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:

\( NE_{rc}(x,R,K) := {NP_{rc}(x, K) \over ME_{rc}(x,R,K)} \)

\( ME_{rc}(x,R,K) := {{SUM(x) / SUM(R)} \over NP_{rc}(R, K)} \)

with \( NP_{rc}(v, K) \) defined as the Convolution of \( v \) using kernel \( K \).

thus, ignoring the cases where \( ME_{rc}(x,R,K) = 0 \) ):

\( NE_{rc}(x,R,K) := {SUM(R) \over SUM(x)} \times {NP_{rc}(x, K) \over NP_{rc}(R, K)} \)

Note that:
 * 1) if no values are cut off at the raster boundaries, then \( \sum\limits_{rc} NP_{rc}(v, K) = \sum\limits_{rc} v_{rc} \times \sum\limits_{rc} K_{rc} \), therefore the values unit of \( NE_{rc}(x,R,K) \) is defined as the values unit of \( {SUM(R) \over SUM(x)} \) multiplied with the values unit of \( {NP_{rc}(x, K) \over  NP_{rc}(R, K)} \), which is 1.
 * 2) assuming \( 0 \le x_{rc} \le R_{rc} \) and 0 \( \le K_{kl} \), it follows that \( 0 \le {NP_{rc}(x, K) \over  NP_{rc}(R, K)} \le 1 \) and \( 0 \le NE_{rc}(x,R,K) \le {SUM(R) \over SUM(x)} \)