## 2 Proposed MCNNs-IFI
### Proposed Improved Fuzzy Integral  
The fuzzy density in traditional fuzzy integral is often set according to user experiences. Many optimization algorithms can be used to determine the optimal fuzzy density, such as particle swarm optimization (PSO), genetic algorithm (GA), differential evolution (DE), firefly algorithm (FA), and artificial bee colony (ABC). Without crossover and mutation steps, PSO is relatively simple, with fewer parameters to adjust, and easy to implement in software and hardware. Therefore, in this study, the proposed improved fuzzy integral uses PSO to automatically determine the optimal fuzzy density. The hypothesis \(\mathrm{X}=\left\{x_{i}\right\}_{i=\ 1\ :\ m}\) represents a set of m classifiers which means the numbers of CNN used in this study and \(\mathrm{g}(x_{i})\) indicates the fuzzy measure and is regarded as the worth of the subset \(x_{i}\). That is, \(x_{i}\) is the output from \(i^{th}\) classifier and \(\mathrm{g}(x_{i})\) is the confidence degree of this class. The fuzzy measure value is [0,1]; if the value is 1, then it indicates that this output can be fully trusted. If the value is 0, then the output has no reference value. The fuzzy measure properties are shown as follows:  
1. \(\mathrm{g}(\mathrm{X})=1\) indicates that when all classifier outputs are consistent, the results must be trusted. 2. \(\mathrm{g}(0)=0\) indicates that outputs of all classifiers are not considered. 3. Fuzzy measure should be an incremental monotonic function: \[\mathrm{A}\subset\mathrm{B}\subset\mathrm{X}\,,\text{then}\ 0\leq\mathrm{g}(A)\leq\mathrm{g}(B)\leq 1\] (3)  
Figure 4: Max pooling. Figure 5: Average pooling. To get the fuzzy measure it should start at fuzzy density which has only one element in g(X) for instance g(\(\{x_{1}\}\)) and g(\(\{x_{2}\}\)). The relationship between fuzzy measures and fuzzy density is presented in Figure 6. In this figure, g(\(\{x_{1}\}\)), g(\(\{x_{2}\}\)), and g(\(\{x_{3}\}\)) are called fuzzy density and g(\(\{x_{1},x_{2},x_{3}\}\)) is the confidence degree of the specific class combining three CNNs. Figure 7 illustrates the flow of fuzzy density and fuzzy measures with PSO algorithm. PSO [17] is a type of an optimization algorithm and it is based on the concept of foraging birds. Birds flying in the air for foraging can be seen as several particles in a solution space of a problem. Then, through multiple iterations, the best solution can be obtained. The fuzzy densities are n \(\times\) k float numbers from 0 to 1, where n is the number of classifiers (i.e., the number of CNNs), and k is the number of categories in the classification problem (i.e., the number of class); therefore, the dimensions of the solution space are n \(\times\) k dimensions and range from 0 to 1. In the initialization phase, \(y\) particles are randomly generated in the solution space, and each particle \(p\) denotes a set of fuzzy densities, as depicted in Figure 8.  
Figure 6: Relationship between fuzzy measures and fuzzy density. Figure 7: Flow of fuzzy density. Then, the fitness value of each particle is evaluated by calculating the precision of the verified data as follows:  
\[\text{Fitness}=\frac{\textit{TP}}{\textit{TP}+\textit{FP}} \tag{4}\]  
where _TP_ is the number of samples that are predicted correctly, and _FP_ is the number of samples that are predicted incorrectly. In the particle swarm, the position of the overall optimal fitness value was defined as _Gbest_; the position of each particle's best fitness value was defined as _Pbest_\({}_{i}\), where i = {1, 2....,j}. When updating a particle's position, the current fitness was evaluated to check whether the current fitness value of the particle is better than _Gbest_ and _Pbest_\({}_{i}\). If yes, then the particle's location was updated and recorded.