# An FCM-GABPN Ensemble Approach for Material Feeding Prediction of Printed Circuit Board Template
## 3 Methodology
### Attributes Selection for Each BPN Prediction Model  
It is necessary to remove irrelevant and redundant attributes to reduce the complexity of analysis and the generated models, and also improve the efficiency of the whole modelling processes [2; 32]. In this study, LC [20], MIC [21], RFE [22], LR [23], lasso regression [24], ridge regression [23], and RFR [24] seven feature selection approaches were employed to select critical attributes related to the scrap rate for each category of samples. The scrap rate can be taken as the dependent variable, and the independent variables are the attributes with No. 1-36, 38, 39, 46, and 47, given in Table 1. The score of independent variables obtained by each feature selection method were calculated and whose average score is greater than a certain threshold (e.g., 0.15) were taken as the input attribute of the prediction model.  
The LC uses the linear correlation coefficient \(lcc(x,y)=\text{cov}(x,y)/\sqrt{\text{var}(x)\text{var}(y)}\) to measure the relationship between the (independent) variable \(x\) and variable \(y\), where var is the variance of a variable and \(\text{cov}(x,y)\) denotes the covariance between \(x\) and \(y\) (namely scrap rate here) [20]. MIC is based on the idea that if a relationship exists between two variables, then a grid can be drawn on the scatterplot of the two variables that partition the data to encapsulate the relationship. To calculate the MIC of a set of two-variable data, all grids up to a maximal grid resolution are explored by computing for every pair of integers (\(x\), \(y\)) the largest possible mutual information achievable by any \(x\)-by-\(y\) grid applied to the data. Then these mutual information (MI) values are normalized to ensure a fair comparison between grids of different dimensions and to obtain modified values between 0 and 1. Finally, the highest normalized MI achieved by any \(x\)-by-\(y\) grid as the value of MIC [21]. The main idea of RFE is to train an estimator based on the initial set of variables and weights are assigned to each one of them at first. Then, variables whose absolute weights are the smallest are pruned from the current set of variables. That procedure is recursively repeated on the pruned set until the desired number of variables to select is eventually reached [22].  
The LR is to establish the regression equation of the dependent variable based on the independent variables, in which the importance of independent variables will be determined according to \(F\)-test. The smaller the value of \(F\)-test, the more important the variable is to the regression equation [21]. The lasso regression is a regularized LR by putting a L1 norm penalty on the regression coefficients.  
Lasso regression will drive more coefficients of weak correlated independent variables to zero, and then facilitate the selection of variables with strong correlation [24]. The ridge regression is similar to lasso regression by putting a L2 norm penalty on the regression to penalize the weak correlated variables for the regression model establishment [25]. RFR is an ensemble of unpruned classification or regression trees, in which each branch of the trees will calculate the importance of each unused attribute in previous steps and then facilitate important-attribute selection simultaneously [26]. The above seven approaches were realized by the encapsulated functions in the machine learning library \"sklearn\" in this study.