Tune Fuzzy Trees
You can tune the parameters of a FIS tree using a similar two-step process as shown in Tuning Fuzzy Inference Systems.
Learn and tune the rules of the FISs in the tree.
Learn the MF parameters of the FISs in the tree.
Create a FIS tree to model for as shown in the following figure. For more information on creating FIS trees, see Fuzzy Trees.
Create fis1 as a Sugeno type FIS, which results in a faster tuning process due to computationally efficient defuzzification method. Add two inputs, both with range the [0, 10] and with three MFs each. Use a smooth differentiable MF, such as gaussmf, to match the characteristics of the data type you are modeling.
fis1 = sugfis('Name','fis1'); fis1 = addInput(fis1,[0 10],'NumMFs',3,'MFType','gaussmf'); fis1 = addInput(fis1,[0 10],'NumMFs',3,'MFType','gaussmf');
Add an output with the range [–1.5, 1.5] having nine MFs corresponding to the nine possible input MF combinations. Doing so provides maximum granularity for the FIS rules. Set the output range according to the possible values of .
fis1 = addOutput(fis1,[-1.5 1.5],'NumMFs',9);Create fis2 as a Sugeno type FIS. Add two inputs. Set the range of the first input to [-1.5, 1.5], which matches the range of the output of fis1. The second input is the same as the inputs of fis1. Therefore, use the same input range, [0, 10]. Add three MFs for each of the inputs.
fis2 = sugfis('Name','fis2'); fis2 = addInput(fis2,[-1.5 1.5],'NumMFs',3,'MFType','gaussmf'); fis2 = addInput(fis2,[0 10],'NumMFs',3,'MFType','gaussmf');
Add an output with range [0 1] and nine MFs. The output range is set according to the possible values of .
fis2 = addOutput(fis2,[0 1],'NumMFs',9);Connect the inputs and the outputs as shown in the diagram. Output 1 of fis1 connects to input 1 of fis2, inputs 1 and 2 of fis1 connect to each other, and input 2 of fis1 connects to input 2 of fis2.
con1 = ["fis1/output1" "fis2/input1"]; con2 = ["fis1/input1" "fis1/input2"]; con3 = ["fis1/input2" "fis2/input2"];
Finally, create a FIS tree using the specified FISs and connections.
fisT = fistree([fis1 fis2],[con1;con2;con3]);
Add an additional output to the FIS tree to access the output of fis1.
fisT.Outputs = ["fis1/output1";fisT.Outputs];Generate input and output training data.
x = (0:0.1:10)'; y1 = sin(x)+cos(x); y2 = y1./exp(x); y = [y1 y2];
Tune the FIS tree parameters in two steps. First, learn the rules of the FIS tree using a global optimization method (particle swarm for this example).
options = tunefisOptions('Method','particleswarm','OptimizationType','learning');
This tuning step uses a small number of iterations to learn a rule base without overfitting the training data. The rule base provides an educated initial condition for the second step to optimize all the FIS tree parameters together. Set the maximum iteration number to 5, and learn the rule base.
options.MethodOptions.MaxIterations = 5; rng('default') % for reproducibility fisTout1 = tunefis(fisT,[],x,y,options);
Best Mean Stall
Iteration f-count f(x) f(x) Iterations
0 100 0.6682 0.9395 0
1 200 0.6682 1.023 0
2 300 0.6652 0.9308 0
3 400 0.6259 0.958 0
4 500 0.6259 0.918 1
5 600 0.5969 0.9179 0
Optimization ended: number of iterations exceeded OPTIONS.MaxIterations.
Next, to tune all the FIS tree parameters, use a local optimization method (pattern search for this example). Local optimization is generally faster than global optimization and can produce better results when the input fuzzy system parameters are already consistent with the training data.
Use the patternsearch method for optimization. Set the number of iterations to 25.
options.Method = 'patternsearch';
options.MethodOptions.MaxIterations = 25;Use getTunableSettings to obtain input, output, and rule parameter settings from the FIS tree.
[in,out,rule] = getTunableSettings(fisTout1);
Tune the FIS tree parameters.
rng('default') % for reproducibility fisTout2 = tunefis(fisTout1,[in;out;rule],x,y,options);
Iter Func-count f(x) MeshSize Method
0 1 0.596926 1
1 3 0.551284 2 Successful Poll
2 13 0.548551 4 Successful Poll
3 20 0.546331 8 Successful Poll
4 33 0.527482 16 Successful Poll
5 33 0.527482 8 Refine Mesh
6 61 0.511532 16 Successful Poll
7 61 0.511532 8 Refine Mesh
8 92 0.505355 16 Successful Poll
9 92 0.505355 8 Refine Mesh
10 128 0.505355 4 Refine Mesh
11 175 0.487734 8 Successful Poll
12 212 0.487734 4 Refine Mesh
13 265 0.487734 2 Refine Mesh
14 275 0.486926 4 Successful Poll
15 328 0.486926 2 Refine Mesh
16 339 0.483683 4 Successful Poll
17 391 0.483683 2 Refine Mesh
18 410 0.442624 4 Successful Poll
19 462 0.442624 2 Refine Mesh
20 469 0.44051 4 Successful Poll
21 521 0.44051 2 Refine Mesh
22 542 0.435381 4 Successful Poll
23 594 0.435381 2 Refine Mesh
24 614 0.398872 4 Successful Poll
25 662 0.398385 8 Successful Poll
26 698 0.398385 4 Refine Mesh
Maximum number of iterations exceeded: increase options.MaxIterations.
The optimization cost reduces from 0.59 to 0.39 in the second step.
Alternatively, you can tune the specific fuzzy systems within a FIS tree. For this example, after learning the rule base of the FIS tree, separately tune fis1 and fis2 parameters.
To obtain parameter settings of a FIS within the FIS tree, use getTunableSettings, specifying the FIS name. First, get the parameter settings for fis1.
[in,out,rule] = getTunableSettings(fisTout1,"FIS","fis1");
Tune the parameters of fis1.
rng('default')
fisTout2 = tunefis(fisTout1,[in;out;rule],x,y,options);Iter Func-count f(x) MeshSize Method
0 1 0.596926 1
1 3 0.551284 2 Successful Poll
2 18 0.510362 4 Successful Poll
3 28 0.494804 8 Successful Poll
4 56 0.494804 4 Refine Mesh
5 84 0.493422 8 Successful Poll
6 107 0.492883 16 Successful Poll
7 107 0.492883 8 Refine Mesh
8 136 0.492883 4 Refine Mesh
9 171 0.492883 2 Refine Mesh
10 178 0.491534 4 Successful Poll
11 213 0.491534 2 Refine Mesh
12 229 0.482682 4 Successful Poll
13 264 0.482682 2 Refine Mesh
14 279 0.446645 4 Successful Poll
15 313 0.446645 2 Refine Mesh
16 330 0.44657 4 Successful Poll
17 364 0.44657 2 Refine Mesh
18 384 0.446495 4 Successful Poll
19 418 0.446495 2 Refine Mesh
20 461 0.445938 4 Successful Poll
21 495 0.445938 2 Refine Mesh
22 560 0.422421 4 Successful Poll
23 594 0.422421 2 Refine Mesh
24 597 0.397265 4 Successful Poll
25 630 0.397265 2 Refine Mesh
26 701 0.390338 4 Successful Poll
Maximum number of iterations exceeded: increase options.MaxIterations.
In this case, the optimization cost is improved by tuning only fis1 parameter values.
Next, obtain the parameter settings for fis2 and tune the fis2 parameters.
[in,out,rule] = getTunableSettings(fisTout2,"FIS","fis2"); rng('default') fisTout3 = tunefis(fisTout2,[in;out;rule],x,y,options);
Iter Func-count f(x) MeshSize Method
0 1 0.390338 1
1 2 0.374103 2 Successful Poll
2 5 0.373855 4 Successful Poll
3 10 0.356619 8 Successful Poll
4 33 0.356619 4 Refine Mesh
5 43 0.350715 8 Successful Poll
6 65 0.349417 16 Successful Poll
7 65 0.349417 8 Refine Mesh
8 87 0.349417 4 Refine Mesh
9 91 0.349356 8 Successful Poll
10 112 0.349356 4 Refine Mesh
11 138 0.346102 8 Successful Poll
12 159 0.346102 4 Refine Mesh
13 172 0.345938 8 Successful Poll
14 193 0.345938 4 Refine Mesh
15 222 0.342721 8 Successful Poll
16 244 0.342721 4 Refine Mesh
17 275 0.342721 2 Refine Mesh
18 283 0.340727 4 Successful Poll
19 312 0.340554 8 Successful Poll
20 335 0.340554 4 Refine Mesh
21 366 0.340554 2 Refine Mesh
22 427 0.337873 4 Successful Poll
23 457 0.337873 2 Refine Mesh
24 521 0.33706 4 Successful Poll
25 551 0.33706 2 Refine Mesh
26 624 0.333193 4 Successful Poll
Maximum number of iterations exceeded: increase options.MaxIterations.
The optimization cost is further reduced by tuning the fis2 parameter values. To avoid overfitting of individual FIS parameter values, you can further tune both fis1 and fis2 parameters together.
[in,out,rule] = getTunableSettings(fisTout3);
rng('default')
fisTout4 = tunefis(fisTout3,[in;out;rule],x,y,options);Iter Func-count f(x) MeshSize Method
0 1 0.333193 1
1 8 0.326804 2 Successful Poll
2 91 0.326432 4 Successful Poll
3 116 0.326261 8 Successful Poll
4 154 0.326261 4 Refine Mesh
5 205 0.326261 2 Refine Mesh
6 302 0.326092 4 Successful Poll
7 352 0.326092 2 Refine Mesh
8 391 0.325964 4 Successful Poll
9 441 0.325964 2 Refine Mesh
10 478 0.32578 4 Successful Poll
11 528 0.32578 2 Refine Mesh
12 562 0.325691 4 Successful Poll
13 612 0.325691 2 Refine Mesh
14 713 0.229273 4 Successful Poll
15 763 0.229273 2 Refine Mesh
16 867 0.22891 4 Successful Poll
17 917 0.22891 2 Refine Mesh
18 1036 0.228688 4 Successful Poll
19 1086 0.228688 2 Refine Mesh
20 1212 0.228688 1 Refine Mesh
21 1266 0.228445 2 Successful Poll
22 1369 0.228441 4 Successful Poll
23 1381 0.227645 8 Successful Poll
24 1407 0.226125 16 Successful Poll
25 1407 0.226125 8 Refine Mesh
26 1447 0.226125 4 Refine Mesh
Maximum number of iterations exceeded: increase options.MaxIterations.
Overall, the optimization cost is smaller after using the three tuning steps.