# Investigation of Temperature Effects into Long-Span Bridges via Hybrid Sensing and Supervised Regression Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works and Challenges

#### 1.2. Objectives

#### 1.3. Contributions

## 2. Long-Span Bridges

#### 2.1. Dashengguan Bridge

#### 2.2. Lupu Bridge

#### 2.3. Rainbow Bridge

## 3. Correlation Analysis

_{xy}is the characteristic matrix containing the highest normalized mutual information achieved by any x-by-y grid, defined here as m

_{xy}. The entry in the characteristic matrix can be expressed as follows:

_{G}denotes the mutual information of the probability distribution induced on the boxes of G, where the probability of a box is proportional to the number of data points falling inside the box. On this basis, the MIC is the maximum of m

_{xy}such that x

_{i}y

_{i}< B, where B = n

^{0.6}.

## 4. Supervised Regression Models

#### 4.1. Linear Regression Model

_{k}is the kth coefficient; β

_{0}denotes the constant term in the model; and n is the number of predictor and response data. x

_{k}is the k

^{th}sample of the predictor data. Moreover, e is the noise or random error term in the linear regression, which is the same residual function. The coefficients of the regression model are estimated by minimizing the mean squared error (MSE) between the prediction $\widehat{y}$ and the real response y. Apart from this criterion, the R-squared (R

^{2}) value is a statistical measure that presents a goodness-of-fit of the regression models. This statistic indicates the rate of variance in the dependent variable (output) that the independent variable explains collectively. On this basis, it can help to understand how well the model fits the data and represent the strength of the relationship between the linear model and the output in the range of 0–1. The R-squared value assesses the scatter of the data samples around the fitted regression model/line. In this regard, a higher R-squared value represents a smaller difference between the observed data and the fitted values. Having considered the real and predicted data, the R-squared value was formulated as follows:

^{2}= 1. Once the best model has been developed, which leads to the best regression coefficients, the predicted data can be derived from the following equation:

#### 4.2. Gaussian Process Regression

_{i},x

_{j}) is the main kernel function (matrix) for i,j = 1, …, n. The kernel function models the dependence between the function values at different input points. The selection of a proper kernel function lies in assumptions such as smoothness and patterns to be expected in the predictor data. Hence, one can realize that the GPR is a parametric supervised regressor; therefore, some unknown parameters (i.e., hyperparameters), such as the type of kernel function and the kernel coefficients, should be determined.

_{S}), rational quadratic kernel (κ

_{R}), and exponential kernel (κ

_{E}). Equations (16)–(18) mathematically express these functions:

^{T}refers to the transpose operation in the mathematics. Note that the input data x are univariate, the expression (x

_{i}− x

_{j})

^{T}(x

_{i}− x

_{j}) is equivalent to (x

_{i}− x

_{j})

^{2}. In these equations, σ

_{S}, σ

_{R}, and σ

_{E}as well as λ

_{S}, λ

_{R}, and λ

_{E}denote the standard deviations and kernel scales of the squared exponential, rational quadratic, and exponential kernel functions, respectively. Furthermore, in Equation (17), η is the scale mixture parameter of the rational quadratic function. Once the GPR model has been developed, one can predict the output data and determine the prediction data $\widehat{y}$, which is the output of the function f(x). Note that training and test ratios equal to 80% and 20% were considered to train the GPR model and predict the displacement data.

#### 4.3. Supervised Vector Regression

_{L}), Gaussian kernel (κ

_{G}), and polynomial kernel (κ

_{P}) functions, which are expressed in Equations (22) and (23), respectively:

_{G}and the polynomial order of κ

_{P}, respectively. Moreover, each of the kernel functions in these equations is equivalent to ϕ(x). Once the SVR model is established, one can predict the response data as $\widehat{y}$ = w

^{T}ϕ(x) + b. Note that training and test ratios equal to 80% and 20% were considered to train the SVR model and predict the displacement data.

## 5. Results

#### 5.1. Dashengguan Bridge

#### 5.2. Lupu Bridge

^{2}were also inserted in this figure to assess the goodness-of-fit of the regression modeling. Unlike the LRM regarding the bridge dome, as shown in Figure 15a, one can see that the LRM of the main span could reasonably model the relationship between the displacement and temperature data. Using the fitted LRMs, Figure 16 compares the real and predicted displacement points of the Lupu Bridge. From Figure 16a, it is clear that the real and predicted samples were not consistent with each other, implying the poor prediction performance of the LRM, as its low R

^{2}value also confirms this conclusion. Nonetheless, the prediction performance of the LRM of the bridge span in Figure 16b was roughly reliable, indicating that temperature was the influential environmental factor affecting the bridge span.

_{S}and λ

_{S}were identical to 12.08 and 2.72 regarding the bridge dome and 18.61 and 9.08 related to the bridge span. In addition, both SVR models are designed using the polynomial kernel function (${\kappa}_{P}$) with the kernel parameter (q) equal to 3. For these models, the optimum number of support vectors and the model bias values corresponded to 2 and −1.79 for the bridge dome and 38 and 11.22 at the bridge span, respectively. Using the optimized hyperparameters, the GPR and SVR models were trained to predict the displacement responses as shown in Figure 17 and Figure 18. In relation to the bridge dome, Figure 17a and Figure 18a show that there were discrepancies between the real and predicted displacement samples, implying poor prediction performances. In contrast, the GPR and SVR models had better performances in predicting the displacement points of the main span of the Lupu Bridge, as can be observed in Figure 17b and Figure 18b, compared to the corresponding points related to the bridge dome.

#### 5.3. Rainbow Bridge

_{S}related to Piers 2 and 3 correspond to 0.0011 and 0.0052, and the kernel parameter λ

_{S}concerning Pier 4 is identical to 0.0001. Moreover, the kernel parameter q concerning the optimized polynomial kernel function is equal to 3 at all elements. Based on the optimized hyperparameters of the GPR models, Figure 27 and Figure 28 compare the real and predicted displacement samples of the four piers and three spans of the Rainbow Bridge, respectively. The same outputs regarding the SVR models are shown in Figure 29 and Figure 30. From Figure 27a, one can perceive that the GPR model fitted to the displacement samples of the first pier could not predict them properly. In Figure 27d, the predicted values are constant, which may be related to the performance of Bayesian hyperparameter optimization. Nevertheless, it is obvious that the GPR models fitted to the displacement data of the second and third piers of the Rainbow Bridge have accurately predicted the real data. On the other hand, the results of the GPR modeling related to the displacement data of the spans, as shown in Figure 28, resemble the LRMs.

#### 5.4. Discussions on Sufficiency of Environmental/Operational Sensors

## 6. Conclusions

- (1)
- When any environmental data are available, it is necessary to perform a correlation analysis to realize relationships between the structural responses. Based on the four correlation analysis methods investigated in this paper, the proposed MIC method provided more reasonable results than the other ones due to its consideration of both the linear and nonlinear correlation patterns.
- (2)
- In the problem of the Dashengguan Bridge, where the single measured environmental factor (temperature) and SAR-based displacement data had a strong linear correlation, Kendall’s correlation coefficient could not yield appropriate outputs as good as the MIC, Pearson’s, and Spearman’s correlation coefficient methods. Hence, this correlation measure can be disregarded in further applications.
- (3)
- The supervised regression techniques could perform well when there is a high correlation between the displacement and temperature data. These techniques failed in providing accurate and reliable results when the temperature and displacement data had a low correlation. This was most likely due to the fact that the other unmeasured environmental and/or operational conditions or even structural damage impacted the displacement data. Since such conditions were not incorporated into the supervised regression models, those could not properly predict the measured (real) displacement data.
- (4)
- The low correlation rates and poor prediction performances mean that the environmental/operational sensors (i.e., temperature sensors in this research) in a bridge structure are not sufficient, and one needs to consider further sensors for measuring other environmental and/or operational conditions such as humidity, wind speed and direction, traffic, etc. Moreover, it is important to investigate the possibility of existing any structural damage by visual inspection or tried-and-test techniques for early damage assessment.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The Dashengguan Bridge: (

**a**) limited displacement data from remote sensing, (

**b**) temperature data from contact sensing.

**Figure 4.**The limited displacements of the Lupu Bridge from 38 SAR images of TerraSAR-X between 2013–2016: (

**a**) the dome, (

**b**) the main span, (

**c**) the temperature data.

**Figure 6.**The limited displacements of the Lupu Bridge from 53 SAR images of Sentinel-1A between 2015–2017: (

**a**) Piers #1–4, (

**b**) Spans #1–3, (

**c**) temperature data.

**Figure 7.**Scatter plots of the displacement versus temperature data in the problem of the Dashengguan Bridge: (

**a**) Pier #4, (

**b**) Pier #5, (

**c**) Pier #6, (

**d**) Pier #8, (

**e**) Pier #9, (

**f**) Pier #10.

**Figure 8.**LRMs fitted to the displacement and temperature data of the Dashengguan Bridge: (

**a**) Pier 4, (

**b**) Pier 5, (

**c**) Pier 6, (

**d**) Pier 8, (

**e**) Pier 9, (

**f**) Pier 10.

**Figure 9.**Bayesian hyperparameter optimization of the kernel parameters of ${\kappa}_{S}$ related to the GPR models of Piers 4–6 and 8–10 of the Dashengguan Bridge: (

**a**) σ

_{S}, (

**b**) λ

_{S}.

**Figure 10.**Bayesian hyperparameter optimization of the SVR models related to Piers 4–6 and 8–10 of the Dashengguan Bridge: (

**a**) the number of support vectors, (

**b**) the model bias values.

**Figure 11.**Real and predicted displacement data of the Dashengguan Bridge based on the LRMs: (

**a**) Pier 4, (

**b**) Pier 5, (

**c**) Pier 6, (

**d**) Pier 8, (

**e**) Pier 9, (

**f**) Pier 10.

**Figure 12.**Real and predicted displacement data of the Dashengguan Bridge based on the GPR models: (

**a**) Pier 4, (

**b**) Pier 5, (

**c**) Pier 6, (

**d**) Pier 8, (

**e**) Pier 9, (

**f**) Pier 10.

**Figure 13.**Real and predicted displacement data of the Dashengguan Bridge based on the SVR models: (

**a**) Pier 4, (

**b**) Pier 5, (

**c**) Pier 6, (

**d**) Pier 8, (

**e**) Pier 9, (

**f**) Pier 10.

**Figure 14.**Scatter plots of the displacement and temperature data of the Lupu Bridge: (

**a**) the dome, (

**b**) the main span.

**Figure 15.**LRMs fitted to the displacement and temperature data of the Lupu Bridge: (

**a**) the dome, (

**b**) the main span.

**Figure 16.**Real and predicted displacement data of the Lupu Bridge based on the LRMs: (

**a**) the dome, (

**b**) the main span.

**Figure 17.**Real and predicted displacement data of the Lupu Bridge based on the GPR models: (

**a**) the dome, (

**b**) the main span.

**Figure 18.**Real and predicted displacement data of the Lupu Bridge based on the SVR models: (

**a**) the dome, (

**b**) the main span.

**Figure 19.**Scatter plots of the displacement and temperature data of the Rainbow Bridge: (

**a**) Pier 1, (

**b**) Pier 2, (

**c**) Pier 3, (

**d**) Pier 4.

**Figure 20.**Scatter plots of the displacement and temperature data of the Rainbow Bridge: (

**a**) Span 1, (

**b**) Span 2, (

**c**) Span 3.

**Figure 21.**LRMs fitted to the displacement and temperature data of the Rainbow Bridge: (

**a**) Pier 1, (

**b**) Pier 2, (

**c**) Pier 3, (

**d**) Pier 4.

**Figure 22.**LRMs fitted to the displacement and temperature data of the Rainbow Bridge: (

**a**) Span 1, (

**b**) Span 2, (

**c**) Span 3.

**Figure 23.**Real and predicted displacement data of the Rainbow Bridge based on the LRMs: (

**a**) Pier 1, (

**b**) Pier 2, (

**c**) Pier 3, (

**d**) Pier 4.

**Figure 24.**Real and predicted displacement data of the Rainbow Bridge based on the LRMs: (

**a**) Span 1, (

**b**) Span 2, (

**c**) Span 3.

**Figure 25.**Bayesian hyperparameter optimization of the kernel parameters of ${\kappa}_{S}$ related to the GPR models of Piers 1–4 and Spans 1–3 of the Rainbow Bridge: (

**a**) σ

_{S}, (

**b**) λ

_{S}.

**Figure 26.**Bayesian hyperparameter optimization of the SVR models related to Piers 1–4 and Spans 1–3 of the Rainbow Bridge: (

**a**) the number of support vectors, (

**b**) the model bias values.

**Figure 27.**Real and predicted displacement data of the Rainbow Bridge based on the GPR models: (

**a**) Pier 1, (

**b**) Pier 2, (

**c**) Pier 3, (

**d**) Pier 4.

**Figure 28.**Real and predicted displacement data of the Rainbow Bridge based on the GPR models: (

**a**) Span 1, (

**b**) Span 2, (

**c**) Span 3.

**Figure 29.**Real and predicted displacement data of the Rainbow Bridge based on the SVR models: (

**a**) Pier 1, (

**b**) Pier 2, (

**c**) Pier 3, (

**d**) Pier 4.

**Figure 30.**Real and predicted displacement data of the Rainbow Bridge based on the SVR models: (

**a**) Span 1, (

**b**) Span 2, (

**c**) Span 3.

**Table 1.**Correlation analysis between limited displacement and temperature data regarding the Dashengguan Bridge.

Pier No. | MIC | Correlation Coefficient Metrics | ||
---|---|---|---|---|

Pearson | Spearman | Kendall | ||

4 | 1.00 | −0.9928 | −0.9931 | −0.9507 |

5 | 1.00 | −0.9899 | −0.9896 | −0.9310 |

6 | 1.00 | −0.9776 | −0.9822 | −0.9064 |

8 | 1.00 | 0.9850 | 0.9901 | 0.9359 |

9 | 1.00 | 0.9943 | 0.9940 | 0.9507 |

10 | 1.00 | 0.9877 | 0.9876 | 0.9211 |

**Table 2.**Correlation analysis between limited displacement and temperature data regarding the Lupu Bridge.

Component | Correlation Coefficient Metrics | |||
---|---|---|---|---|

MIC | Pearson | Spearman | Kendall | |

Dome | 0.56 | −0.4555 | −0.4925 | −0.3285 |

Span | 0.90 | −0.8689 | −0.8483 | −0.6557 |

**Table 3.**Correlation analysis between the limited displacement data related to the piers and spans of the Rainbow Bridge and temperature records.

Elements | MIC | Correlation Coefficient Metrics | ||
---|---|---|---|---|

Pearson | Spearman | Kendall | ||

Pier 1 | 0.72 | 0.6081 | 0.6289 | 0.4296 |

Pier 2 | 0.61 | 0.4194 | 0.4151 | 0.2946 |

Pier 3 | 0.33 | 0.2521 | 0.2439 | 0.1625 |

Pier 4 | 0.39 | 0.3745 | 0.3612 | 0.2481 |

Span 1 | 0.75 | −0.7140 | −0.7184 | −0.5326 |

Span 2 | 0.70 | −0.6793 | −0.7005 | −0.5195 |

Span 3 | 0.74 | −0.7165 | −0.7281 | −0.5442 |

Bridge Name | Elements | Correlation Rate | Prediction Accuracy | Decision | |
---|---|---|---|---|---|

Linear | Nonlinear | ||||

Dashengguan | Piers 4–6 & 8–10 | High | High | High | Sufficient |

Lupu | Dome | Low | Low | Low | Insufficient |

Span | High | High | High | Sufficient | |

Rainbow | Pier 1 | Low | Low | Low | Insufficient |

Pier 2 | Low | Low | High * | Sufficient * | |

Pier 3 | Low | Low | High * | Sufficient * | |

Pier 4 | Low | Low | Low | Insufficient | |

Span 1 | High | High | Low | Insufficient ** | |

Span 2 | High | High | Low | Insufficient ** | |

Span 3 | High | High | Low | Insufficient ** |

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## Share and Cite

**MDPI and ACS Style**

Behkamal, B.; Entezami, A.; De Michele, C.; Arslan, A.N.
Investigation of Temperature Effects into Long-Span Bridges via Hybrid Sensing and Supervised Regression Models. *Remote Sens.* **2023**, *15*, 3503.
https://doi.org/10.3390/rs15143503

**AMA Style**

Behkamal B, Entezami A, De Michele C, Arslan AN.
Investigation of Temperature Effects into Long-Span Bridges via Hybrid Sensing and Supervised Regression Models. *Remote Sensing*. 2023; 15(14):3503.
https://doi.org/10.3390/rs15143503

**Chicago/Turabian Style**

Behkamal, Bahareh, Alireza Entezami, Carlo De Michele, and Ali Nadir Arslan.
2023. "Investigation of Temperature Effects into Long-Span Bridges via Hybrid Sensing and Supervised Regression Models" *Remote Sensing* 15, no. 14: 3503.
https://doi.org/10.3390/rs15143503