# A Performance Prediction Method Utilizing Time-Dependent Subsystem Transfers between Family Systems
## 3 Cluster, Family, and Subsystems  
### Basic Concepts of Each Level System  
Based on the research and analysis of complex systems in Section 2, it is evident that the establishment of mathematical models and parameterization for complex systems poses significant challenges. Complex systems exhibit inherent uncertainty in their states, operate within intricate and dynamic environments, and are susceptible to unpredictable environmental disturbances. Traditional performance prediction methods rely on statistical data with limited samples and necessitate the accumulation of a substantial amount of fault data. To address these concerns, this study introduces the concepts of cluster systems, family systems, and subsystems.  
A cluster system is defined as a system composed of three or more identical or similar subsystems that perform the same task, represented by the symbol \(C\).  
If the number of subsystems is \(n\), then  
\[C=\{s_{1},s_{2},\cdots,s_{i},\ldots,s_{n}\} \tag{1}\]  
where \(n\in Z\), \(n\geq 3\), and \(i\in[1,n]\).  
Applying cluster system theory to complex systems can simplify them into several simple subsystems, enabling the analysis of the working principle, composition structure, and performance trends. In a cluster system, subsystems do not necessarily have to be identical; as long as they are similar in composition, structure, principle, or function, they can be considered alike. This facilitates the simplification of complex systems.  
In a cluster system, the response of each subsystem to factors such as load and environment exhibits a certain distribution pattern. To address the issue of insufficient reference samples in reliability engineering research on complex systems, subsystems with similar performance or state results can be classified into the same family of classes and used as reference samples. Aggregating similar subsystems is highly significant for identifying influencing factors and mechanisms in cluster system analysis and research.  
A family system refers to the subsystems within a cluster system that share the closest key performance parameters. These subsystems can be grouped together based on their common characteristics, thus forming a distinct population. As subsystems within a family system demonstrate similar performance or state outcomes within the same cluster of systems, each subsystem can serve as a reference sample for other subsystems within the same family system. Therefore, the concept of family systems holds great significance in analyzing and researching cluster systems. The family system is represented by the symbol \(F\), the number of family systems is \(k\), and the intermediate value is represented by \(l\):  
\[F=\{F_{l}|l=1,2,\ldots,k\} \tag{2}\]  
where \(k\in[1,n]\) and \(l\in[1,k]\).  
The definitions of subsystems and cluster systems are cross-referenced within the definition of cluster systems, adhering to the hierarchical relationship between these terms. A cluster system encompasses a subsystem when one system encapsulates another, while the enclosed system is denoted as a subsystem. Subsystems can serve as integral functional components within a cluster system or operate independently to fulfill specific functions and actions.  
Here, we introduce the concept of subsystems based on cluster system theory, wherein multiple subsystems serve as reference samples for each other. In a cluster system, the presence of three or more identical or similar entities is essential to define them as subsystems. The symbol of the subsystem is \(s\) and the number of feature elements is \(m\):  
\[s_{i}=s_{i1},s_{i2},\ldots,s_{im} \tag{3}\]  
where \(m\in Z\) and \(m\geq 1\).