## 3 Methodology
### Aggregation of Fuzzy Forecasts in Fuzzy Collaborative Forecasting
Fuzzy intersection combines \(n\) fuzzy forecasts in the following manner:  
\[\mu_{I(\widetilde{c}_{t}(1),\ \widetilde{c}_{t}(2)\  
The result of this step is a polygon-shaped fuzzy number that specifies the narrowest range of the fuzzy forecast. However, in practical applications a crisp forecast is usually required. Therefore, a crisp forecast has to be generated from the polygon-shaped fuzzy number. For this purpose, a variety of defuzzification methods are applicable [33]. Once the defuzzified value is obtained it is compared with the actual value to evaluate the accuracy. However, among the existing defuzzification methods, no one method is better than all the other methods in every case. In addition, the most suitable defuzzification method for a fuzzy variable is often chosen from the existing methods, and thus the optimality of the chosen method cannot be guaranteed. Also, the shape of the polygon-shaped fuzzy number is special. These phenomena are reasons for proposing a tailored defuzzification method. In this study, a back propagation network is applied, because theoretically a well-trained back propagation network (without being stuck in a local minima) with a good selected topology can successfully map any complex distribution.  
The configuration of the back propagation network used is established as follows:  
1. Inputs: \(2m\) parameters corresponding to the \(m\) corners of the polygon-shaped fuzzy number and the membership function values of these corners. The reason is that simple-aggregation results in a convex domain and each point in it can be expressed with the combination of corners. The fuzzy intersection of \(L\) fuzzy forecasts will have at most \(2\)-\((2^{L}+2)\) corners. All input parameters have to be normalized into a range narrower than \([0\ 1]\) before they are fed into the network.
2. Single hidden layer: Generally one or two hidden layers are more beneficial for the convergence property of the back propagation network.
3. The number of neurons in the hidden layer is chosen from \(1\)\(\sim\)\(4m\) according to a preliminary analysis, considering both effectiveness (forecasting accuracy) and efficiency (execution time).
4. Output: the crisp forecast.
5. Network learning rule: Delta rule.
6. Network learning algorithms: There are many advanced algorithms for training a back propagation network, e.g. the Fletcher-Reeves algorithm, the Broydon-Fletcher-Goldfarb-Shanno algorithm, the Levenberg-Marquardt algorithm, and the Bayesian regularization method [34]. In this study, the Levenberg-Marquardt algorithm is applied.  
Figure 1: The fuzzy intersection of two triangular fuzzy numbers.  
* Number of epochs per replication: 10,000.
* Activation function: Log-sigmoid function.
* Number of initial conditions/replications: Because the performance of a back propagation network is sensitive to the initial condition, the training process will be repeated many times with different initial conditions that are randomly generated. Among the results, the best one is chosen for the subsequent analyses.  
The Levenberg-Marquardt algorithm was designed for training with second-order speed without having to compute the Hessian matrix. It uses approximation and updates the network parameters in a Newton-like way. When training a back propagation network, the Hessian matrix can be approximated as:  
\[H=J^{T}J \tag{59}\]  
and the gradient can be computed as:  
\[g=J^{T}e \tag{60}\]  
where \(J\) is the Jacobian matrix containing the first derivatives of the network errors with respect to the weights and biases; \(e\) is the vector of the network errors. The Levenberg-Marquardt algorithm uses this approximation and updates the network parameters in a Newton-like way:  
\[x_{ep+1}=x_{ep}-[J^{T}J+\mu J]^{-1}J^{T}e \tag{61}\]  
where \(ep\) is the epoch number. [PERSON]'s method is faster and more accurate near an error minimum, so the Levenberg-Marquardt algorithm's purpose is to move as quickly as possible to [PERSON]'s method.