arcgis 10.1 - How to find Entropy in land use (Voronoi polygons)?

I have a shapefile of land use in Montreal. It is composed of polygons. There are seven land uses: Residential, Commercial, Industrial, Waterbody, Parks, Open Area and Government. I need to find Entropy from this layer. Can anyone suggest how to do that? I am using Arcmap 10.1

Answer

I would imagine that the easiest way would be to convert the polygons to a raster and then calculate entropy within a NxN moving window. Since you have only seven landcover types you can do this manually but it will be arduous.

Entropy is calculated as -sum(Pi*ln(Pi)) where; Pi is the probability of each landcover. In this case "probability" is referring to proportion of the class in relation to all the other classes. To calculate this you will need a raster for each class that represents the proportion of that class within the specified window. You would then apply a raster algebra statement to calculate entropy (H).

H = -1*( (lc1 * Ln(lc1) + (lc2 * Ln(lc2) + ...)

The expected maximized entropy is where a window is equally split between all classes or Ln(m) where; m is number of classes. You can get tricky with this method and test variable scales (window sizes).

In R parlance the function is such (and can easily be translated to Python):

  entropy <- function(x) { 

    p=vector() 
      if ( length(unique(x)) <= 1) { return(0) }
        nv <- length(unique(x))  
      for( i in unique(x) ) { 
        p <- append(p, ( length(x[x == i]) / nv * log(length(x[x == i]) / nv) ) ) 
      }
    return( -sum(p) )
    }