## 2 Development of a Servo Valve Prediction Model
### Rationale for Expectancy Projections
#### 2.2.2 ARIMA Theoretical Foundations  
The promising ARIMA model, also known as the Box-Jenkins methodology, consists of a series of activities for identifying time series data, as well as estimating and diagnosing ARIMA models. In an ARIMA model, the future value of a variable can be understood as being the correlated property of past values and past errors, expressed by the following equation [26]:  
\[Y_{t}=b_{0}+b_{1}Y_{t-1}+b_{2}Y_{t-2}+\ldots+b_{p}Y_{t-p}+\delta_{1}-b_{1} \delta_{t-1}-b_{2}\delta_{t-2}-\ldots-b_{q}\delta_{t-q} \tag{3}\]  
where \(Y_{t}\) is the actual value; \(t\) and \(\delta\) are the random errors at \(t\); \(b\) is the coefficient; and \(p\) and \(q\) are integers, often referred to as the autoregressive and moving average, respectively. Building an ARIMA predictive model includes model identification, parameter estimation, and diagnostic checks.  
The main advantage of ARIMA forecasting is that it only requires data on time series. This function is suitable if a large number of time series are to be predicted, in addition to avoiding some of the problems that tend to arise in multivariate models, for which the timeliness of the data is also essential. If a large structural model is constructed containing variables that are only released over a long period, forecasts using this model are conditional forecasts based on the prediction of unavailable observations, thus increasing the potential for forecast uncertainty.  
The ARIMA(_p,d,q_) model is a linear model suitable for dealing with random series. It originates from the autoregressive model AR(_p_), the moving average model MA(_q_) and the association of AR(_p_) and MA(_q_), and the ARMA(_p,q_) model. ARIMA(_p,d,q_) models are usually organized in the following form:  
\[\phi(B)\
abla^{d}x_{t}=\theta(B)\delta_{T} \tag{4}\]  
\[E(\delta_{t})=0,V_{ar}(\delta_{t})=\sigma_{s}^{2},E(\delta_{t},\delta_{s})=0,s\
eq t \tag{5}\]  
\[Ex_{s}\delta_{t}=0,\forall s\leq 1 \tag{6}\]  
where _p,q_ is the order of the AR and MA models and \(d\) is the number of serial differences. \(p\), \(d\), \(q\) are all integers. \(\delta_{t}\) is the estimated residual for each time period. If the model is optimal, it should be independent and distributed as a normal random variable with a mean of 0. \(\sigma_{s}^{2}\) is the variance of the residuals.  
\[\
abla^{d}=(1-B)^{d} \tag{7}\]  
\[\phi(B)=1-\phi_{1}B-\ldots-\phi_{p}B^{p} \tag{8}\]  
\[\theta(B)=1-\theta_{1}B-\ldots-\theta_{q}B^{q} \tag{9}\]  
where \(\phi(B)\) and \(\theta(B)\) are polynomials of degree \(p\) and \(q\) in \(B\). \(B\) is the back-shift operator.  
The data prediction of the experimental data obtained in Section 3 was carried out by means of exponential smoothing algorithm models and ARIMA algorithm models, among others. The data for each performance parameter under each set of operating conditions was divided into two parts, the training set and the test set, by time before and after, e.g., the first 70% of a certain data was classified as the training set and the last 30% was classified as the test set. We then used the prediction model to make subsequent predictions, compare the predicted results with the remaining 30% of the test set data, and perform error calculations for model validation.