An Example of Using the Interpretability Improvement

interpretability, it was applied to the problem of modelling the mechanical property UTS of steel (15-input and 1-output; 2820 data). The initial number of fuzzy rules is set to 20 and Th1 - Th5 are set to 0.001, 0.1, 0.6, 0.9 and 0.95, respectively.

Table 5-7 shows the main parameters of the fuzzy models, which were obtained following the different operation steps during the interpretability improvement process. In this table, rule length refers to the total number of antecedent conditions; it can be seen that the interpretability improvement approach succeeded in reducing the number of fuzzy rules, the number of fuzzy sets and in generally simplifying the structure of the fuzzy rules.

Table 5-7. The main parameters of the UTS fuzzy models following the different stages of the interpretability improvement

Fuzzy model Number of fuzzy rules

Number of fuzzy sets in every input and output

dimension

Rule length of every fuzzy rule

RMSEof training data Before the interpretability improvement 15 Inputs: [15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15] Output: 15 [15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15] 31.04 After Step 5.5.1.2 12 Inputs: [12; 12; 12; 12; 12; 12; 12; 12; 12; 12; 12; 12; 12; 12; 12] Output: 12 [15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15; 15] 35.87 After Step 5.5.1.3 12 Inputs: [10; 10; 10; 11; 11; 9; 9; 12; 10; 10; 9; 8; 10; 7; 10] Output: 12 [12; 9; 13; 14; 11; 15; 13; 9; 13; 12; 12; 13] 47.01 After Step 5.5.1.4 12 Inputs: [4; 6; 6; 7; 6; 6; 3; 8; 7; 7; 4; 4; 7; 7; 5] Output: 8 [12; 9; 13; 14; 11; 15; 12; 9; 13; 12; 12; 13] 66.75

Chapter 5: FM-HCMO

Figure 5-18 shows the membership functions of Input 14 and Input 15 before the interpretability improvement, and Figure 5-19 shows the same membership functions after the interpretability improvement. Comparing these figures, it can be seen that the distributions of membership functions have improved significantly.

(a) (b)

Figure 5-18. The membership functions of Inputs 14 and 15 before the interpretability improvement

(a) (b)

Figure 5-19. The membership functions of Inputs 14 and 15 after the interpretability improvement

5.5.2.2 Effects of the Thresholds of the Interpretability Improvement Approach

In the proposed interpretability improvement approach, there are five thresholds that need to be set in the 4-step operation. To inspect their effects on the system performance, a set of experiments have been carried out. These experiments are based on the UTS data (2820 data) with the initial number of fuzzy rules being set to 12.

1. The first step of the interpretability improvement is to remove the redundant fuzzy rules. Th1 is used to define whether one rule is redundant or not. Th1 was set to be variable in the range 0 Th1 0.05 and other thresholds and parameters were fixed at Th2 = 1; Th3 = 1; Th4 = 1; Th5 = 1. Figure 5-20 shows the system performance with different Th1 values.

From Figure 5-20, it can be seen that, with the increase of Th1, the RMSE of the obtained system increases while the number of rules, the number of fuzzy sets and the total rule length decrease. This means that, with the increase of Th1, the resulting model accuracy decreases and its interpretability increases.

Chapter 5: FM-HCMO

(a) (b)

(c) (d)

Figure 5-20. The performance of fuzzy models following the interpretability improvement with different Th1: (a) RMSE versus Th1; (b) the number of rules versus Th1; (c) the number of fuzzy sets versus Th1; (d) the total length of rules versus Th1

2. The second step of the interpretability improvement is to merge similar fuzzy rules. Th2 is used to define whether two rules are similar enough to be merged. In this experiment, Th2 was set to be variable in the range 0.01

Th2 1 and other thresholds and parameters were fixed at Th1 = 0; Th3 = 1;Th4 = 1; Th5 = 1. Figure 5-21 shows the model performance with various

Th2 values.

values of the obtained fuzzy model tend to decrease while the number of rules, the number of fuzzy sets and the total rule length tend to increase, which means that the obtained model accuracy tends to increase and its interpretability tends to decrease.

(a) (b)

(c) (d)

Figure 5-21. The performance of fuzzy models following the interpretability improvement with different Th2: (a) RMSE versus Th2; (b) the number of rules versus Th2; (c) the number of fuzzy sets versus Th2; (d) the total length of rules versus Th2

3. The third step of the interpretability improvement is to remove redundant fuzzy sets. Th3 is used to define whether one fuzzy set is redundant. In this experiment, Th3 was set to be variable in the range 0.5 Th3 1 and other

Chapter 5: FM-HCMO

thresholds and parameters were fixed at Th1 = 0; Th2 = 1; Th4 = 1; Th5 = 1. Figure 5-22 shows the system performance with different Th3.

From this figure, it can be seen that, with the increase of Th3, the RMSE of the result model tends to decrease; the number of rules does not change; the number of fuzzy sets and the total rule length increase. This means that the obtained model accuracy tends to increase while its interpretability decreases.

(a) (b)

(c) (d)

Figure 5-22. The performance of fuzzy models following the interpretability improvement with different Th3: (a) RMSE versus Th3; (b) the number of rules versus Th3; (c) the number of fuzzy sets versus Th3; (d) the total length of rules versus Th3

4. The last step of interpretability improvement is to merge similar fuzzy sets.

Th4 and Th5 are used to define whether two fuzzy sets are similar enough to be merged. Because Th4 and Th5 have the same effect, only Th4 was tested in this experiment. Also, it was set to be variable in the range 0.8 Th4 1 and other thresholds and parameters were fixed at Th1 = 0; Th2 = 1; Th3 = 1;Th5 = 1. Figure 5-23 shows the model performance with different Th4.

From this figure, it can be seen that, with the increase of Th4, the RMSE of the obtained model has the tendency of decreasing; the number of rules and the total rule length do not change; the number of fuzzy sets increases, which means that the obtained model accuracy tends to increase while its interpretability decreases.

From these experiments, it can be seen that the thresholds Th1 ~ Th5 can greatly affect the system performance in terms of accuracy as well as interpretability.

Chapter 5: FM-HCMO

(a) (b)

(c) (d)

Figure 5-23. The performance of fuzzy models following the interpretability improvement with different Th4: (a) RMSE versus Th4; (b) the number of rules versus Th4; (c) the number of fuzzy sets versus Th4; (d) the total length of rules versus Th4.