Seasonal forecasting of lake water quality and algal bloom risk using a continuous Gaussian Bayesian network

Freshwater management is challenging, and advance warning that poor water quality was likely, a season ahead, could allow for preventative measures to be put in place. To this end, we developed a Bayesian network (BN) for seasonal lake water quality prediction. BNs have become popular in recent years, but the vast majority are discrete. Here we developed a Gaussian Bayesian network (GBN), a simple class of continuous BN. The aim was to forecast, in spring, total phosphorus (TP), chlorophyll-a (chl-a), cyanobacteria biovolume and water colour for the coming growing season (May–October) in lake Vansjø in southeast Norway. To develop the model, we first identified controls on inter-annual variability in water quality using correlations, scatterplots, regression tree based feature importance analysis and process knowledge. Key predictors identified were lake conditions the previous summer, a TP control on algal variables, a colour-cyanobacteria relationship, and weaker relationships between precipitation and colour and between wind and chl-a. These variables were then included in the GBN and conditional probability densities were fitted using observations (≤ 39 years). GBN predictions had R2 values of 0.37 (cyanobacteria) to 0.75 (colour) and classification errors of 32 % (TP) to 13 % (cyanobacteria). For all but lake colour, including weather nodes did not improve predictive performance (assessed through cross validation). Overall, we found the GBN approach to be well-suited to seasonal water quality forecasting. It was straightforward to produce probabilistic predictions, including the probability of exceeding management-relevant thresholds. The GBN could be purely parameterised using observed data, despite the small dataset. This wasn’t possible using a discrete BN, highlighting a particular advantage of using GBNs when sample sizes are small. Although low interannual variability and high temporal autocorrelation in the study lake meant the GBN performed similarly to a seasonal naïve forecast, we believe the forecasting approach presented could be useful in areas with higher sensitivity to catchment nutrient delivery and seasonal climate, and for forecasting at shorter time scales (e.g. daily to monthly). Despite the parametric constraints of GBNs, their simplicity, together with the relative accessibility of BN software with GBN handling, means they are a good first choice for BN development, particularly when datasets for model training are small.

Handling Overlaps When Lifting Gaussian Bayesian Networks

Gaussian Bayesian networks are widely used for modeling the behavior of continuous random variables. Lifting exploits symmetries when dealing with large numbers of isomorphic random variables. It provides a more compact representation for more efficient query answering by encoding the symmetries using logical variables. This paper improves on an existing lifted representation of the joint distribution represented by a Gaussian Bayesian network (lifted joint), allowing overlaps between the logical variables. Handling overlaps without grounding a model is critical for modelling real-world scenarios. Specifically, this paper contributes (i) a lifted joint that allows overlaps in logical variables and (ii) a lifted query answering algorithm using the lifted joint. Complexity analyses and experimental results show that - despite overlaps - constructing a lifted joint and answering queries on the lifted joint outperform their grounded counterparts significantly.

Extracting Job Title Hierarchy from Career Trajectories: A Bayesian Perspective

A job title usually implies the responsibility and the rank of a job position. While traditional job title analysis has been focused on studying the responsibilities of job titles, this paper attempts to reveal the rank of job titles. Specifically, we propose to extract job title hierarchy from employees' career trajectories. Along this line, we first quantify the Difficulty of Promotion (DOP) from one job title to another by a monotonic transformation of the length of tenure based on the assumption that a longer tenure usually implies a greater difficulty to be promoted. Then, the difference of two job title ranks is defined as a mapping of the DOP observed from job transitions. A Gaussian Bayesian Network (GBN) is adopted to model the joint distribution of the job title ranks and the DOPs in a career trajectory. Furthermore, a stochastic algorithm is developed for inferring the posterior job title rank by a given collection of DOPs in the GBN. Finally, experiments on more than 20 million job trajectories show that the job title hierarchy can be extracted precisely by the proposed method.