Slides are here

In this tutorial, we highlight the Shapley and interaction index.

First, clear your path and figures.

		  

close all; 
clear all;

Next, lets make the ChI become the minimum operator.

		  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OWA and min
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[g] = fi_owa( [ 0 0 1 ] )'
[ ShapleyVector ] = fi_shapley( g )
[InteractionIndex] = fi_interaction_index( g )

The result is the following (see below). Note, the Shapley values are equal, which is true for any OWA/LCOS.

		  

g =

     0     0     0     0     0     0     1

ShapleyVector =

    0.3333    0.3333    0.3333

InteractionIndex =

    1.0000    0.5000    0.5000
    0.5000    1.0000    0.5000
    0.5000    0.5000    1.0000

Next, lets make the ChI become the maximum operator.

		  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OWA and max
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[g] = fi_owa( [ 1 0 0 ] )'
[ ShapleyVector ] = fi_shapley( g )
[InteractionIndex] = fi_interaction_index( g )

The result is the following (see below). Note, the Shapley values are again equal, which is true for any OWA/LCOS.

		  

g =

     1     1     1     1     1     1     1

ShapleyVector =

    0.3333    0.3333    0.3333

InteractionIndex =

    1.0000   -0.5000   -0.5000
   -0.5000    1.0000   -0.5000
   -0.5000   -0.5000    1.0000

Next, lets make the ChI become the average operator.

		  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% OWA and average
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[g] = fi_owa( [ 1 1 1 ] ./ 3 )'
[ ShapleyVector ] = fi_shapley( g )
[InteractionIndex] = fi_interaction_index( g )

The result is the following (see below). Note, the Shapley values are again equal, which is true for any OWA/LCOS. Furthermore, the off diagional interaction terms are 0, which is true for additivity.

		  

g =

  Columns 1 through 5

    0.3333    0.3333    0.6667    0.3333    0.6667

  Columns 6 through 7

    0.6667    1.0000

ShapleyVector =

    0.3333    0.3333    0.3333

InteractionIndex =

    1.0000    0.0000    0.0000
    0.0000    1.0000    0.0000
    0.0000    0.0000    1.0000

Last, we do a Sugeno lambda fuzzy measure example.

		  

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Sugeno lambda FM-based
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[g, lambda] = fi_sugeno_lambda_measure( [0.2 0.3 0.1] )
[ ShapleyVector ] = fi_shapley( g )
[InteractionIndex] = fi_interaction_index( g )

The result is the following (see below).

		  

g =

  Columns 1 through 5

    0.2000    0.3000    0.6865    0.1000    0.3622

  Columns 6 through 7

    0.4933    1.0000

lambda =

    3.1091

ShapleyVector =

    0.3437    0.4592    0.1971

InteractionIndex =

    1.0000    0.2155    0.0912
    0.2155    1.0000    0.1223
    0.0912    0.1223    1.0000