LISA: Local Indicator of Spatial Autocorrelation
What is spatial autocorrelation?
Spatial autocorrelation is a statistical phenomenon that occurs when the values of a variable are correlated with the values of that same variable in neighboring areas. In other words, it is the tendency of similar values to be found near each other in space. This can be observed in many different types of data, including economic, demographic, and environmental data.
For example, imagine you are studying the population density of different neighborhoods in a city. You might find that neighborhoods with high population densities (lots of people living in a small area) tend to be located near other neighborhoods with high population densities. This is an example of spatial autocorrelation. Simply, the values of the population density variable (high or low) are correlated with the values of that same variable in neighboring areas. This can be measured and studied using statistical techniques.
What questions can spatial autocorrelation help answer?
Spatial autocorrelation can be useful for answering a wide range of questions. For example, it can help researchers understand the spatial patterns and clusters of certain phenomena, such as the distribution of diseases, crime rates, or economic indicators. It can also help identify areas where certain conditions are more or less prevalent, and can be used to develop spatial models and predictions. Additionally, spatial autocorrelation can help identify areas that may be in need of further study or intervention. For example, if a high Moran's I value is observed for a particular variable, such as air pollution levels, it may indicate that there is a cluster of high pollution areas that warrants further investigation. Because of this, spatial autocorrelation can be helpful indicators in exploratory spatial data analysis (ESDA) or other analysis seeking insights into the spatial patterns and relationships in a variety of data sets.
Indicators of spatial autocorrelation
A common measure of spatial autocorrelation is the Local Moran's I index, a measure first suggest by Luc Anselin (1). This index measures the degree to which the values of a variable are clustered together in space. A high Moran's I value indicates that there is a strong spatial autocorrelation in the data, while a low Moran's I value indicates that there is little or no spatial autocorrelation. Another measure of spatial autocorrelation is the Geary's C index, which is similar to Moran's I but takes into account the distance between the data points. Many other measures of spatial autocorrelation that can be used, depending on the specific data and research question at hand.
Moran's I index is calculated as a ratio of the sum of the products of the deviations from the mean of each pair of observations, divided by the sum of the squared deviations of each observation from the mean. This can be expressed as:
Moran's I = (n * sum(xi * yi)) / (sum(xi) * sum(yi))
- n is the number of observations
- xi and yi are the values of the variable for each observation
To calculate Moran's I, you first need to compute the mean of the variable, then subtract the mean from each observation to get the deviation from the mean for each observation. Next, you calculate the product of the deviations for each pair of observations, then sum those products. Finally, you divide that sum by the sum of the squared deviations of each observation from the mean. The resulting value is Moran's I.
Moran's I can range from -1 to 1. A value of -1 indicates a perfect negative spatial autocorrelation, where high values of the variable tend to be located near low values and vice versa. A value of 0 indicates no spatial autocorrelation, while a value of 1 indicates a perfect positive spatial autocorrelation, where high values tend to be located near other high values and low values tend to be located near other low values.
To find more on LISA and spatial autocorrelation, take a look at these resources:
(1) Anselin, Luc. 1995. “Local Indicators of Spatial Association — LISA.” Geographical Analysis 27: 93–115.