## 4 Simulation Experiments
### The Simulation Experiments of Indicators  
Based on the simulation settings above, 100 sets of simulation signals of each type anomaly are generated as samples to test the performances of indicators and calculate the boundaries. Every set of simulation signals has only one anomaly with the random amplitude, and the time when the anomalies occur is random too. For the constant drift fault, noise fault, and multiplicative fault, the anomaly parameters \(\mathbf{d}\), \(\mathbf{e}\), and \(\mathbf{K_{e}}\) are fixed in a single simulation. According to Figure 6, the first step is to use the proportion of error points \(r\) to analyze the detection function. The details of different anomalies are shown in Figure 8 and Table 2. The boundary thresholds, \(T_{\text{r1}}\) and \(T_{\text{r2}}\), are calculated by the genetic algorithm and equal to 0.186 and 0.936, respectively. The anomalies can be treated as false points or outliers once \(r\) is under \(T_{\text{r1}}\); the anomalies may be outlier patches, multiplicative faults in a low dynamic situation, transient faults, or noise faults, once \(r\) is between \(T_{\text{r1}}\) and \(T_{\text{r2}}\); the anomalies may be constant drift faults, multiplication faults in a dynamic situation, or other permanent hard faults once \(r\) exceeds over \(T_{\text{r1}}\). Figure 8: The distribution of the proportion of error points in different situations. Figure 7: The output signal and residual value of a single gyro. Firstly, we analyze the situations where \(r\) is between \(T_{\text{r1}}\) and \(T_{\text{r2}}\). In this kind of situation, the second step is to identify the multiplicative faults. The distribution of \(D_{K}\) of transient faults, noise faults, constant drift faults, and multiplicative faults are shown in Figure 9 and Table 3. The boundary threshold \(T_{\text{Dk}}\) calculated by the genetic algorithm is 0.0552. The anomalies can be regarded as multiplicative faults once \(D_{K}\) is under \(T_{\text{Dk}}\). In Figure 9, the Y-axis range is from 0 to 0.9. Besides, for other anomaly types, some values of \(D_{\text{k}}\) are greater than 0.9 and up to \(5\times 10^{3}\), which are not drawn in the figure. When the real state values are small, the values of \(r\) in multiplicative faults are close to outlier patches. For the outlier patches, normal points take a larger part in the diagnosis period, so the ratio of measurement data to real values fluctuates around 1. As a result, the \(D_{K}\) will also remain at a low level. It is difficult to distinguish the multiplicative faults and outlier patches in such a situation. The mutation degree is designed to solve this problem. The distributions of the mutation degree of multiplicative faults and outlier patches are shown in Figure 10 and Table 4. The boundary threshold \(T_{\text{h}}\) calculated by the genetic algorithm is 58.83. The anomalies can be regarded as outlier patches once the mutation degree exceeds over \(T_{\text{h}}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Data Anomalies Type**} & \multicolumn{3}{c}{**Distribution**} \\ \cline{2-4}  & **0.186\textless** & **0.186\textless{}0.936** & **0.936\textless{}** \\ \hline Isolated Outlier & 100 & 0 & 0 \\ Outlier Patch & 97 & 3 & 0 \\ Transient Fault & 3 & 96 & 1 \\ Noise Fault & 0 & 99 & 1 \\ Constant Drift Fault & 0 & 1 & 99 \\ Multiplicative Fault & 3 & 68 & 29 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The distribution of the proportion of error points in different situations. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Data Anomalies Types**} & \multicolumn{2}{c}{**Distribution**} \\ \cline{2-3}  & **\textless{}0.0552** & **0.0552\textless{}** \\ \hline Multiplicative Fault & 100 & 0 \\ Noise Fault & 0 & 100 \\ Constant Drift Fault & 1 & 99 \\ Transient Fault & 0 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The distribution of the proportional coefficient variance of different anomalies.