# Damage Identification of Turnout Rail through a Covariance-Based Condition Index and Quantitative Pattern Analysis

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Formulation of Condition Index

^{th}period. The response matrix $C{A}_{n}$ containing covariance-based and amplitude-based segments can be expressed as:

#### 2.2. Construction of Reference Models for Damage Detection

**V**. Sparse Bayesian regression is adopted for diminishing the overfitting problem, and $f\left(x,{x}_{V}\right)$ is a combination of kernel functions, as shown in Equation (5).

^{th}set of damage-sensitive interrelationships ($r=1\dots R$), the residual between the current observed interrelationship and the predicted inter-relationship is formulated as Equation (7).

## 3. Case Study

## 4. Discussions

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**A typical response segment and its components after wavelet packet analysis. (

**a**) A segment of a typical response. (

**b**) The four components of a typical response.

**Figure 4.**The comparison between observed and predicted historical patterns for model validation (the first scenario). (

**a**) The relationship between covariance-based member 1 and amplitude-based member 2 in the historical CI matrix. (

**b**) The relationship between covariance-based member 4 and amplitude-based member 2 in the historical CI matrix.

**Figure 5.**The comparison between observed and predicted new patterns for anomaly detection (the second scenario). (

**a**) The relationship between covariance-based member 1 and amplitude-based member 2 in the new CI matrix. (

**b**) The relationship between covariance-based member 4 and amplitude-based member 2 in the new CI matrix.

**Figure 6.**The comparison between observed and predicted current patterns for anomaly detection (the third scenario). (

**a**) The relationship between covariance-based member 1 and amplitude-based member 2 in the current CI matrix. (

**b**) The relationship between covariance-based member 4 and amplitude-based member 2 in the current CI matrix.

Bayes Factor (Probability of Damage) for the Interrelationship 1 | Bayes Factor (Probability of Damage) for the Interrelationship 2 | Synthetic Bayes Factor (Synthetic Probability) | |
---|---|---|---|

Scenario 2 | 0.1 (1.1%) | 0.6 (5.3%) | 0.3 (2.9%) |

Scenario 3 | 10.5 (50.9%) | 40.4 (78.6%) | 26.2 (65.4%) |

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**MDPI and ACS Style**

Wang, J.-F.; Lin, J.-F.; Xie, Y.-L.
Damage Identification of Turnout Rail through a Covariance-Based Condition Index and Quantitative Pattern Analysis. *Infrastructures* **2023**, *8*, 176.
https://doi.org/10.3390/infrastructures8120176

**AMA Style**

Wang J-F, Lin J-F, Xie Y-L.
Damage Identification of Turnout Rail through a Covariance-Based Condition Index and Quantitative Pattern Analysis. *Infrastructures*. 2023; 8(12):176.
https://doi.org/10.3390/infrastructures8120176

**Chicago/Turabian Style**

Wang, Jun-Fang, Jian-Fu Lin, and Yan-Long Xie.
2023. "Damage Identification of Turnout Rail through a Covariance-Based Condition Index and Quantitative Pattern Analysis" *Infrastructures* 8, no. 12: 176.
https://doi.org/10.3390/infrastructures8120176