Local Inverse Simpson's Index (iLISI & cLISI)¶
Description¶
The Local Inverse Simpson's Index (LISI) measures local mixing by estimating the effective number of classes in local neighborhoods of cells and constitutes a fundamental metric for evaluating single-cell data integration quality. Two main variants exist :
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iLISI (integration LISI) which denotes the effective number of datasets in a neighborhood to assess batch mixing. Higher iLISI values indicate better batch mixing.
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cLISI (conservation LISI) which measures the preservation of cell types after integration. Higher cLISI values suggest better preservation of biological structure.
The LISI of a cell is defined as the effective number of batches, properly scaled, among its k nearest neighbors. This metric allows objective quantification of integration algorithm performance by simultaneously measuring batch effect correction and biological information conservation.
Formulas¶
LISI is based on the inverse Simpson index applied locally. For a given cell :
Local Simpson Index Calculation :¶
where \(p_j\) is the proportion of cells from category \(j\) (batch for iLISI, cell type for cLISI) in the k-nearest neighbor neighborhood of cell \(i\).
LISI (Local Inverse Simpson Index) :
Interpretation :
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iLISI: effective number of batches in the local neighborhood
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Minimum value = 1 (homogeneous neighborhood, single batch)
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Maximum value = total number of batches (perfect mixing)
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cLISI: effective number of cell types in the local neighborhood
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Low value = good conservation (homogeneous neighborhood for one cell type)
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High value = potential over-mixing (loss of biological structure)
Normalized Version :
By default, this function returns a value scaled between 0 and 1 instead of the original LISI range from 0 to the number of batches.
where \(N_{categories}\) is the total number of categories (batches or cell types).
Sources¶
Korsunsky, I., Millard, N., Fan, J. et al. Fast, sensitive and accurate integration of single-cell data with Harmony. Nat Methods 16, 1289–1296 (2019). & Github