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Fig. 10. Example—2 Clean set: target versus predicted test values. The test outputs belong to algorithm T2HFIT™ that has the test RMSE E; = 0.0058. — Figure 10 Example—2 Clean set: target versus predicted test values. The test outputs belong to algorithm T2HFIT™ that has the test RMSE E; = 0.0058.

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Fig. 1. Membership functions. (a) Type-1 MF (5) with mean m = 5.0 and o = 2.0 (b) Type-2 Fuzzy MF with fixed o = 2.0 and means m, = 4.5 and m2 = 5.5. LMF by (2) as per (8) is on the dotted line and UMF fi ; (a) as per (9) is on the solid line. In this work, the LMF was defined as [4]:

Fig. 3. Hierarchical fuzzy inference system. (a) Complete tree with three nodes Ni, No, and N3 and with inputs 71, v2, 3, £4, and x5. (b) Illustration of the i-th node N; that has n’ inputs 24 € {x1,...,an} for j = 1 to n’ and output y;. Fig. 2. Two-phase construction of a hierarchical fuzzy inference tree.

1) Initial Population: Two fitness measures were consid- ered: approximation error minimization and parameter count minimization. To simultaneously optimize these objectives during the structure-tuning phase, an initial population W° of randomly generated HFITs was formed and sorted according to their nondominance.

Fig. 5. MOGP Operations: (a) Crossover between two parent trees a and b. (b) A total five types of mutation of the parent tree.

Fig. 6. FIS fitness versus FIS parameter mapping across the Pareto-front. This graph was created during a multiobjective training of the example—2 mentioned in Section V-B.

PARAMETER SETTING FOR THE EXPERIMENTS

DESCRIPTIONS OF THE EXISTING FIS ALGORITHMS ACQUIRED FOR THE PERFORMANCE COMPARISONS TABLE II Similarly, the proposed T2FIS algorithms T2HFITS and T2HFIT™ offered better performance compared to the al- gorithms T2FLS (Singleton), T2FLS (TSK), TSCIT2FNN, T2TSKFNS, T2FNN, SIT2FNN, RIT2NFS-WB, MRI2NFS. The algorithm SEIT2FNN reported test RMSE EF, = 0.0022 and the parameter count was 84; whereas, in comparison to SEIT2FNN, the algorithm T2HFIT™ offered a slightly higher test RMSE £; = 0.0028, but had a lower parameter count c(w), ie., 72.

PERFORMANCE EVALUATION ON SYSTEM IDENTIFICATION (EXAMPLE- 1)

*This is approximately calculated. It may be larger. PERFORMANCE EVALUATION ON CLEAN SET OF NOISY CHAOTIC TIME SERIES PREDICTION (EXAMPLE-2) mean correlation coefficient r,, of the training set of all algo- rithms is 1.00, and the mean correlation coefficient r; of the test set of the algorithms TIHFITS, TIHFIT™, T2HFITS, and T2HFIT™ are 0.9858, 0.9864, 0.9783, and 0.9912 respectively. That is, the test correlation coefficients are closer to 1.00 (a high positive correlation between target and predicted outputs). Such performance indicates that the algorithms consistently performed with a high accuracy, and the obtained low values of STDs are evidence of this fact (Table [V(a)).

Fig. 7. Example—1: designed HFIT, where the shaded nodes indicate T2FIS. The training RMSE of models (a), (b), (c) and (d) are 0.0043, 0.0041, 0.0034, and 0.0028, respectively.

Fig. 8. Example-1: target versus predicted test values. The test outputs belong to algorithm T2HFIT™, which has the test RMSE FE; = 0.0028.

Fig. 9. Example—2 Clean set: designed HFIT. The shaded nodes are T2FIS. The training RMSE of models (a), (b), (c) and (d) are 0.0115, 0.0115, 0.0108, and 0.0032, respectively.

EXAMPLE 2-NOISY SET: PERFORMANCE COMPARISON

(b) Performance Comparison

(a) Performance Statistics (10 repetitions)

PERFORMANCE EVALUATION ON ABALONE AGE PREDICTION PROBLEM (EXAMPLE-4)

Fig. 12. Example—5: designed HFIT, where the shaded nodes are T2FIS. The training RMSE of models (a), (b), (c) and (d) are 0.2455, 0.2838, 0.2767, and 0.2840, respectively. Fig. 11. Example—4: designed HFIT, where the shaded nodes are T2FIS. The training RMSE of models (a), (b), (c) and (d) are 2.1097, 2.2857, 2.1154, and 2.1275, respectively.

(a) Performance Statistics (10 Repetitions) PERFORMANCE EVALUATION ON BOX-JENKINS GAS CONCENTRATION PROBLEM (EXAMPLE-5)

complex problem since a vast number of factors govern its dissolution rate profile and it has a high noise and redundancy because the dataset was obtained from various experimental measurements and instruments. As per the dataset provided in [97], [98], this problem has 747 samples and a total of 300 input features, which influence the PLGA protein particle’s dissolution rate [99]. The input features are categorized into five groups: protein descriptor, formulation characteristics, plasticizer, emulsifier, and time delay, which has 85, 17, 98, 99, and | features, respectively.

Fig. 13. Example—6: Designed HFIT, where the shaded nodes are T2FIS. EXAMPLE 6: PERFORMANCE COMPARISON Using the parameter setting menti ioned in Table I and using 10-fold cross-validation, the proposed algorithm T1HFIT™ was able to select seven input features and was able to approximate a test RMSE of E; = 18.66. The selected features were: phase polyvinyl alcohol Mw (90), ASA (2122), pH 8 msdon (%192), aromatic bond count (%204), a(xx) (%218), PH 12 msacc (%281), time days (2299). Similarly, the proposed algorithm T2HFIT was able to approximate a test RMSE of &, = 15.259 with only four input features: aromatic atom count (266), phase polyvinyl alcohol concentration inner phase (%gg), PH 1 msdon (#285), time days (2299). Additionally, T2HFIT provided a simple mode (i.e., c(w) = 108) com- pared to TIHFIT™ that had a model’s complexity c(w) = 156. Moreover, T2HFIT™ takes 7.16 minutes of training time compared to the 45.7 minutes of 1 T1HFIT. This difference in time is due to the difference between the number of input features being selected by T2HFIT™ and TIHFIT™.

PERFORMANCE SUMMARY ON BENCHMARK EXAMPLES: SINGLE OBJECTIVE VERSUS MULTIOBJECTIVE AND TYPE-|1 VERSUS TYPE-2 T2HFIT™ was much less than RIT2NFS-WB. For example— 4, T2HFIT™ outperformed all its counterparts in both accu- racy and complexity. For example—5, the training RMSE of T2HFIT™ was close to SEIT2FNN, but on model’s complexity and training time, TZHFIT™ outperformed SEIT2FNN by a comfortable margin. Therefore, it may be concluded that the proposed HFIT versions performed efficiently against other algorithms found in the literature. TABLE X