# Wavelet Packet Decomposition to Characterize Injection Molding Tool Damage +
## 3 Results and Discussion
### Classification of All Input Vectors Based on AE Signals  
For the purposes of presenting all feature vectors \(Z\), we introduced principal component analysis (PCA) as an unsupervised feature reduction method. PCA offers us transformation of high-dimensional feature vector \(Z\) to a much lower dimensional vector _i.e._, \(P\). PCA is mathematically defined [25] as an orthogonal linear transformation that transforms the data into a new coordinate system such that the greatest variance by some projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Figure 8 shows the distribution of feature vectors \(Z\) in a 3D coordinate system of the first three principal components. Feature vectors \(Z\) are divided into five classes; Class 1--the closing of the mold, Class 2--the injection  
Figure 7: A bottom-up tree structure for feature selection [24]. of the melt with maximum pressure, Class 3--holding (packing) stage, Class 4--opening of the tool, Class 5--AE signals characterizing damage of the tool. Based on the measured AE bursts, we set 10-dimensional feature subset vector \(Z_{s}\) with instances that are real-valued descriptors (variables). Vector \(Z_{s}\) offers us defining points in appropriate multidimensional space to characterize the process. We used a classification to identify the categories or sub-populations based on a training dataset, containing instances whose category membership is known. The training data consist of a set of 704 input vectors \(Zs\) that have been designated based on the time moment of their acquisition during the injection molding cycle and unsupervised FCM for vectors of the damaged tool. For neural network pattern recognition we used a feed-forward network with the F Tan-Sigmoid transfer function in both the hidden and the output layer. We used 10 neurons in the hidden layer and five output neurons, because there are five target categories associated with each input vector. The network is trained with scaled conjugate gradient back-propagation. For training of our network, data were randomly distributed into three subsets. A training subset (70% of the data) is used for computing the gradient and updating the network weights \(W\) and biases \(B\). In the validation subset (15% of the data) the error is monitored during the training process. \begin{table}
\begin{tabular}{c||c|c|c|c|c||c} \hline \hline  & \multicolumn{6}{c}{**All Confusion Matrix**} \\ \hline \multirow{6}{*}{**Output Class**} & \multirow{2}{*}{1} & 8 & \multirow{2}{*}{0} & 1 & \multirow{2}{*}{0} & 3 \\  & & 1.1\% & & 0.1\% & 0 & 0.4\% \\ \cline{3-7}  & & & & & \\ \cline{3-7}  & \multirow{2}{*}{2} & 27 & \multirow{2}{*}{0} & 4 & \multirow{2}{*}{0} & 12.9\% \\  & & 3.8\% & & 0.6\% & & \\ \cline{3-7}  & \multirow{2}{*}{3} & 32 & \multirow{2}{*}{0} & 378 & 8 & 4 \\  & & 4.5\% & & 53.7\% & & 0.6\% & \\ \cline{3-7}  & \multirow{2}{*}{4} & 3 & 1 & 26 & \multirow{2}{*}{0} & 13.3\% \\  & & 0.4\% & & 0.1\% & & 206 \\ \cline{3-7}  & \multirow{2}{*}{5} & 2 & \multirow{2}{*}{0} & 1 & \multirow{2}{*}{0} & 29.3\% & 1.4\% \\  & & 0.3\% & & 0.1\% & & 29.3\% & \\ \cline{3-7}  & \multirow{2}{*}{\(E\)} & 81.0\% & 10.0\% & 0.8\% & 31.6\% & 3.3\% & 8.4\% \\ \cline{3-7}  & \multirow{2}{*}{o.c./t.c.} & 1 & 2 & 3 & 4 & 5 & \(E\) \\ \cline{3-7}  & \multicolumn{6}{c}{**Target Class**} \\ \hline \hline \end{tabular} \(E\)—Means error.  
\end{table} 
Table 1: Classification of all input vectors in all confusion matrix. Figure 8: PCA visualization of all feature vectors \(Z\) during injection molding cycles. The network weights and biases are saved at the minimum of the validation subset error. The testing subset (15% of the data) has no effect on training but provides an independent measure of network performance during and after training. The results of classification of all input data are shown in Table 1. The table has five output classes (o.c.) and five target classes (t.c.) with a number of vectors classified into appropriate classes.