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The formula for global Moran's I is: where i is an index of analysis units (basically, measurement units of of your map, or in your case pixels in the raster) and j is an index of the neighbors of each map unit. The formula for local Moran's I is extremely similar, except that since local Moran's I is calculated separately for each analysis unit indexed ...


I would first make a join of the two tables crime and population. Then I would add another column where I calculate the crime case per population, simply divide the crime case column by the population coloumn. Works if the numbers refer to the exact same areas. Then you can visualize this column in a choropleth map and immediately see where the crime is ...


You can also use: the Python module PySAL: Spatial Autocorrelation. You can use it in the Python console of QGIS or in a script in the Processing Toolbox R (Applied Spatial Data Analysis with R: analysis with R, “The Problem of Spatial Autocorrelation:” forty years on, Geary's C test for spatial autocorrelation, for example, but there are many others) ...


You might want to take a look at this question (How to implement Spatial Autocorrelation using QGIS or PostgreSQL (or any free application)?) and look at programmatic solutions like R. QGIS may be another option as well. This website outlines different formulas and measures for spatial autocorrelation that may also be of interest to you

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