## 3 Data-Driven Recurrence Quantification Analysis (RQA)
### Data-Driven RQA
#### 3.2.3 The Statistical RQA Measures  
The statistical RQA measures proposed in this case are based on the computation of an ensemble of surrogates, which are representative stationary and linear realizations of the null hypothesis. A statistical measure is computed for the original given time series and the surrogates. If the statistic is significantly different from the values obtained for the surrogate set, the null hypothesis can be rejected. The details of the analysis process is as follows.  
1. The stationary surrogate technique and linear surrogate technique are adopted separately to generate N groups of stationary and linear surrogate signals.
2. Improved-RQA is carried out for each group of stationary and linear signals and the RQA measure matrix corresponding to the target recurrence rate are obtained. Each column of the RQA measure matrix represents a set of index vectors of the surrogate signals.
3. Calculate the mean and the variance of the RQA measures to obtain the non-stationarity testing and nonlinearity testing confidence interval \(\xi_{S}=\left[\begin{array}{cc}u_{S}-3\sigma_{S}&u_{S}+3\sigma_{S}\\ \end{array}\right]\), \(\xi_{L}=\left[\begin{array}{cc}u_{L}-3\sigma_{L}&u_{L}+3\sigma_{L}\\ \end{array}\right]\).
4. Improved-RQA analysis is carried out for the original given signal. Considering the strong noise test environment of the bridge structure, the noise influence factor is introduced to avoid the strong noise submerging the signal and cover up of the real recursive topology of the signal. The mathematical expression of the RQA measures is as follows. \[RQAs_{ori}=a_{noise}\cdot RQAs_{improved-RQA}\] (26) where \(a_{noise}\) is the noise influence factor. \(a_{noise}=1\) if the given signal is pure. Otherwise, \(a_{noise}\) is selected according to the SNR level.
5. If each element \(RQAs_{ori,i}\) of the given signal is within the confidence interval, the null hypothesis will be accepted. The nonstationary element \(\Xi_{S,i}=0\) and the nonlinear element \(\Xi_{L,i}=0\). Otherwise, \(\Xi_{S,i}=1\), \(\Xi_{L,i}=1\). Let the total size of the RQAs be \(N\) (\(i=1,2,\cdots,N\)), the number of the nonstationary elements and nonlinear elements, which are within the confidence interval, be \(n_{s},n_{l}\). When the ratios of \(n_{s}/N,n_{l}/N\) are greater than the threshold \(\eta\), the null hypothesis is accepted. Otherwise, the null hypothesis is rejected. Let the binary non-stationarity statistical RQA measure be SNL and nonlinearity statistical RQA measure index be SNL. SNS and SNL can be calculated as follows. \[\Xi_{S,i}=\left\{\begin{array}{cc}0,&if\ RQAs_{ori,i}\in\xi_{S}\\ 1,&if\ RQAs_{ori,i}\
otin\xi_{S}\\ \end{array}\right.\] (27) \[\Xi_{L,i}=\left\{\begin{array}{cc}0,&if\ RQAs_{ori,i}\in\xi_{L}\\ 1,&if\ RQAs_{ori,i}\
otin\xi_{L}\\ \end{array}\right.\] (28) \[n_{s}=\sum\Xi_{S,i},\ n_{l}=\sum\Xi_{L,i}\] (29) \[\text{SNS}=\left\{\begin{array}{cc}0,&if\ n_{s}/N\leq\eta_{S}\\ 1,&if\ n_{s}/N>\eta_{S}\\ \end{array}\right.\] (30) \[\text{SNL}=\left\{\begin{array}{cc}0,&if\ n_{l}/N\leq\eta_{L}\\ 1,&if\ n_{l}/N>\eta_{L}\\ \end{array}\right.\] (31)
6. Let the median value of RQAs be RQAm, then the element pair (RQAm,SNS) and (RQAm,SNL) will be set up as the statistical nonstationary and nonlinear test measures. The flow chart of the data-driven RQA algorithm is shown in Figure 5.