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Probabilistic Evaluation of Causal Relationship between Variables for Water Quality Management

J. He*

    Department of Civil Engineering, Schulich School of Engineering, University of Calgary, 2500 University Drive, Calgary, T2N 1N4, Canada

*Corresponding author. Tel: +1-403-2204112 Fax: +1-403-2827026 Email:


In aquatic environments, a complex interplay exists among physical, chemical, and biological water quality characteristics, which are constantly influenced by exogenous factors such as hydrological, meteorological and geological conditions. Due to the spatial and temporal variations of exogenous factors, the relationship between the water quality parameters and these factors hence becomes complicated and challenging. Given the large data matrix, one type of methods frequently seen in the literature belongs to the multivariate analysis which generates a qualitative measure of the relationships among variables in a geometrically intuitive way. However, a quantitative evaluation from a probabilistic perspective is favorable since it defines a measurable causality among variables so that more efficient water management strategies can be formulated. This paper illustrates a new way to discover the relationship between two variables by estimating their joint distribution which fully interprets the statistical dependence. A multivariate Gaussian mixture model was employed to describe the data. The model parameters were determined using the previously developed estimation approach, which is capable of dealing with both multivariate variables and censored data. The joint distribution and the conditional distribution were computed and used to describe the statistical distribution of water quality parameters, which are subject to the effects of hydro-meteorological conditions. The method was demonstrated by a case study on the Bow River in Alberta, Canada. The results shed light on how one variable affects the distribution of the other variable under complex environments in a probabilistic context.

Keywords: joint distribution, conditional probability, water quality management, mixture model, censored water quality data

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