%% extend_nary2.m
%% This function uses Zadeh's Extension Principle (Section 7.2 in the
%% book Uncertain Rule-Based Fuzzy Logic Systems: Introduction and
%% New Directions, by Jerry M. Mendel, and published by Prentice-Hall,
%% 2000.) to extend an "n"-ary operation "op_n" to "n" type-1 fuzzy sets.
%% Written by Nilesh N. Karnik - July 22,1998
%% For use with MATLAB 5.1 or higher.
%% NOTE : This program uses a much smaller amount of memory
%% compared to "extend_nary1.m"; and, correspondingly, is quite slow.
%% It can, however, be used for any values of "m" and "n", in general.
%% For performing a weighted averaging operation, a better program -
%% one which uses even a smaller amount of memory - is
%% "weighted_avg.m".
%% Outputs : "out" and "mem" (column vectors) are, respectively,
%% the domain and the memberships of the result of the extended
%% "op_n" operation.
%% Inputs : "X" is an "n x m" matrix, containing the domain elements
%% of the "n" type-1 sets (the domain of each set is assumed to
%% contain "m" elements). "Y" has the same size as "X" and contains
%% the memberships of the points in "X". "op_n" is a string (entered
%% in inverted commas, as " 'op_n' ") containing the name of the
%% function which defines the n-ary operation to be extended (the
%% function should be stored in the file "op_n.m"). If "tnorm" < 0
%% (scalar), minimum t-norm is used, otherwise product t-norm is used.
%% "step" (scalar) is an optional parameter. If it is not provided, the
%% "n" type-1 sets are assumed to have discrete domains and the output
%% is presented just by sorting the result of the extended "op_n"
%% operation. If "step" is provided, it is assumed that the "n" type-1
%% sets actually have continuous domains and are discretized for
%% computing purposes. The resulting type-1 set, in this case, has
%% equispaced domain points : "[minx : step : maxx]", where "minx" and
%% "maxx" are, respectively, the minimum and the maximum of the results
%% of the "op_n" operation between domain elements of the participating
%% type-1 sets.
%% Note 1 : The t-conorm used is maximum.
%% Note 2 : If "op_n" is a built-in function, you may have to use the
%% MATLAB function "builtin" rather than "feval".
%% Note 3 : The function "op" should be written so that it can work
%% with matrices, and perform the n-ary operation on each column of the
%% matrix, similar to the MATLAB functions "min(X)" or "prod(X)", where
%% "X" is a matrix. As an example, see the function "wt_avg_op.m".
%% Uses function "trimvec.m" .
function[out,mem] = extend_nary2(X,Y,op_n,tnorm,step)
[n,m] = size(X) ;
X1 = X ;
Y1 = Y ;
lm = m^(n-1) ;
temp1 = zeros(1,m) ;
temp2 = temp1 ;
xout = zeros(1,lm*m) ;
memout = zeros(1,lm*m) ;
for i1 = 1 : lm,
xout(m*(i1-1)+1 : m*i1) = feval(op_n,X1) ;
if tnorm < 0,
memout(m*(i1-1)+1 : m*i1) = min(Y1) ;
else
memout(m*(i1-1)+1 : m*i1) = prod(Y1) ;
end %% if minflag
temp1(1:m-1) = X1(n,2:m) ;
temp1(m) = X1(n,1) ;
X1(n,:) = temp1 ;
temp2(1:m-1) = Y1(n,2:m) ;
temp2(m) = Y1(n,1) ;
Y1(n,:) = temp2 ;
for i2 = 1 : n-1,
if mod(i1,m^i2) == 0,
temp1(1:m-1) = X1(n-i2,2:m) ;
temp1(m) = X1(n-i2,1) ;
X1(n-i2,:) = temp1 ;
temp2(1:m-1) = Y1(n-i2,2:m) ;
temp2(m) = Y1(n-i2,1) ;
Y1(n-i2,:) = temp2 ;
end %% if mod(i1,m^i2)
end %%% for i2
end %%% for i1
if nargin == 4,
[out,mem] = trimvec(xout,memout) ;
else
minx = min(xout) ;
maxx = max(xout) ;
eps = step/2 ;
i2 = 1 ;
for k = minx : step : maxx,
out(i2) = k ;
in1 = find(abs(xout-k) <= eps) ;
if ~isempty(in1),
mem(i2) = max(memout(in1)) ;
else
mem(i2) = mem(i2-1) ;
end %%% if
i2 = i2 + 1;
end %% for k
if k ~= maxx,
out(i2) = maxx ;
in1 = find(abs(xout-maxx) <= eps) ;
if ~isempty(in1),
mem(i2) = max(memout(in1)) ;
else
mem(i2) = mem(i2-1) ;
end %%% if ~isempty
end % if k
end % if nargin
return ;
