What does first requirement for LISA statistic mean?

Question

According to Anselin, a LISA statistic have to satisfy two requirements:

1. The LISA for each observation gives an indicator of the extent of significant spatial clustering of similar values around that observation.
2. The sum of LISAs for all observations is proportional to a global indicator of spatial association.

The second requirement is very straightforward but I cannot understand mathematically the first requirement.

For example, he said in his paper:

The L_i should be such that it is possible to infer the statistical significance of the patter of spatial association at location i. More formally, this requires the operationalization of a statement such as Prob[L_i>δ_i]≤α_i...

Is this the first requirement?

Answer 1

The key of the first requirement is related to the word significant. This implies that the observed pattern (i.e. a cluster / outlier) is not the result of a random process.

To understand this, you can think of each element on your geographic dataset as a piece of a puzzle. You throw all the pieces in a table. What are the odds of ending with a set of pieces with high (or low) values close to each other? If the odds are beyond a confidence level, you can say that the observed pattern is significant.

Software packages such as Geoda compute the p value for a LISA by permuting the values of the features x number of times and check the proportion of these times that the pattern is observed.