# A Time Correlation Based Clustering Method for a Design of a Transformable Product
## 3 Proposed Method
According to Equations (1) and (2), the time independence between needs can be obtained using Equation (3):  
\[r_{ij}=\frac{w_{N_{i}N_{j}}}{Max\{\omega_{N_{i}N_{j}}\}}\times\frac{\partial (C_{N_{i}}u_{3})}{\partial(C_{N_{j}}u_{3})}, \tag{3}\]  
where \(Max\{w_{N_{i}N_{j}}\}\) is the correction coefficient for values in the matrix easily distinguishable. According to Equation (3), the relationship matrix of needs can be established using Equation (4):  
\[R=\begin{array}{ccccc}N_{1}&\left[ \begin{array}{ccccc}r_{11}&r_{12}&&\\ N_{2}&&r_{22}&&\\ &\vdots&&\ddots&\\ &&&&&r_{mm}\\ &&N_{1}&N_{2}&\cdots&N_{m}\\ \end{array} \right]. \tag{4}\]  
The graph theory uses a graph as its object of study to transform the relationship matrix into a graph [56]. According to the relationship matrix, we can draw a need-related undirected graph, in which nodes are needs and edges are the relationship between the needs. Segmentations of the undirected graph are decided by setting an appropriate threshold. Each interconnected subgraph forms a kind of need with the time dependence. Each kind of need corresponds to a potential state of the product. Function design is conducted for each product form to obtain a function chain of the product in each use state. Function chains express sub-functions and their relationships. A directed graph can represent the logical relationship between nodes [56]. Function chains can be expressed by directed graphs, for example, \(G=(F,L)\), where \(F=\{F_{1},F_{2},\ldots,F_{n}\}\), \(L=\{L_{1},L_{2},\ldots,L_{n}\}\). \(G\) is a function chain, \(F\) is a set of sub-functions of function chain G, and L is a set of relationships among sub-functions. Figure 4a shows a directed graph of a function chain in which a product performs one function. In practice, a product function can be used many times. A chain of multiple functions is shown in Figure 4b. When a product needs to perform multiple functions, it needs to be transited from one function chain to another. Much research has been done on the transition from one graph to another [57; 58; 59; 60]. The cost of a transition is directly related to its suitability. The lower the cost of the transition, the higher the possibility that the two function chains can be converted to each other. The higher the similarity between the two function chains, the easier conversion of the chains. The transition from one function chain to another requires a series of operations, including the node converting, deleting, adding, and edge deletion and addition. Two function chains G\({}_{1}\) and G\({}_{2}\) are shown in Figure 5.  
The sum of costs of all operations from \(G_{1}\) to \(G_{2}\) is called the distance of transition from \(G_{1}\) to \(G_{2}\), expressed by \(Trans\{G_{1},G_{2}\}\). The similarity between \(G_{1}\) and \(G_{2}\) is expressed as \(Sim\{G_{1},G_{2}\}\) using Equation (5):  
\[Sim\{G_{1},G_{2}\}=1-\frac{Trans[G_{1},G_{2}]}{\sum[G_{i}]}, \tag{5}\]  
where \(|G_{i}|\) is the total number of nodes and edges in \(G_{1}\) and \(G_{2}\). The distance of transition from \(G_{1}\) to \(G_{2}\) is calculated by Equation (6):  
\[Trans\{G_{1},G_{2}\}=con(F_{1},F_{4})+add(F_{5})+add(L_{6})+add(L_{7})+add(L_{ 3}), \tag{6}\]  
where \(con(F_{i},F_{j})\) converts node \(F_{i}\) to node \(F_{j}\), \(del(F_{i})\) deletes node \(F_{i}\), \(add(F_{i})\) adds node \(F_{i}\), \(del(L_{i})\) deletes edge \(L_{i}\), and add(\(L_{i}\)) adds edge \(L_{i}\). The cost of deleting and adding one node or edge can be assigned as one unit currency. Because there may be some similarity between two nodes, the cost of converting nodes is designed as \(1\)-\(Sim\{F_{i},\)\(F_{j}\}\). \(Sim\{F_{i},\)\(F_{j}\}\) is the similarity between nodes \(F_{i}\) and \(F_{j}\). The higher the similarity, the lower the cost of conversion. Semantic similarity computation has many applications in information retrieval, information extraction, word sense disambiguation, and machine translation [61; 62]. In this paper, we use it to calculate similarity between two nodes based on the semantic similarity.