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August & September  2019, 12(4&5): 1091-1100. doi: 10.3934/dcdss.2019075

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

Yuan Li , , Lei Yan , Lingbo Wang and  Wei Hou

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.

Keywords: Carbonation depth, normal distribution, least squares method, Bayesian Quantile method, Maximum Likelihood method.
Mathematics Subject Classification: 37A50.
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, 2019, 12 (4&5) : 1091-1100. 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.  Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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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.   Google Scholar

[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.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

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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.   Google Scholar

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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. Google Scholar

[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.  Google Scholar

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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.   Google Scholar

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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.   Google Scholar

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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) Google Scholar

[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.   Google Scholar

[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) Google Scholar

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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. Google Scholar

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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.   Google Scholar

[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) Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

M. Y. Sulaiman, A. M. Akaak and M. A. Wahab, et al., Wind characteristics of Oman, Energy, 27 (2002), 35-46. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Xia, X. X. Dai and Y. Feng, Bayesian-MCMC-based parameter estimation of stealth aircraft RCS models, Chinese Physics, 24 (2015), 129501. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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) Google Scholar

[31]

C. F. Yuan, D. T. Niu and C. T. Sun, Carbonation depth prediction of Songhu River Highway Bridge, Concrete, 6 (2009), 46-48.   Google Scholar

[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. Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

G. Canavos, Applied Probability Statistical Methods, New York: Little & Brown Company, 1998. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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) Google Scholar

[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.   Google Scholar

[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) Google Scholar

[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. Google Scholar

[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.   Google Scholar

[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) Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[24]

M. Y. Sulaiman, A. M. Akaak and M. A. Wahab, et al., Wind characteristics of Oman, Energy, 27 (2002), 35-46. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

W. Xia, X. X. Dai and Y. Feng, Bayesian-MCMC-based parameter estimation of stealth aircraft RCS models, Chinese Physics, 24 (2015), 129501. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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) Google Scholar

[31]

C. F. Yuan, D. T. Niu and C. T. Sun, Carbonation depth prediction of Songhu River Highway Bridge, Concrete, 6 (2009), 46-48.   Google Scholar

[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. Google Scholar

Table 1.  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
Table Options

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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
Table 2.  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
Table Options

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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
Table 3.  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
Table Options

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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
Table 4.  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
Table Options

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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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