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## Estimation of normal distribution parameters and its application to carbonation depth of concrete girder bridges

 Department of Bridge Engineering in School of Highway, Chang'an University, Xi'an 710064, China

* Corresponding author: Yuan Li

Received  July 2017 Revised  January 2018 Published  November 2018

Taking carbonation depth uncertainty into account is key to approach durability analysis of concrete girder bridges in a probabilistic way. The Normal distribution has been widely used to represent the probability distribution of carbonation depth. In this study, two new methods such as Least Squares method and Bayesian Quantile method, are used to estimate the parameters of the Normal distribution. These two considered methods are also compared with the commonly used Maximum Likelihood method via an extensive numerical simulation and three real carbonation depth data examples based on performance measures such as, K-S test, RMSE and ${\text{R}}^{2}$. The numerical study reveals that the Least Squares method is the best one for estimating the parameters of the Normal distribution. Statistical analysis of real carbonation depth data sets are presented to demonstrate the applicability and the conclusion of the simulation results.

Citation: Yuan Li, Lei Yan, Lingbo Wang, Wei Hou. Estimation of normal distribution parameters and its application to carbonation depth of concrete girder bridges. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019075
##### References:
 [1] P. Biswabrata and K. Debasis, Bayes estimation and prediction of the two-parameter gamma distribution, Journal of Statistical Computation & Simulation, 81 (2011), 1187-1198. doi: 10.1080/00949651003796335. [2] P. Biswabrata and K. Debasis, Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions, Journal of Statistical Computation and Simulation, 86 (2016), 170-182. doi: 10.1080/00949655.2014.1001759. [3] G. Canavos, Applied Probability Statistical Methods, New York: Little & Brown Company, 1998. [4] M. J. Diamantopoulou, R. Özçelik and F. Crecente-Campo, Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods, Biosystems Engineering, 133 (2015), 33-45. [5] H. L. Gan and X. L. Xie, Carbonation life prediction of service reinforced concrete bridge based on reliability theory of durability, Concrete, 3 (2013), 48-51. [6] X. Guan, D. T. Niu and J. B. Wang, Carbonation service life prediction of coal boardwalks bridges based on durability testing, Journal of Xi'an University of Architecture and Technology, 47 (2015), 71-76. [7] H. P. Hong, S. H. Li and T. G. Mara, Performance of the generalized least-squares method for the Gumbel distribution and its application to annual maximum wind speeds, Journal of Wind Engineering and Industrial Aerodynamics, 119 (2013), 121-132. [8] S. Y. Huang, Wavelet based empirical Bayes estimation for the uniform distribution, Statistics & Probability Letters, 32 (1997), 141-146. doi: 10.1016/S0167-7152(96)00066-1. [9] M. T. Liang, R. Huang and S. A. Fang, Carbonation service life prediction of existing concrete viaduct/bridge using time-dependent reliability analysis, Journal of Marine Science and Technology, 21 (2013), 94-104. [10] H. L. Lu and S. H. Tao, The estimation of Pareto distribution by a weighted least square method, Quality & Quantity, 41 (2007), 913-926. [11] B. Miladinovic and C. P. Tsokos, Ordinary, Bayes, empirical Bayes, and non-parametric reliability analysis for the modified Gumbel failure model, Nonlinear Analysis, 71 (2009), 1426-1436. [12] U. J. Na, S. J. Kwon, S. R. Chaudhuri, et al., Stochastic model for life prediction of RC structures exposed to carbonation using random field simulation, KSCE Journal of Civil Engineering, 16 (2012), 133-143. [13] J. Nabakumar, K. Somesh and C. Kashinath, Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models, Journal of Applied Statistics, 43 (2016), 2697-2712. doi: 10.1080/02664763.2016.1142950. [14] D. T. Niu, Y. Q. Chen and S. Yu, Model and reliability analysis for carbonation of concrete structures, Journal of Xi'an University of Architecture and Technology, 27 (1995a), 365-369. [15] D. T. Niu, Y. C. Shi and Y. S. Lei, Reliability analysis and probability model of concrete carbonation, Journal of Xi'an University of Architecture and Technology, 27 (1995b), 252-256. [16] D. T. Niu, Z. P. Dong and Y. X. Pu, Fuzzy prediction on carbonation life of concrete structures, Proceedings of the Ninth Conference of Civil Engineering Society, Hanzhou, (1999a), 367-370. (in Chinese) [17] D. T. Niu, Z. P. Dong and Y. X. Pu, Random model of predicting the carbonated concrete depth, Industrial Construction, 29 (1999b), 41-45. [18] D. T. Niu, C. F. Yuan and C. F. Wang, et al., Carbonation service life prediction of reinforced concrete railway bridges based on durability testing, Journal of Xi'an University of Architecture and Technology, 43 (2011), 160-165. (in Chinese) [19] T. B. M. J. Ouarda, C. Charron and J. Y. Shin, et al., Probability distributions of wind speed in the UAE, Energy Conversion & Management, 93 (2015), 414-434. [20] J. X. Peng and J. R. Zhang, Incremental process based carbonation depth prediction model of concrete structures and its probability analysis, Journal of Highway and Transportation Research and Development, 29 (2012), 54-83. [21] F. Ren, J. Y. Liu and X. Y. Pei, et al., Reliability analysis of bridge durability based on concrete carbonation, Journal of Highway and Transportation Research and Development, 21 (2004), 71-80. (in Chinese) [22] P. K. Singh, S. K. Singh and U. Singh, Bayes estimator of Inverse Gaussian parameters under general entropy loss function using Lindley's approximation, Communications in Statistics - Simulation and Computation, 37 (2008), 1750-1762. doi: 10.1080/03610910701884054. [23] A. A. Soliman, Comparison of linex and quadratic Bayes estimators for the Rayleigh distribution, Communications in Statistics-theory and Methods, 29 (2000), 95-107. [24] M. Y. Sulaiman, A. M. Akaak and M. A. Wahab, et al., Wind characteristics of Oman, Energy, 27 (2002), 35-46. [25] F. J. Torres, Estimation of parameters of the shifted Gompertz distribution using least squares, maximum likelihood and moments methods, Journal of Computational & Applied Mathematics, 255 (2014), 867-877. doi: 10.1016/j.cam.2013.07.004. [26] J. W. Wu, W. L. Hung and C. H. Tsai, Estimation of parameters of the Gompertz distribution using the least squares method, Applied Mathematics and Computation, 158 (2004), 133-147. doi: 10.1016/j.amc.2003.08.086. [27] W. Xia, X. X. Dai and Y. Feng, Bayesian-MCMC-based parameter estimation of stealth aircraft RCS models, Chinese Physics, 24 (2015), 129501. [28] S. H. Xu, D. T. Niu and Q. L. Wang, The determination of concrete cover depth under atmospheric condition, China Civil Engineering Journal, 38 (2005), 45-68. [29] Z. T. Yu and D. J. Han, Carbonation reliability assessment of existing reinforced concrete girder bridges, Journal of South China University of Technology, 32 (2004), 50-66. [30] C. F. Yuan, D. T. Niu and Q. S. Gai, et al., Durability testing and carbonation life prediction of Songhu River Bridge, Bridge Construction, 2 (2010), 21-24. (in Chinese) [31] C. F. Yuan, D. T. Niu and C. T. Sun, Carbonation depth prediction of Songhu River Highway Bridge, Concrete, 6 (2009), 46-48. [32] J. Z. Zhou, E. Erdem and G. Li, et al., Comprehensive evaluation of wind speed distribution models: A case study for North Dakota sites, Energy Conversion and Management, 51 (2010), 1449-1458.

show all references

##### References:
 [1] P. Biswabrata and K. Debasis, Bayes estimation and prediction of the two-parameter gamma distribution, Journal of Statistical Computation & Simulation, 81 (2011), 1187-1198. doi: 10.1080/00949651003796335. [2] P. Biswabrata and K. Debasis, Bayes estimation for the Block and Basu bivariate and multivariate Weibull distributions, Journal of Statistical Computation and Simulation, 86 (2016), 170-182. doi: 10.1080/00949655.2014.1001759. [3] G. Canavos, Applied Probability Statistical Methods, New York: Little & Brown Company, 1998. [4] M. J. Diamantopoulou, R. Özçelik and F. Crecente-Campo, Estimation of Weibull function parameters for modelling tree diameter distribution using least squares and artificial neural networks methods, Biosystems Engineering, 133 (2015), 33-45. [5] H. L. Gan and X. L. Xie, Carbonation life prediction of service reinforced concrete bridge based on reliability theory of durability, Concrete, 3 (2013), 48-51. [6] X. Guan, D. T. Niu and J. B. Wang, Carbonation service life prediction of coal boardwalks bridges based on durability testing, Journal of Xi'an University of Architecture and Technology, 47 (2015), 71-76. [7] H. P. Hong, S. H. Li and T. G. Mara, Performance of the generalized least-squares method for the Gumbel distribution and its application to annual maximum wind speeds, Journal of Wind Engineering and Industrial Aerodynamics, 119 (2013), 121-132. [8] S. Y. Huang, Wavelet based empirical Bayes estimation for the uniform distribution, Statistics & Probability Letters, 32 (1997), 141-146. doi: 10.1016/S0167-7152(96)00066-1. [9] M. T. Liang, R. Huang and S. A. Fang, Carbonation service life prediction of existing concrete viaduct/bridge using time-dependent reliability analysis, Journal of Marine Science and Technology, 21 (2013), 94-104. [10] H. L. Lu and S. H. Tao, The estimation of Pareto distribution by a weighted least square method, Quality & Quantity, 41 (2007), 913-926. [11] B. Miladinovic and C. P. Tsokos, Ordinary, Bayes, empirical Bayes, and non-parametric reliability analysis for the modified Gumbel failure model, Nonlinear Analysis, 71 (2009), 1426-1436. [12] U. J. Na, S. J. Kwon, S. R. Chaudhuri, et al., Stochastic model for life prediction of RC structures exposed to carbonation using random field simulation, KSCE Journal of Civil Engineering, 16 (2012), 133-143. [13] J. Nabakumar, K. Somesh and C. Kashinath, Bayes estimation for exponential distributions with common location parameter and applications to multi-state reliability models, Journal of Applied Statistics, 43 (2016), 2697-2712. doi: 10.1080/02664763.2016.1142950. [14] D. T. Niu, Y. Q. Chen and S. Yu, Model and reliability analysis for carbonation of concrete structures, Journal of Xi'an University of Architecture and Technology, 27 (1995a), 365-369. [15] D. T. Niu, Y. C. Shi and Y. S. Lei, Reliability analysis and probability model of concrete carbonation, Journal of Xi'an University of Architecture and Technology, 27 (1995b), 252-256. [16] D. T. Niu, Z. P. Dong and Y. X. Pu, Fuzzy prediction on carbonation life of concrete structures, Proceedings of the Ninth Conference of Civil Engineering Society, Hanzhou, (1999a), 367-370. (in Chinese) [17] D. T. Niu, Z. P. Dong and Y. X. Pu, Random model of predicting the carbonated concrete depth, Industrial Construction, 29 (1999b), 41-45. [18] D. T. Niu, C. F. Yuan and C. F. Wang, et al., Carbonation service life prediction of reinforced concrete railway bridges based on durability testing, Journal of Xi'an University of Architecture and Technology, 43 (2011), 160-165. (in Chinese) [19] T. B. M. J. Ouarda, C. Charron and J. Y. Shin, et al., Probability distributions of wind speed in the UAE, Energy Conversion & Management, 93 (2015), 414-434. [20] J. X. Peng and J. R. Zhang, Incremental process based carbonation depth prediction model of concrete structures and its probability analysis, Journal of Highway and Transportation Research and Development, 29 (2012), 54-83. [21] F. Ren, J. Y. Liu and X. Y. Pei, et al., Reliability analysis of bridge durability based on concrete carbonation, Journal of Highway and Transportation Research and Development, 21 (2004), 71-80. (in Chinese) [22] P. K. Singh, S. K. Singh and U. Singh, Bayes estimator of Inverse Gaussian parameters under general entropy loss function using Lindley's approximation, Communications in Statistics - Simulation and Computation, 37 (2008), 1750-1762. doi: 10.1080/03610910701884054. [23] A. A. Soliman, Comparison of linex and quadratic Bayes estimators for the Rayleigh distribution, Communications in Statistics-theory and Methods, 29 (2000), 95-107. [24] M. Y. Sulaiman, A. M. Akaak and M. A. Wahab, et al., Wind characteristics of Oman, Energy, 27 (2002), 35-46. [25] F. J. Torres, Estimation of parameters of the shifted Gompertz distribution using least squares, maximum likelihood and moments methods, Journal of Computational & Applied Mathematics, 255 (2014), 867-877. doi: 10.1016/j.cam.2013.07.004. [26] J. W. Wu, W. L. Hung and C. H. Tsai, Estimation of parameters of the Gompertz distribution using the least squares method, Applied Mathematics and Computation, 158 (2004), 133-147. doi: 10.1016/j.amc.2003.08.086. [27] W. Xia, X. X. Dai and Y. Feng, Bayesian-MCMC-based parameter estimation of stealth aircraft RCS models, Chinese Physics, 24 (2015), 129501. [28] S. H. Xu, D. T. Niu and Q. L. Wang, The determination of concrete cover depth under atmospheric condition, China Civil Engineering Journal, 38 (2005), 45-68. [29] Z. T. Yu and D. J. Han, Carbonation reliability assessment of existing reinforced concrete girder bridges, Journal of South China University of Technology, 32 (2004), 50-66. [30] C. F. Yuan, D. T. Niu and Q. S. Gai, et al., Durability testing and carbonation life prediction of Songhu River Bridge, Bridge Construction, 2 (2010), 21-24. (in Chinese) [31] C. F. Yuan, D. T. Niu and C. T. Sun, Carbonation depth prediction of Songhu River Highway Bridge, Concrete, 6 (2009), 46-48. [32] J. Z. Zhou, E. Erdem and G. Li, et al., Comprehensive evaluation of wind speed distribution models: A case study for North Dakota sites, Energy Conversion and Management, 51 (2010), 1449-1458.
Comparison of the estimation methods
 Maximum likelihood method Bayesian Quantile method Least Squares method $n$ Parameter $\mu$ $\sigma$ $\mu$ $\sigma$ $\mu$ $\sigma$ 10 mean 0.12214 1.15672 0.11672 1.16318 0.09491 1.08113 RMSE 0.26513 0.35772 0.25617 0.36147 0.19817 0.27136 KS 0.35337 0.32109 0.24578 R$^{2}$ 0.83298 0.84576 0.88978 20 mean 0.07571 1.10291 0.06984 1.08983 0.05116 1.05886 RMSE 0.18364 0.24536 0.19225 0.22139 0.14281 0.18775 KS 0.26355 0.28776 0.19771 R$^{2}$ 0.90137 0.89516 0.92335 30 mean 0.05319 1.06572 0.05187 1.07102 0.04785 1.04213 RMSE 0.15361 0.21369 0.14793 0.20398 0.11251 0.15720 KS 0.15367 0.13476 0.09877 R$^{2}$ 0.95226 0.96237 0.97562 50 mean 0.04367 1.05318 0.04412 1.05277 0.03918 1.03889 RMSE 0.11623 0.15617 0.10987 0.15726 0.08273 0.12918 KS 0.12981 0.13287 0.08726 R$^{2}$ 0.96314 0.96512 0.98715 100 mean 0.03647 1.04891 0.03265 1.04912 0.02797 1.03276 RMSE 0.07629 0.13912 0.07292 0.14021 0.05172 0.09885 KS 0.08398 0.08203 0.06512 R$^{2}$ 0.97651 0.97261 0.99143 200 mean 0.02674 1.03628 0.02556 1.03719 0.02102 1.01493 RMSE 0.05728 0.07635 0.05276 0.07682 0.04729 0.05112 KS 0.06729 0.07102 0.05112 R$^{2}$ 0.98112 0.98372 0.99557 300 mean 0.01839 1.02987 0.01821 1.02898 0.01315 1.01011 RMSE 0.03672 0.05729 0.03629 0.05827 0.02791 0.03174 KS 0.05237 0.05311 0.04986 R$^{2}$ 0.99108 0.99203 0.99778 500 mean 0.00587 1.00532 0.00526 1.00516 0.00338 1.00201 RMSE 0.02392 0.03738 0.02371 0.03276 0.01818 0.01679 KS 0.03129 0.03063 0.02701 R$^{2}$ 0.99536 0.99277 0.99913 1000 mean 0.00161 1.00114 0.00108 1.00112 0.00036 1.00008 RMSE 0.01307 0.02119 0.01298 0.02101 0.00737 0.00082 KS 0.01112 0.01134 0.00601 R$^{2}$ 0.99821 0.99903 0.99996
 Maximum likelihood method Bayesian Quantile method Least Squares method $n$ Parameter $\mu$ $\sigma$ $\mu$ $\sigma$ $\mu$ $\sigma$ 10 mean 0.12214 1.15672 0.11672 1.16318 0.09491 1.08113 RMSE 0.26513 0.35772 0.25617 0.36147 0.19817 0.27136 KS 0.35337 0.32109 0.24578 R$^{2}$ 0.83298 0.84576 0.88978 20 mean 0.07571 1.10291 0.06984 1.08983 0.05116 1.05886 RMSE 0.18364 0.24536 0.19225 0.22139 0.14281 0.18775 KS 0.26355 0.28776 0.19771 R$^{2}$ 0.90137 0.89516 0.92335 30 mean 0.05319 1.06572 0.05187 1.07102 0.04785 1.04213 RMSE 0.15361 0.21369 0.14793 0.20398 0.11251 0.15720 KS 0.15367 0.13476 0.09877 R$^{2}$ 0.95226 0.96237 0.97562 50 mean 0.04367 1.05318 0.04412 1.05277 0.03918 1.03889 RMSE 0.11623 0.15617 0.10987 0.15726 0.08273 0.12918 KS 0.12981 0.13287 0.08726 R$^{2}$ 0.96314 0.96512 0.98715 100 mean 0.03647 1.04891 0.03265 1.04912 0.02797 1.03276 RMSE 0.07629 0.13912 0.07292 0.14021 0.05172 0.09885 KS 0.08398 0.08203 0.06512 R$^{2}$ 0.97651 0.97261 0.99143 200 mean 0.02674 1.03628 0.02556 1.03719 0.02102 1.01493 RMSE 0.05728 0.07635 0.05276 0.07682 0.04729 0.05112 KS 0.06729 0.07102 0.05112 R$^{2}$ 0.98112 0.98372 0.99557 300 mean 0.01839 1.02987 0.01821 1.02898 0.01315 1.01011 RMSE 0.03672 0.05729 0.03629 0.05827 0.02791 0.03174 KS 0.05237 0.05311 0.04986 R$^{2}$ 0.99108 0.99203 0.99778 500 mean 0.00587 1.00532 0.00526 1.00516 0.00338 1.00201 RMSE 0.02392 0.03738 0.02371 0.03276 0.01818 0.01679 KS 0.03129 0.03063 0.02701 R$^{2}$ 0.99536 0.99277 0.99913 1000 mean 0.00161 1.00114 0.00108 1.00112 0.00036 1.00008 RMSE 0.01307 0.02119 0.01298 0.02101 0.00737 0.00082 KS 0.01112 0.01134 0.00601 R$^{2}$ 0.99821 0.99903 0.99996
Parameter estimates, RMSE, KS and R$^{2}$ for the first data set
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 14.7500 1.2923 0.2677 0.1912 0.8826 Bayesian Quantile method 14.6534 1.4505 0.2301 0.2171 0.8755 Least Squares method 14.5703 1.2197 0.1329 0.1162 0.9283
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 14.7500 1.2923 0.2677 0.1912 0.8826 Bayesian Quantile method 14.6534 1.4505 0.2301 0.2171 0.8755 Least Squares method 14.5703 1.2197 0.1329 0.1162 0.9283
Parameter estimates, RMSE, KS and R$^{2}$ for the second data set
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 24.5556 9.5808 1.0122 0.1175 0.9218 Bayesian Quantile method 24.6528 10.3198 0.9526 0.1013 0.9427 Least Squares method 23.5642 10.6848 0.7128 0.0816 0.9577
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 24.5556 9.5808 1.0122 0.1175 0.9218 Bayesian Quantile method 24.6528 10.3198 0.9526 0.1013 0.9427 Least Squares method 23.5642 10.6848 0.7128 0.0816 0.9577
Parameter estimates, RMSE, KS and R$^{2}$ for the third data set
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 2.9852 0.5702 0.0441 0.0966 0.9761 Bayesian Quantile method 3.0127 0.5985 0.0412 0.0843 0.9788 Least Squares method 2.9697 0.6770 0.0391 0.0498 0.9916
 Estimated parameters Method $\mu$ $\sigma$ RMSE KS R$^{2}$ Maximum Likelihood method 2.9852 0.5702 0.0441 0.0966 0.9761 Bayesian Quantile method 3.0127 0.5985 0.0412 0.0843 0.9788 Least Squares method 2.9697 0.6770 0.0391 0.0498 0.9916
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