# Gyro Error Compensation in Optoelectronic Platform Based on a Hybrid ARIMA-Elman Model
## 4 Hybrid ARIMA-Elman Model
### Specific Steps  
The specific steps are as follows:  
Step 1: Acquire fiber-optic gyro error data. Step 2: Decompose the original gyro error data into a white noise sequence and a drift error sequence, using the m-layer lifting wavelet transform. After the constant zero offset is compensated in the gyro static output, the signal output will only contain the drift error with low frequency and white noise with high frequency. Considering both performance of the algorithm and the actual requirements of the system, Harr wavelet and Daubechies wavelet basis functions are good choices. One reason is that these two wavelet basis functions have the characteristics of orthogonality and tight support. The other reason is that both the transform and inverse transform of these two wavelets can be explained succinctly by a linear orthogonal transform in matrix form, which is suitable for system programming. In this paper, in order to reduce the signal loss, a Daubechies wavelet with a second-order vanishing moment has been chosen. In theory, the higher the decomposition scale of the wavelet, the stronger the ability of characterizing the local characteristics of the signal will be. However, the amount of calculation is much greater at the same time. Therefore, we choose the db2 wavelet as a wavelet base, and 4 as the wavelet decomposition scale. For the reason that the drift error is a narrow-band signal with low frequency, and the white noise is a wide-band signal with high frequency, it can be determined that, after a suitable \(p\)-time lifting wavelet decomposition, the drift will be mainly decomposed into \(c(k)\), and the white noise will be mainly decomposed into \(d(k)\). Thus, if the reconstruction is performed separately for \(c(k)\) and \(d(k)\) using the lifting wavelet reconstruction algorithm, the drift error and white noise can be obtained, respectively. All of the above steps make it possible to separate the drift error from the white noise. Step 3: Perform a graying strategy on the decomposed drift error; that is, perform one or more accumulation operations. Grey system theory is a new method to study small-sample problems, poor-information problems, and other uncertain problems. This method realizes the correct understanding and effective control of system behavior and evolution laws by generating and developing some known information and extracting valuable information. The ashing operation generates the new data through accumulating the original data, which can weaken the randomness of the original data and enhance its regularity to a certain extent. Therefore, it makes it simple to get the change law of the data. Then, a standard Jarque-Bear test is used to judge the grayed output data. In general, to diagnose whether a given sequence is normally distributed or not, the Jarque-Bear normality test can be performed, as illustrated in Equation (18). If the value is close to 3, the sequence is Gaussian distributed and linear, which can be approximated by an ARIMA model. Otherwise, it is nonlinear and is suitable to be approximated by an Elman neural network. In the test, whether the kurtosis of the data is 3 or not is based on the equation:  
\[kurtosis =\frac{E\{(x-E\{x\})^{4}}}{(E\{(x-E\{x\})^{4}\}^{2}}, \tag{19}\]  
where \(x\) is the random sequence and \(E\) is the expectation operation. Step 4: The values close to 3 (through test equation Jarque-Bear) are modeled using the ARIMA model, and the remainder is approximated using the Elman neural network. Meanwhile, use the differential evolution algorithm to determine the ARIMA model parameters and neural network weight values. It is well known that the parameters of the ARIMA and Elman neural network models are difficult to determine. For example, the values of AR factor coefficients (i.e., \(a_{1}\cdots a_{m}\)) and values of MA factors (i.e., \(b_{1}\cdots b_{n}\)) should be determined.