Neighbourhood enrichment

Neighbourhood Enrichment (aka NE) of \( x \) in \( R \) using kernel \( K \) is defined (in the context of the EuClueScanner, or more generic in the context of (dyna-)Clue) as:

Neighbourhood Potential (the convolution of a specific land use type with a specific kernel defining a distance decay) divided by the MeanEnrichment for that type and kernel (and zero if MeanEnrichment is zero).

MeanEnrichment of a Land Use Type and Kernel is defined as the sum of the occurrence of a Land Use Type divided by the sum of the potential to any land use using the same Kernel, AKA the potential potential.

Or mathematically:

$$ \begin{align} NE_i(x,R,K) &:= {NP_i(x, K) \over ME_i(x,R,K)}      \\ \\ ME_i(x,R,K) &:= {{SUM(x) / SUM(R)} \over NP_i(R, K)} \\ \end{align} $$

with \( NP(v, K) \) defined as the Convolution of \( v \) using kernel \( K \).

thus:

\( NE_i(x,R,K) := {SUM(R) \over SUM(x)} \times {NP_i(x, K) \over NP_i(R, K)} \)

Note that if no values are cut off at the raster boundaries, then \( \sum Convolution(v, K) = \sum v \times \sum K \), from which follows that
 * 1) the unit of \( \sum\limits_i NP_i(v, K) \) is equal to the unit of the \( \sum\limits_i v \) multiplied with the unit of \( \sum\limits_i K \)
 * 2) the unit of \( NP_i(x, K) \) is equal to the unit of x multiplied with the unit of SUM(K)
 * 3) the unit of \( NE_i(x,R,K) \) is 1.