# A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Recurrence Plot Method

**Figure 1.**An example of the recurrence plot method: (

**a**) a segment of a tonnage signal collected from progressive stamping process, (

**b**) the trajectory of the tonnage signal in a three dimensional space, and (

**c**) the corresponding RP plot of the tonnage signal.

## 3. Process Monitoring Scheme

#### 3.1. Template Recurrence Plot

#### 3.2. Continuous-Scale Recurrence Plot and Monitoring Statistics

- (1)
- More information of process data has been preserved in the template RP and the continuous-scale RP. The values in the template RP and the continuous-scale RP are varying in [0, 1]. These values can be considered as the probabilities that the corresponding elements are equal to 1. More methods can be used to analyze the process data with the new proposed concepts.
- (2)
- The relationship between a new process data and the in-control data can be directly represented in the continuous-scale RP. The values have directly meanings to represent the differences between the new process data and the in-control data. A large value in continuous-scale RP means the largest difference while a small value means little difference.

- (i)
- Hard threshold:$${A}_{New}={{\displaystyle \sum}}_{i=1}^{N}{\displaystyle \sum}_{j=1}^{N}{R}_{diff}{\left(i,j\right)}^{2}*I({R}_{diff}{\left(i,j\right)}^{2}\ge {T}_{constant})$$
- (ii)
- Soft threshold:$${A}_{New}={{\displaystyle \sum}}_{i=1}^{N}{\displaystyle \sum}_{j=1}^{N}{R}_{diff}{\left(i,j\right)}^{2}*I({R}_{diff}{\left(i,j\right)}^{2}\ge {T}_{s}\left(i,j\right))$$

#### 3.3. Parameter Settings

- (1)
- When the information related to the process condition changes is unavailable, the hard threshold approach is preferred to use. We can determine the appropriate value of ${T}_{constant}$ based on the empirical distribution of ${R}_{diff}{\left(i,j\right)}^{2}$, which can be estimated from a set of in-control profile data.
- (2)
- When we have enough in-control and out-of-control profile data, the soft threshold is preferred to use. Here, we should notice that it is difficult to use the soft threshold in practice due to information and data limitation. The prerequisite of using the soft threshold to derive the monitoring statistics is that we should have numerous in-control and out-of-control process data. If we have enough in-control and out-of-control profile data, we can calculate the probability of the element value distribution in ${R}_{diff}$ as:$$P\left({R}_{diff}{\left(i,j\right)}^{2}\ge {T}_{s}\left(i,j\right)\right)\ge {P}_{0}$$

#### 3.4. RP-Based Bootstrap Control Chart

- Obtain $k$ in-control profile data, and calculate the corresponding monitoring statistics $\left\{{A}_{1},{A}_{2},\cdots ,{A}_{k}\right\}$.
- Draw a random sample of size ${n}_{c}$ with replacement from the monitoring statistics $\left\{{A}_{1},{A}_{2},\cdots ,{A}_{k}\right\}$, and obtain a bootstrap sample $\left\{{A}_{1}^{*},{A}_{2}^{*},\cdots ,{A}_{{n}_{c}}^{*}\right\}$, here ${n}_{c}$ should be much smaller than $k$.
- Compute the mean of the bootstrap sample $\overline{{A}^{*}}\left({n}_{c}\right)$.
- Repeat the step 2 and step 3 a large amount of times, say $B$ times (e.g., $B$=1000). Then we can derive $B$ bootstrap sample means as $\overline{{A}_{1}^{*}}\left({n}_{c}\right),\overline{{A}_{2}^{*}}\left({n}_{c}\right),\cdots ,\overline{{A}_{B}^{*}}\left({n}_{c}\right)$.
- Sort the $B$ bootstrap sample means in an increasing order to obtain a new $B$ bootstrap sample means denoted as ${S}_{A}=\left\{\overline{{A}_{\left(1\right)}^{*}}\left({n}_{c}\right),\overline{{A}_{\left(2\right)}^{*}}\left({n}_{c}\right),\cdots ,\overline{{A}_{\left(B\right)}^{*}}\left({n}_{c}\right)\right\}$.
- Define a constant $\alpha $ to represent the false alarm rate of the bootstrap control chart. Then, find the upper control limit (UCL) and the lower control limit (LCL) based on the formulae $P\left({S}_{A}<UCL\right)=1-\alpha /2$ and $P\left({S}_{A}<LCL\right)=\alpha /2$.

#### 3.5. Monitoring Scheme

- Obtain two groups of in-control profile data. One group data is used to estimate the template RP plot ${\overline{R}}_{n}$ by using Equation (3).
- Calculate the continuous-scale RP plot ${R}_{diff}$ by using Equation (4) based on the other group of in-control data and derive the monitoring statistics by using Equation (6) via the top-r approach.
- Estimate the control limit of the proposed bootstrap control chart following the procedures introduced in Subsection 3.4.
- When a new profile is obtained, calculate the continuous-scale RP plot ${R}_{diff}$ and the monitoring statistic of this profile. Then, test the monitoring statistic by the constructed bootstrap control chart. If this statistic falls outside the control limit estimated from step 3, we can conclude that the process is out-of-control. Otherwise, the process is in-control.

## 4. Performance Comparisons

${T}_{constant}$ | ${\delta}_{F}$ | Our method | Recurrence Rate (RR) | Determinism (DET) | Entropy (ENT) | Laminarity (LAM) | Trapping Time (TT) |
---|---|---|---|---|---|---|---|

0.9 | 0.5 | 66.31 | 165.43 | 183.21 | 178.32 | 185.26 | 183.61 |

1.0 | 1 | 12.27 | 35.57 | 116.21 | 41.23 | 27.13 | |

1.5 | 1 | 1.348 | 4.513 | 11.32 | 4.39 | 2.752 | |

0.8 | 0.5 | 102.35 | 171.83 | 191.23 | 184.85 | 186.93 | 172.97 |

1.0 | 1 | 11.28 | 37.87 | 110.32 | 43.94 | 25.98 | |

1.5 | 1 | 1.043 | 4.832 | 12.12 | 4.231 | 2.793 | |

0.7 | 0.5 | 67.25 | 167.85 | 180.23 | 175.92 | 187.63 | 188.76 |

1.0 | 1 | 12.87 | 34.51 | 115.81 | 41.53 | 32.08 | |

1.5 | 1 | 1.287 | 4.639 | 11.84 | 4.372 | 2.873 | |

0.6 | 0.5 | 68.78 | 163.86 | 197.35 | 177.38 | 193.23 | 185.34 |

1.0 | 1 | 11.65 | 35.82 | 115.73 | 42.85 | 27.62 | |

1.5 | 1 | 1.493 | 4.892 | 11.76 | 2.38 | 2.87 |

${T}_{constant}$ | ${\delta}_{F}$ | Our method | Recurrence Rate (RR) | Determinism (DET) | Entropy (ENT) | Laminarity (LAM) | Trapping Time (TT) |
---|---|---|---|---|---|---|---|

0.9 | 0.5 | 8.763 | 188.92 | 189.32 | 187.32 | 185.43 | 147.87 |

1.0 | 1.342 | 168.83 | 126.28 | 183.98 | 118.23 | 66.25 | |

1.5 | 1 | 153.21 | 62.32 | 173.52 | 67.42 | 28.09 | |

0.8 | 0.5 | 13.87 | 187.76 | 188.21 | 198.12 | 193.87 | 152.54 |

1.0 | 1.212 | 172.12 | 131.23 | 185.91 | 123.32 | 72.89 | |

1.5 | 1 | 163.31 | 69.89 | 180.19 | 69.72 | 34.42 | |

0.7 | 0.5 | 16.87 | 182.2 | 183.24 | 188.53 | 173.19 | 150.32 |

1.0 | 1.19 | 164.83 | 121.92 | 176.59 | 121.73 | 63.13 | |

1.5 | 1 | 157.37 | 58.24 | 163.73 | 59.39 | 30.28 | |

0.6 | 0.5 | 18.73 | 182.82 | 179.31 | 178.73 | 176.48 | 156.36 |

1.0 | 1.142 | 165.76 | 125.65 | 170.12 | 118.86 | 65.62 | |

1.5 | 1 | 154.83 | 53.32 | 161.28 | 57.39 | 30.83 |

## 5. Progressive Stamping Processes Analysis

#### 5.1. Introduction of Progressive Stamping Processes

#### 5.2. Template RP Plot

**Figure 4.**Tonnage signals and their recurrence plot (RP) plots under normal and faulty conditions: (

**a**) Normal and faulty tonnage signal. (

**b**) The RP plot of the normal signal. (

**c**) The RP plot of the faulty signal.

**Figure 5.**(

**a**) The template recurrence plot (RP) plot based on the normal signals; (

**b**) The template RP plot based on the faulty signals.

**Figure 6.**Continuous-scale recurrence plot (RP) plots: (

**a**) Continuous-scale RP plot under normal condition. (

**b**) Continuous-scale RP plot under faulty condition.

#### 5.3. Continuous-Scale RP Plot

#### 5.4. Process Monitoring Results

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Zhou, C.; Zhang, W.
A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot. *Entropy* **2015**, *17*, 6379-6396.
https://doi.org/10.3390/e17096379

**AMA Style**

Zhou C, Zhang W.
A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot. *Entropy*. 2015; 17(9):6379-6396.
https://doi.org/10.3390/e17096379

**Chicago/Turabian Style**

Zhou, Cheng, and Weidong Zhang.
2015. "A New Process Monitoring Method Based on Waveform Signal by Using Recurrence Plot" *Entropy* 17, no. 9: 6379-6396.
https://doi.org/10.3390/e17096379