## 4 Simulation Experiment
\[F_{D}(f_{gy})+F_{D}(f_{hy})=F_{I}(f_{gy},f_{hy})=F_{I}(f_{hy},f_{gy}) \tag{32}\]  
The detectability evaluation value of the compound fault \(f_{1}\) is equal to the modulus of the difference between the two detectability evaluation values of the single faults \(f_{gy}\) and \(f_{hy}\). \[|F_{D}(f_{gy})-F_{D}(f_{hy})|=F_{D}(f_{1}) \tag{33}\]  
For verification, we set multiple compound faults with different parameters. From Table 3, it can be noticed that in the new cases, there is indeed some relationship between the diagnosability indexes based on the FID, despite the numeral error that exists because of the limits of decimal digits. However, the diagnosability indexes obtained with the KLD do not have this connection. The four representative geodesic lines in Figure 2 are selected and displayed separately. It can be seen in Figure 5 that among these geodesic lines (there are theoretically infinite ones), there are two special geodesic lines, which are the shortest geodesic lines in the \"unit circle\"; in addition, the FIDs of the two special geodesic lines are equal, and the other geodesic lines are symmetrical about these two special geodesic lines. According to the research in [38], they are named the \"symmetry lines\", and \"symmetry lines\" have extremely important properties in the information space. In the figure, the two geodesic \"symmetry lines\" appear to be \"straight lines\" and have unequal lengths because they can only be displayed through the Euclidean plane. However, with the Riemann metric, these two special geodesics are both typical curves and  
Figure 5: Representative geodesics of \(f_{1}\) on the fault information manifold. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Fault** & \(\mathbf{F_{D}}\) & \(\mathbf{F_{I}(\_,f_{gy})}\) & \(\mathbf{F_{I}(\_,f_{hy})}\) \\ \hline \(f_{2}(4,3)\) & 112.811393 & 121.653057 & 9.351245 \\ \(f_{3}(1,1)\) & 64.618801 & 70.140214 & 5.539564 \\ \(f_{4}(1,3)\) & 116.118182 & 121.653057 & 5.539564 \\ \(f_{5}(3,1)\) & 61.390573 & 70.140216 & 8.809759 \\ \(f_{6}(3,7)\) & 177.282673 & 186.091232 & 8.809759 \\ \(f_{7}(5,5)\) & 148.024455 & 157.201838 & 9.686858 \\ \(f_{8}(5,9)\) & 201.904966 & 211.050089 & 9.686858 \\ \(f_{9}(7,5)\) & 147.632920 & 157.201838 & 10.078330 \\ \(f_{10}(9,7)\) & 176.307810 & 186.091233 & 10.298639 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Diagnosability indexes on a fault information manifold for compound faults. have the same length. According to [38], a \"symmetry line\" embodies a certain symmetry and conservation principle in the information space. With the Riemannian metric, free particles travel unequal distances along different geodesics to accumulate the same amount of energy. The geodesics between two points in a Riemannian manifold are not unique, and the FID between two points corresponds to the geodesics of the shortest length. Since on different geodesics, free particles have the same energy that is accumulated in a certain period of time, according to the \"principle of lowest energy\", free particles must choose to travel the shortest path. In the information space, the geodesics corresponding to the FID also correspond to the shortest path for accumulating energy from the \"initial state\" endpoint to the \"final state\" endpoint, which can also be regarded as the inevitable path of information. For the fault information space, after a certain fault occurs in the system, this state can also be understood as an \"initial state\". In the absence of the injection of new faults, the \"initial state\" will follow the geodesic line and move to reach a certain \"final state\". The geodesic corresponding to the FID reflects the developmental trajectory of the fault. **Remark 1**. :
1. _There are special geodesic \"symmetry lines\" that exist on a fault information manifold, and they can represent the diagnosability properties of a fault. A \"symmetry line\" is an inevitable path for the development and evolution of a fault after setting the \"initial state\"._
2.