# Statistical Model Checking in Process Mining: A Comprehensive Approach to Analyse Stochastic Processes

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Process Mining

#### 2.1.1. Traces and Event Logs

#### 2.1.2. Petri Net

#### 2.2. Statistical Model Checking

## 3. Related Work

## 4. Proposed Framework

- Process-mining model.
- Modelling and verification.
- Analysis.

#### 4.1. Process-Mining Model

#### 4.1.1. Process Discovery

#### 4.1.2. Conformance Checking

#### 4.2. Modelling and Verification

#### 4.3. Analysis

## 5. Case Study

#### 5.1. Event Log and Process-Mining Algorithms

#### 5.2. Verification

#### 5.2.1. Estimation with Confidence Level of 99%

#### 5.2.2. Estimation with Confidence Level of 95%

#### 5.2.3. Hypothesis Testing

#### 5.3. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Process mining with discovery and conformance [24].

**Table 1.**Model size of generated DTMC models. For some process discovery algorithms, the transition numbers differ among different replay algorithms.

PM Algorithm | Num. of States | Num. of Transitions |
---|---|---|

Alpha Miner | 26 | $[178,254]$ |

Discover using Decomposition | 26 | 178 |

Mine for Heuristic Net | 27 | $[172,219]$ |

Mine Transition System | 26 | $[178,201]$ |

**Table 2.**Property $P=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}[F\le 10\phantom{\rule{4pt}{0ex}}(s=x\left)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.105 | 0.092 | 0.053 | 0.202 | 0.091 | 0.096 | 0.101 | 0.104 |

Discover using Decomposition | 0.1 | 0.101 | 0.104 | 0.105 | 0.098 | 0.107 | 0.102 | 0.106 |

Mine for Heuristic Net | 0.161 | 0.271 | 0.154 | 0.274 | 0.252 | 0.187 | 0.238 | 0.275 |

Mine Transition System | 0.266 | 0.101 | 0.105 | 0.107 | 0.132 | 0.103 | 0.24 | 0.103 |

**Table 3.**Property $P=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}[\neg \phantom{\rule{4pt}{0ex}}(s=x)\phantom{\rule{4pt}{0ex}}U\le 5\phantom{\rule{4pt}{0ex}}(s=y\left)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.0133 | 0.0617 | 0.0381 | 0.0392 | 0.0620 | 0.0199 | 0.0128 | 0.0142 |

Discover using Decomposition | 0.0131 | 0.0118 | 0.0129 | 0.0136 | 0.0140 | 0.0125 | 0.0121 | 0.0133 |

Mine for Heuristic Net | 0.1354 | 0.1575 | 0.1328 | 0.1200 | 0.1415 | 0.1526 | 0.1261 | 0.1194 |

Mine Transition System | 0.0192 | 0.0152 | 0.0131 | 0.0126 | 0.0180 | 0.0133 | 0.0139 | 0.0125 |

**Table 4.**Property ${R}_{cost}=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}\left[F(s=x)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 97.808 | 51.731 | 54.089 | 54.106 | 51.509 | 98.993 | 98.231 | 97.697 |

Discover using Decomposition | 99.026 | 99.882 | 98.851 | 98.215 | 98.301 | 98.557 | 99.060 | 98.143 |

Mine for Heuristic Net | 42.123 | 31.69 | 23 | 29.92 | 30.62 | 31.377 | 47.430 | 37.896 |

Mine Transition System | 99.467 | 97.626 | 98.595 | 98.781 | 98.636 | 99.596 | 98.181 | 97.89 |

**Table 5.**Property $P=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}\left[X\right(s=x\left)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.6471 | 0.6521 | 0.5545 | 0.6408 | 0.6476 | 0.5541 | 0.6471 | 0.6472 |

Discover using Decomposition | 0.6502 | 0.6476 | 0.6454 | 0.6447 | 0.6464 | 0.6487 | 0.6462 | 0.6490 |

Mine for Heuristic Net | 0.6494 | 0.6462 | 0.6524 | 0.6480 | 0.6456 | 0.6470 | 0.6486 | 0.6515 |

Mine Transition System | 0.6471 | 0.6299 | 0.6255 | 0.6478 | 0.5660 | 0.6055 | 0.6459 | 0.5516 |

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.105 | 0.086 | 0.05 | 0.21 | 0.086 | 0.092 | 0.105 | 0.093 |

Discover using Decomposition | 0.111 | 0.118 | 0.104 | 0.099 | 0.109 | 0.106 | 0.114 | 0.094 |

Mine for Heuristic Net | 0.157 | 0.264 | 0.153 | 0.271 | 0.135 | 0.148 | 0.158 | 0.132 |

Mine Transition System | 0.275 | 0.104 | 0.097 | 0.111 | 0.104 | 0.093 | 0.109 | 0.106 |

**Table 7.**Property $P=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}[\neg \phantom{\rule{4pt}{0ex}}(s=x)\phantom{\rule{4pt}{0ex}}U\le 5(s=y\left)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.013 | 0.063 | 0.031 | 0.039 | 0.052 | 0.010 | 0.017 | 0.014 |

Discover using Decomposition | 0.015 | 0.015 | 0.016 | 0.015 | 0.021 | 0.013 | 0.016 | 0.018 |

Mine for Heuristic Net | 0.130 | 0.158 | 0.031 | 0.120 | 0.134 | 0.136 | 0.201 | 0.104 |

Mine Transition System | 0.022 | 0.016 | 0.014 | 0.014 | 0.015 | 0.017 | 0.016 | 0.015 |

**Table 8.**Property ${R}_{cost}\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}\left[F(s=x)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 98.041 | 51.128 | 52.511 | 54.189 | 51.151 | 97.429 | 96.365 | 94.495 |

Discover using Decomposition | 97.740 | 99.061 | 98.222 | 97.266 | 97.708 | 98.691 | 99.396 | 99.029 |

Mine for Heuristic Net | 62.779 | 31.619 | 22.963 | 29.833 | 32.974 | 32.487 | 47.621 | 38.138 |

Mine Transition System | 99.435 | 98.210 | 98.915 | 97.469 | 97.295 | 97.486 | 97.155 | 98.469 |

**Table 9.**Property $P=\phantom{\rule{4pt}{0ex}}?\phantom{\rule{4pt}{0ex}}\left[X\right(s=x\left)\right]$.

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 0.651 | 0.635 | 0.555 | 0.642 | 0.651 | 0.651 | 0.645 | 0.643 |

Discover using Decomposition | 0.660 | 0.656 | 0.649 | 0.644 | 0.643 | 0.635 | 0.649 | 0.655 |

Mine for Heuristic Net | 0.649 | 0.649 | 0.655 | 0.654 | 0.648 | 0.660 | 0.646 | 0.647 |

Mine Transition System | 0.659 | 0.653 | 0.657 | 0.653 | 0.654 | 0.656 | 0.644 | 0.651 |

**Table 10.**Property $P\ge 0.5\phantom{\rule{4pt}{0ex}}[\neg \phantom{\rule{4pt}{0ex}}(s=x)\phantom{\rule{4pt}{0ex}}U\phantom{\rule{4pt}{0ex}}\u201cvisited\u201d]$; samples required = ?

Process Discovery Algorithm | Replay Algorithm 1 | Replay Algorithm 2 | Replay Algorithm 3 | Replay Algorithm 4 | Replay Algorithm 5 | Replay Algorithm 6 | Replay Algorithm 7 | Replay Algorithm 8 |
---|---|---|---|---|---|---|---|---|

Alpha Miner | 28 | 29 | 25 | 27 | 29 | 28 | 28 | 28 |

Discover using Decomposition | 28 | 30 | 27 | 30 | 29 | 27 | 29 | 29 |

Mine for Heuristic Net | 22 | 29 | 34 | 28 | 28 | 29 | 34 | 24 |

Mine Transition System | 25 | 31 | 29 | 27 | 29 | 26 | 28 | 26 |

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**MDPI and ACS Style**

Mangi, F.A.; Su, G.; Zhang, M.
Statistical Model Checking in Process Mining: A Comprehensive Approach to Analyse Stochastic Processes. *Future Internet* **2023**, *15*, 378.
https://doi.org/10.3390/fi15120378

**AMA Style**

Mangi FA, Su G, Zhang M.
Statistical Model Checking in Process Mining: A Comprehensive Approach to Analyse Stochastic Processes. *Future Internet*. 2023; 15(12):378.
https://doi.org/10.3390/fi15120378

**Chicago/Turabian Style**

Mangi, Fawad Ali, Guoxin Su, and Minjie Zhang.
2023. "Statistical Model Checking in Process Mining: A Comprehensive Approach to Analyse Stochastic Processes" *Future Internet* 15, no. 12: 378.
https://doi.org/10.3390/fi15120378