Nonparametric link functions with shape constraints in stochastic degradation processes: Application to emerging contaminants

Abstract

Effective oxidation-based elimination of emerging contaminants (ECs) requires a good understanding of the effects of treatment conditions, such as the kinds and dosages of reagents, on the EC degradation rate. Due to limited knowledge on the complex reaction mechanism and the multiple covariates to represent the treatment conditions, it is generally hard to parametrically quantify the relation between these covariates and the degradation rate. On the other hand, qualitative analysis based on chemical mechanisms often provides shape information of the covariate–rate relation, such as monotonicity and local concavity in each coordinate. Based on the chemical kinetics, we use stationary stochastic processes for parametric modeling of log-transformed EC degradation under each combination of the treatment conditions. The tensor-product Bernstein bases are then used to approximate the covariate–rate relation. The shape information is naturally translated to constraints on the coefficients of the bases functions. Likelihood-based inference procedures are developed for both point and interval estimation in the proposed models. Simulation results show that the use of shape information significantly improves the accuracy of estimates. The proposed method is successfully applied to EC degradation data collected from real experiments.

Keywords:

• Bernstein bases
• semi-parametric model
• stochastic process models
• water treatment

Additional information

Funding

Dr. Tan’s research was supported by General Research Fund projects No. 11226716 and No. 11205118 sponsored by the Research Grants Council of Hong Kong, and Strategic Research Grant 7004522 funded by City University of Hong Kong. Dr. Ye’s research was supported by the National Research Foundation Singapore under its Campus for Research Excellence and Technological Enterprise (CREATE), the National Science Foundation of Jiangsu Province under grant BK20180232, and Singapore MOE Tier 2 under Grant R-266-000-125-112.

Notes on contributors

Lanqing Hong

Lanqing Hong received a B.Eng. degree (2014) from Shanghai Jiao Tong University. She received her Ph.D. degree (2018) in Industrial System Engineering and Management from National University of Singapore. Her current research interests include reliability engineering, degradation modeling, stochastic process models, environmental risk evaluation and data analysis for environmental sustainability problems. She is also working on small sample learning problems for artificial intelligence.

Matthias Hwai Yong Tan

Matthias Hwai Yong Tan is an assistant professor in the School of Data Science at City University of Hong Kong. He received his B.Eng. degree in Mechanical-Industrial Engineering from the Universiti Teknologi Malaysia, an M.Eng. degree in Industrial and Systems Engineering from the National University of Singapore, and a Ph.D. degree in Industrial and Systems Engineering from Georgia Institute of Technology. His research interests include uncertainty quantification, design and analysis of computer experiments, and applied statistics.

Zhi-Sheng Ye

Zhi-Sheng Ye received a joint B.E. (2008) in Material Science & Engineering, and Economics from Tsinghua University. He received his Ph.D. degree (2012) from National University of Singapore. He is currently an assistant professor in the Department of Industrial Systems Engineering and Management, National University of Singapore. His research interests include reliability engineering, complex systems modeling, and industrial statistics.