# Change Point Enhanced Anomaly Detection for IoT Time Series Data

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## Abstract

**:**

## 1. Introduction

- $\left({Q}_{1}\right)$
- Can we minimize the water consumption and losses using a rule-based decision system that alerts users in real-time?
- $\left({Q}_{2}\right)$
- Can we enhance anomaly detection techniques using change point detection?
- $\left({Q}_{3}\right)$
- Can we remove the false positives that anomaly detection techniques find using change point detection?
- $\left({Q}_{4}\right)$
- Can we enable predictive and proactive maintenance using a rule-based decision support system for water management?

- (C1)
- A new rule-based decision system for anomaly detection in IoT time series in order to answer $\left({Q}_{1}\right)$;
- (C2)
- A new confidence metric based on the support for a point to be an anomaly and the support for a point to be a change point to remove false positives in order to answer $\left({Q}_{2}\right)$;
- (C3)
- A new pipeline that automatically manages to detect real anomalies and remove the false positives introduced by change point using the confidence score in order to answer $\left({Q}_{3}\right)$;
- (C4)
- Extensive experiments on real-world multivariate time series using five anomaly detection and five change point detection algorithms in order to enable predictive and proactive maintenance and answer $\left({Q}_{4}\right)$.

## 2. Related Work

## 3. Methodology

#### 3.1. Time Series Definition

- (i)
- ${m}_{t}$: the trend component represents variations of low frequency and can be determined by the moving averages or spectral smoothing methods;
- (ii)
- ${s}_{t}$: the seasonal component is a function that represents normal fluctuations that are more or less stable after a known period (or lag) h;
- (iii)
- ${y}_{t}$: the noise (residual) are used to check if a analysis model has correctly determined the information in the data points and can help to predict future values [31].

#### 3.2. Time Series Outlier Detection

#### 3.2.1. Gaussian Distribution

#### 3.2.2. K-Means

#### 3.2.3. Isolation Forest

#### 3.2.4. One-Class Support Vector Machine

#### 3.2.5. Autoencoders

#### 3.3. Time Series Change Point Detection

#### 3.3.1. Window-Based Segmentation Model

#### 3.3.2. Binary Segmentation Model

#### 3.3.3. Bottom-up Segmentation Model

#### 3.3.4. Pruned Exact Linear Time Model

#### 3.3.5. Exact Segmentation Dynamic Programming Model

## 4. Proposed Solution

#### 4.1. Anomaly Detection Module

#### 4.2. Change Point Detection Module

#### 4.3. Change Point Enhanced Anomaly Detection Module

#### 4.4. Decision Module

- ${A}_{1}$
- When the support for anomaly detection is high, and there is no or low support for change point detection, then the system automatically marks the point as an anomaly.
- ${A}_{2}$
- When the support for change point detection is high, and there is no or low support for anomaly detection, the system marks the point as a change point and not an anomaly.
- ${A}_{3}$
- When the support for change point detection is low, and there is no support for anomaly detection, the system marks the point as a change point and not an anomaly.

- ${H}_{1}$
- When the anomaly support is low and no change point is detected or the change point support is low, regardless if $suppor{t}_{AD}\left(x\right)>suppor{t}_{CPD}\left(x\right)$ or $suppor{t}_{AD}\left(x\right)<suppor{t}_{CPD}\left(x\right)$.
- ${H}_{2}$
- When both the anomaly support and change point support are high, regardless if $suppor{t}_{AD}\left(x\right)>suppor{t}_{CPD}\left(x\right)$ or $suppor{t}_{AD}\left(x\right)<suppor{t}_{CPD}\left(x\right)$.
- ${H}_{3}$
- When the anomaly support is equal to the change point support, regardless if both are high or low.

## 5. Experimental Results

#### 5.1. Dataset

#### 5.2. Results

#### 5.2.1. Evaluation

#### 5.2.2. Anomaly Detection

#### 5.2.3. Change Point Detection

#### 5.2.4. Change Point Enhanced Anomaly Detection

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

IoT | Internet of Things |

GD | Gaussian Distribution |

OC-SVM | One-Class Support Vector Machine |

IF | Isolation Forest |

AE | Autoencoder |

WinSeg | Window Segmentation Model |

BinSeg | Binary Segmentation Model |

BottomUp | Bottom-Up Segmentation Model |

PELT | Pruned Exact Linear Time Model |

OPT | Exact Segmentation Dynamic Programming Model |

MAE | Mean Absolute Value |

CNN | Convolutional Neural Networks |

LSTM | Long Short-Term Neural Networks |

ARIMA | Auto Regressive Integrated Moving Average |

## References

- Sahal, R.; Breslin, J.G.; Ali, M.I. Big data and stream processing platforms for Industry 4.0 requirements mapping for a predictive maintenance use case. J. Manuf. Syst.
**2020**, 54, 138–151. [Google Scholar] [CrossRef] - Obaidat, S.; Liao, H. Integrated decision making for attributes sampling and proactive maintenance in a discrete manufacturing system. Int. J. Prod. Res.
**2020**, 1–23. [Google Scholar] [CrossRef] - Antzoulatos, G.; Mourtzios, C.; Stournara, P.; Kouloglou, I.O.; Papadimitriou, N.; Spyrou, D.; Mentes, A.; Nikolaidis, E.; Karakostas, A.; Kourtesis, D.; et al. Making urban water smart: the SMART-WATER solution. Water Sci. Technol.
**2020**, 82, 2691–2710. [Google Scholar] [CrossRef] - Liu, J.; Wang, P.; Jiang, D.; Nan, J.; Zhu, W. An integrated data-driven framework for surface water quality anomaly detection and early warning. J. Clean. Prod.
**2020**, 251, 119145. [Google Scholar] [CrossRef] - Gonzalez-Vidal, A.; Cuenca-Jara, J.; Skarmeta, A.F. IoT for Water Management: Towards Intelligent Anomaly Detection. In Proceedings of the 2019 IEEE 5th World Forum on Internet of Things (WF-IoT), Limerick, Ireland, 15–18 April 2019; pp. 858–863. [Google Scholar] [CrossRef]
- Fahim, M.; Sillitti, A. Anomaly Detection, Analysis and Prediction Techniques in IoT Environment: A Systematic Literature Review. IEEE Access
**2019**, 7, 81664–81681. [Google Scholar] [CrossRef] - Moleda, M.; Momot, A.; Mrozek, D. Predictive Maintenance of Boiler Feed Water Pumps Using SCADA Data. Sensors
**2020**, 20, 571. [Google Scholar] [CrossRef][Green Version] - Cristea, V.; Mocanu, M.; Anton, S.; Apostol, E.; Dobre, C.; Leordeanu, C.; Pop, F. Insights and Views in Smart Data and e-Services for Water Management; Politehnica Press: Bucharest, Romania, 2018; pp. 1–97. [Google Scholar]
- Kieu, T.; Yang, B.; Guo, C.; Jensen, C.S. Outlier Detection for Time Series with Recurrent Autoencoder Ensembles. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, Macao, China, 10–16 August 2019; pp. 2725–2732. [Google Scholar] [CrossRef][Green Version]
- Vishwakarma, G.K.; Paul, C.; Elsawah, A. An algorithm for outlier detection in a time series model using backpropagation neural network. J. King Saud Univ. Sci.
**2020**, 32, 3328–3336. [Google Scholar] [CrossRef] - Li, J.; Pedrycz, W.; Jamal, I. Multivariate time series anomaly detection: A framework of Hidden Markov Models. Appl. Soft Comput.
**2017**, 60, 229–240. [Google Scholar] [CrossRef] - Carreño, A.; Inza, I.; Lozano, J.A. Analyzing rare event, anomaly, novelty and outlier detection terms under the supervised classification framework. Artif. Intell. Rev.
**2019**, 53, 3575–3594. [Google Scholar] [CrossRef][Green Version] - Laptev, N.; Amizadeh, S.; Flint, I. Generic and Scalable Framework for Automated Time-series Anomaly Detection. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Sydney, Australia, 10–13 August 2015; pp. 1939–1947. [Google Scholar] [CrossRef]
- Cheng, Y.; Xu, Y.; Zhong, H.; Liu, Y. HS-TCN: A Semi-supervised Hierarchical Stacking Temporal Convolutional Network for Anomaly Detection in IoT. In Proceedings of the 2019 IEEE 38th International Performance Computing and Communications Conference (IPCCC), London, UK, 29–31 October 2019; pp. 1–7. [Google Scholar] [CrossRef]
- Sarvari, H.; Domeniconi, C.; Prenkaj, B.; Stilo, G. Unsupervised Boosting-Based Autoencoder Ensembles for Outlier Detection. In Advances in Knowledge Discovery and Data Mining; Lecture Notes in Computer Science; Springer: Berlin/Heidelberg, Germany, 2021; pp. 91–103. [Google Scholar] [CrossRef]
- Ahmed, M.; Mahmood, A.N.; Islam, M.R. A survey of anomaly detection techniques in financial domain. Future Gener. Comput. Syst.
**2016**, 55, 278–288. [Google Scholar] [CrossRef] - Guo, T.; Xu, Z.; Yao, X.; Chen, H.; Aberer, K.; Funaya, K. Robust Online Time Series Prediction with Recurrent Neural Networks. In Proceedings of the 2016 IEEE International Conference on Data Science and Advanced Analytics (DSAA), Montreal, QC, Canada, 17–19 October 2016; pp. 816–825. [Google Scholar] [CrossRef][Green Version]
- Ebrahimzadeh, Z.; Zheng, M.; Karakas, S.; Kleinberg, S. Deep Learning for Multi-Scale Changepoint Detection in Multivariate Time Series. arXiv
**2019**, arXiv:1905.06913. [Google Scholar] - Saurav, S.; Malhotra, P.; TV, V.; Gugulothu, N.; Vig, L.; Agarwal, P.; Shroff, G. Online anomaly detection with concept drift adaptation using recurrent neural networks. In Proceedings of the ACM India Joint International Conference on Data Science and Management of Data, Goa, India, 11–13 January 2018; pp. 78–87. [Google Scholar] [CrossRef]
- Maleki, S.; Maleki, S.; Jennings, N.R. Unsupervised anomaly detection with LSTM autoencoders using statistical data-filtering. Appl. Soft Comput.
**2021**, 108, 107443. [Google Scholar] [CrossRef] - Ribeiro, A.H.; Tiels, K.; Aguirre, L.A.; Schön, T. Beyond exploding and vanishing gradients: Analysing RNN training using attractors and smoothness. In International Conference on Artificial Intelligence and Statistics; PMLR: Boston, MA, USA, 2020; pp. 2370–2380. [Google Scholar]
- Hochreiter, S.; Schmidhuber, J. Long Short-Term Memory. Neural Comput.
**1997**, 9, 1735–1780. [Google Scholar] [CrossRef] - Munir, M.; Siddiqui, S.A.; Dengel, A.; Ahmed, S. DeepAnT: A Deep Learning Approach for Unsupervised Anomaly Detection in Time Series. IEEE Access
**2018**, 7, 1991–2005. [Google Scholar] [CrossRef] - Kieu, T.; Yang, B.; Jensen, C.S. Outlier Detection for Multidimensional Time Series Using Deep Neural Networks. In Proceedings of the 2018 19th IEEE International Conference on Mobile Data Management (MDM), Aalborg, Denmark, 25–28 June 2018; pp. 125–134. [Google Scholar] [CrossRef]
- Zhang, M.; Li, X.; Wang, L. An Adaptive Outlier Detection and Processing Approach Towards Time Series Sensor Data. IEEE Access
**2019**, 7, 175192–175212. [Google Scholar] [CrossRef] - Kant, N.; Mahajan, M. Time-Series Outlier Detection Using Enhanced K-Means in Combination with PSO Algorithm. In Engineering Vibration, Communication and Information Processing; Springer: Singapore, 2018; pp. 363–373. [Google Scholar] [CrossRef]
- Feremans, L.; Vercruyssen, V.; Cule, B.; Meert, W.; Goethals, B. Pattern-based anomaly detection in mixed-type time series. In Machine Learning and Knowledge Discovery in Databases; Springer: Berlin/Heidelberg, Germany, 2019; pp. 240–256. [Google Scholar] [CrossRef]
- Yeh, C.C.M.; Zhu, Y.; Ulanova, L.; Begum, N.; Ding, Y.; Dau, H.A.; Silva, D.F.; Mueen, A.; Keogh, E. Matrix Profile I: All Pairs Similarity Joins for Time Series: A Unifying View That Includes Motifs, Discords and Shapelets. In Proceedings of the 2016 IEEE 16th International Conference on Data Mining (ICDM), Barcelona, Spain, 12–15 December 2016; pp. 1317–1322. [Google Scholar] [CrossRef]
- Brockwell, P.J.; Davis, R.A. Time Series: Theory and Methods; Springer: Berlin/Heidelberg, Germany, 1991. [Google Scholar] [CrossRef]
- Shumway, R.H.; Stoffer, D.S. Time Series Analysis and Its Applications: With R Examples; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef]
- Brockwell, P.J.; Davis, R.A. Introduction to Time Series and Forecasting; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar] [CrossRef]
- Gupta, M.; Gao, J.; Aggarwal, C.C.; Han, J. Outlier Detection for Temporal Data: A Survey. IEEE Trans. Knowl. Data Eng.
**2014**, 26, 2250–2267. [Google Scholar] [CrossRef] - Cheng, Z.; Zou, C.; Dong, J. Outlier detection using isolation forest and local outlier factor. In Proceedings of the Conference on Research in Adaptive and Convergent Systems, Chongqing, China, 24–27 September 2019; pp. 161–168. [Google Scholar] [CrossRef]
- Jin, B.; Chen, Y.; Li, D.; Poolla, K.; Sangiovanni-Vincentelli, A. A One-Class Support Vector Machine Calibration Method for Time Series Change Point Detection. In Proceedings of the 2019 IEEE International Conference on Prognostics and Health Management (ICPHM), San Francisco, CA, USA, 17–20 June 2019; pp. 1–5. [Google Scholar] [CrossRef][Green Version]
- Aminikhanghahi, S.; Cook, D.J. A survey of methods for time series change point detection. Knowl. Inf. Syst.
**2016**, 51, 339–367. [Google Scholar] [CrossRef][Green Version] - Górecki, T.; Horváth, L.; Kokoszka, P. Change point detection in heteroscedastic time series. Econom. Stat.
**2018**, 7, 63–88. [Google Scholar] [CrossRef][Green Version] - Qu, Z.; Perron, P. Estimating and Testing Structural Changes in Multivariate Regressions. Econometrica
**2007**, 75, 459–502. [Google Scholar] [CrossRef] - Han, C.; Taamouti, A. Partial Structural Break Identification. Oxf. Bull. Econ. Stat.
**2017**, 79, 145–164. [Google Scholar] [CrossRef] - Davis, J.V.; Kulis, B.; Jain, P.; Sra, S.; Dhillon, I.S. Information-theoretic metric learning. In Proceedings of the 24th International Conference on Machine Learning—ICML ’07, Corvalis, OR, USA, 20–24 June 2007; pp. 209–216. [Google Scholar] [CrossRef]
- Zou, C.; Yin, G.; Feng, L.; Wang, Z. Nonparametric maximum likelihood approach to multiple change-point problems. Ann. Stat.
**2014**, 42, 970–1002. [Google Scholar] [CrossRef][Green Version] - Lung-Yut-Fong, A.; Lévy-Leduc, C.; Cappé, O. Homogeneity and change-point detection tests for multivariate data using rank statistics. J. Société Française Stat.
**2015**, 156, 133–162. [Google Scholar] - Harchaoui, Z.; Cappe, O. Retrospective Mutiple Change-Point Estimation with Kernels. In Proceedings of the 2007 IEEE/SP 14th Workshop on Statistical Signal Processing, Madison, WI, USA, 26–29 August 2007; pp. 768–772. [Google Scholar] [CrossRef]
- Kifer, D.; Ben-David, S.; Gehrke, J. Detecting Change in Data Streams. In Proceedings of the 30th International Conference on Very Large Data Bases, Toronto, ON, Canada, 29 August–3 September 2004; pp. 180–191. [Google Scholar]
- Liu, S.; Wright, A.; Hauskrecht, M. Change-point detection method for clinical decision support system rule monitoring. Artif. Intell. Med.
**2018**, 91, 49–56. [Google Scholar] [CrossRef][Green Version] - Bai, J. Estimating Multiple Breaks One at a Time. Econom. Theory
**1997**, 13, 315–352. [Google Scholar] [CrossRef][Green Version] - Fryzlewicz, P. Wild binary segmentation for multiple change-point detection. Ann. Stat.
**2014**, 42, 2243–2281. [Google Scholar] [CrossRef] - Keogh, E.; Chu, S.; Hart, D.; Pazzani, M. An online algorithm for segmenting time series. In Proceedings of the 2001 IEEE International Conference on Data Mining, San Jose, CA, USA, 29 November–2 December 2001; pp. 289–296. [Google Scholar] [CrossRef]
- Fryzlewicz, P. Unbalanced Haar Technique for Nonparametric Function Estimation. J. Am. Stat. Assoc.
**2007**, 102, 1318–1327. [Google Scholar] [CrossRef][Green Version] - Killick, R.; Fearnhead, P.; Eckley, I.A. Optimal Detection of Changepoints With a Linear Computational Cost. J. Am. Stat. Assoc.
**2012**, 107, 1590–1598. [Google Scholar] [CrossRef] - Rigaill, G. A pruned dynamic programming algorithm to recover the best segmentations with 1 to K_max change-points. J. Société Française Stat.
**2015**, 156, 180–205. [Google Scholar] - Assent, I. Clustering high dimensional data. WIREs Data Min. Knowl. Discov.
**2012**, 2, 340–350. [Google Scholar] [CrossRef] - Cakir, M.; Guvenc, M.A.; Mistikoglu, S. The experimental application of popular machine learning algorithms on predictive maintenance and the design of IIoT based condition monitoring system. Comput. Ind. Eng.
**2021**, 151, 106948. [Google Scholar] [CrossRef]

## Short Biography of Authors

Elena-Simona Apostol is an Associate Professor of Computer Science at the Computer Science and Engineering Department, Faculty of Automatic Control and Computers, University Politehnica of Bucharest (UPB). She received her Ph.D. (2014) from University Politehnica of Bucharest, Romania. She was a postdoctoral researcher at Microsoft Research Center in Paris in collaboration with INRIA (The French Institute for Research in Computer Science and Automation) where she worked on state of the art Big Data Analysis, Multi-Site Cloud Computing, and Bioinformatics. She was an invited researcher during her Ph.D. studies at INRIA Rennes, France, working within the joint research team between KerData at INRIA and UPB on Big Data management and analytics. During her bachelor and master studies, she was an intern and junior research engineer at the Fraunhofer FOKUS Institute, Berlin, Germany where she worked on Computer Networking and Telecommunications with a focus on mobile and service orientated architectures. Her research focuses on Big Data, Data Management, Parallel and Distributed algorithms, Machine Learning, and Data Science. | |

Ciprian-Octavian Truică is an Assistant Professor of Computer Science at the Computer Science and Engineering Department, Faculty of Automatic Control and Computers, University Politehnica of Bucharest, Romania. He received his Ph.D. (2018) in Data Management and Text Mining from University Politehnica of Bucharest. He holds an M.Sc. Degree (2013) in Computer Science Engineering and Information Technology from University Politehnica of Bucharest, a B.Sc. degree (2013) in Computer Science and Mathematics from the University of Bucharest, and a B.Sc. (2011) degree in Computer Science and Electrical Engineering from University Politehnica of Bucharest. He was a postdoctoral researcher (2019–2020) in the Data-Intensive Systems group at the Department of Computer Science, Aarhus University, Aarhus, Denmark where he worked Big Data Analytics for time series. During his Ph.D. studies, he was an invited researcher (2015–2016) at the ERIC laboratory, Université de Lyon, France where he worked on Data Management, Machine Learning, and Natural Language Processing. His research interests mainly relate to Big Data, Data Management, Machine Learning, Text Mining, Natural Language Processing, and Time Series Analysis. | |

Florin Pop is an Professor of Computer Science at the Computer Science and Engineering Department, Faculty of Automatic Control and Computers, University Politehnica of Bucharest (UPB). He received his PhD (2008) on Distributed Scheduling techniques from UPB. He has authored or coauthored more than 150 publications (books, chapters, and papers in international journals and well-established and ranked conferences). His research interests include scheduling and resource management, multi-criteria optimization methods, grid middleware tools, application development, prediction methods, self-organizing systems, and contextualized services in distributed systems. He is a Senior Member of ACM and euroCRIS. In 2012, he received the IBM Faculty Award for the project CloudWay (Improving Resource Utilization for a Smart Cloud Infrastructure). He was awarded the Prize for Excellence from IBM (2008) and Oracle (2009). In 2011 he received the Best Young Researcher in Software Services Award from the FP7 SPRERS Project (Strengthening the Participation of Romania in European Research and Development in Software Services) and two best paper awards. He was involved either as a coordinator or member in several international (i.e., EGEE III, SEE-GRID-SCI, ERRIC, and Data4Water) and national research projects in the distributed systems field. | |

Christian Esposito received the Ph.D. degree in computer engineering and automation from the University of Napoli Federico II, in 2009. He is currently a tenured Assistant Professor with the University of Salerno, and was a non-tenured Assistant Professor with the University of Napoli Federico II, and a Research Fellow with the University of Salerno and the Institute for High Performance Computing and Networking, The National Research Council (ICAR-CNR). He has been involved in the organization of about 40 international conferences workshops. His research interests include reliable and secure communications, middleware, distributed systems, positioning systems, multi-objective optimization, and game theory. He has served as a Reviewer and the Guest Editor for several international journals and conferences (with about 200 completed reviews). He is also an Associate Editor of IEEE ACCESS. |

**Figure 3.**Change Point Detection (Note the line is the time series and the dot is the change point).

Case | Rule | Decision |
---|---|---|

${A}_{1}$ | $suppor{t}_{AD}\left(x\right)\to 1\wedge (suppor{t}_{CPD}\left(x\right)=0\vee suppor{t}_{CPD}\left(x\right)\to 0)\Rightarrow confidence\left(x\right)\to 1$ | Automatic response |

${A}_{2}$ | $(suppor{t}_{AD}\left(x\right)=0\vee suppor{t}_{AD}\left(x\right)\to 0)\wedge suppor{t}_{CPD}\left(x\right)\to 1\Rightarrow confidence\left(x\right)\to 0$ | Automatic response |

${A}_{3}$ | $suppor{t}_{AD}\left(x\right)=0\wedge suppor{t}_{CPD}\left(x\right)\to 0\Rightarrow confidence\left(x\right)=0$ | Automatic response |

${H}_{1}$ | $suppor{t}_{AD}\left(x\right)\to 0\wedge (suppor{t}_{CPD}\left(x\right)=0\vee suppor{t}_{CPD}\left(x\right)\to 0)\Rightarrow confidence\left(x\right)\to 0.5$ | Human intervention |

${H}_{2}$ | $suppor{t}_{AD}\left(x\right)\to 1\wedge suppor{t}_{CPD}\left(x\right)=1\Rightarrow confidence\left(x\right)\to 0.5$ | Human intervention |

${H}_{3}$ | $suppor{t}_{AD}\left(x\right)=suppor{t}_{CPD}\left(x\right)\Rightarrow confidence\left(x\right)=0.5$ | Human intervention |

Anomaly Detection | Change Point Detection | ||
---|---|---|---|

Algorithm | MAE | Algorithm | MAE |

GD | 0.027 ± 0.001 | WinSeg | 0.017 ± 0.001 |

K-Means | 0.031 ± 0.004 | BinSeg | 0.092 ± 0.003 |

OC-SVM | 0.018 ± 0.001 | BottomUp | 0.092 ± 0.002 |

IF | 0.024 ± 0.003 | PELT | 0.095 ± 0.005 |

AE | 0.163 ± 0.010 | OPT | 0.095 ± 0.002 |

Anomaly Detection | Change Point Detection | Anomaly Confidence | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

IDX | GD | K-Means | OC-SVM | IF | AE | Support | WinSeg | BinSeg | BottomUp | PELT | OPT | Support | |

110 | 0.00 | ✓ | ✓ | ✓ | ✓ | 0.80 | 0.00 | ||||||

112 | ✓ | 0.20 | 0.00 | 1.00 | |||||||||

375 | ✓ | 0.20 | ✓ | ✓ | ✓ | 0.60 | 0.25 | ||||||

410 | ✓ | 0.20 | ✓ | ✓ | 0.40 | 0.33 | |||||||

424 | ✓ | ✓ | ✓ | 0.60 | 0.00 | 1.00 | |||||||

520 | 0.00 | ✓ | ✓ | ✓ | ✓ | ✓ | 1.00 | 0.00 | |||||

885 | ✓ | ✓ | ✓ | ✓ | 0.80 | ✓ | ✓ | ✓ | ✓ | 0.80 | 0.50 | ||

891 | ✓ | ✓ | ✓ | ✓ | ✓ | 1.00 | 0.00 | 1.00 | |||||

895 | ✓ | 0.20 | ✓ | ✓ | ✓ | ✓ | 0.80 | 0.20 | |||||

925 | ✓ | 0.20 | ✓ | ✓ | ✓ | 0.60 | 0.25 |

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## Share and Cite

**MDPI and ACS Style**

Apostol, E.-S.; Truică, C.-O.; Pop, F.; Esposito, C.
Change Point Enhanced Anomaly Detection for IoT Time Series Data. *Water* **2021**, *13*, 1633.
https://doi.org/10.3390/w13121633

**AMA Style**

Apostol E-S, Truică C-O, Pop F, Esposito C.
Change Point Enhanced Anomaly Detection for IoT Time Series Data. *Water*. 2021; 13(12):1633.
https://doi.org/10.3390/w13121633

**Chicago/Turabian Style**

Apostol, Elena-Simona, Ciprian-Octavian Truică, Florin Pop, and Christian Esposito.
2021. "Change Point Enhanced Anomaly Detection for IoT Time Series Data" *Water* 13, no. 12: 1633.
https://doi.org/10.3390/w13121633