Geographically Weighted Regression

Geographically Weighted Regression (GWR) is a statistical method used to analyze data with a geographic component or element. It focuses on how the relationships between variables change from one location to another. It helps us understand how and why relationships between variables might vary across a geography or geographical area.

GWR helps identify and understand how local geographical factors influence the relationships between variables. It is widely used in geography, urban planning, environmental science, epidemiology, and other fields where location matters. Additionally, it finds applications in research that deals with public health, housing prices, land use, agricultural production, nature conservation, economics, and many other fields.

• Geographically Weighted Regression is a modeling technique used to examine and estimate spatial variations in the relationships between variables. 
• GWR takes into account the geographic context and enables the study of different variables and their relationships in different locations within an area, keeping in view that the same variables may present different connections across varied locations.
• GWR is valuable in urban planning, public health, environmental management, and other fields where local policies and interventions have a significant impact. It allows decision-makers to tailor strategies to specific geographical areas.
• GWR is ideal when exploring localized spatial patterns and variations, while Spatial Regression is more appropriate for interpreting global relationships over larger areas.

Geographically Weighted Regression (GWR) is akin to using a special map to understand how certain elements change as the location changes. Instead of assuming that relationships between the factors under study are static across locations, GWR highlights how relationships vary from one location to another.

The Geographically Weighted Regression model is an advanced extension of Ordinary Least Squares (OLS) regression that offers a more sophisticated modeling approach. It identifies and allows the relationships between independent and dependent variables to vary based on geographic locations.

While GWR is primarily used as an exploratory technique, its effectiveness as a prediction tool is a matter of debate due to the following reasons:

• The quality of the data gathered for research affects GWR results; it typically requires data that has not been corrupted in any way.
• The dataset size and complexity of the task at hand impact GWR since the methodology focuses on complex spatial data. It works for both simple and complex data but gives more reliable results when complex data is analyzed. Similarly, large local datasets improve the accuracy of GWR-driven predictions.
• The Kernel function chosen to analyze the distance or difference between various datasets affects the results.

Its primary strength lies in its ability to visualize how the relationships between variables change across space, accounting for spatial autocorrelation. In essence, GWR constructs separate OLS regression equations for each location in a dataset, considering only the data points within a specific spatial bandwidth.

The bandwidth, which defines the size and shape of the area/neighborhood, can either be determined manually based on prior knowledge or automatically by the software, typically using an adaptive approach.

Though the GWR methodology can be complex when several local datasets are involved, researchers and practitioners are now able to apply it extensively due to advanced Geographically Weighted Regression software and computational models.

It is a dynamic tool that can outline extensive and intensive spatial patterns, trends, and relationships. The Kernel function, which calculates the distance decay, facilitates and steers GWR-based research.

GWR helps researchers and decision-makers execute key projects and initiatives to resolve geography-based problems. Hence, such endeavors are important to address macro-level problems, devise effective policies, and drive growth.

Some key assumptions associated with GWR have been listed below:

• Spatial Non-stationarity: The fundamental GWR assumption is that the relationships between variables are not constant across the study area. It assumes that the relationship between two variables of the same type or variety can be different in different locations.
• Spatial Autocorrelation: GWR accounts for spatial autocorrelation, which means that nearby locations tend to have more similar values than areas located farther apart. It assumes that the errors or residuals in the model may exhibit spatial patterns.
• Local Effects: GWR assumes there are local, context-specific factors that can modify the relationships between variables. It acknowledges that different places have unique characteristics that influence these relationships.
• Independence of Observations: GWR, like standard regression, assumes that observations are independent of each other. This means that the value of the dependent variable for one location does not depend on the value for another location. However, it is important to understand that in some cases, there may be certain interlinks or connections between observations noted at different locations.
• Bandwidth Selection: GWR assumes that an appropriate bandwidth (the neighborhood or spatial extent used for each local model) has been selected. The choice of bandwidth can significantly affect the results. An acutely local bandwidth will give highly localized results. The bandwidth should be chosen based on a theoretical interpretation of the problem, or it should be determined through statistical methods.

The process of Geographically Weighted Regression (GWR) can be explained in three broad steps.

Step #1

First, Ordinary Least Squares (OLS) regression models are used to calculate global regression coefficients (β) for the independent variables. This is done for the entire study area without considering spatial variations. The OLS model estimates the relationships between the dependent variable (yi) and the independent variables (x1i, x2i, ..., xn).

The global coefficients are determined using the formula β’ = (XT X)^-1 XT Y, where X represents the independent variables, and Y is the dependent variable.

Step #2

After obtaining global coefficients and identifying which independent variables to include, GWR comes into play. This step is taken when a theoretical basis exists to believe that relationships between variables differ across space.

GWR creates localized regression models for each location in the study area. These localized models are expressed as yi = β0 + β1x1i + β2x2i + ... + βnxi + Ɛi. The key difference here is that for each location, the coefficients (β') are now determined with a spatial matrix of weights, W(i), specific to that location.

This weighting matrix assigns higher importance to data points closer to the location of interest, reflecting the spatial influence.

Step #3

GWR begins with a global analysis using OLS, which is then fine-tuned to create localized regression models that consider spatial variations and utilize weighted data points to understand better how relationships between variables change from place to place.

This approach is particularly valuable when spatial patterns and local influences play a significant role in the data being analyzed.

Let us study some examples to decode the concept further.

Example #1

In a hypothetical scenario, Jane is researching the connection between temperature and ice cream sales in various cities. While a conventional global regression model assumes a consistent relationship between the two variables, Jane opts for Geographically Weighted Regression (GWR) to explore potential local variations.

With GWR, she creates individual regression models for each city, taking into account factors like local preferences, culture, and unique weather patterns, which may influence the temperature-ice cream sales relationship differently in each location.

This approach allows Jane to gain localized insights, recognizing that the factors affecting the relationship can differ from one place to another, making GWR a valuable tool for understanding the spatial heterogeneity of her data.

Based on this information, Jane can modify her sales strategy to boost sales in specific geographical areas.

Example #2

This is a GWR study published in 2021 stated that GWR is a method using which the spatial heterogeneity of the relationship between Land Surface Temperature (LST) and the indirect impact of urbanization on Net Primary Productivity (NPP) can be assessed in the city of Kunming.

GWR is a powerful tool for analyzing spatially varying relationships between variables, making it an essential technique for interpreting how the UHI effect influences vegetation in different parts of the city.

The key points related to the use of GWR in this study were:

• Spatial Variation Analysis: GWR is employed to account for the spatial variation in the relationship between LST and NPP_ind (indirect impact of NPP). Unlike traditional regression models like Ordinary Least Squares (OLS), which assume a constant relationship across the study area, GWR allows for varying relationships based on geography. This is particularly important in urban areas, which exhibit complex spatial patterns in variables like LST and NPP.
• Localized Relationships: GWR provides localized regression coefficients for each grid cell within the study area. It allows for a more detailed understanding of how the UHI effect influences NPP in specific regions. In this study, the GWR model identifies areas where higher LST positively impacts NPP, providing a more fine-grained perspective.
• Higher Fitting Accuracy: By considering the spatial context and localized relationships, GWR often provides a better fitting model compared to OLS. In this study, GWR shows a higher R-squared value, indicating a better fit to the data. The AIC statistic also suggests that the GWR model is more appropriate for the data, further emphasizing its accuracy.
• Quantifying Spatial Changes: GWR reveals how the impact of the UHI effect on NPP_ind changes across the city over time. It demonstrates that the areas positively influenced by LST are expanding, and this expansion is quantified. This information is crucial for urban planning and environmental management decisions.
• Policy Implications: The GWR results offer valuable insights to policymakers. They can identify specific areas where interventions to mitigate the UHI effect and promote urban vegetation growth are most needed. This targeted approach can lead to more effective and sustainable urban development strategies.

The differences between geographically weighted regression and spatial regression are listed in the following table.

This article has been a guide to what is Geographically Weighted Regression. We explain its examples, assumptions, equation, and comparison with spatial regression. You may also find some useful articles here -

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