Tune Fuzzy Rules and Membership Function Parameters

This example shows how to tune a fuzzy inference system (FIS). Using the tunefis function, you can:

• Tune membership function parameters for input and output variables.

• Learn fuzzy rules.

• Tune the antecedent and consequent parameters of fuzzy rules.

For more information on tuning a FIS, see Tuning Fuzzy Inference Systems.

Tune Membership Function Parameters

For both type-1 and type-2 FISs, you can specify tunable parameter settings for the input and output MFs and tune the values of the selected parameters. You can tune the parameters for any combination of input and output MFs. This example shows an example workflow using a type-1 FIS. For an example that tunes a type-2 FIS, see Predict Chaotic Time Series Using Type-2 FIS.

Create a FIS.

fis = mamfis;
fis = addInput(fis,[0 10],'NumMFs',3);
fis = addOutput(fis,[0 1],'NumMFs',3);
fis = addRule(fis,[1 1 1 1;1 1 1 1;1 1 1 1]);

Extract input and output parameter settings from the FIS.

[in,out] = getTunableSettings(fis)

in = 
  VariableSettings with properties:

                   Type: "input"
           VariableName: "input1"
    MembershipFunctions: [1×3 fuzzy.tuning.MembershipFunctionSettings]
                FISName: "fis"

out = 
  VariableSettings with properties:

                   Type: "output"
           VariableName: "output1"
    MembershipFunctions: [1×3 fuzzy.tuning.MembershipFunctionSettings]
                FISName: "fis"

The parameter settings are represented by VariableSettings objects that include the FIS name, variable type, variable name, and MF parameter settings. Examine the parameter settings of MF 1 of input 1.

in(1).MembershipFunctions(1).Parameters

ans = 
  NumericParameters with properties:

    Minimum: [-Inf -Inf -Inf]
    Maximum: [Inf Inf Inf]
       Free: [1 1 1]

For each parameter value of an input or output MF, you can specify its availability for tuning and its minimum and maximum values. By default, all MF parameters are free for tuning and their ranges are set to [-Inf, Inf].

You can specify the tunability of all parameters in an MF using the setTunable function. For example, make MF 1 of input 1 nontunable.

in(1).MembershipFunctions(1) = setTunable(in(1).MembershipFunctions(1),false);

You can also specify the tunability of individual MF parameters. Make the first parameter of MF 2 of input 1 nontunable.

in(1).MembershipFunctions(2).Parameters.Free(1) = false;

For each parameter, you can specify the range of possible values. For example, set the minimum values for the second and third parameters of MF 3 of input 1 to 0.

in(1).MembershipFunctions(3).Parameters.Minimum(2:3) = 0;

Similarly, set the maximum values for second and third parameters of MF 3 of input 1 to 15.

in(1).MembershipFunctions(3).Parameters.Maximum(2:3) = 15;

The tuning process sets the default minimum and maximum range values of tunable MF parameters to their corresponding input or output ranges.

Finally, make the MF parameters of all output membership functions nontunable.

out = setTunable(out,false);

Specify input and output training data. For this example, generate training data using the following function.

$$\mathit{y}=\left|\frac{\sin \left(2\mathit{x}\right)}{{\mathit{e}}^{\mathit{x}/5}}\right|$$

x = (0:0.1:10)';
y = abs(sin(2*x)./exp(x/5));

Specify options for tunefis. For this example, use the genetic algorithm tuning method. Use a maximum of five generations for optimization.

options = tunefisOptions("Method","ga");
options.MethodOptions.MaxGenerations = 5;

If you have Parallel Computing Toolbox™ software, you can improve the speed of the tuning process by setting options.UseParallel to true. If you do not have Parallel Computing Toolbox software, set options.UseParallel to false, which is the default value.

By default, tunefis uses the root mean squared error (RMSE) for cost calculation. You can change the cost function to norm1 or norm2 by setting options.DistanceMetric.

options.DistanceMetric = "norm1";

Tune fis using the parameter settings, training data, and tuning options.

rng('default') % for reproducibility
[fisout,optimout] = tunefis(fis,[in;out],x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              100           32.84           32.84        0
    2              147           32.84           32.84        1
    3              194           32.84           32.84        2
    4              241           32.84           32.84        3
    5              288           32.84           32.84        4
Optimization terminated: maximum number of generations exceeded.

fisout includes the updated parameter values. optimout provides additional outputs of the optimization method and any error messages that are returned during the update process of the input fuzzy system using the optimized parameter values.

optimout

optimout = struct with fields:
    tuningOutputs: [1×1 struct]
    totalFcnCount: 288
     totalRuntime: 2.7322
     errorMessage: []

optimout.tuningOutputs

ans = struct with fields:
             x: [5 9.1667 5.8333 10 14.1667]
          fval: 32.8363
      exitflag: 0
        output: [1×1 struct]
    population: [50×5 double]
        scores: [50×1 double]

You can optionally tune fis using either the input or output parameter settings only. In this example, since you set the output parameter settings to nontunable, tuning the FIS with just the input parameter settings produces the same results.

rng('default')
[fisout,optimout] = tunefis(fis,in,x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              100           32.84           32.84        0
    2              147           32.84           32.84        1
    3              194           32.84           32.84        2
    4              241           32.84           32.84        3
    5              288           32.84           32.84        4
Optimization terminated: maximum number of generations exceeded.

optimout

optimout = struct with fields:
    tuningOutputs: [1×1 struct]
    totalFcnCount: 288
     totalRuntime: 2.4521
     errorMessage: []

optimout.tuningOutputs

ans = struct with fields:
             x: [5 9.1667 5.8333 10 14.1667]
          fval: 32.8363
      exitflag: 0
        output: [1×1 struct]
    population: [50×5 double]
        scores: [50×1 double]

Tune Fuzzy Rules

In addition to tuning membership function parameters, you can tune the antecedent and consequent parameters of the rules in a fuzzy system.

Obtain rule parameter settings from a fuzzy system using getTunableSettings.

[~,~,rule] = getTunableSettings(fis)

rule=3×1 object
  3×1 RuleSettings array with properties:

    Index
    Antecedent
    Consequent
    FISName

Each rule parameter setting includes the FIS name, the index of the rule in the FIS, and parameter settings for the rule antecedent and consequent (the rule clauses).

For a rule clause, you can set the following parameters settings.

• AllowNot — Allow the use of NOT logic, that is, negative MF indices. By default, rules do not allow NOT logic.

• Free — Make the input/output MF indices available for tuning. By default, clause parameters are available for tuning.

• AllowEmpty — Allow the absence of input/output variables, that is, zero MF indices. By default, the absence of a variable is allowed.

rule(1).Antecedent(1)

ans = 
  ClauseParameters with properties:

      AllowNot: 0
    AllowEmpty: 1
          Free: 1

Allow NOT logic in the antecedent of rule 1.

rule(1).Antecedent.AllowNot = true;

Make the consequent of rule 1 not available for tuning.

rule(1).Consequent.Free = 0;

Do not allow absence of a variable in the consequent of rule 2.

rule(2).Consequent.AllowEmpty = false;

Set rule 3 as nontunable.

rule(3) = setTunable(rule(3),false);

Set options.DistanceMetric to norm2.

options.DistanceMetric = "norm2";

Tune fis using the rule parameter settings.

rng('default')  % for reproducibility
fisout = tunefis(fis,rule,x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              100           1.648           2.575        0
    2              147           1.648           2.448        1
    3              194           1.648           2.212        2
    4              241           1.648           2.052        3
    5              288           1.648           1.874        4
Optimization terminated: maximum number of generations exceeded.

Since you specified rule 3 as nontunable, you can exclude rule 3 when you tune fis. Doing so produces the same tuning result.

rng('default')  % for reproducibility
fisout = tunefis(fis,rule(1:2),x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              100           1.648           2.575        0
    2              147           1.648           2.448        1
    3              194           1.648           2.212        2
    4              241           1.648           2.052        3
    5              288           1.648           1.874        4
Optimization terminated: maximum number of generations exceeded.

Learn Fuzzy Rules

You can configure tunefis to learn the rules of a fuzzy system. To do so, set the OptimizationType option of tunefisOptions to "learning".

fisin = fis;
fisin.Rules = [];
options.OptimizationType = "learning";

Set the maximum number of rules in the tuned FIS to 3.

options.NumMaxRules = 3;

The size of the tuned rule base might be less than NumMaxRules, because tunefis removes duplicate rules from the tuned FIS. If you do not specify NumMaxRules, then tunefis adds the maximum number of rules determined by the possible combinations of input MFs. The default input MF combinations include zero MF indices, which allow absence of variables. The default combinations exclude negative MF indices, so that NOT logic is not allowed.

Set options.DistanceMetric to "rmse" and tune the FIS.

options.DistanceMetric = "rmse";
rng('default')  % for reproducibility
fisout = tunefis(fisin,[],x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              400           0.165          0.2956        0
    2              590           0.165          0.2805        1
    3              780           0.165          0.2578        2
    4              970           0.165          0.2393        3
    5             1160           0.165          0.2322        4
Optimization terminated: maximum number of generations exceeded.

During the tuning process, the FIS automatically learns rules after cost optimization with the training data. Examine the tuned rules.

fisout.Rules

ans = 
  1×3 fisrule array with properties:

    Description
    Antecedent
    Consequent
    Weight
    Connection

  Details:
                   Description           
         ________________________________

    1    "input1==mf3 => output1=mf1 (1)"
    2    "input1==mf1 => output1=mf2 (1)"
    3    "input1==mf2 => output1=mf1 (1)"

You can remove some of the existing rules and learn additional rules.

fisout.Rules(2:end) = [];
rng('default')  % for reproducibility
fisout = tunefis(fisin,[],x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              400           0.165          0.2956        0
    2              590           0.165          0.2805        1
    3              780           0.165          0.2578        2
    4              970           0.165          0.2393        3
    5             1160           0.165          0.2322        4
Optimization terminated: maximum number of generations exceeded.

fisout.Rules

ans = 
  1×3 fisrule array with properties:

    Description
    Antecedent
    Consequent
    Weight
    Connection

  Details:
                   Description           
         ________________________________

    1    "input1==mf3 => output1=mf1 (1)"
    2    "input1==mf1 => output1=mf2 (1)"
    3    "input1==mf2 => output1=mf1 (1)"

You can also tune the antecedents and consequents of existing rules and learn new rules. To do so, obtain the tunable rule parameter settings and pass them to the tunefis function.

fisout.Rules(2:end) = [];
fisout.Rules(1).Antecedent = 1;
fisout.Rules(1).Consequent = 1;
[~,~,rule] = getTunableSettings(fisout);
rng('default')
fisout = tunefis(fisin,rule,x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              400           0.165          0.3063        0
    2              590           0.165          0.2845        1
    3              780           0.165          0.2549        2
    4              970           0.165          0.2344        3
    5             1160           0.165          0.2153        4
Optimization terminated: maximum number of generations exceeded.

fisout.Rules

ans = 
  1×3 fisrule array with properties:

    Description
    Antecedent
    Consequent
    Weight
    Connection

  Details:
                   Description           
         ________________________________

    1    "input1==mf1 => output1=mf2 (1)"
    2    "input1==mf2 => output1=mf1 (1)"
    3    "input1==mf3 => output1=mf1 (1)"

Tune MF and Rule Parameters

You can tune all MF and rule parameters simultaneously. First, obtain all parameter settings for the FIS.

[in,out,rule] = getTunableSettings(fis);

Configure the tuning options.

options = tunefisOptions('Method','ga');
options.MethodOptions.MaxGenerations = 5;

Tune the MF and rule parameters of the FIS.

rng('default')  % for reproducibility
fisout = tunefis(fis,[in;out;rule],x,y,options);

                                  Best           Mean      Stall
Generation      Func-count        f(x)           f(x)    Generations
    1              400           0.165           0.296        0
    2              590          0.1638          0.2821        0
    3              780          0.1625          0.2697        0
    4              970          0.1625          0.2616        1
    5             1160          0.1604          0.2512        0
Optimization terminated: maximum number of generations exceeded.

For a large fuzzy system, if you tune all FIS parameters in the same tuning process, obtaining the expected results can take several iterations. To improve the tuning time, you can tune parameters using the following two steps.

1. Tune or learn rule parameters only.

2. Tune both MF and rule parameters.

Learning and tuning rules is less computationally expensive that uning the MF parameters, due to the small number of rule parameters. Therefore, the first step quickly converges to a fuzzy rule base during training. In the second step, using the rule base from the first step as an initial condition improves convergence of the parameter tuning process.

Generate FIS from Data and Tune

When you manually create a FIS for tuning, you must either manually create an initial rule base or learn the initial rules. Alternatively, you can generate a FIS using the genfis function, which creates an initial rule base based on your training data. You can then optimize the FIS using tunefis. In this approach, the tuning process can employ a local optimization method because the rule base is derived from the training data.

This example uses the same training data as the preceding examples.

Create options for genfis that specify five MFs, a Gaussian MF for the input, and a constant MF for the output.

goptions = genfisOptions('GridPartition','NumMembershipFunctions',5, ...
    'InputMembershipFunctionType','gaussmf', ...
    'OutputMembershipFunctionType','constant');

Generate the initial FIS and get its parameter settings.

fisin = genfis(x,y,goptions);
[in,out,rule] = getTunableSettings(fisin);

Use the pattern search method for optimization, setting the maximum number of iterations to 25, and tune the FIS.

toptions = tunefisOptions('Method','patternsearch');
toptions.MethodOptions.MaxIterations = 25;
rng('default')
fisout = tunefis(fisin,[in;out],x,y,toptions);

Iter     Func-count       f(x)      MeshSize     Method
    0           1       0.346649             1      
    1          19       0.346649           0.5     Refine Mesh
    2          37       0.273812             1     Successful Poll
    3          38       0.236413             2     Successful Poll
    4          39       0.190794             4     Successful Poll
    5          40       0.182142             8     Successful Poll
    6          47       0.182142             4     Refine Mesh
    7          49       0.162927             8     Successful Poll
    8          56       0.162927             4     Refine Mesh
    9          67       0.162927             2     Refine Mesh
   10          69       0.159539             4     Successful Poll
   11          80       0.159539             2     Refine Mesh
   12          92       0.159539             1     Refine Mesh
   13          94       0.159421             2     Successful Poll
   14         106       0.159373             4     Successful Poll
   15         117       0.159373             2     Refine Mesh
   16         125       0.159185             4     Successful Poll
   17         136       0.159185             2     Refine Mesh
   18         151       0.159185             1     Refine Mesh
   19         152        0.15914             2     Successful Poll
   20         167        0.15914             1     Refine Mesh
   21         170       0.158914             2     Successful Poll
   22         185       0.158914             1     Refine Mesh
   23         187       0.158839             2     Successful Poll
   24         202       0.158839             1     Refine Mesh
   25         206       0.158366             2     Successful Poll
   26         215       0.158121             4     Successful Poll
Maximum number of iterations exceeded: increase options.MaxIterations.

You can increase the number of iterations to further optimize the cost.