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:

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

with NP_rc(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 ∑Convolution(v,K) = ∑v × ∑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.