Deakin University
Browse

File(s) under permanent embargo

On the derivatives of set functions in matrix representation

journal contribution
posted on 2021-01-01, 00:00 authored by Gleb BeliakovGleb Beliakov
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 sciences

Volume

546

Pagination

469 - 480

Publisher

Elsevier

Location

Amsterdam, The Netherlands

ISSN

0020-0255

Language

eng

Publication classification

C1 Refereed article in a scholarly journal

Usage metrics

    Research Publications

    Categories

    No categories selected

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC