In this tutorial, we demonstrate the interval-valued fuzzy integral. For example, an interval-valued integrand is h(xi)=[a,b], where a is less than or equal to b. We do not provide custom slides for this tutorial. If the reader would like to learn more, two related articles include "D. T. Anderson, T. C. Havens, C. Wagner, J. M. Keller, M. F. Anderson, D. J. Wescott, Extension of the Fuzzy Integral for General Fuzzy Set-Valued Information, IEEE Transactions on Fuzzy Systems, 2014" and "D. T. Anderson, P. Elmore, F. Petry, T. Havens, "Fuzzy Choquet Integration of Homogeneous Possibility and Probability Distributions," Information Sciences, 2016".

First, clear your path and figures.


close all; 
clear all;

Next, create an average fuzzy measure/ChI.


[g] = fi_owa( [ 1 1 1 ] ./ 3 )'    %3 inputs

Next, since this is a tutorial, we create some random interval-valued data (you can enter in specific values if you like!). Each row is an input and each input has two columns, index 1 is the left interval endpoint and index 2 is the right interval endpoint.


% make up some random interval-valued data
N = 3; % number of inputs
inputs = rand(N,2);
% random interval width, max of value 1
inputs(:,2) = (1 - inputs(:,1)) .* rand(N,1) + inputs(:,1); 
inputs(:,2) = min(inputs(:,2),1);

Last, we call the interval-valued FI with plotit=1 so we can see the results.


% do we want to plot it?
plotit = 1;
% call the interval-valued FI
[ interval ] = fi_choquet_integral_interval( inputs , g , plotit )

interval =

    0.6158    0.9288

Where the interval-valued ChI breaks down into two real-valued ChIs (see below).

function [ interval ] = fi_choquet_integral_interval( inputs , FM )
	% result
	interval = zeros(1,2);	
	% do on left most endpoints
	interval(1) = fi_choquet_integral_h_and_g_form( inputs(:,1)' , FM );
	% do on right most endpoints
	interval(2) = fi_choquet_integral_h_and_g_form( inputs(:,2)' , FM );

Example output is as follows. Note, in the plot the x-axis is the X domain and the y-axis just shows our inputs staggered for comparative analysis.