This example shows how to tune parameters of a FIS tree, which is a collection of connected fuzzy inference systems. This example uses particle swarm and pattern search optimization, which require Global Optimization Toolbox™ software.

Automobile fuel consumption prediction in miles per gallon (MPG) is a typical nonlinear regression problem. It uses several automobile profile attributes to predict fuel consumption. The training data is available in the University of California at Irvine Machine Learning Repository and contains data collected from automobiles of various makes and models.

This example uses the following six input data attributes to predict the output data attribute MPG with a FIS tree:

Number of cylinders

Displacement

Horsepower

Weight

Acceleration

Model year

Load the data. Each row of the dataset obtained from the repository represents a different automobile profile.

data = loadgas;

`data`

contains 7 columns, where the first six columns contain the following input attributes.

Number of cylinders

Displacement

Horsepower

Weight

Acceleration

Model year

The seventh column contains the output attribute, MPG.

Create separate input and output data sets, `X`

and `Y`

, respectively.

X = data(:,1:6); Y = data(:,7);

Partition the input and output data sets into training data (odd-indexed samples) and validation data (even-indexed samples).

trnX = X(1:2:end,:); % Training input data set trnY = Y(1:2:end,:); % Training output data set vldX = X(2:2:end,:); % Validation input data set vldY = Y(2:2:end,:); % Validation output data set

Extract the range of each data attribute, which you will use for input/output range definition during FIS construction.

dataRange = [min(data)' max(data)'];

For this example, construct a FIS tree using the following steps:

Rank the input attributes based on their correlations with the output attribute.

Create multiple FIS objects using the ranked input attributes.

Construct a FIS tree from the FIS objects.

**Rank Inputs According to Correlation Coefficients**

Calculate the correlation coefficients for the training data. In the final row of the correlation matrix, the first six elements show the correlation coefficients between the six put data attributes and the output attribute.

c1 = corrcoef(data); c1(end,:)

`ans = `*1×7*
-0.7776 -0.8051 -0.7784 -0.8322 0.4233 0.5805 1.0000

The first four input attributes have negative values, and the last two input attributes have positive values.

Rank the input attributes that have negative correlations in descending order by the absolute value of their correlation coefficients.

Weight

Displacement

Horsepower

Number of cylinders

Rank the input attributes that have positive correlations in descending order by the absolute value of their correlation coefficients.

Model year

Acceleration

These rankings show that the weight and model year have the highest negative and positive correlations with MPG, respectively.

**Create Fuzzy Inference Systems**

For this example, implement a FIS tree with the following structure.

The FIS tree uses multiple two-input-one-output FIS objects to reduce the total number of rules used In the inference process. `fis1`

, `fis2`

, and `fis3`

directly take the input values and generate intermediate `MPG`

values, which are further combined using `fis4`

and `fis5`

.

Input attributes with negative and positive correlation values are paired up to combine both positive and negative effects on the output for prediction. The inputs are grouped according to their ranks as follows:

Weight and model year

Displacement and acceleration

Horsepower and number of cylinders

The last group includes only inputs with negative correlation values since there are only two inputs with positive correlation values.

This example uses Sugeno-type FIS objects for faster evaluation during the tuning process as compared to Mamdani systems. Each FIS includes two inputs and one output, where each input contains two default triangular membership functions (MFs), and the output includes 4 default constant MFs. Specify the input and output ranges using the corresponding data attribute ranges.

The first FIS combines the weight and model year attributes.

fis1 = sugfis('Name','fis1'); fis1 = addInput(fis1,dataRange(4,:),'NumMFs',2,'Name',"weight"); fis1 = addInput(fis1,dataRange(6,:),'NumMFs',2,'Name',"year"); fis1 = addOutput(fis1,dataRange(7,:),'NumMFs',4);

The second FIS combines the displacement and acceleration attributes.

fis2 = sugfis('Name','fis2'); fis2 = addInput(fis2,dataRange(2,:),'NumMFs',2,'Name',"displacement"); fis2 = addInput(fis2,dataRange(5,:),'NumMFs',2,'Name',"acceleration"); fis2 = addOutput(fis2,dataRange(7,:),'NumMFs',4);

The third FIS combines the horsepower and number of cylinder attributes.

fis3 = sugfis('Name','fis3'); fis3 = addInput(fis3,dataRange(3,:),'NumMFs',2,'Name',"horsepower"); fis3 = addInput(fis3,dataRange(1,:),'NumMFs',2,'Name',"cylinders"); fis3 = addOutput(fis3,dataRange(7,:),'NumMFs',4);

The fourth FIS combines the outputs of the first and second FIS.

fis4 = sugfis('Name','fis4'); fis4 = addInput(fis4,dataRange(7,:),'NumMFs',2); fis4 = addInput(fis4,dataRange(7,:),'NumMFs',2); fis4 = addOutput(fis4,dataRange(7,:),'NumMFs',4);

The final FIS combines the outputs of third and fourth FIS and generates the estimated MPG. This FIS has the same input and output ranges as the fourth FIS.

fis5 = fis4; fis5.Name = 'fis5'; fis5.Outputs(1).Name = "mpg";

**Construct FIS Tree**

Connect the fuzzy systems (`fis1`

, `fis2`

, `fis3`

, `fis4`

, and `fis5`

) according to the FIS tree diagram.

fisTin = fistree([fis1 fis2 fis3 fis4 fis5],[ ... "fis1/output1" "fis4/input1"; ... "fis2/output1" "fis4/input2"; ... "fis3/output1" "fis5/input2"; ... "fis4/output1" "fis5/input1"])

fisTin = fistree with properties: FIS: [1x5 sugfis] Connections: [4x2 string] Inputs: [6x1 string] Outputs: "fis5/mpg" DisableStructuralChecks: 0 See 'getTunableSettings' method for parameter optimization.

Tuning is performed in two steps.

Learn the rule base while keeping the input and output MF parameters constant.

Tune the parameters of the input/output MFs and rules.

The first step is less computationally expensive due to the small number of rule parameters, and it 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 provides fast convergence of the parameter tuning process.

**Learn Rules**

To learn a rule base, first specify tuning options using a `tunefisOptions`

object. Global optimization methods (genetic algorithm or particle swarm) are suitable for initial training when all the parameters of a fuzzy system are untuned. For this example, tune the FIS tree using the particle swarm optimization method (`'particleswarm'`

).

To learn new rules, set the `OptimizationType`

to `'learning'`

. Restrict the maximum number of rules to 4. The number of tuned rules of each FIS can be less than this limit, since the tuning process removes duplicate rules.

options = tunefisOptions('Method','particleswarm',... 'OptimizationType','learning', ... 'NumMaxRules',4);

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`

.

Set the maximum number of iterations to 50. To reduce training error in the rule learning process, you can increase the number of iterations. However, using too many iterations can overtune the FIS tree to the training data, increasing the validation errors.

options.MethodOptions.MaxIterations = 50;

Since particle swarm optimization uses random search, to obtain reproducible results, initialize the random number generator to its default configuration.

`rng('default')`

Tune the FIS tree using the specified tuning data and options. Set the input order of the training data according to the FIS tree connections as follows: `weight`

, `year`

, `displacement`

, `acceleration`

, `horsepower`

, and `cylinders`

.

inputOrders1 = [4 6 2 5 3 1]; orderedTrnX1 = trnX(:,inputOrders1);

Learning rules with `tunefis`

function takes approximately 4 minutes. For this example, enable tuning by setting `runtunefis`

to `true`

. To load pretrained results without running `tunefis`

, you can set `runtunefis`

to `false`

.

runtunefis = false;

Parameter settings can be empty when learning new rules. For more information, see `tunefis`

.

if runtunefis fisTout1 = tunefis(fisTin,[],orderedTrnX1,trnY,options); %#ok<UNRCH> else tunedfis = load('tunedfistreempgprediction.mat'); fisTout1 = tunedfis.fisTout1; fprintf('Training RMSE = %.3f MPG\n',calculateRMSE(fisTout1,orderedTrnX1,trnY)); end

Training RMSE = 3.399 MPG

The `Best f(x)`

column shows the training root-mean-squared-error (RMSE).

The learning process produces a set of new rules for the FIS tree.

`fprintf("Total number of rules = %d\n",numel([fisTout1.FIS.Rules]));`

Total number of rules = 17

The learned system should have similar RMSE performance for both the training and validation data sets. To calculate the RMSE for the validation data set, evaluate `fisout1`

using validation input data set `vldX`

. To hide run-time warnings during evaluation, set all the warning options to `none`

.

Calculate the RMSE between the generated output data and the validation output data set `vldY`

. Since the training and validation errors are similar, the learned system does not overfit the training data.

orderedVldX1 = vldX(:,inputOrders1); plotActualAndExpectedResultsWithRMSE(fisTout1,orderedVldX1,vldY)

**Tune All Parameters**

After learning the new rules, tune the input/output MF parameters along with the parameters of the learned rules. To obtain the tunable parameters of the FIS tree, use the `getTunableSettings`

function.

[in,out,rule] = getTunableSettings(fisTout1);

To tune the existing FIS tree parameter settings without learning new rules, set the `OptimizationType`

to `'tuning'`

.

`options.OptimizationType = 'tuning';`

Since the FIS tree already learned rules using the training data, use a local optimization method for fast convergence of the parameter values. For this example, use the pattern search optimization method (`'patternsearch'`

).

`options.Method = 'patternsearch';`

Tuning the FIS tree parameters takes more iterations than the previous rule-learning step. Therefore, increase the maximum number of iterations of the tuning process to 75. As in the first tuning stage, you can reduce training errors by increasing the number of iterations. However, using too many iterations can overtune the parameters to the training data, increasing the validation errors.

options.MethodOptions.MaxIterations = 75;

To improve pattern search results, set method option `UseCompletePoll`

to true.

options.MethodOptions.UseCompletePoll = true;

Tune the FIS tree parameters using the specified tunable settings, training data, and tuning options.

Tuning parameter values with `tunefis`

function takes several minutes. To load pretrained results without running `tunefis`

, you can set `runtunefis`

to `false`

.

rng('default') if runtunefis fisTout2 = tunefis(fisTout1,[in;out;rule],orderedTrnX1,trnY,options); %#ok<UNRCH> else fisTout2 = tunedfis.fisTout2; fprintf('Training RMSE = %.3f MPG\n',calculateRMSE(fisTout2,orderedTrnX1,trnY)); end

Training RMSE = 3.037 MPG

At the end of the tuning process, the training error reduces compared to the previous step.

Validate the performance of the tuned FIS tree, `fisout2`

, using the validation input data set `vldX`

.

Compare the expected MPG obtained from the validation output data set `vldY`

and actual MPG generated using `fisout2`

. Compute the RMSE between these results.

plotActualAndExpectedResultsWithRMSE(fisTout2,orderedVldX1,vldY)

Tuning the FIS tree parameters improves the RMSE compared to the results from the initial learned rule base. Since the training and validation errors are similar, the parameters values are not overtuned.

To gain insight into the operation of your fuzzy tree, you can add the outputs of the component fuzzy systems as outputs of your FIS tree. For this example, to access the intermediate FIS outputs, add three additional outputs to the tuned FIS tree.

fisTout3 = fisTout2; fisTout3.Outputs(end+1) = "fis1/output1"; fisTout3.Outputs(end+1) = "fis2/output1"; fisTout3.Outputs(end+1) = "fis3/output1";

To generate the additional outputs, evaluate the augmented FIS tree, `fisTout3`

.

actY = evaluateFIS(fisTout3,orderedVldX1); figure,plot(actY(:,[2 3 4 1])),xlabel("Input dataset index"),ylabel("MPG"),axis([1 200 0 55]) legend(["Output of fis1" "Output of fis2" "Output of fis3" "Output of fis5"],... 'Location','NorthEast','NumColumns',2) title("Intermediate and Final Outputs")

The final output of the FIS tree (`fis5`

output) appears to be highly correlated with the outputs of `fis1`

and `fis3`

. To validate this assessment, check the correlation coefficients of the FIS outputs.

c2 = corrcoef(actY(:,[2 3 4 1])); c2(end,:)

`ans = `*1×4*
0.9541 0.8245 -0.8427 1.0000

The last row of the correlation matrix shows that the outputs of `fis1`

and `fis3`

(first and third column, respectively) have higher correlations with the final output as compared to the output of `fis2`

(second column). This result indicates that simplifying the FIS tree by removing `fis2`

and `fis4`

and can potentially produce similar training results compared to the original tree structure.

Remove `fis2`

and `fis4`

from the FIS tree and connect the output of `fis1`

to the first input of `fis5`

. When you remove a FIS from a FIS tree, any existing connections to that FIS are also removed.

fisTout3.FIS([2 4]) = []; fisTout3.Connections(end+1,:) = ["fis1/output1" "fis5/input1"]; fis5.Inputs(1).Name = "fis1out";

To make the number of FIS tree outputs match the number of outputs in the training data, remove the FIS tree outputs from `fis1`

and `fis3`

.

fisTout3.Outputs(2:end) = [];

Update the input training data order according to the new FIS tree input configuration.

inputOrders2 = [4 6 3 1]; orderedTrnX2 = trnX(:,inputOrders2);

Since the FIS tree configuration is changed, you must rerun both the learning and tuning steps. In the learning phase, the existing rule parameters are also tuned to fit the new configuration of the FIS tree.

options.Method = "particleswarm"; options.OptimizationType = "learning"; options.MethodOptions.MaxIterations = 50; [~,~,rule] = getTunableSettings(fisTout3); rng('default') if runtunefis fisTout4 = tunefis(fisTout3,rule,orderedTrnX2,trnY,options); %#ok<UNRCH> else fisTout4 = tunedfis.fisTout4; fprintf('Training RMSE = %.3f MPG\n',calculateRMSE(fisTout4,orderedTrnX2,trnY)); end

Training RMSE = 3.380 MPG

In the training phase, the parameters of the membership function and rules are tuned.

options.Method = "patternsearch"; options.OptimizationType = "tuning"; options.MethodOptions.MaxIterations = 75; options.MethodOptions.UseCompletePoll = true; [in,out,rule] = getTunableSettings(fisTout4); rng('default') if runtunefis fisTout5 = tunefis(fisTout4,[in;out;rule],orderedTrnX2,trnY,options); %#ok<UNRCH> else fisTout5 = tunedfis.fisTout5; fprintf('Training RMSE = %.3f MPG\n',calculateRMSE(fisTout5,orderedTrnX2,trnY)); end

Training RMSE = 3.049 MPG

At the end of the tuning process, the FIS tree contains updated MF and rule parameter values. The rule base size of the new FIS tree configuration is smaller than the previous configuration.

`fprintf("Total number of rules = %d\n",numel([fisTout5.FIS.Rules]));`

Total number of rules = 11

Evaluate the updated FIS tree using the four input attributes of the checking dataset.

orderedVldX2 = vldX(:,inputOrders2); plotActualAndExpectedResultsWithRMSE(fisTout5,orderedVldX2,vldY)

The simplified FIS tree with four input attributes produces better results in terms of RMSE as compared to the first configuration, which uses six input attributes. Therefore, it shows that a FIS tree can be represented with fewer number of inputs and rules to generalize the training data.

You can further improve the training error of the tuned FIS tree by:

Increasing number of iterations in both the rule-learning and parameter-tuning phases. Doing so increases the duration of the optimization process and can also increase validation error due to overtuned system parameters with the training data.

Using global optimization methods, such as

`ga`

and`particleswarm`

, in both rule-learning and parameter-tuning phases.`ga`

and`particleswarm`

perform better for large parameter tuning ranges since they are global optimizers. On the other hand,`patternsearch`

and`simulannealbnd`

perform better for small parameter ranges since they are local optimizers. If rules are already added to a FIS tree using training data, then`patternsearch`

and`simulannealbnd`

may produce faster convergence as compared to`ga`

and`particleswarm`

. For more information on these optimization methods and their options, see`ga`

,`particleswarm`

,`patternsearch`

, and`simulannealbnd`

.Changing the FIS properties, such as the type of FIS, number of inputs, number of input/output MFs, MF types, and number of rules. For fuzzy systems with a large number of inputs, a Sugeno FIS generally converges faster than a Mamdani FIS since a Sugeno system has fewer output MF parameters (if

`constant`

MFs are used) and faster defuzzification. Small numbers of MFs and rules reduce the number of parameters to tune, producing a faster tuning process. Furthermore, a large number of rules may overfit the training data.Modifying tunable parameter settings for MFs and rules. For example, you can tune the support of a triangular MF without changing its peak location. Doing so reduces the number of tunable parameters and can produce a faster tuning process for specific applications. For rules, you can exclude zero MF indices by setting the

`AllowEmpty`

tunable setting to`false`

, which reduces the overall number of rules during the learning phase.Changing FIS tree properties, such as number of fuzzy systems and connections between the fuzzy systems.

Using different ranking and grouping of the inputs to the FIS tree.

function plotActualAndExpectedResultsWithRMSE(fis,x,y) % Calculate RMSE bewteen actual and expected results [rmse,actY] = calculateRMSE(fis,x,y); % Plot results figure subplot(2,1,1) hold on bar(actY) bar(y) bar(min(actY,y),'FaceColor',[0.5 0.5 0.5]) hold off axis([0 200 0 60]) xlabel("Validation input dataset index"),ylabel("MPG") legend(["Actual MPG" "Expected MPG" "Minimum of actual and expected values"],... 'Location','NorthWest') title("RMSE = " + num2str(rmse) + " MPG") subplot(2,1,2) bar(actY-y) xlabel("Validation input dataset index"),ylabel("Error (MPG)") title("Difference Between Actual and Expected Values") end function [rmse,actY] = calculateRMSE(fis,x,y) % Evaluate FIS actY = evaluateFIS(fis,x); % Calculate RMSE del = actY - y; rmse = sqrt(mean(del.^2)); end function y = evaluateFIS(fis,x) % Specify options for FIS evaluation persistent evalOptions if isempty(evalOptions) evalOptions = evalfisOptions("EmptyOutputFuzzySetMessage","none", ... "NoRuleFiredMessage","none","OutOfRangeInputValueMessage","none"); end % Evaluate FIS y = evalfis(fis,x,evalOptions); end

`fistree`

| `getTunableSettings`

| `sugfis`

| `tunefis`