## 4 Consistency Analysis
### Consistency Analysis Based on Hierarchical Clustering Method  
Clustering is an unsupervised knowledge discovery process that groups a set of data such that the intracluster similarity is maximized and the inter-cluster similarity is minimized [33]. The distance is a measure of the similarity between samples when the samples are clustered. Assuming that the \"P\" is the number of feature quantities for each cell, and the \"\(x_{ij}\)\" represents the \(j\)-th feature of the No. \"\(i\)\" cell, each cell can be viewed as a point in the vector space defined as \(\mathbf{R}^{P}\); therefore, the number of cell is \"\(n\)\", and correspondingly the number of point is \"\(n\)\" in \(\mathbf{R}^{P}\). A kind of distance should be defined in \(\mathbf{R}^{P}\); the \"\(d_{ij}\)\" represents the distance between the \(i\)-th cell and the \(j\)-th cell during the clustering process, and the closer points tend to be clustered as a cluster or vice versa. The distance commonly used in clustering is Euclidean distance, which is defined as follows:  
\[d_{ij}=\left[\sum_{i=1}^{p}\left(x_{ik}-x_{jk}\right)^{2}\right]^{1/2}. \tag{25}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample Number** & **Capacity (mAh)** & **DC Resistance (mf)** & **q** & **A (Vs)** & **P1 (V)** & **P2 (V)** & **P3 (V)** \\ \hline
25 & 2979.1 & 37.56 & 0.2076 & 12564.5 & 3.607 & 3.644 & 4.139 \\
26 & 2987.2 & 37.27 & 0.2033 & 12586.6 & 3.603 & 3.638 & 4.136 \\
27 & 2988 & 36.66 & 0.1990 & 12587.7 & 3.602 & 3.648 & 4.132 \\
28 & 3000.4 & 36.67 & 0.1950 & 12646.5 & 3.6 & 3.65 & 4.139 \\
29 & 2997.4 & 37.61 & 0.1983 & 12626.9 & 3.607 & 3.651 & 4.141 \\
30 & 2985.7 & 36.10 & 0.2007 & 12574.7 & 3.596 & 3.646 & 4.134 \\
31 & 3045.4 & 36.52 & 0.1908 & 12836.2 & 3.6 & 3.636 & 4.134 \\
32 & 3064.6 & 35.69 & 0.1852 & 12928.2 & 3.587 & 3.637 & 4.136 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Sample Number** & **Capacity (mAh)** & **DC Resistance (mf)** & **q** & **A (Vs)** & **P1 (V)** & **P2 (V)** & **P3 (V)** \\ \hline
1 & 2925.9 & 37.63 & 0.2222 & 12331.6 & 3.646 & 3.677 & 4.151 \\
2 & 2907.4 & 38.87 & 0.2200 & 12240.2 & 3.657 & 3.685 & 4.127 \\
3 & 2927.6 & 38.67 & 0.2117 & 12320.8 & 3.651 & 3.678 & 4.133 \\
4 & 2854.1 & 38.63 & 0.2228 & 12036.6 & 3.689 & 3.708 & 4.143 \\
5 & 2913.9 & 37.40 & 0.2157 & 12264.1 & 3.655 & 3.684 & 4.141 \\
6 & 2913.4 & 38.43 & 0.2208 & 12256.3 & 3.656 & 3.686 & 4.133 \\
7 & 2920.5 & 38.03 & 0.2190 & 12319.9 & 3.651 & 3.682 & 4.153 \\
8 & 2930.2 & 35.57 & 0.2116 & 12340.4 & 3.643 & 3.673 & 4.12 \\
9 & 2904.3 & 37.83 & 0.2226 & 12241.4 & 3.632 & 3.664 & 4.157 \\
11 & 2923.8 & 37.20 & 0.2151 & 12306.2 & 3.626 & 3.655 & 4.134 \\
12 & 2902.2 & 38.43 & 0.2189 & 12210.9 & 3.638 & 3.657 & 4.13 \\
13 & 2890.3 & 37.83 & 0.2174 & 12161.0 & 3.631 & 3.663 & 4.124 \\
14 & 2951.9 & 38.87 & 0.2114 & 12403.4 & 3.637 & 3.664 & 4.116 \\
15 & 2908.9 & 37.20 & 0.220 Due to the different units of cell features, the measurement range varies greatly; therefore, it is necessary to normalize data of each feature. The following is the commonly used normalization. \[x_{ij}^{\prime}=\frac{x_{ji}-x_{j}}{\sqrt{s_{j}}},i=1,2,...,n,j=1,2,...,p, \tag{26}\]  
where the \(x_{j}=\frac{1}{n}\sum_{i=1}^{n}x_{ij}\) and the \(s_{jj}=\frac{1}{n-1}\sum_{i=1}^{n}\bigl{(}x_{ij}-x_{j}\bigr{)}^{2}\) is the sample mean and variance of the \(j\)-th feature, respectively. The basic idea of hierarchical clustering is that, at first, each cell is considered as a class, and then the nearest distance samples are merged into a new class; at the same time, calculating the distance between the new class and other classes, repeating two recent classes of mergers, until all cells are merged into one class. Therefore, the distance between samples before and after vibration could be calculated.