# A Time Correlation Based Clustering Method for a Design of a Transformable Product
## 3 Proposed Method
All words are stored on a word tree according to the semantic relationship. The similarity of words can be calculated based on their distance in the word tree. The similarity of nodes is calculated by numbering them for their distance on the word tree using Equations (7)-(10) [63]. Figure 4: Function chains (**a**) function chain that performs one function; (**b**) function chain that performs multiple functions. Figure 5: Two function chains (**a**) function chain G\({}_{1}\); (**b**) function chain G\({}_{2}\). (1) If two words are not in the same tree. \[Sim\Big{\{}F_{i_{i}}F_{j}\Big{\}}=0.1. \tag{7}\]  
(2) If two words are in the same tree. \[Sim[F_{i_{i}}F_{j}]=\cos\!\left(n\times\frac{\pi}{180}\right)\!\left(\frac{n-k+1 }{n}\right)\times\left\{ \begin{array}{ll}0.65&\mbox{\it In the second branch;}\\ 0.80&\mbox{\it In the third branch;}\\ 0.90&\mbox{\it In the fourth branch;}\\ 0.96&\mbox{\it In the fifth branch;}\end{array} \right. \tag{8}\]  
where \(n\) is the total number of nodes in the branch layer, and \(k\) is the distance between two branches. (3) When numbers are same and the end number is '\(=\)'. \[Sim\Big{\{}F_{i_{i}}F_{j}\Big{\}}=1, \tag{9}\]  
(4) When numbers are same and the end number is '\(\#\)'. \[Sim\Big{\{}F_{i_{i}}F_{j}\Big{\}}=0.5. \tag{10}\]  
Using the semantic similarity calculation software of the synonym forest, we can calculate the similarity of two words according to the above definitions [64]. After calculating transition distances of function chains, the transition distances are weighted according to Equation (11):  
\[w_{S_{p}S_{q}}=\frac{\sum w_{N_{i}}+\sum w_{N_{j}}}{z-1}\quad N_{i}\in S_{p} \quad N_{j}\in S_{q}, \tag{11}\]  
where \(z\) is the number of function chains, divided by \(z-1\) to make the sum of the weights be \(1\). The weighted transition distance of two function chains is calculated by Equation (12):  
\[WSim[G_{p},G_{q}]=\frac{w_{S_{p}S_{q}}Sim\Big{\{}G_{p},G_{q}\Big{\}}}{Max\Big{\{} w_{S_{p}S_{q}}\Big{\}}}, \tag{12}\]  
where \(Max\{\omega_{S_{p}S_{q}}\}\) is the correction coefficient to make values in the matrix easily distinguishable. For an undirected graph \(A=(G,\,W)\), where \(G=\{G_{1},\,G_{2},\,...,\,G_{n}\}\), \(W=\{W_{1},\,W_{2},\,\ldots,\,W_{m}\}\). \(G\) is a set of function chains, and \(W\) is a set of similarity between function chains. Using threshold value \(W_{0}\) removes edges less than \(W_{0}\) in the undirected graph and deletes independent nodes. A new subgraph \(A_{0}\) is generated. Interconnected nodes in subgraph \(A_{0}\) are the function chains of the transformable product. The connection between nodes indicates that the function chains represented by these nodes can be transformed each other. Each interconnected node represents a state of a transformable product, and these nodes form the concept of a transformable product. As it is difficult to map the functional domain to physical domain directly, physical components can be designed by refining functions in the mapping process, as shown in Figure 6.  
For a certain function, the solution to realize the function can be selected by designers firstly. Components for implementing the function are then determined. The number of identical components in each state represents the space and material savings of the transformable product. The effectiveness of the method can be evaluated according to the proportion of shared components, as shown in Equation (13):  
\[P=\frac{NCT}{\sum NCS_{i}}, \tag{13}\]  
where \(P\) is the proportion of shared components, \(NCT\) is the number of components in the transformable product, and \(NCS_{i}\) is the number of components in the \(i\)-th state. The trend of technological evolutionsis used to improve the level of idealization [65].