## 3 Results
### Framework for Integrating Cutting-Edge Technologies into Aircraft Health Management
#### 3.13.1 Functional Relationship Layer  
The functional relationship layer of the multi-level aircraft health management system (Figure 8) represents how data flow through various levels of processing and analysis, ultimately supporting maintenance decisions. Below is a description of the model components and equations, elaborating on how each step transforms the data from raw sensor inputs to actionable intelligence:  
1. Sensor data collection is the lowest level of the system. It involves the collection of raw data from various sensors embedded throughout the aircraft. These sensors monitor a wide range of parameters such as engine health, structural integrity, system performance, and environmental conditions. 2. At the local processing level, data collected by sensors are preliminarily processed within the aircraft's onboard systems. This processing might include filtering noise from data, normalization, and basic anomaly detection. The goal here is to prepare data for more sophisticated analysis either onboard or after transmission to ground-based systems. 3. Data that have been locally processed are then sent to the MRO center where they are further integrated and analyzed. This integration may involve aggregating data from individual aircraft over time or combining them with data from other aircraft to detect patterns or trends that may not be apparent from single aircraft data. 4. Virtual integrated processing centers are the highest level. At this level, data integrated at MRO centers are further analyzed in a virtual integrated processing center. This center utilizes data from the entire fleet to perform comprehensive analytics, such as predictive maintenance modeling, fleet optimization, and long-term trend analysis. This stage leverages advanced data analytics, machine learning, and possibly simulation models. For each component, we can define the data transformation:  
* \(x_{lt}\) represents a single data point collected from sensors at time \(t\). \(\{x_{1t}\}\) denotes a set of sensor data points, which could include data from various sensors and possibly from various times up to \(t\). In the context of fleet-wide analysis, it would encompass sensor data from all aircraft in the fleet. * \(y_{2t}\) symbolizes the processed data at time \(t\) after the first level of local processing. * \(\{y_{2t}\}\) includes all processed data points that have been locally processed on the aircraft and are now being considered for further integrated analysis. * \(z_{3t}\) are data integrated and analyzed at the MRO center at time \(t\). * \(w_{4t}\) is the fleet-wide advanced analytics output at time \(t\). * The local processing function \(y_{2t}=f_{1}(x_{1t})\) could involve filtering, normalization, and preliminary statistical analysis aimed at preparing data for more detailed analysis. * The MRO integration function \(z_{3t}=f_{2}(x_{1t},y_{2t})\) could include more sophisticated statistical methods, trend analysis, and integration with historical data to form a comprehensive picture of the aircraft's health. It may also involve cross-referencing data from similar types or models of aircraft. * The fleet-wide analysis function \(w_{4t}=f_{3}(z_{3t},\{x_{1t}\},\{y_{2t}\})\) represents the use of advanced machine learning algorithms, predictive modeling, and possibly simulation to make strategic maintenance decisions. This function takes into account not just current and past data from one aircraft but integrates information across the entire fleet. Example Application For instance, if the sensor data \(x_{1t}\) include engine temperature and pressure readings, \(y_{2t}\) might represent these readings adjusted for flight altitude and environmental conditions. \(z_{3t}\) could then integrate these readings with historical data to assess engine wear, while \(w_{4t}\) might predict when the engine will require maintenance based on similar patterns seen across the fleet.