## 4 Simulation Experiment
This set of geodesics describes the unit circle on the fault information manifold with the same FID centered at the location of the compound fault, the point (4,4). The FID unit circle of a statistical manifold does not usually correspond to a circle in a Euclidean space, and vice versa. In fact, on the fault information manifold, the endpoints of these 128 geodesics form a \"unit circle\", but when mapped in Euclidean space, the shape of this unit circle is distorted, forming a shape similar to that of a \"comet\". Figure 2 also expresses the response of the FID to the fault development process in this fault form. Because the two single faults are set to the x and y rectangular coordinate axes, the manifold, which was originally a curved surface, is \"leveled\", and the FID, which was originally a unit circle on the manifold, is distorted in this process. In this coordinate system, the evaluation of the diagnosis of faults based on the FID can be succinctly and intuitively described and studied. As mentioned above, the X-axis (\(f_{gy}\)) and Y-axis (\(f_{hy}\)) represent two single-fault scenarios, and the first quadrant represents a compound fault scenario. Now, three types of faults are displayed in the coordinate system: a single gyro fault \(f_{gy}\), a single infrared sensor fault \(f_{hy}\), and a compound fault \(f_{1}\):(4,4). Figure 2: The FID unit circle of the compound fault \(f_{1}\). According to the definition of detectability in this paper, the detectability indexes of single faults \(f_{gy}\) and \(f_{hy}\) and of \(f_{1}\) are shown in Figure 3, indicating FID of the line between the coordinates where the fault is located and the origin. It should be noted that the \"line\" here is actually the geodesic on the manifold, not a line mapped in Euclidean space. The isolability between the single faults \(f_{gy}\) and \(f_{hy}\) and the compound fault \(f_{1}\) is shown in the Figure 4, indicating the FIDs of the geodesics between the two faults. The line shown in Figure 4a is obviously not the shortest line between two points in Euclidean space, but it is the shortest path between them on a manifold. Table 1 shows the numerical results of the detectability and isolability indexes for the three types of faults. Firstly, the coordinates of the compound fault \(f_{1}\) under the fault information manifold can be expressed as (9.351245, 141.061290), indicating that the isolability between the compound fault and \(f_{hy}\) (single infrared fault) is stronger than that of \(f_{gy}\) (single gyro fault). It can also be stated that the coupling between \(f_{gy}\) and the compound fault \(f_{1}\) is deep and difficult to isolate. Secondly, it can be found that \(F_{I}\) on the right is a symmetric matrix. This is because the FID is a true distance measure with symmetry, and it is fundamentally more scientific and accurate than diagnosability evaluation methods based on the KLD. The principle of the diagnosability evaluation method based on the KLD is stated briefly as follows. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Fault** & \(\mathbf{F_{D}}\) & \(\mathbf{F_{I}(\boldsymbol{,f_{gy}})}\) & \(\mathbf{F_{I}(\boldsymbol{,f_{hy}})}\) & \(\mathbf{F_{I}(\boldsymbol{,f_{1}})}\) \\ \hline \(f_{gy}\) & 9.351268 & 0 & 9.351245 & 150.237868 \\ \(f_{hy}\) & 141.061220 & 9.351268 & 0 & 141.061290 \\ \(f_{1}\) & 131.709952 & 150.237868 & 141.061290 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Diagnosability indexes on the fault information manifold for the three types of faults. Figure 4: Fault isolability indexes on the fault information manifold. (**a**) \(F_{I}(f_{gy},f_{hy})\); (**b**) \(F_{I}(f_{1},f_{gy})\); (**c**) \(F_{I}(f_{1},f_{hy})\). Figure 3: Fault detectability indexes on the fault information manifold. (**a**) \(F_{D}(f_{gy})\); (**b**) \(F_{D}(f_{hy})\); (**c**) \(F_{D}(f_{1})\). The distance between two distributions (in this paper, two faults) \(p(x|\theta_{i})\) and \(p(x|\theta_{j})\) can be approximated with various alternative approximations.