Metric that uses the Inverse Simpson’s Index to calculate the diversification within a specified neighbourhood. The Simpson index describes the probability that two entities are taken at random from the dataset and its inverse represent the effective number of batches in a neighbourhood. The inverse Simpson index has been proposed as a diversity score for batch mixing in single cell RNAseq by Korunsky et al.. They provide a distance-based neighbourhood weightening in their Lisi R package. A simplified way of distance-based neighbour weightening by \(\frac{1}{d^{2}}\) with d representing the euclidean distance in pricipal component space and an unweighted version are implemented as wisi and isi score in the CellMixS R package.
Inverse Simpson Index with \(p(b)\) representing the batch probabilities in the local neighborhood distributions:
\[\frac{1}{\sum_{b=1}^{B} p(b)}\]