A feature reconstruction-based multi-task regression model for cyanobacterial distribution forecasting along the water column

Highlights

• Cyanobacterial water pollution threatens the cleaner ecosystem and human health.

• Data on cyanobacteria cell density at 11 water depths were collected for modelling.

• A multi-task regression model has been built to share knowledge among water depths.

• ∼20% forecasting error has been reduced compared to nonlinear single-task models.

• The model applicability has been demonstrated by real-world data of a tropical lake.

Abstract

Cyanobacterial water pollution has been threatening the cleaner ecosystem and urban sustainability due to the harmfulness to aquatic ecosystems and human health, which triggers the development of an effective forecasting tool for cyanobacterial blooms. Along the water column, the variations in cyanobacteria cell densities show various distribution patterns and are influenced by multiple environmental factors. Most data-driven models treat cyanobacteria forecasting at a specific water depth as a single task, which fails to share knowledge amongst water depths, resulting in unfavourable forecasting accuracy. This is why an increasing number of nonlinear black-box models have been built for cyanobacteria forecasting but at the expense of model interpretability. This study aims to investigate whether forecasting accuracy and model interpretability can be enhanced by (i) using easily accessible predictors and (ii) developing a feature reconstruction-based multi-task regression model with knowledge sharing amongst water depths. Real-world data from a tropical lake are used to evaluate the effectiveness of the model. For the studied lake, the highest average cyanobacteria cell density occurs at 1.0 m, after which it decreases by over 30% at 5.5 m. The correlation coefficients of time-serial cyanobacteria cell densities between adjacent water depths are greater than 0.95 (P < 0.001). The forecasting results indicate that, compared to single-task nonlinear models, 20.59%, 16.25%, and 22.70% error reductions, measured by the mean square error, are achieved for one-day-ahead, two-day-ahead, and three-day-ahead cyanobacterial bloom forecasts. The accurate bloom and non-bloom signals under the proposed model are up to 94.81% and 98.28%. Based on the proposed model, the relative importance of predictors, the sparsity of regression coefficients, and the covariance relationship of regression coefficients can interpret the model adequately and elucidate the mechanism of knowledge sharing and forecasting accuracy improvement.

Introduction

Harmful cyanobacterial blooms have raised worldwide concerns as they are threatening the cleaner ecosystem (Nuamah et al., 2020) and human health (Burford et al., 2020). Cyanobacterial blooms not only lead to the degradation of water quality and the death of aquatic life (Huisman et al., 2018) but also produce toxins, e.g. microcystins, that threaten the sustainability of drinking water and food safety (Liu et al., 2018). As a measure of the sustainable management of freshwater resources, the accurate forecasting of cyanobacterial blooms in freshwater bodies assists decision-makers in taking proactive action to mitigate risks (Davidson et al., 2016) and helps residents to mitigate the potentially adverse effects of using contaminated water (Chen et al., 2015). The distribution of cyanobacteria depends on the water depth (Ni et al., 2018). Compared to cyanobacteria forecasting in surface water, the vertical distribution forecasting for cyanobacteria cell densities is even more crucial as freshwater at different water depths contributes to aquatic production (e.g. aquaculture and water intakes for waterworks) and recreational activities (e.g. fishing, boating, and swimming). The variations in cyanobacteria cell densities along the water column are influenced by multiple factors. Such variations and multivariate aquatic system make cyanobacterial blooms challenging to be forecasted.

Conventional forecasting models for cyanobacterial blooms include process-based models and data-driven models (Rousso et al., 2020). Process-based models (Qin et al., 2015) strongly depend on the domain knowledge of cyanobacterial blooms, and such models have many hyper-parameters to calibrate. They are commonly used for scenario analysis (Chen et al., 2015). The representative data-driven models include double exponential smoothing (DES) (Shuhaibar and Riffat, 2008), multiple linear regression (MLR) (Rajaee and Boroumand, 2015), auto-regression integrated moving average (ARIMA) (Chen et al., 2015), fuzzy logic (Kim et al., 2014), Bayesian hierarchical modelling (Obenour et al., 2014), hidden Markov model (Jiang et al., 2016), evolutionary algorithms (Cao et al., 2016), artificial neural networks (ANN) (Tian et al., 2017), and support vector regression (SVR) (Lou et al., 2017). ANN and SVR models have been recently used to assess water eutrophication with algae blooms (Nieto et al., 2019) and simulate cyanobacterial blooms in the tidal freshwater (Shen et al., 2019). These data-driven models normally focus on forecasting at a specific water depth. Individual models for different water depths need to be built when using such models for vertical distribution forecasting, which not only increases modelling workload but also neglects knowledge sharing amongst water depths. This is why increasingly complicated nonlinear black-box models have been employed to improve forecasting accuracy but at the expense of model interpretability. For example, the adaptive particle swarm optimisation-support vector regression (APSO-SVR) has been developed (Yao et al., 2020) to enhance the forecasting accuracy of a basic SVR model. Its counterpart for classification/prediction—namely, adaptive particle swarm optimisation-support vector machine (Moodi et al., 2020)—has also presented favourable performance. Although these black-box data-driven models, including ANN- and SVR-based models, offer passable performance via mining some nonlinear rules, the weak interpretability restricts the applicability (Zhang et al., 2020).

The spatial-temporal information could enhance the understanding of a system, which has broad applications, such as the understanding of the spatial-temporal characteristic of water quality (Zhang and Chen, 2011) and the analysis of a multi-event system (Dubey et al., 2017). Except for the time dimension of data monitoring, the water column can be naturally regarded as a spatial dimension. As an early attempt, Torres et al. (2011) built an ANN model with three outputs to forecast cyanobacteria cell density at three water depths, including the surface, euphotic zone limit, and bottom. Mattei et al. (2018) built a depth-resolved ANN model to forecast marine phytoplankton production at six fixed depths. The depth-resolved model was found superior in forecasting accuracy compared to the depth-integrated model with multiple co-predictors and one depth-integrated phytoplankton output (Mattei et al., 2018). The phenomenon of accuracy improvement in these models is attributed to the multi-output mechanism (Xu et al., 2019) with shared hidden neurons, which helps integrating common information to improve model performance. Although such a knowledge-sharing method is useful, the practical requirements of model interpretability and output stability may weaken the applications of multi-output ANN models, which trigger the development of a simple model with competitive accuracy and better interpretability. Due to the complicated nonlinear functions, the ANN models lack the most intuitive and quantitative interpretation of knowledge sharing amongst water depths. The uncertainties in multiple hyper-parameters and the random seeds used in the ANN models generally result in nonstable training results, which may not decrease the forecasting accuracy but could cause different interpretations for such models. Inspired by the knowledge-sharing experience amongst water depths (Mattei et al., 2018), instead of using complicated nonlinear models, such as extreme learning models (Albadra and Tiuna, 2017), deep learning models (Wang et al., 2020), and reinforcement learning models (Chen et al., 2020), this study moves forward a step to investigate whether forecasting accuracy and model interpretability can be enhanced by (i) using easily accessible predictors and (ii) developing a linear multi-task learning model with knowledge sharing amongst water depths.

The problem characteristics of multiple water depths and common water-quality predictors naturally correspond to a type of intelligent models, namely, multi-task learning. Multi-task learning is a transfer learning paradigm in the machine learning field (Pan and Yang, 2009). The core idea of multi-task learning lies in sharing the domain information of related tasks to improve learning performance (Zhang and Yang, 2017). It has been widely used in many fields, such as spatial-temporal event forecasting at multiple sites (Zhao et al., 2015), water quality prediction for urban waste-water stations (Liu et al., 2016), lake water quality prediction at macroscales (Collins et al., 2019), and energy demand forecasting (Tan et al., 2020). A multi-task regression can be implemented either by linear or nonlinear models, in which knowledge (e.g. predictors, features, and parameters) can be shared either implicitly or explicitly (Pan and Yang, 2009). In this study, a linear multi-task regression model with explicit knowledge sharing is extended to improve model interpretability. To date, the multi-task modelling paradigm has not been developed for cyanobacterial distribution forecasting along the water column. In this study, unlike traditional single-task regression modelling that focuses only on a single depth without any relations to other depths, a feature reconstruction-based multi-task regression (FRMTR) model is developed to fuse data, share knowledge amongst water depths, and forecast the distribution of cyanobacteria cell density along the water column. In the proposed FRMTR model, high-level features are constructed from easily accessible predictors to enhance the forecasting ability. More details are explained in Section 2.2. The forecasting accuracy of the FRMTR model is compared with advanced models, including ANN and APSO-SVR. For evaluating the applicability of the proposed model, a whole year data of 11 water depths of a tropical lake in Singapore are first provided by Singapore’s National Water Agency. Such data are employed for both the model evaluation and the mechanism understanding of the forecasting model and cyanobacterial distributions along the water column.