regulation, culture and transport systems alter how people respond to distance, price and opportunity. One city’s behavioural relationship should not automatically be exported to another.
7. Scale Can Manufacture Apparent Stability
Large-scale aggregation can smooth away local variation, producing one clean relationship that seems universal. Zooming in can reveal several distinct local regimes.
8. Local Variation Can Also Be Noise
Not every changing local coefficient reflects a real geographic mechanism. Small samples, measurement error, collinearity and model instability can create apparent nonstationarity. The pattern must be tested, not merely displayed.
9. Geographically Weighted Regression Is One Tool
Geographically weighted regression allows relationships to vary across space by giving greater weight to nearby observations. It can reveal local structure, but it is exploratory and sensitive to bandwidth, variable choice and spatial dependence. It should not be treated as a machine that automatically discovers truth.
10. Multiscale Relationships Matter
Different variables may operate at different spatial scales. Transit accessibility may matter locally, labour markets regionally and national policy at a much broader scale. A model that forces every relationship to operate at the same spatial bandwidth can miss the structure.
11. Primary Geography: Why Does Shade Work Differently?
Ask children to compare two shaded locations: one breezy, one enclosed. The same shade can produce different thermal comfort because surrounding conditions differ. That is the intuitive seed of nonstationarity.
12. Secondary Geography: Compare the Same Cause Across Two Places
Students can ask why heavy rainfall causes severe flooding in one city but limited damage in another. The answer may lie in drainage, land cover, preparedness and settlement pattern. The cause–outcome relationship is geographically conditioned.
13. Advanced Geography: Do Not Confuse Local Association With Local Cause
A strong local coefficient may still reflect confounding or selection. This article therefore complements Spatial Association and Causation: spatial variation in relationships still requires causal discipline.
14. Singapore Example: Transit and Property
The value of proximity to rapid transit can vary by neighbourhood because service quality, alternative modes, land use, school access and redevelopment potential differ. One island-wide average can therefore hide local property-market responses.
15. Singapore Example: Heat and Greenery
Tree cover generally contributes cooling, but the magnitude can vary with canopy form, surrounding buildings, wind, moisture and time of day. Geography needs the local context around the variable.
16. Flood Example
The same rainfall intensity can produce different runoff across soil types and urban forms. A model calibrated in one catchment may therefore fail when transferred elsewhere.
17. Health Example
The relationship between heat and health outcomes can vary with age structure, air-conditioning access, housing form and baseline health. A national average can conceal local vulnerability.
18. Economic Example
Distance to a city centre may matter strongly for one industry and weakly for another. Even within an industry, digitalisation and logistics can change the geography of location advantage over time.
19. Time Can Change the Spatial Relationship Too
A relationship that varies across space may also change through time. A station opening, new regulation or technological shift can alter local coefficients. Geography therefore often needs both spatial and temporal nonstationarity.
20. Hostile Test: “The National Model Fits Well, So Local Differences Do Not Matter”
A strong overall fit does not prove parameter stability. Residual maps, local diagnostics and out-of-area validation can reveal places where the global relationship fails.
21. Where Nonstationarity Reasoning Breaks
- Local-coefficient worship: treating every map variation as substantive truth.
- Small-sample instability: confusing noise with local process.
- Scale blindness: ignoring the spatial scale at which each mechanism operates.
- Causal overreach: interpreting local association as local causation.
- Model-transfer error: assuming relationships calibrated in one region automatically apply elsewhere.
- Static-map error: ignoring temporal change in relationships.
22. Ten Questions for Spatial Nonstationarity
- Which relationship is being estimated?
- Why might it vary geographically?
- Does effect size change across space?
- Does direction change?
- Which local conditions could explain the variation?
- Could sample size or measurement error create the pattern?
- At what spatial scale does each variable operate?
- Do residuals reveal geographic failure zones?
- Does the relationship change over time too?
- Can the model predict successfully outside the place where it was estimated?
23. Where This Fits
Spatial Heterogeneity owns geographic difference broadly. Spatial Autocorrelation owns spatial dependence among nearby observations. This article owns the question of whether relationships themselves vary across space.
The Idea to Keep
A law of geography becomes much more trustworthy after we have asked where it stops behaving like a law.