Local Moran’s I Calculation Example

01-26-2023 11:03 AM
New Contributor II

I am attempting to manually replicate results obtained from the Cluster and Outlier Analysis tool (ArcGIS Pro v2.9). I have not had any luck, so I’m hoping that someone in the community can help me figure out where I’ve gone wrong. I used the Cluster and Outlier Analysis tool configured as attached. For DeKalb County (highlighted in my attached spreadsheet) the tool gives a Local Moran’s I Index equal to 1.65866.

After running the tool, I tried to replicate the DeKalb County result using the formula shown here: https://pro.arcgis.com/en/pro-app/latest/tool-reference/spatial-statistics/h-how-cluster-and-outlier...

After running through the math, I keep calculating a different value for Local Moran’s I Index for DeKalb County: 1.700519. Below are my calculations based on the dataset attached. What am I missing?

X bar = 41.509374

x_i (DeKalb County) = 47.009175

n = 75

Dekalb County neighbors and their respective x_j values:

Fulton County: x_j = 48.810092

Clayton County: x_j = 43.978097

Henry County: x_j = 47.661295

Rockdale County: x_j = 45.341663

Gwinnett County: x_j = 46.796238


The spatial weight of each of the five neighbors is 0.2, and the other counties in the dataset have a spatial weight of zero.

The above neighbors and weights yield the sum (w_i,j*(x_j-xbar)) = 5.008103.

The sum (x_j-xbar)^2 for all non-DeKalb County features is 1,198.589782, which divided by (n-1) gives an S_i^2 value of 16.197159.

Everything above gives a Local Moran’s I statistic of [(47.009175-41.509374)/16.197159]*5.008103 = 1.700519

Am I mis-understanding or mis-calculating part of the equation for Local Moran’s I? The other features on the map have similar discrepancies between my math and the ArcGIS tool output. I’ve been scratching my head on this one for a few days, so I figured it is time to call in the experts. Thank you!

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2 Replies
Esri Contributor

Hi Josh, 

Thanks for the question. I think your math looks solid and it would be good for us to take a closer look and replicate the results you're seeing. 

The zipped file has a single .shp file. By any chance would you be able to zip the shapefile once more with all the files it requires? Alternatively you can share it as a feature class in a zipped file geodatabase if that works.  


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New Contributor II

Alberto - Thanks for looking into this! Here's the shapefile zipped with all the necessary files.

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