# Spatial Regression

*What is spatial regression?*

Spatial regression is a statistical technique used to analyze the relationship between a dependent variable and one or more independent variables while taking into account the spatial autocorrelation of the data. Spatial autocorrelation refers to the phenomenon of nearby observations being more similar to each other than observations that are farther apart.

In a GIS context, spatial regression can be used to analyze patterns in spatial data and make predictions about the values of the dependent variable based on the values of the independent variables. For example, a spatial regression analysis could be used to predict the value of a property based on its location, surrounding land use, and other factors.

There are several different types of spatial regression techniques, including ordinary least squares (OLS) regression, geographically weighted regression (GWR), and spatial autoregressive models (SAR). The choice of which technique to use will depend on the specific characteristics of the data and the research question being addressed.

*What sort of questions can spatial regression help answer?*

Some examples of questions that spatial regression can help to answer include:

- What is the relationship between a dependent variable (such as property values) and one or more independent variables (such as location, land use, or transportation access)?
- How does the relationship between the dependent and independent variables vary across different regions or locations?
- How accurately can the dependent variable be predicted based on the values of the independent variables?
- What factors have the greatest influence on the dependent variable?

Spatial regression is applied in a variety of fields, including geography, economics, sociology, and environmental science, among others. It can be particularly useful for identifying patterns and trends in data that may not be immediately apparent when looking at the data in isolation.

*Example*

Suppose you are studying the relationship between the distance of a house from a school and its value. You collect data on the location and value of houses in your neighborhood, as well as the location of the nearest school.

To analyze this data using spatial regression, you could create a scatterplot showing the value of each house on the y-axis and the distance of each house from the school on the x-axis. If there is a relationship between these two variables, you would expect to see a pattern in the data points on the scatterplot. For example, you might see that houses that are closer to the school tend to have higher values, or that houses that are farther from the school tend to have lower values.

To formally analyze this relationship, you could then use a statistical technique like ordinary least squares (OLS) regression to fit a line to the data points on the scatterplot. This line represents the best fit for the data, and can be used to make predictions about the value of a house based on its distance from the school.

For example, suppose you find that the line has a slope of $100,000 and an intercept of $300,000. This means that, on average, the value of a house decreases by $100,000 for every mile that it is farther from the school. Using this information, you could make a prediction about the value of a house that is two miles from the school:

$Value = 300,000 - (2 \times 100,000) = 100,000$

This prediction indicates that the value of the house is likely to be around $100,000.

This is just a simple example of how spatial regression can be used to analyze the relationship between two variables. In practice, spatial regression can be used to analyze more complex relationships involving multiple independent variables, as well as to account for spatial autocorrelation in the data.