Non-parametric Bayesian modeling for risk-based management of Bathing Water Quality

Bayesian non-parametric models are rarely used for predictive modeling of recreational waters. In the present study, we use a Dirichlet Process Gaussian Mixture Model (DPMM) for model-based clustering of hydrologic data collected at three river bathing sites (3 rivers, N = 256, N = 281, N = 1170). The three sites differ in their climatic conditions. Rivers 1 and 3 are continentally influenced (highly unbalanced dataset with few but severe contamination episodes); River 2 is more maritime-influenced (regular rainfall leads to balanced data set with regularly occurring pollution episodes); DP models can be used for model-based clustering, where the number of clusters does not have to be pre-defined but is inferred from the dataset itself. For each new observation x­_I, the probability of belonging to an already existing cluster as well as the probability of belonging to a new cluster is calculated. We used this property to identify unknown, i.e. high-risk situations, at the individual river sites.

We first applied the DPMM to the available hydraulic training data for model training before conditionally updating a predefined lognormal prior for each cluster, representing the E.coli concentration in the river. For prediction, we first evaluated whether a new observation belongs to an existing cluster or whether it constitutes a new cluster. Based on this evaluation, we used either the posterior predictive distribution or the prior predictive distribution for cases where a new cluster was identified. The water quality assessment was subsequently based on the 90^th and 95^th percentiles of the individual predictive distribution. Model performance was evaluated by means of calculating four criteria: (i) the root mean squared error (RMSE), (ii) the percentage coverage of predictive intervals in relation to the test data (80%), (iii) the detection rate of confirmed contaminations (E.coli > 1800 MPN/100 mL), and (iv) the number of predicted bathing days in the test data. The ratio between training and test data was incrementally altered from 10-70%. We compared the DPMM model with four alternative data-driven algorithms: (i) an intercept-only model (zero model), (ii) a multiple linear regression based on stepwise variable selection (stepwise), (iii) a quantile random forest (QRM) and (iv) a Bayesian updating approach, where individual clusters were predetermined manually based on hydrologic characteristics instead of being inferred by the DPMM. The results show that especially for River 1 and 3, only the Bayesian models could predict over 90% of observed contaminations. Through its ability to identify unknown hydraulic situations and its combination with a prior predictive distribution, the DPMM algorithm can predict high-risk periods without the need to be trained on a dataset that includes this specific contamination information. This is achieved as it identified new hydrologic information as anomalies related to the training set. Thereby, the approach is especially suitable as a precautionary approach for recreational waters, where information-rich datasets are often missing.