# Optimizing the De-Noise Neural Network Model for GPS Time-Series Monitoring of Structures

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## Abstract

**:**

## 1. Introduction

## 2. Identification Models

#### 2.1. Back-Propagation Neural Networks (BPN)

_{i}, i = 1,2,…n) assume an input unit and transmits these signals to the hidden units. Each hidden unit (Z

_{j}, j = 1,2,…m) calculates the sum of the weighted input signals, applies an activation function (f) and sends their result to the output unit [24]. The process of input signals to the hidden layer and output hidden layers can be represented as follow:

_{i}are the bias on the hidden node and weight factor from input to hidden units.

_{j}are the bias on the output node and weight factor from hidden to the output units.

_{k}(n) and δ

_{i}(n) are the error coefficients at the output and hidden nodes, respectively.

#### 2.2. Cascade- Forward Back-Propagation Neural Network (CFN)

#### 2.3. Adaptive Filter Neural Network (ADFN)

#### 2.4. Extended Kalman Filter Neural Network (EKFN)

_{p}and w

_{0}are coefficients and biases, these are lumped together into the weight vector w.

_{t}be the error made in approximating f by the NN, implementing the mapping λ for the input signal x

_{t}. Such a network can be represented as:

_{t}of the NN at times t = 1, 2,…n are as follow:

_{t}is the Kalman gain matrix, C

_{t}is a matrix of derivatives of λ

_{i}, R

_{t}and Q

_{t}are the covariance matrix for the process noise and measurement errors as shown in Equations (1) and (2). Herein, in this model assumed P

_{0}and R

_{t}symmetric positive definite matrices, Q

_{t}a symmetric positive semi definite matrix, and Equation (13) is initialized with a given w

_{0}[29].

## 3. Results and Discussions

#### 3.1. Simulation Noise Results

Models | BPN | CFN | ADFN | EKFN |
---|---|---|---|---|

MAE | 0.0093 | 8.65 × 10^{−4} | 0.0024 | 0.0037 |

MSE | 1.22 × 10^{−4} | 1.206 × 10^{−5} | 7.98 × 10^{−6} | 3.174 × 10^{−5} |

R-Square | 0.987 | 0.986 | 0.997 | 0.992 |

^{−6}) values are shown with ADFN model. Therefore, it can be considered that the ADFN is the best model to predict the true position of the sensors’ monitoring for the noisy signals. Meanwhile, it can be shown that the models CFN and EKFN performed as second best, due to high fitting shown and low absolute and mean square errors for the two models. In addition, the CFN gives a quite low value of MAE (8.65 × 10

^{−4}). Therefore, the CFN, ADFN and EKFN models have shown better results than BPN which is recommended by Chen et al. [19]. In addition, the results show clearly that the NN model filter can be efficiently used to remove the white noise signals for the random process of structures. It implies the multi-step filtering procedure can be constrained to de-noise the noisy sensor recordings for the structural oscillations and to determine its oscillation amplitude and modal frequency; this outcome complies with the results obtained by Moschas and Stiros [1]. Based on these results, it can be concluded that the ADFN served most effectively to predict the accurate monitoring for the noisy signals, therefore, the de-noised signals can be used to evaluate the movements of structures. The next section contains a case study to investigate this conclusion.

**Figure 5.**Comparison of the simulation results (

**a**) Back-Propagation Neural Network (BPN); (

**b**) Cascade- Forward Back-Propagation Neural Network (CFN); (

**c**) Adaptive Filter Neural Network (ADFN); (d) Extended Kalman Filter Neural Network (EKFN).

#### 3.2. GPS-Bridge Movement Application: Case Study

**Figure 8.**Dynamic performance for z direction. (

**a**) Long period displacement; (

**b**) Short period displacement; (

**c**) Multi filter method (MFM) de-noise frequency contents; (

**d**) ADFN de-noise frequency contents.

Parameter | Original Signal | MFM Model | ADFN Model |
---|---|---|---|

MAE (mm) | 5.789 × 10^{−3} | 5.119 × 10^{−3} | 4.018 × 10^{−3} |

MSE (mm) | 7.575 × 10^{−5} | 5.872 × 10^{−5} | 3.742 × 10^{−5} |

STD (mm) | 8.703 × 10^{−3} | 7.663 × 10^{−3} | 6.117 × 10^{−3} |

Absolute Max (mm) | 8.861 × 10^{−2} | 5.611 × 10^{−2} | 4.773 × 10^{−2} |

## 4. Conclusions

- (1)
- The multi-step filtering procedure can be constrained to de-noise the noisy measurements of the oscillations of structures, and to determine its oscillation amplitude and modal frequency. In addition, it is concluded that the ADFN is the best model and thus suggested for use to de-noise the GPS measurements. In addition, the ADFN model is observed to be more effective and accurate than MFM model for de-noising short-period components of displacements of GPS real monitoring data in time and frequency domains.
- (2)
- The apparent displacement measurements contain noises that can be considered as vibration and background noises. Moreover, the ADFN model increased the accuracy of short-period displacement by 83.3% in the x and y directions and by 93.8% in the z direction.
- (3)
- The de-noised short-period displacement component of the measurements has decreased the power spectrum density to 98%. This means that the noise has high effect on the high and low frequency vibration modes of structures. From the frequency modes calculations, it is assumed that the low frequency modes of the short-period displacement component values are between 0–0.2 Hz.
- (4)
- The GPS measurements with a sampling rate of 1.0 Hz may in fact underestimate the amplitude of displacement components, this problem, however, can be expected to overcome with the modern high-frequency GPS.
- (5)
- The de-noised short-period displacement component based on NN predictive models is expected to have significant implications in the SHM and the design of structures in the low-frequency content.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kaloop, M.R.; Hu, J.W. Optimizing the De-Noise Neural Network Model for GPS Time-Series Monitoring of Structures. *Sensors* **2015**, *15*, 24428-24444.
https://doi.org/10.3390/s150924428

**AMA Style**

Kaloop MR, Hu JW. Optimizing the De-Noise Neural Network Model for GPS Time-Series Monitoring of Structures. *Sensors*. 2015; 15(9):24428-24444.
https://doi.org/10.3390/s150924428

**Chicago/Turabian Style**

Kaloop, Mosbeh R., and Jong Wan Hu. 2015. "Optimizing the De-Noise Neural Network Model for GPS Time-Series Monitoring of Structures" *Sensors* 15, no. 9: 24428-24444.
https://doi.org/10.3390/s150924428