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

Yuan Li; Lei Yan; Lingbo Wang; Wei Hou

doi:10.3934/dcdss.2019075

Article Contents

2019, Volume 12, Issue 4&5: 1091-1100. Doi: 10.3934/dcdss.2019075 open access

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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
* Corresponding author: Yuan Li

Received: June 30, 2017

Revised: December 31, 2017

Published: March 2019

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Abstract

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.

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Mathematics Subject Classification: 37A50.

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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

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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

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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

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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

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