# Nonlinear and Non-Stationary Detection for Measured Dynamic Signal from Bridge Structure Based on Adaptive Decomposition and Multiscale Recurrence Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Improved Ensemble Empirical Mode Decomposition

#### 2.1. Ensemble Empirical Mode Decomposition

- Generate ${x}^{i}\left[n\right]=x\left[n\right]+{w}^{i}\left[n\right]$, where $x\left[n\right]$ is the original signal and ${w}^{i}\left[n\right]\left(i=1,\cdots ,I\right)$ are different realizations of added white noise.
- Each ${x}^{i}\left[n\right]\left(i=1,\cdots ,I\right)$ is fully decomposed by EMD and get their modes $IM{F}_{k}^{i}\left[n\right]$, where $k=1,\cdots ,K$ indicates the modes.
- Assign ${\overline{IMF}}_{k}$ as the k-th mode of $x\left[n\right]$, obtained as the average of the corresponding $IM{F}_{k}^{i}$:$${\overline{IMF}}_{k}\left[n\right]=\frac{1}{I}{{\displaystyle \sum}}_{i=1}^{I}IM{F}_{k}^{i}\left[n\right]$$

#### 2.2. Improved Ensemble Empirical Mode Decomposition

- Add N Gaussian white noise ${w}^{i}\left(t\right)$ $\left(i=1,2,\cdots ,N\right)$ in signal $x\left(t\right)$:$${x}^{i}\left(t\right)=x\left(t\right)+{w}^{i}\left(t\right)\left(i=1,2,\cdots ,N\right)$$
- Obtain the first residue ${r}_{1}\left(t\right)$ and the first mode ${\overline{IMF}}_{1}$ by EEMD:$${\overline{IMF}}_{1}=\frac{1}{N}{{\displaystyle \sum}}_{i=1}^{N}{E}_{j}\left({x}^{i}\left(t\right)\right)$$$${r}_{1}\left(t\right)=x\left(t\right)-{\overline{IMF}}_{1}$$
- Perform EMD on Gaussian white noise ${w}^{i}\left(t\right)$ and generate the IMFs and residue of ${w}^{i}\left(t\right)$:$${w}^{i}\left(t\right)={\displaystyle \sum}_{j=1}^{m}{E}_{j}\left({w}^{i}\left(t\right)\right)+{r}_{mi}^{\prime}\left(t\right)$$
- Take ${r}_{1}\left(t\right)$ as the signal to be decomposed, add ${E}_{1}\left({w}^{i}\left(t\right)\right)$ $\left(i=1,2,\cdots ,N\right)$ and generate ${r}_{1i}^{k}\left(t\right)$.$${r}_{1i}^{k}\left(t\right)={r}_{1}\left(t\right)+{\epsilon}_{1}{E}_{1}\left({w}^{i}\left(t\right)\right)$$
- Perform EMD on ${r}_{1i}^{k}\left(t\right)$ and obtain the ${\overline{IMF}}_{2}$ of signal $x\left(t\right)$ by averaging the N IMFs.$${\overline{IMF}}_{2}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{j}\left({r}_{1i}^{k}\left(t\right)\right)$$
- For $k=2,\cdots ,K$, calculate the k-th residue.$${r}_{k}\left(t\right)={r}_{\left(k-1\right)}\left(t\right)-{\overline{IMF}}_{k}\left(t\right)$$
- Decompose ${r}_{k}\left(t\right)+{\epsilon}_{k}{E}_{k}\left({w}^{i}\left(t\right)\right)$, $i=1,\cdots ,N$, until their first EMD mode and define the (k+1)-th mode using the equation below.$${\overline{IMF}}_{k+1}=\frac{1}{N}{\displaystyle \sum}_{i=1}^{N}{E}_{1}\left({r}_{k}\left(t\right)+{\epsilon}_{k}{E}_{k}\left({w}^{i}\left(t\right)\right)\right)$$
- Go to step 6 for the next k.
- Step 6 to 8 are performed until the obtained residue is no longer feasible to be decomposed (the residue does not have at least two extrema). The final residue is expressed using the equation below.$$R\left(t\right)=x\left(t\right)-{\displaystyle \sum}_{j=1}^{K}{\overline{IMF}}_{j}$$
- Let us define that each IMF contains two parts (i.e., the useful characteristic signal and noise signal), and set ${\overline{IMF}}_{j}={s}_{j}\left(t\right)+{n}_{j}\left(t\right).$ ${s}_{j}\left(t\right),$ which stands for the characteristic signal component, and ${n}_{j}\left(t\right)$ stands for the noise component. The cross-correlation function between different IMFs is shown below.$${R}_{ij}\left(\tau \right)={R}_{{s}_{i}{s}_{j}}\left(\tau \right)+{R}_{{s}_{i}{n}_{j}}\left(\tau \right)+{R}_{{s}_{j}{n}_{i}}\left(\tau \right)+{R}_{{n}_{i}{n}_{j}}\left(\tau \right)\left(i\ne j\right)$$
- It is known that corresponding IMFs of different series of noise have no correlation with each other. Then, we can set ${R}_{{s}_{i}{n}_{j}}\left(\tau \right)={R}_{{s}_{j}{n}_{i}}\left(\tau \right)={R}_{{n}_{i}{n}_{j}}\left(\tau \right)=0,$ and Equation (11) can be expressed by the formula below.$${R}_{ij}\left(\tau \right)={R}_{{s}_{i}{s}_{j}}\left(\tau \right)\left(i\ne j\right)$$
- Normalized cross correlation coefficient ${r}_{ij}\left(\tau \right)$ is introduced to quantify the correlation between two different IMFs, which can be expressed by the equation below.$$\left|{r}_{ij}\left(\tau \right)\right|=\frac{\left|{R}_{{s}_{i}{s}_{j}}\left(\tau \right)\right|}{\sqrt{{R}_{{s}_{i}}\left(0\right){R}_{{s}_{j}}\left(0\right)}}\le 1\hspace{1em}\left(i\ne j\right)$$

#### 2.3. Testing and Verification

^{−13}and that EEMD is in the order of 1. Clearly, the decomposition accuracy of IEEMD is much higher than that of the original EEMD method.

## 3. Data-Driven Recurrence Quantification Analysis (RQA)

#### 3.1. Brief Description on Recurrence Plot (RP) and RQA

#### 3.2. Data-Driven RQA

#### 3.2.1. Improved RQA Method

- Set the initial value of recurrence threshold ${\epsilon}_{0}$ and target recurrence rate vector $R{R}_{tar}$, the value of ${\epsilon}_{0}$ can be set up based on the equation [39]: ${\epsilon}_{0}=0.5\xb7\sqrt{{{\displaystyle \sum}}_{j}{\left|{v}_{k,j}\right|}^{2}}/N,\left(k=1,\cdots ,N;j=1,\cdots ,M\right)$, where ${v}_{k,j}$ is a delay matrix consisting of embedding the dimension and delay time.
- Set ${\epsilon}_{i}={\epsilon}_{i-1}$ and construct a recurrence matrix $R{P}_{i}$ based on Equation (16). Obtain the $R{R}_{i}$ value based on Equation (18).
- If $R{R}_{i}$ and $R{R}_{tar,i}$ satisfy the set tolerance, record the target threshold value ${\epsilon}_{tar,i}$. If not, the solution process is performed iteratively until the allowable error is satisfied.
- Repeat step 2 and step 3 until the threshold vector ${\epsilon}_{tar}$ corresponding to $R{R}_{tar}$ is obtained.
- According to every value of ${\epsilon}_{tar}$, the RQA is performed to obtain the RQA measure vector $RQAs$.

#### 3.2.2. Selection of Other RQA Parameters and Surrogate Techniques

#### 3.2.3. The Statistical RQA Measures

- The stationary surrogate technique and linear surrogate technique are adopted separately to generate N groups of stationary and linear surrogate signals.
- 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.
- Calculate the mean and the variance of the RQA measures to obtain the non-stationarity testing and nonlinearity testing confidence interval ${\xi}_{S}=\left[\begin{array}{cc}{u}_{S}-3{\sigma}_{S}& {u}_{S}+3{\sigma}_{S}\end{array}\right]$, ${\xi}_{L}=\left[\begin{array}{cc}{u}_{L}-3{\sigma}_{L}& {u}_{L}+3{\sigma}_{L}\end{array}\right]$.
- 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.$$RQA{s}_{ori}={a}_{noise}\xb7RQA{s}_{improved-RQA}$$
- If each element $RQA{s}_{ori,i}$ of the given signal is within the confidence interval, the null hypothesis will be accepted. The nonstationary element ${\mathsf{\Xi}}_{S,i}=0$ and the nonlinear element ${\mathsf{\Xi}}_{L,i}=0$. Otherwise, ${\mathsf{\Xi}}_{S,i}=1$, ${\mathsf{\Xi}}_{L,i}=1$. Let the total size of the RQAs be N $\left(i=1,2,\cdots ,N\right)$, 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 SNS and nonlinearity statistical RQA measure index be SNL. SNS and SNL can be calculated as follows.$${\mathsf{\Xi}}_{S,i}=\{\begin{array}{c}0,ifRQA{s}_{ori,i}\in {\xi}_{S}\\ 1,ifRQA{s}_{ori,i}\notin {\xi}_{S}\end{array}$$$${\mathsf{\Xi}}_{L,i}=\{\begin{array}{c}0,ifRQA{s}_{ori,i}\in {\xi}_{L}\\ 1,ifRQA{s}_{ori,i}\notin {\xi}_{L}\end{array}$$$${n}_{s}={\displaystyle \sum}{\mathsf{\Xi}}_{S,i},\text{}{n}_{l}={\displaystyle \sum}{\mathsf{\Xi}}_{L,i}$$$$\mathrm{SNS}=\{\begin{array}{c}0,if{n}_{s}/N\le {\eta}_{S}\\ 1,if{n}_{s}/N{\eta}_{S}\end{array}$$$$\mathrm{SNL}=\{\begin{array}{c}0,if{n}_{l}/N\le {\eta}_{L}\\ 1,if{n}_{l}/N{\eta}_{L}\end{array}$$
- 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.

## 4. Multiscale Recurrence Analysis of a Model Cable-Stayed Bridge

#### Test Description

## 5. Results and Discussions

## 6. Conclusions

## 7. Future Works

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The flow chart of ensemble empirical mode decomposition (EEMD) and improved ensemble empirical mode decomposition (IEEMD): (

**a**) EEMD and (

**b**) IEEMD.

**Figure 7.**Photographs of the model cable-stayed bridge: (

**a**) general view of the model bridge, (

**b**) damage of the connecting plate, and (

**c**) damage of the stay-cable.

**Figure 12.**The time delay, embedding dimension, and phase space of IMF1: (

**a**) Time delay. (

**b**) Embedding dimension. (

**c**) Phase space.

**Figure 16.**The interquartile range of the estimated bi-coherence values of IMF1 and IMF7: (

**a**) IMF1. (

**b**) IMF7.

**Figure 19.**Time-varying diagram of the first two-order frequency, damping ratio of the curved cable-stayed model bridge: (

**a**) first order, (

**b**) second order, (

**c**) first order, and (

**d**) second order.

Parameters | IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF7 | IMF8 | IMF9 | IMF10 | IMF11 |
---|---|---|---|---|---|---|---|---|---|---|---|

3 | 3 | 5 | 12 | 16 | 18 | 28 | 35 | 45 | 40 | 23 | |

m | 8 | 9 | 6 | 5 | 3 | 2 | 2 | 2 | 2 | 2 | 2 |

IMFs | DET | ENTR | Lmax | LAM | VENTR | Vmax | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Z1 | D1 | Z2 | D2 | Z3 | D3 | Z4 | D4 | Z5 | D5 | Z6 | D6 | |

IMF1 | 0.61 | 0 | 0.69 | 0 | 86 | 0 | 0.34 | 0 | 0.44 | 0 | 93 | 0 |

IMF2 | 0.32 | 0 | 0.37 | 0 | 43 | 0 | 0.46 | 0 | 0.57 | 0 | 45 | 0 |

IMF3 | 0.64 | 1 | 0.79 | 1 | 260 | 1 | 0.82 | 1 | 1.14 | 1 | 80 | 1 |

IMF4 | 0.86 | 1 | 1.84 | 1 | 1501 | 1 | 0.91 | 1 | 2.51 | 1 | 256 | 1 |

IMF5 | 0.92 | 1 | 3.79 | 1 | 3667 | 1 | 0.93 | 1 | 3.93 | 1 | 1201 | 1 |

IMF6 | 0.95 | 1 | 4.21 | 1 | 4123 | 1 | 0.94 | 1 | 4.23 | 1 | 764 | 1 |

IMF7 | 0.98 | 1 | 4.46 | 1 | 5012 | 1 | 0.98 | 1 | 4.58 | 1 | 664 | 1 |

IMF8 | 0.98 | 1 | 4.75 | 1 | 5056 | 1 | 0.98 | 1 | 5.14 | 1 | 458 | 1 |

IMF9 | 0.98 | 1 | 4.99 | 1 | 5072 | 1 | 0.99 | 1 | 5.25 | 1 | 489 | 1 |

IMF10 | 0.99 | 1 | 5.43 | 1 | 5030 | 1 | 0.99 | 1 | 5.74 | 1 | 547 | 1 |

IMF11 | 0.99 | 1 | 5.64 | 1 | 5206 | 1 | 0.99 | 1 | 6.51 | 1 | 552 | 1 |

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**MDPI and ACS Style**

Zhang, E.; Shan, D.; Li, Q.
Nonlinear and Non-Stationary Detection for Measured Dynamic Signal from Bridge Structure Based on Adaptive Decomposition and Multiscale Recurrence Analysis. *Appl. Sci.* **2019**, *9*, 1302.
https://doi.org/10.3390/app9071302

**AMA Style**

Zhang E, Shan D, Li Q.
Nonlinear and Non-Stationary Detection for Measured Dynamic Signal from Bridge Structure Based on Adaptive Decomposition and Multiscale Recurrence Analysis. *Applied Sciences*. 2019; 9(7):1302.
https://doi.org/10.3390/app9071302

**Chicago/Turabian Style**

Zhang, Erhua, Deshan Shan, and Qiao Li.
2019. "Nonlinear and Non-Stationary Detection for Measured Dynamic Signal from Bridge Structure Based on Adaptive Decomposition and Multiscale Recurrence Analysis" *Applied Sciences* 9, no. 7: 1302.
https://doi.org/10.3390/app9071302