## 3 The Detection and Isolation Method of Data Anomalies  
### Simulation and Analysis of Anomaly Signal  
To demonstrate the differences among the anomalies, a set of gyro measurement signals are simulated, and six typical anomalies given in Table 1 are injected into those signals. The dynamic gyro signal generator is built by Simulink (shown in Figure 3). The gyro redundant unit uses 6-gyro-dodecahedron configuration. The simulation only considers two main random noises, angle random walk (ARW) and rate random walk (RRW), and the determinate error and other random noise are ignored. The sampling time is 100 s; sampling frequency is 100 Hz. The frequency of the sinusoidal signal is \(1/20\pi\) Hz; the amplitude is \(10^{\circ}\)/s. The false alarm rate is 1%; the detection threshold is 16.812; and \(\sigma\approx 1.1489\). Several typical anomalies are injected into the original signal in turn (shown in Figure 4). Among them, the isolated outliers, outlier patches, and transient faults are injected into normal observation signals every 10 s. The noise faults, constant drift faults, and multiplicative faults are injected at 50 th second. The amplitudes of isolated outliers and outlier patches are a series of random numbers greater than 3\(\sigma\), and the amplitudes of transient faults are a series of random numbers greater than 4\(\sigma\). The parameters of constant drift fault, noise fault, and multiplicative fault are fixed in each turn, and the parameters are given by \(\mathbf{e}=[0,\cdots,\alpha_{e},\cdots 0]^{T}\), \(E[\mathbf{e}^{T}\mathbf{e}]=\sigma^{2}\); \(\mathbf{d}=[0,\cdots,\sigma^{2},\cdots 0]^{T}\); \(\mathbf{K}_{e}=diag(0,\cdots,0.2,\cdots,0)\). The diagnosis period \(N\) is introduced here. The definition of diagnosis period refers to the number of sampling points in the duration from the first moment when the detection function exceeds the threshold to the moment to get the diagnosis result. The data during the period is analyzed to recognize the anomaly mode. It is important to take a suitable diagnosis period. If the period is too short, the information is not enough to get accurate recognition results. Conversely, an unduly long period will lead to a delay, and cause the diagnosis sensitivity to decrease. The SVD-based method is used to detect those signals. The detection function is shown in Figure 5a. Figure 5b shows the detection function values in a single diagnosis period. The diagnosis period \(N\) is 100, and the X-axis represents the number of sampling points. Note that, in Figure 5, the red line is the detection threshold, and the blue line represents the detection function. Once the blues line exceeds over the red line, it can be considered that the anomalies appear. Figure 4: Gyro simulation signals. Figure 3: Gyro signal generator. As Figure 5a shows, the improved method has several useful features of the fault detection: First, the detection function values exceed the detection threshold whenever data anomalies occur; second, the values and trends of the detection function are consistent with the anomalies deterioration degree. Because of the consistency between anomalies and detection function, analyzing detection function is an eligible approach for anomaly recognition. From inspection of Figure 5b, the detection function values of different anomalies present different distributions and trends in a single diagnosis period. The diversity among different anomaly modes is the key to recognize them. For the distributions, detection functions of short-duration auto-recovery anomalies exceed the threshold fewer times, while the detection functions of sustained anomalies exceed the threshold more times for its longer durations. On the other hand, the anomaly points of short-duration auto-recovery anomalies only make up a small part, and they deviate seriously from the trend than other normal points present. However, for sustained anomalies, the anomaly points take up large proportions, so the trend of normal points is indistinct and unstable.