File(s) under permanent embargo
On the derivatives of set functions in matrix representation
This paper formalises the operations on capacities in matrix algebra framework. Various quantities that characterise the importance of the inputs and their dependencies are expressed through capacity derivatives, obtained through matrix–vector multiplication. Many relations between the set functions derivatives are established and found to be the consequences of the Divergence Theorem. New formulas for Shapley values and nonmodularity indices are found. The sums of the Shapley interaction indices are found to be related to lower order derivatives at the top and bottom elements of the respective power sets. The presented methods simplify many calculations and will facilitate efficient software implementations and applications of the capacity-based decision making methods.
History
Journal
Information sciencesVolume
546Pagination
469 - 480Publisher
ElsevierLocation
Amsterdam, The NetherlandsPublisher DOI
ISSN
0020-0255Language
engPublication classification
C1 Refereed article in a scholarly journalUsage metrics
Categories
No categories selectedKeywords
Licence
Exports
RefWorks
BibTeX
Ref. manager
Endnote
DataCite
NLM
DC