## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Passenger Flow Prediction of URT
#### 2.1.3 ARMA-RBFNN Algorithm  
The data processed by the change-point algorithm and wavelet transform method are non-stationary time series. The non-stationary time series are transformed into stationary time series by difference computing. However, the ARMA model could not predict the nonlinearity of these data. The ARMA-RBFNN algorithm is constructed to reduce the prediction error. The specific steps are described as follows:  
_Step 1_: The data are input after secondary processing.  
_Step 2_: The stationarity, as well as the value of both the autocorrelation function and the partial autocorrelation function, are calculated. This assumes that the covariances of two random variables are defined as \(x_{t}\) and \(x_{t-k}\), separately, in \(b(w)\) with \(k\) period intervals. The autocovariance function (defined as \(\text{cov}_{k}\)), with the lag of \(k\) period, is calculated by Equation (7).  
\[\text{cov}_{k}=\text{cov}[b(w),b(w-k)]=E[(b(w)-\overline{u})(b(w-k)-\overline{u })] \tag{7}\]  
where \(\overline{u}\) is the mean value of the passenger flow time series.  
The autocorrelation function of the passenger flow time series is shown as \(\rho_{k}=\frac{\text{cov}_{k}}{\hat{\sigma}_{x_{t-k}}\hat{\sigma}_{x_{t}}}=E[x _{t}-E(x_{t})]\). When the passenger flow time series is stationary, the autocorrelation function is defined as \(\rho_{k}=\frac{\text{cov}_{k}}{\hat{\sigma}_{0}}\).  
The data sample is presented as \(\hat{\rho}_{k}=\frac{\sum\limits_{t=1}^{n-k}[x_{t}-\overline{x}][x_{t-k}- \overline{x}]}{\sum\limits_{t=1}^{n}[x_{t}-\overline{x}]^{2}}\), where \(\overline{x}=\frac{\sum\limits_{t=1}^{n}\overline{x}}{n}\). The partial autocorrelation degree of the data sample is defined as \(\hat{\phi}_{kk}=\begin{cases}\hat{\rho}_{k},&k=1\\ \frac{\hat{\rho}_{k}-\sum\limits_{t=1}^{k-1}\hat{\phi}_{k-1,t}\hat{\rho}_{k- 1}}{1-\sum\limits_{t=1}^{k-1}\hat{\phi}_{k-1,t}\hat{\rho}_{k-1}},&k=2,3,\cdots \end{cases}\)  
If the graph has tailing characteristics, it belongs to a stationary time series, and _Step 4_ should be completed. Otherwise, it belongs to a non-stationary time series, and _Step 3_ should be completed.  
_Step 3_: Difference computing transforms non-stationary time series into stationary time series. Due to the periodicity of the passenger flow entering the station, periodic differences are established for non-stationary time series.  
\[y_{t}=x_{t+s}-x_{t} \tag{8}\]  
_Step 4_: It is assumed that the passenger flow time series consists of linear and nonlinear components, i.e., \(L(t)\) is the linear part and \(N(t)\) is the nonlinear part.  
\[y_{t}=L(t)+N(t) \tag{9}\]  
_Step 5_: \(p\) and \(q\) are determined. The rank of ARMA algorithm is determined by the AIC criterion.  
_Step 6_: Aiming at the linear part of the passenger flow time series, the ARMA model is used for passenger flow prediction. The prediction results are defined as \(p(t)\), and the nonlinear prediction error (defined as \(e(t)\)) is calculated as follows:  
\[e(t)=y_{t}-p(t) \tag{10}\]  
_Step 7_: \(e(w)\) is randomly divided into two parts: 75% error for the training set and 25% for the test set.  
_Step 8_: The K-MEANS clustering algorithm is applied to determine the radial basis function center. The average method is used for the radial basis function widths so as to determine the core parameters of the RBFNN algorithm.  
_Step 9_: The training set is used to train the RBFNN algorithm. The RBFNN algorithm is used to predict the results of the test set.  
_Step 10_: If the prediction value of RBF is defined as \(\hat{N}_{t}\), the equation represents the prediction result.  
\[\hat{y}_{t}=\hat{L}_{t}+\hat{N}_{t} \tag{11}\]