Slides are here

In this tutorial, we demonstrate two common ways to impute the full (all variables) fuzzy measure from just the measure defined on the singletons (aka the densities).

First, clear your path and figures.

		  

close all; 
clear all;

First, pass in the densities, g(x1)=0.2, g(x2)=0.3 and g(x3)=0.1, and get lambda for the Sugeno lambda fuzzy measure and use it to impute the values for the rest of the fuzzy measure (cardinality of size greater than 1).

		  

[g, lambda] = fi_sugeno_lambda_measure( [0.2 0.3 0.1] )
fi_choquet_integral_h_and_g_form( [0.4 0.2 0.1] , g )

 
g =

  Columns 1 through 4

    0.2000    0.3000    0.6865    0.1000

  Columns 5 through 7

    0.3622    0.4933    1.0000

lambda =

    3.1091

ans =

    0.2087
		

In this example, use a S-decomp, where S=max operator here. For each variable of cardinality greater than one, it takes the maximum of the corresponding densities.

		  

[g] = fi_s_decomp( [0.2 0.1 0.9] )
fi_choquet_integral_h_and_g_form( [0.4 0.2 0.1] , g )

 
g =

  Columns 1 through 4

    0.2000    0.1000    0.2000    0.9000

  Columns 5 through 7

    0.9000    0.9000    1.0000

ans =

    0.1600