# Recurrence Plot Based Damage Detection Method by Integrating Control Chart

^{*}

## Abstract

**:**

^{2}control chart, a multivariate statistical process control technique, is used to monitor these features. The T

^{2}control chart technique, however, has the assumption that all the data should follow a normal distribution. The RP based T

^{2}bootstrap control chart is proposed to estimate the control chart parameters. The performance of the proposed RP based T

^{2}bootstrap control chart is evaluated by a simulation study and compared with other univariate bootstrap control charts based on recurrence plot features. A real case study of rolling element bearing fault detection demonstrates that the proposed fault detection method achieves a very good performance.

## 1. Introduction

^{2}control chart technique in this paper. The Hotelling’s T

^{2}control chart, which is introduced by Hotelling, is a multivariate statistical process control technique that can monitor vibration signals by using all the features extracted from the recurrence plot [20]. The T

^{2}control chart requires that the features should follow multivariate normal distribution. However, we do not have the distribution information of these features extracted by the RQA method and we cannot conclude that these features follow multivariate normal distribution.

^{2}bootstrap control chart. The bootstrap, a kind of resampling method, can be used to estimate the sampling distribution of a statistic while assuming only that the observations are independent and identically distributed. Bajgier proposed a bootstrap control chart to monitor the mean of a process [21]. In this paper, we extend the Bajgier’s bootstrap control chart to the T

^{2}control chart case to develop a T

^{2}bootstrap control chart. However, Seppala pointed out that there is an obvious limitation in that Bajgier’s bootstrap control chart implicitly assumes that the process is stable and in-control when the control limits are computed [22]. If this assumption is violated, the control limits computed will be too wide. For this reason, a pre-process method is developed to remove the outliers based on the RP method before conducting the bootstrap method.

^{2}control chart technique to monitor process conditions. Then, we compare our proposed damage detection scheme with other RP based univariate control charts method to show the performance of our proposed method. At last, a real case of rolling element bearing damage detection is studied to demonstrate our proposed method and conclusions are given.

## 2. Review of the RP Method

#### 2.1. Introduction of RP Method

**X**= [X

_{1}, X

_{2}, …, X

_{N}]

^{T}as a series of vibration signals with N points each. Then a series of d -dimensional vectors $\left\{{\overrightarrow{\mathit{x}}}_{i}\right\}$ can be constructed from the one-dimensional signal {X

_{i}} by using Equation (1).

**R**is called recurrence matrix with N′ columns and N′ rows. R(i, j) represents the element with row i and column j. If the distance between ${\overrightarrow{\mathit{x}}}_{i}$ and ${\overrightarrow{\mathit{x}}}_{j}$ is less than the threshold ξ, R

_{i,j}= 1, otherwise R

_{i,j}= 0. If we plot the element “1” as black dot and plot the element ‘0’ as white dot, then we can conclude that the R can be visualized as a binary image, which is named as RP graph is this paper. Figure 1 is an example to show the RP method. Figure 1a shows a simulated signal, and Figure 1b is its RP graph.

#### 2.2. Recurrence Quantification Analysis

_{min}is the minimal length of the diagonal lines, P(v) is the histogram of the vertical lines with the lengths v, and v

_{min}is the minimal length of the vertical lines.

- (1)
- Recurrence Rate (RR)RR is a feature to measure the density of the black dots in the RP graph. A larger RR value means more black dots in the RP graph. The RR can characterize the signal’s stationarity, periodicity and complexity.$$RR=\frac{1}{{N}^{2}}\cdot {\displaystyle \sum _{i,j=1}^{N}{R}_{i,j}}$$
- (2)
- Determinism (DET)DET represents the proportion of the black dots forming the diagonal lines which are longer than l
_{min}. According to Marwan [13], the diagonal line refers to the recurrences of the original signal in the corresponding signal segments. The DET measures the occurrence of diagonal lines with different lengths in the RP plot.$$DET=\frac{{\displaystyle {\sum}_{l={l}_{\mathit{min}}}^{N}l\cdot P(l)}}{{\displaystyle {\sum}_{i,j}^{N}{R}_{i,j}}}$$ - (3)
- Entropy (ENT)ENT represents the Shannon information entropy of the selected diagonal lines whose lengths are longer than l
_{min}, and it reflects the complexity of the RP in respect of the diagonal lines.$$ENT=-{\displaystyle \sum _{l={l}_{\mathit{min}}}^{N}P(l)}\cdot \mathit{ln}P(l)$$ - (4)
- Laminarity (LAM)LAM measures the proportion of the black dots forming the vertical lines which are longer than v
_{min}. The vertical lines have the same interpretation as the horizontal lines which are caused by the slow changes from a certain signal value in the original signal. Hence, LAM is a measure of the occurrence of vertical lines in the RP plot.$$LAM=\frac{{\displaystyle {\sum}_{v={v}_{\mathit{min}}}^{N}v\cdot P(v)}}{{\displaystyle {\sum}_{v=1}^{N}v\cdot P(v)}}$$ - (5)
- Trapping time (TT)TT calculates the average length of the vertical lines which are longer than v
_{min}, and it represents the average length of the vertical lines in RP plots.$$TT=\frac{{\displaystyle {\sum}_{v={v}_{\mathit{min}}}^{N}v}\cdot P(v)}{{\displaystyle {\sum}_{v={v}_{\mathit{min}}}^{N}P(v)}}$$

**Y**= [

**RR**,

**DET**,

**ENT**,

**LAM**,

**TT**] as the feature to represent the signal. In the following section, the

**T**control chart is constructed based on this vector

^{2}**Y**to monitor the signal.

## 3. Methodology Development

^{2}control chart technique. First, the RP method is conducted on the collected signals and five features are extracted. The T

^{2}control chart is then introduced and used to monitor the five features. After that, the nonparametric bootstrap method is introduced to estimate the control limit of the T

^{2}control chart.

#### 3.1. T^{2} Control Chart Based on the RP Method

^{2}control chart, which is also called Hotelling T

^{2}control chart, is widely used in multivariate statistical process control [20]. We assume we have p variables x

_{1}, x

_{2}, …, x

_{p}and then we obtain a group of samples

**X**= [

**X**

_{1},

**X**

_{2}, …,

**X**

_{p}]. Assuming the samples follow a multivariate normal distribution, then the sample mean vector is $\overline{\mathit{x}}=1/N\cdot {\displaystyle {\sum}_{j=1}^{N}{x}_{j}}$ and the sample covariance matrix is $\mathit{S}=1/(N-1)\cdot {\displaystyle {\sum}_{j=1}^{N}({\mathit{x}}_{\mathit{j}}-\overline{\mathit{x}}){({\mathit{x}}_{\mathit{j}}-\overline{\mathit{x}})}^{\prime}}$, where N is the sample number of each variable. Thus, the T

^{2}statistic is

_{1}, u

_{2}, … u

_{N}} is a group of vibration signals. Then the RP graph {R

_{1}, R

_{2}, R

_{N}} can be derived and RR, DET, ENT, LAM, TT can be estimated by RQA. Defining

**Y**= [

**RR**,

**DET**,

**ENT**,

**LAM**,

**TT**], a T

^{2}control chart based on the RP method can be constructed based on the

**Y**. The T

^{2}statistic based on RP method can be obtained as

^{2}control chart needs to be estimated here. Usually, there are two distinct phases, which are Phase I and Phase II when we use the control chart techniques. In Phase I, the main goals are estimating the control chart parameters and removing the outliers. We do not have any prior information on deriving appropriate control limits. Therefore, after a group of process data is collected, some pre-process algorithms or de-noising methods are used to remove the outliers. Then estimate the distribution of the data and the control chart parameters to build a control chart. After that, test the data by using the control chart. Then remove the out-of-control data and rebuild the control chart until all data is in-control. In Phase II, the major goal is to detect changes in the newly observed data. The estimated in-control data distribution from the Phase I dataset is used. The performance of a Phase II procedure is often measured from the view point of the Average Run Length (ARL), which is defined as the average number of samples taken before the chart triggers a signal [28]. Usually, the in-control ARL is controlled at a specific level with a defined false alarm rate α. For example, people often set the in-control ARL of the $\overline{X}$ control chart to ARL=370 with the false alarm rate α = 0.027 when the in-control data follows normal distribution. The out-of-control ARL is used to measure the performance of the control chart so that the control chart performs better if the out-of-control ARL is smaller when detecting a given change. Please refer to Montgomery to see a detailed introduction about Phase I and Phase II analysis [29]. Traditionally, if we assume that the observed data follows multivariate normal distribution, then the Phase I control limit of the T

^{2}control chart is UCL = (m − 1)(n − 1)p/(m(n − 1) + 1 − p) · F

_{α}(p, m(n − 1) + 1 − p) and the Phase II control limit is UCL = (m + 1)(n − 1)p/(m(n − 1) + 1 − p) · F

_{α}(p, m(n − 1) + 1 − p), where m is the samples number used for training the control limit in Phase I. The parameter α is the false alarm rate, which means the probability of T

^{2}> UCL is α when the data is in-control. However, we cannot obtain the distribution information of the defined variable

**Y**. Therefore the nonparametric bootstrap method is used to estimate the control limit.

#### 3.2. Bootstrap Method

^{2}statistic and control limits are estimated with the assumption that the samples follow multivariate normal distribution. However, we cannot ensure that the features vector

**Y**= [

**RR**,

**DET**,

**ENT**,

**LAM**,

**TT**] is multivariate with a normal distribution. To address this problem, the bootstrap method is used to estimate the T

^{2}statistic and the control limit of the T

^{2}control chart. First, we apply the concept of the nonparametric bootstrap to estimate the sample mean and the sample covariance of the vector

**Y**. Then the T

^{2}statistic is obtained and the control limit of T

^{2}control chart is estimated.

**X**= (x

_{1}, x

_{2}, … x

_{n}) and the underlying cumulative distribution function of

**X**is F(x). Assigning a probability of 1/n to each value in

**X**, the empirical distribution function can be written as:

**X**follows normal distribution if n → ∞.

**Y**= (

**Y**

_{1},

**Y**

_{2}, …,

**Y**

_{n}) follows a multivariate normal distribution if we can obtain enough sample numbers. Then the T

^{2}statistic can be obtained and the control limit of the T

^{2}bootstrap control chart can be estimated based on Bajgier’s method.

^{2}bootstrap control chart are provided as follows.

- Assume we obtain a group of in-control process signals {x
_{1}, x_{2}, … x_{n}}. Conduct the RP method to analyze the signals to obtain the recurrence plots {R_{1}, R_{2}, … R_{n}}. - Conduct the RQA to extract features of the {R
_{1}, R_{2}, … R_{n}} in order to obtain the vectors {**Y**,_{1}**Y**, …_{2}**Y**_{n}} - Obtain the T
^{2}statistic according to ${T}_{i}^{2}=n{({\mathit{Y}}_{\mathit{i}}-{\overline{\mathit{Y}}}^{*})}^{T}{\mathit{S}}^{-1}({\mathit{Y}}_{\mathit{i}}-{\overline{\mathit{Y}}}^{*})$ to get a group of T^{2}statistic values $\{{T}_{1}^{2},{T}_{2}^{2},\dots ,{T}_{n}^{2}\}$, where ${\overline{\mathit{Y}}}^{*}$ and**S**are the sample mean and sample covariance of the vectors {**Y**_{1},**Y**_{2}, …,**Y**_{n}}, respectively. - Draw a random sample of size n
_{c}, such as n_{c}= 100,000, with replacement, from the samples $\{{T}_{1}^{2},{T}_{2}^{2},\dots ,{T}_{n}^{2}\}$, then get the sample $\{{T}_{1}^{2},{T}_{2}^{2},\dots ,{T}_{{n}_{c}}^{2}\}$, which is a bootstrap sample. - Sort the bootstrap T
^{2}statistic $\{{T}_{1}^{2},{T}_{2}^{2},\dots ,{T}_{{n}_{c}}^{2}\}$ in ascending order to derive $\{{{T}^{\prime}}_{1}^{2},{{T}^{\prime}}_{2}^{2},\dots ,{{T}^{\prime}}_{{n}_{c}}^{2}\}$. Find the value ${{T}^{\prime}}_{i}^{2}$ that ${{T}^{\prime}}_{i}^{2}\ge \alpha \cdot {n}_{c}$, where i = 1, 2, …, n_{c}. Thus, the T^{2}bootstrap control chart’s control limit is $UCL={{T}^{\prime}}_{i}^{2}$.

^{2}control chart can be evaluated by a simulation study. If we simulate n

_{c}samples each time, we can then obtain the coverage probability p

_{i}=P(T

^{2}< UCL) of these samples. After repeating this simulation for n

_{s}times, such as n

_{s}= 5000, we can estimate the in-control average run length (ARL) of the proposed bootstrap T

^{2}control chart as ${\text{ARL}}_{{T}^{2}}=1/{n}_{s}\cdot {\displaystyle {\sum}_{i=1}^{{n}_{s}}1/(1-{p}_{i})}$.

#### 3.3. Damage Detection Scheme

^{2}control chart. Following the notations introduced earlier, the procedures of the proposed damage detection scheme are stated as follows. Figure 2 shows the framework of the proposed monitoring scheme.

- Collect a group of system vibration signals {u
_{1}, u_{2}, … u_{n}} under normal conditions. - Conduct the RP method to analyze the vibration signals and extract the five features RR, DET, ENT, LAM, TT by RQA and obtain the feature vector
**Y**_{i}= [**RR**,**DET**,**ENT**,**LAM**,**TT**], i = 1,2, ⋯, n. - Estimate the sample mean ${\overline{\mathit{Y}}}^{*}$ and sample covariance
**S**of the vectors**Y**_{i}, and then obtain the T^{2}statistic according to the equation ${T}_{i}^{2}=n{({\mathit{Y}}_{\mathit{i}}-{\overline{\mathit{Y}}}^{*})}^{T}{\mathit{S}}^{-1}({\mathit{Y}}_{\mathit{i}}-{\overline{\mathit{Y}}}^{*})$. - Estimate the control limit UCL of the proposed T
^{2}bootstrap control chart according to the procedures introduced in section bootstrap method. - When a new system vibration signal is collected, use the RP method and RQA method to obtain the feature vector
**Y**_{new}= [**RR**,**DET**,**ENT**,**LAM**,**TT**]. - Obtain the T
^{2}statistic according to the equation ${T}_{new}^{2}=n{({\mathit{Y}}_{\mathit{new}}-{\overline{\mathit{Y}}}^{*})}^{T}{\mathit{S}}^{-1}({\mathit{Y}}_{\mathit{new}}-{\overline{\mathit{Y}}}^{*})$. Monitor this T^{2}statistic using the proposed RP-based T^{2}bootstrap control chart. If ${T}_{new}^{2}\ge UCL$, we can conclude that the signal is out-of-control. It means a process fault occurred in the system. If ${T}_{new}^{2}<UCL$ we can conclude that the signal is in-control and the system is in a normal condition.

## 4. Performance Evaluation

^{2}bootstrap control chart with other RP-based univariate control charts which are the RQA-based univariate control charts in terms of the average run length (ARL). Here, the RQA-based univariate control charts are constructed based on the five RP features introduced in Section RQA introduction. These five features are RR, DET, ENT, LAM and TT. For simplicity, the RQA-based control charts are named the RR-based control chart, the DET-based control chart, the ENT-based control chart, the LAM-based control chart and the TT-based control chart. Without loss of generality, we will compare the monitoring performance of the proposed RP-based T

^{2}bootstrap control chart and the RQA-based control charts in order to analyze the simulated signals. The out-of-control ARLs are compared when detecting the mean shift case and the frequency change case. The steps for obtaining the out-of-control ARLs of the RP-based T

^{2}bootstrap control chart and the RQA-based control charts are stated as follows.

- Step 1.
- According to the procedures introduced in section
**T**control chart based on the RP method, obtain the control limits of the RP-based T^{2}^{2}bootstrap control chart and the RQA-based control charts of the simulated in-control signals. - Step 2.
- Simulate groups of out-of-control signals with mean shift or frequency change. Obtain the five features (RR, DET, ENT, LAM and TT) by RQA and the T
^{2}statistic ${T}_{Y}^{2}$. - Step 3.
- Estimate the run length (RL) of the RP-based T
^{2}bootstrap control chart and the RQA-based control charts with the false alarm rate α = 0.05. - Step 4.
- Repeat the above steps n
_{p}= 5000 times to obtain the out-of control ARLs of the RP-based T^{2}bootstrap control chart and the RQA-based control charts for the mean shift case or the frequency change case.

_{0}= D + ξ as the in-control signal, f′ = f

_{0}+ F as the out-of control signal, D as the main process effect under in-control condition, F as the signal change component and ξ as the process noise. Without loss of generality, we assume the process noise ξ follows a normal distribution and can be represented as ξ ~ σ

_{Noise}· N(0,1). Denote A

_{F}as the amplitude of F, and then define δ

_{F}= A

_{F}/σ

_{Noise}to measure the magnitude of the signal change component. Figures 3–5 are examples to show the original signals and their RP graphs of the in-control case, the mean shift case with δ

_{F}= 1.5 and the frequency change case with δ

_{F}= 1.5. We can see that there are some differences among the three in-control and out-of-control RP graphs although the original signals of the in-control and out-of-control cases are seen as the same. We will conduct our proposed damage detection scheme on the simulated signals to show that our proposed method is effective for detecting small or large signal change.

_{0}· δ

_{F}measures the magnitude of the signal shift component. Table 1 shows the out-of control ARLs in detecting the mean shift case.

_{0}· δ

_{F}measures the magnitude of the frequency change component. Table 2 shows the out-of-control ARLs in detecting the frequency change case.

^{2}bootstrap control chart are small and will be even smaller when δ

_{F}is increasing in both the mean shift case and frequency change case. Second, the out-of-control ARLs of the proposed RP-based T

^{2}bootstrap control chart are always the smallest when compared with the other RQA based control charts. Therefore, we can conclude that: (1) the proposed damage detection scheme can detect both small and large signal changes effectively and efficiently; (2) the proposed RP-based T

^{2}bootstrap control chart uniformly performs better than any other RQA based control charts.

## 5. Case Study

#### 5.1. Data Introduction

_{2}bootstrap control chart [30]. The experiment equipment consists of a motor (left), a torque transducer/encoder (center), a dynamometer (right), and control electronics. The test bearings support the motor shaft. Single point faults were introduced into the test bearings with different fault diameters. Vibration data was collected using accelerometers, which were attached to the housing by use of magnetic bases. Data was collected for normal bearings, a single point fault occurring on the inner race, on the outer race and on the rolling element. Please refer Bearing Data Center website

^{26}for more instructions about the data set.

#### 5.2. Feature Extraction

#### 5.3. Training Control Limit

_{2}bootstrap control chart can be obtained based on the procedures mentioned before. Here we use 250 samples in group “NB”as training group to train the control chart. 50 samples are used as testing group to test the RP based T

_{2}bootstrap control chart. Here, we set the false alarm rate α = 0.05. Figure 8 shows the testing result of the RP based T

^{2}bootstrap control chart. The horizontal axis is the sample index and the vertical axis is the derived T

_{2}statistic value of the testing group data. The straight solid line is the control limit of the RP based T

_{2}bootstrap control chart estimated from the training group data with defined false alarm rate α = 0.01 in this case. Each point in the Figure 8 represents the T

_{2}statistic value of each sample in the testing group data. According to the control chart theory, if all statistic values are below the control limit, it means all the samples are in-control. From Figure 8, we can see that all the points are below the solid line which is the control limit of the RP based T

_{2}bootstrap control chart. Therefore, we can conclude that our proposed RP based T

_{2}bootstrap control chart works well to monitor the bearing signals in normal condition. Hence, this RP based T

_{2}bootstrap control chart can be used to detect the conditions of the rolling element bearing.

#### 5.4. Results

_{2}statistics of all samples are calculated. These T

_{2}statistics are tested by the constructed RP based T

_{2}bootstrap control chart as shown in Figure 8. Thus, the conditions of rolling element bearing can be detected by the proposed method. Figure 9a, c show the detection results that the samples are come from the group “FIR”, “FOR” and “FRE”, which means the bearing fault occurred on the inner race, the outer race and the rolling element respectively. In Figure 9, the points marked as red circles represent the fault bearing signals and the points marked as blue stars at the bottom of each figure represent the normal bearing signals, while the blue solid line represents the control limit of the constructed RP based T

_{2}bootstrap control chart. Due to the large difference of the values of the T

_{2}statistics of faulty vibration signals and normal vibration signals, we cannot see blue points and the solid line clearly. Fortunately, all the blue points and the solid line are the same as the blue points and blue line in Figure 8. Therefore, we can consider Figure 8 as enlarged views of the bottom parts in Figure 9. All the red points in Figure 9 are located at the top of the solid line. Therefore, we can conclude that the constructed RP based T

_{2}bootstrap control chart can successfully detect the bearing faults occurred on the inner race, the outer race and the rolling element. Moreover, the values of red points in Figure 9a,b,c are in different intervals, which means the T

_{2}statistic also can be used to diagnose the bearing faults for further research.

_{2}control chart technique performs very well in detecting different types of bearing faulty vibration signals.

## 6. Conclusions

_{2}control chart for condition-based maintenance, structural health monitoring, and so forth. The RP method is a nonparametric and nonlinear signal processing method that can characterize the vibration-based signal to be a two dimensional matrix. Five recurrence plot features can be obtained from the two dimensional matrix and can be used to classify vibration signals in normal conditions and in fault conditions. There are several advantages to analyzing the vibration-based signal by using the RP method: (1) it is not necessary to know the information of signal distribution, (2) the method does not require the signal length, (3) the method is a nonparametric method and it is not necessary to estimate the model parameter, (4) the method does not require consideration of the frequency change or the amplitude change in the signal.

_{2}bootstrap control chart. A simulation study has shown that the proposed damage detection scheme can detect signal shift and frequency change quickly and effectively. A real case study of rolling element bearing faults demonstrates that the proposed damage detection scheme can achieve a very good performance in detecting different types of bearing fault.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**An example of recurrence plot method: (a) a simulated signal and (b) the corresponding recurrence plot (RP) plot.

**Figure 3.**An example of the simulated in-control signals and its RP graph: (a) simulated in-control signal, (b) the RP graph of the in-control signal.

**Figure 4.**An example of the simulated signals and its RP graph of the mean shift case with δ

_{F}= 1.5: (a) simulated out-of-control signal, (b) the RP graph of the out-of-control signal.

**Figure 5.**An example of the simulated signals and its RP graph of the frequency change case with δ

_{F}=1.5: (a) simulated out-of-control signal, (b) the RP graph of the out-of-control signal.

**Figure 7.**The RP graphs of normal and fault bearing vibration signals: (

**a**) normal bearing vibration signal, (

**b**) fault bearing vibration signal in “FRE”group, (

**c**) RP graph of normal bearing vibration signal, (

**d**) RP graph of fault bearing vibration signal in “FRE”group.

**Figure 9.**The detection result of fault bearing vibration signals: (a) the T

_{2}control chart of group “FIR”; (b) the T

_{2}control chart of group “FOR”; (c) the T

_{2}control chart of group “FRE”.

δ_{F} | Bootstrap Control Chart
| |||||
---|---|---|---|---|---|---|

T^{2} | RR | DET | ENT | LAM | TT | |

0.5 | 27.203 | 174.633 | 198.132 | 188.652 | 189.652 | 195.857 |

0.6 | 24.024 | 161.48 | 198.578 | 188.701 | 188.316 | 193.202 |

0.7 | 21.331 | 142.943 | 197.204 | 178.5 | 190.289 | 191.909 |

0.8 | 18.64 | 126.856 | 195.935 | 171.977 | 184.923 | 188.615 |

0.9 | 16.863 | 111.653 | 192.81 | 162.948 | 185.796 | 186.819 |

1 | 14.402 | 96.655 | 190.947 | 154.253 | 180.793 | 182.548 |

1.2 | 10.874 | 75.048 | 189.406 | 142.363 | 182.895 | 181.005 |

1.5 | 7.742 | 52.341 | 173.475 | 137.22 | 162.997 | 159.368 |

1.8 | 6.271 | 41.524 | 149.204 | 138.41 | 143.257 | 136.839 |

2 | 5.397 | 36.482 | 134.857 | 138.01 | 131.723 | 125.794 |

2.5 | 4.27 | 28.509 | 105.045 | 143.724 | 102.997 | 89.348 |

3 | 3.499 | 23.138 | 82.251 | 142.575 | 84.112 | 68.992 |

3.5 | 2.618 | 18.076 | 64.955 | 145.422 | 67.132 | 46.406 |

δ_{F} | Bootstrap Control Chart
| |||||
---|---|---|---|---|---|---|

T^{2} | RR | DET | ENT | LAM | TT | |

0.5 | 22.056 | 136.293 | 197.75 | 181.662 | 184.538 | 192.838 |

0.6 | 19.404 | 112.366 | 193.593 | 172.972 | 182.563 | 186.877 |

0.7 | 16.806 | 96.412 | 192.472 | 167.398 | 179.814 | 183.251 |

0.8 | 14.777 | 80.397 | 187.281 | 153.824 | 175.054 | 181.184 |

0.9 | 13.907 | 65.608 | 179.698 | 145.221 | 168.537 | 175.855 |

1 | 12.448 | 56.265 | 176.104 | 132.161 | 164.166 | 176.251 |

1.2 | 10.247 | 44.514 | 170.879 | 119.805 | 158.505 | 170.777 |

1.5 | 8.502 | 34.46 | 165.137 | 119.498 | 150.905 | 160.905 |

1.8 | 6.877 | 28.493 | 142.494 | 121.26 | 137.88 | 143.475 |

2 | 6.298 | 25.717 | 132.545 | 121.796 | 129.801 | 131.23 |

2.5 | 4.972 | 18.889 | 92.051 | 98.782 | 92.845 | 95.207 |

3 | 4.124 | 14.103 | 97.773 | 82.882 | 90.112 | 99.111 |

3.5 | 3.72 | 11.404 | 99.684 | 66.035 | 87.438 | 101.268 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, C.; Zhang, W.
Recurrence Plot Based Damage Detection Method by Integrating Control Chart. *Entropy* **2015**, *17*, 2624-2641.
https://doi.org/10.3390/e17052624

**AMA Style**

Zhou C, Zhang W.
Recurrence Plot Based Damage Detection Method by Integrating Control Chart. *Entropy*. 2015; 17(5):2624-2641.
https://doi.org/10.3390/e17052624

**Chicago/Turabian Style**

Zhou, Cheng, and Weidong Zhang.
2015. "Recurrence Plot Based Damage Detection Method by Integrating Control Chart" *Entropy* 17, no. 5: 2624-2641.
https://doi.org/10.3390/e17052624