Fuzzy Systems that deal with Linguistic Ambiguities and Uncertainty in the Real-world using Python
<Settings>
output variable tip = { cheap = 10, average = 15}
intput variable x1, x2 = (5, 8)
<Activated Fuzzy Rules>
Rule No.1 : If service = good AND food = rancid, then tip = cheap
Rule No.2 : If service = good AND food = delicous, then tip = average
fuzzy subset F1(x), F2(y) is normalized
<Solution with Sugeno Fuzzy System>
w1 = AND(F1(x1), F2(x2)) = 0.2, z1 = 10 based on Rule No.1
w2 = AND(F1(x1), F2(x2)) = 0.8, z2 = 15 based on Rule No.2
Then, final output union z_n = (w1 * z1 + w2 * z2)/(w1 * w2) = (0.2 * 10 + 0.8 + 15)/(0.2 + 0.8) = 14
<Settings>
Using Max-Min Operation & Sugeno’s Simplified FIE
Input Variable Room Temperature ~ x
Input Variable Outside Temperature ~ y
Representative Values for x & y = [20 22.5 25 27.5 30 32.5 35] (ºC)
<Fuzzy Rules>
Rule No.1 : IF x in LOW AND y is HIGH, THEN z is SLOW ~ 500 rpm
Rule No.2 : IF x in HIGH AND y is LOW, THEN z is MEDIUM ~ 1000 rpm
Rule No.3 : IF x in HIGH AND y is HIGH, THEN z is FAST ~ 1500 rpm
Rule No.4 : IF x in LOW AND y is LOW, THEN z is STOP ~ 0 rpm
Fuzzy c-Means Clustering is soft clustering in which the observation value for each cluster has a possibility or probability(0~1) that is not a truth value(0 or 1). This is different from k-Means Clustering.
%2D Data is initialized from Random Functions following code :
data_n = 300;
dist = 3; % for random wide distribution
data = randn(data_n, 2);
data = [data; randn(data_n,2)+dist*ones(data_n,2)];
data = [data; randn(data_n,2)-dist*ones(data_n,2)];
%optimization stopped until objective function improved by less than 0.001 between the final two iterations
Iteration count = 1, obj. fcn = 7583.608816
Iteration count = 2, obj. fcn = 6109.651661
Iteration count = 3, obj. fcn = 5939.244259
...
Iteration count = 15, obj. fcn = 3037.072359
Iteration count = 16, obj. fcn = 3037.072340
Iteration count = 17, obj. fcn = 3037.072333
%example of probability about 10 points and 2 cluster
0.9357 0.9914 0.9834 0.9689 0.9629 0.9385 0.9842 0.9898 0.9539 0.9450
0.0643 0.0086 0.0166 0.0311 0.0371 0.0615 0.0158 0.0102 0.0461 0.0550
