My observations are points along a transect, irregularly spaced.

I aim at finding the distance values that maximize the clustering of my observation attribute, in order to use it in the following LISA analysis (Local Moran I).

I iteratively run Global Moran I function with PySAL 2.0, recreating a different distance-based weight matrix (binary, assigning 1 to neighbors and 0 to not neighbors) with a search radius 0.5m longer at every iteration.

At every iteration, I save z_sim,p_sim, I statistics, together with the distance at which these stats have been computed.

From these information, what strategy is best to find distances that potentially show underlying spatial processes that (pseudo)-significantly cluster my point data?


  • Esri style: ArcMap Incremental Global Moran I tool identify peaks of z-values where p is significant as interesting distances
  • Literature: I found many papers that simply choose the distance with the higher absolute significant value of I


Because with varying search radius the number of observations considered in the neighborhood change, thus, the weight matrix also change, the I value is not comparable.

1 Answer 1


After a little research, I finally came up with the answer I was looking for.

when using Global Moran's I index (I) with incrementally increasing distance searches (thus, changing the weight matrix at every iteration), only the the z-values are independent from both weight matrices and variable intensity variations, thus, they are comparable across multiple analyses.

The I in Moran's I statistics is not comparable across analyses, i.e, if with distance of 10m I=0.3 and distance 15m I=0.6, we cannot say that with a distance of 15m the clustering strength is double. We could only say that in both cases there is a positive (sign of the I) spatial autocorrelation. For the strengths, we use the z-values.

That is why ESRI plots distances in the x-axis and z-values in the y axis, indicating significant (p-value < than specified signification level) peaks as interesting distances.

For more information, it is clearly explained during a class that Luc Anselin in this Global Autocorrelation class, given in 2016 in Chicago University.


follow from minute 38 when he talks about the permutation approach.

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