#
Structural Health Monitoring for Condition Assessment Using Efficient Supervised Learning Techniques^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Method

^{v×n}is the training set, that consists of $v$-dimensional features in the undamaged and damaged conditions, where $n$ represents the number of sensors mounted on the structure; then, the training set has $v$ AR coefficients or principal components at each sensor location. Next, if

**z**ϵ ℜ

^{v}represents the vector containing the classification labels for each element in the training data set, the classification methods have to classify the test data through the information extracted from the training and class label sets.

#### 2.1. Feature Selection

**θ**= [θ

_{1},θ

_{2},…,θ

_{a}] is the vector of AR coefficients; $a$ is the model order; e(t) is an uncorrelated residual sequence used to quantify the difference between the measured and predicted responses.

**T**ϵ ℜ

^{v}

^{×k}using an additional matrix

**P**ϵ ℜ

^{n}

^{×k}in the following form:

#### 2.2. Classification Methods

**x**from the training set $X$, with mean

**μ**

_{k}and covariance

**Σ**

_{k}, is given by:

**Σ**

_{k}=

**Σ**, and the classification rule is based on a linear score function that is given by:

_{k}is the probability that a randomly selected observation falls in the $k$-th class.

**μ**

_{k},

**Σ**

_{k}, and p

_{k}are again obtained from the training data set.

**z**, DT recursively partitions the space such that the samples with the same labels are grouped together. In order to train the classifier, it becomes essential to specify the number of branch nodes (decision splits), the minimum number of branch and leaf node observations, and the prior probabilities for each class. A full discussion of the classification decision tree is beyond of the scope of this article, and readers are referred to [30] for further details.

## 3. Results and Discussion

**θ**of the AR model.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Capellari, G.; Eftekhar Azam, S.; Mariani, S. Damage detection in flexible plates through reduced-order modeling and hybrid particle-Kalman filtering. Sensors
**2016**, 16, 2. [Google Scholar] [CrossRef] [PubMed] - Eftekhar Azam, S.; Mariani, S.; Attari, N. Online damage detection via a synergy of proper orthogonal decomposition and recursive Bayesian filters. Nonlinear Dyn.
**2017**, 89, 1489–1511. [Google Scholar] [CrossRef] - Eftekhar Azam, S.; Mariani, S. Online damage detection in structural systems via dynamic inverse analysis: A recursive Bayesian approach. Eng. Struct.
**2018**, 159, 28–45. [Google Scholar] [CrossRef] - Sarmadi, H.; Karamodin, A.; Entezami, A. A new iterative model updating technique based on least squares minimal residual method using measured modal data. Appl. Math. Model.
**2016**, 40, 10323–10341. [Google Scholar] [CrossRef] - Farrar, C.R.; Worden, K. Structural Health Monitoring: A Machine Learning Perspective; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 2013. [Google Scholar]
- Entezami, A.; Shariatmadar, H.; Sarmadi, H. Structural damage detection by a new iterative regularization method and an improved sensitivity function. J. Sound Vib.
**2017**, 399, 285–307. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H. An unsupervised learning approach by novel damage indices in structural health monitoring for damage localization and quantification. Struct. Health Monit.
**2017**, 17, 325–345. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H. Structural health monitoring by a new hybrid feature extraction and dynamic time warping methods under ambient vibration and non-stationary signals. Measurement
**2019**, 134, 548–568. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H. Damage localization under ambient excitations and non-stationary vibration signals by a new hybrid algorithm for feature extraction and multivariate distance correlation methods. Struct. Health Monit.
**2019**, 18, 347–375. [Google Scholar] [CrossRef] - Sohn, H.; Czarnecki, J.A.; Farrar, C.R. Structural Health Monitoring Using Statistical Process Control. J. Struct. Eng.
**2000**, 126, 1356–1363. [Google Scholar] [CrossRef] - Fugate, M.L.; Sohn, H.; Farrar, C.R. Vibration-based damage detection using statistical process control. Mech. Syst. Signal Process.
**2001**, 15, 707–721. [Google Scholar] [CrossRef] - Sophian, A.; Tian, G.Y.; Taylor, D.; Rudlin, J. A feature extraction technique based on principal component analysis for pulsed Eddy current NDT. NDT Int.
**2003**, 36, 37–41. [Google Scholar] [CrossRef] - Zhong, A.; Song, H.; Han, B. Extracting structural damage features: Comparison between PCA and ICA. In Intelligent Computing in Signal Processing and Pattern Recognition, Lectures Notes in Control and Informatic; Springer: Berlin/Heidelberg, Germany, 2006; Volume 345, pp. 840–845. [Google Scholar] [CrossRef]
- Entezami, A.; Shariatmadar, H.; Karamodin, A. Data-driven damage diagnosis under environmental and operational variability by novel statistical pattern recognition methods. Struct. Health Monit.
**2018**. [Google Scholar] [CrossRef] - Rezaiee-Pajand, M.; Entezami, A.; Shariatmadar, H. An iterative order determination method for time-series modeling in structural health monitoring. Adv. Struct. Eng.
**2017**, 21, 300–314. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H.; Karamodin, A. An improvement on feature extraction via time series modeling for structural health monitoring based on unsupervised learning methods. Sci. Iran
**2018**. [Google Scholar] [CrossRef] - Sohn, H.; Worden, K.; Farrar, C.R. Statistical damage classification under changing environmental and operational conditions. J. Intell. Mater. Syst. Struct.
**2002**, 13, 561–574. [Google Scholar] [CrossRef] - Niu, G.; Son, J.-D.; Widodo, A.; Yang, B.-S.; Hwang, D.-H.; Kang, D.-S. A comparison of classifier performance for fault diagnosis of induction motor using multi-type signals. Struct. Health Monit.
**2007**, 6, 215–229. [Google Scholar] [CrossRef] - Gaudenzi, P.; Nardi, D.; Chiappetta, I.; Atek, S.; Lampani, L.; Pasquali, M.; Sarasini, F.; Tirilló, J.; Valente, T. Sparse sensing detection of impact-induced delaminations in composite laminates. Compos. Struct.
**2015**, 133, 1209–1219. [Google Scholar] [CrossRef] - Addin, O.; Sapuan, S.; Mahdi, E.; Othman, M. A Naïve-Bayes classifier for damage detection in engineering materials. Mater. Des.
**2007**, 28, 2379–2386. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H.; Mariani, S. Fast unsupervised learning methods for structural health monitoring with large vibration data from dense sensor networks. Struct. Health Monit.
**2019**. [Google Scholar] [CrossRef] - Entezami, A.; Shariatmadar, H.; Mariani, S. A novelty detection method for large-scale structures under varying environmental conditions. In Proceedings of the Sixteenth International Conference on Civil, Structural and Environmental Engineering Computing (Civil-Comp 2019), Riva del Garda, Italy, 16–19 September 2019. [Google Scholar]
- Box, G.E.; Jenkins, G.M.; Reinsel, G.C. Time Series Analysis: Forecasting and Control, 4th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2008. [Google Scholar]
- Mujica, L.E.; Vehí, J.; Ruiz, M.; Verleysen, M.; Staszewski, W.; Worden, K. Multivariate statistics process control for dimensionality reduction in structural assessment. Mech. Syst. Signal Process.
**2008**, 22, 155–171. [Google Scholar] [CrossRef] - Tibaduiza, D.A.; Mujica, L.E.; Rodellar, J.; Güemes, A. Structural damage detection using principal component analysis and damage indices. J. Intell. Mater. Syst. Struct.
**2015**. [Google Scholar] [CrossRef] - Fisher, R.A. The use of multiple measurements in taxonomic problems. Ann. Eugen.
**1936**, 7, 179–188. [Google Scholar] [CrossRef] - McLachlan, G.J. Discriminant Analysis and Statistical Pattern Recognition; Wiley: Hoboken, NJ, USA, 2004. [Google Scholar]
- Alpaydin, E. Introduction to Machine Learning; MIT Press: Cambridge, MA, USA, 2014. [Google Scholar]
- Coppersmith, D.; Hong, S.J.; Hosking, J.R. Partitioning nominal attributes in decision trees. Data Min. Knowl. Discov.
**1999**, 3, 197–217. [Google Scholar] [CrossRef] - Breiman, L.; Friedman, J.; Stone, C.J.; Olshen, R.A. Classification and Regression Trees; CRC Press: Boca Raton, FL, USA, 1984. [Google Scholar]
- Kullaa, J.; Santaoja, K.; Eymery, A. Vibration-based structural health monitoring of a simulated beam with a breathing crack. Key Eng. Mater.
**2013**, 569, 1093–1100. [Google Scholar] [CrossRef] - Yuan, G.-X.; Ho, C.-H.; Lin, C.-J. Recent advances of large-scale linear classification. Proc. IEEE
**2012**, 100, 2584–2603. [Google Scholar] [CrossRef]

**Figure 1.**The numerical benchmark concrete model [31].

**Figure 2.**Comparison among the considered classification methods, using both AR and PCA feature selection approaches: (

**a**) classification accuracy, (

**b**) computing time.

Case | Structural State | Description |
---|---|---|

1 | Undamaged | No crack |

2 | Damaged | Crack length = 10 mm |

3 | Damaged | Crack length = 20 mm |

4 | Damaged | Crack length = 30 mm |

5 | Damaged | Crack length = 50 mm |

6 | Damaged | Crack length = 100 mm |

7 | Damaged | Crack length = 150 mm |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2019 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 (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Entezami, A.; Shariatmadar, H.; Mariani, S.
Structural Health Monitoring for Condition Assessment Using Efficient Supervised Learning Techniques. *Proceedings* **2020**, *42*, 17.
https://doi.org/10.3390/ecsa-6-06538

**AMA Style**

Entezami A, Shariatmadar H, Mariani S.
Structural Health Monitoring for Condition Assessment Using Efficient Supervised Learning Techniques. *Proceedings*. 2020; 42(1):17.
https://doi.org/10.3390/ecsa-6-06538

**Chicago/Turabian Style**

Entezami, Alireza, Hashem Shariatmadar, and Stefano Mariani.
2020. "Structural Health Monitoring for Condition Assessment Using Efficient Supervised Learning Techniques" *Proceedings* 42, no. 1: 17.
https://doi.org/10.3390/ecsa-6-06538