# Wavelet Packet Decomposition to Characterize Injection Molding Tool Damage +
## 3 Results and Discussion
### Unsupervised Fuzzy C Means Clustering  
We used unsupervised fuzzy C means clustering (FCM) for classification of feature vectors \(Z_{j}\) (\(j=1,\,n\)) into \(D=2\) clusters. FCM clustering was found to be an effective algorithm for separating AE patterns composed of multiple features extracted from the AE waveforms [17; 23]. Cluster 2 represents AE bursts during injection of the melt, cluster 3 represents process-orientated AE bursts during the holding stage, and cluster 5 bursts indicate damage during the injection and holding stages of the injection molding process. We used AE bursts measured during the injection molding process with a damaged tool, since we can also acquire AE signals representing cluster 5. We used fuzzy partitioning to connect feature vector \(Z_{j}\) (\(j=1,\,n\)) to cluster \(\omega_{i}\) (\(i=1,\,D\)) with different membership grades \(u_{i}(Z_{j})\) between 0 and 1.  
With FCM we find cluster centers \(V=\left[K_{1}\left|\right.K_{2}\right]\) that minimize the function \(J\):  
\[J\left(U,V\right)=\sum_{j=1}^{n}\sum_{i=1}^{D}\left[u_{i}\left(Z_{j}\right) \right]^{f}d_{e}^{2}\left(Z_{j},K_{i}\right) \tag{13}\]  
\(U\) is the membership matrix with \(D\) lines and \(n\) columns with elements \(u_{i}(Z_{j})\) representing the membership value of the \(j^{\text{th}}\) feature vector \(Z_{j}\) to the cluster \(\omega_{i}\) with a condition:  
\[\sum_{j=1}^{D}\,u_{i}\left(Z_{j}\right)=1 \tag{14}\]\(f\) is a scalar representing the fuzzy degree. The square of Euclidian distance \(d_{e}\) between the feature vector and the cluster is:  
\[{d_{e}}^{2}(Z_{j},K_{i})=(Z_{j}\text{-}K_{i})^{\prime}(Z_{j}\text{-}K_{i}) \tag{15}\]  
The algorithm of FCM has the following steps [17]:  
1. the membership matrix \(U\) is randomly initialized with values between 0 and 1 that represent the membership values \(u_{i}(Z_{j})\) of the \(n\) feature vectors \(Z_{j}\) (\(j=1\), \(n\)) to each cluster \(\omega_{i}\) (\(i=1\), \(D\)).
2. The new cluster centers \(K_{i}\) are calculated according to: \[K_{i}=\frac{\sum_{j=1}^{n}\left[u_{i}\left(x_{j}\right)\right]^{f}x_{j}}{\sum _{j=1}^{n}\left[u_{i}\left(x_{j}\right)\right]^{f}}\] (16)
3. The membership matrix U is updated such that: \[u_{i}\left(Z_{j}\right)=\left[\sum_{k=1}^{D}\left(\frac{d\left(Z_{j},K_{i} \right)}{d\left(Z_{j},K_{k}\right)}\right)^{2\left(f-1\right)}\right]^{-1}\] (17)  
Steps 2 and 3 are iterated until the improvement over the previous iteration is below a threshold \(\theta\) where \(\beta\) are the iteration steps:  
\[\parallel\text{U}^{\beta}\text{-}\text{U}^{\beta-1}\parallel<\ \theta,\ 0<\theta\ <1 \tag{18}\]  
Figure 4 shows examples of signals designated as process-orientated AE burst (cluster 3) and AE burst indicating damage (cluster 5). Figure 5 shows wavelet packet coefficients associated with all 32 nodes on 5 th resolution of the wavelet packet tree for signals represented on Figure 4. With a red color, coefficients associated with nodes 8, 16, and 24 on 5 th resolution are emphasized.  
Figure 4: Signals designated as process-orientated AE burst (**a**) and AE burst indicating damage (**b**).  
Figure 6 show feature vectors representing process-orientated AE bursts (cluster 3) and feature vectors indicating damage (cluster 5) during the holding stage of the injection molding process. Axes of 3D plot are normalization values of wavelet packets energy at 5 th resolution. A black circle indicates the position of the feature vector of the AE burst from Figure 4a and a black square the position of the  
Figure 5: Wavelet packet coefficients associated with all the nodes on 5 th resolution of the wavelet packet tree for above process-orientated AE burst (**a**) and AE burst indicating damage (**b**).  
Figure 6: Visualization of feature vectors designated as cluster 3 and 5.  
feature vector of the AE burst from Figure 4b. Cluster 3 feature vectors are concentrated around the origin of the coordinate system while Cluster 5 feature vectors are spread away from the origin.