Regression with continuous mixture of Gaussian distributions for modeling the memory time of water treatment

In this article, a new model of regression with a continuous mixture of Gaussian distributions is developed. The proposed model is applied for the prediction of the time memory of water treatment corresponding to the minimal conductivity. The model training is performed on the basis of a dataset concerning the conductivity of the water and its variations. In order to estimate the regression parameters, we suggested an extension of the expectation maximization (EM) algorithm. More precisely, we explicitly computed the conditional expectation of the complete data log-likelihood in the EM algorithm. The results of this algorithm provide consistent estimators of the parameters. The calibrated regression model with a continuous mixture of Gaussian distributions outperforms the classical regression model, as shown in the numerical analysis. We prove that the EM algorithm outperforms the two-Stage Least Squares method. These results are presented to illustrate the performance and applicability of this proposed model in water treatment.