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Eutrophication of lakes and the risk of harmful cyanobacterial blooms due is a major challenge for management of aquatic ecosystems, and climate change is expected to reinforce these problems. Modelling of aquatic ecosystems has been widely used to predict effects of altered land use and climate change on water quality, assessed by chemistry and ph...

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... name Unit No. of values Node states 1 2 3 4 5 6 1 Management Ref Worst Best 1 Climate Ref Had 1 Year 1990-1995 1996-2001 2002-2007 2008 series. INCA-P produced daily predictions of discharge and material transport in the river (concentration of suspended solids, soluble reactive P and total P (TP)), which were then passed to the lake model. ...
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... a BN model, each node (variable) is typically defined by a discrete probability distribution across a number of alternative states (i.e., intervals or categories). This structure enables differ- ent types of information to be linked by conditional probability tables (CPT) (see Table 2 and section 3.1). Although continuous variables may also be included in a BN with certain restrictions, this type of nodes are not considered here. ...
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... computing capacity of computers have increased to the extent that even relatively complex and big networks can be built and run ( Lehikoinen et al., 2013), but more complex BNs nevertheless require more data or other information than simpler ones. In this study, we aimed at including only the nodes that were necessary to (i) run the model according to selected scenarios, (ii) represent particular processes that were important Table 2 Examples of conditional probability tables (CPT) for each module of the BN model. Each column contains the probability distribution of a child node for a given combination of states of the parent nodes. ...
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... discrete probability distributions in the CPTs are also obtained by different approaches in the different BN modules. Table 2 contains examples of CPTs for each module, while all CPTs are included in Supplementary data. ...
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... = Reference, Year = 1990-1995, Irradiance = 0-100 and Temperature = 0-10) was determined by the count of predicted chl-a values obtained in this interval for this par- ticular combination of states of the parents nodes (40) divided by the total number of observations for this combination (3015). I.e., the probability is 40/3015 = 0.013 (the upper left cell in Table 2a). Thus, the probability distribution in this column arises from the variability between the 60 MyLake model realisations as well as from the temporal variability during the period 1990-1995. ...
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... Table B1a); here an assumed probability distribution based on the neighbour col- umn was inserted. For the nodes in module 2, where the CPTs had a high number of columns, columns with Experience = 0 were populated with probability distributions from the neighbour col- umn (see example in Table 2a). (Testing showed that the assumed probability distributions in such cases had negligible effects on the posterior probability distributions of the child nodes). ...
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... example, out of the 34 observations of Cyano concentration below 1000 g/L in the May-June season, 10 observations (probability 0.29) came from a year where the CyanoMax in the same year exceeded 2000 g/L. The total number of cyanobacteria samples (90) was relatively low for calculating the 9 frequency distributions in the CPT of Cyano (and of CyanoMax; Table 2c and d). We therefore complemented the temporal data for the target lake with the larger spatial dataset from the regional dataset EUREGI (described in section 2.2.3). ...
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... one part of a dataset should be used for "training" (model calibration) while another part is reserved for evaluation by comparison with model predictions (Chen and Pollino, 2012). However, the data on the most crucial component of this model -Cyanobacteria -could not be divided without com- promising the calibration (construction of CPTs; see Table 2b). Moreover, predictions based on future scenarios could not be compared to real data. ...
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... The CPT for cyanobacteria. Due to the limited number of cyanobacteria observations (Table 2b), to reserve a subset of the cyanobacteria data for evaluation purposes would not be meaningful. Instead, we used the independent EUREGI dataset (see section 2. ...
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... effect of water temperature on cyanobacteria. Moreover, since the conditional probabilities used for calculating posterior probabilities for cyanobacteria are based on very few obser- vations for some of the parent state combinations (Table 2b), it is important to check that these CPTs do not provide spuri- ous results. We therefore inspected more closely relationship between temperature, Chl-a and cyanobacteria by setting evi- dence (fixating probabilities) for the nodes Temperature and Chl-a. ...
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... out of 90 from Lake Vansjø). Nevertheless, the EUREGI dataset gave similar probability distributions in the CPT for cyanobacteria (Table B2) to those from Lake Vansjø (Table 2b-c). Consequently, model version 3 with CPT from the EUREGI dataset predicted effects of climate and management scenarios on ecological status of cyanobacte- ria (Fig. B2e) that were very similar to the default model version (Fig. 5e). ...
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... out of 90 from Lake Vansjø). Nevertheless, the EUREGI dataset gave similar probability distributions in the CPT for cyanobacteria (Table B2) to those from Lake Vansjø (Table 2b-c). Consequently, model version 3 with CPT from the EUREGI dataset predicted effects of climate and management scenarios on ecological status of cyanobacte- ria (Fig. B2e) that were very similar to the default model version (Fig. 5e). ...
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... the com- bined Phytoplankton status, a similar solution was used, with some exceptions: when the status of Cyano was better than or equal to the status of Chl-a, the combined status was set equal to the status of Chl-a (Table A1f). The overall lake status (Table 2d) was set equal to the phytoplankton status when the physico-chemical status was equal or better, and to one lower state when the physico-chemical status was worse. ...
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... B1-B5 . Tables B1 and B2. ...
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... probability table for the two cyanobacteria nodes based on the alternative, larger dataset (EUREGI, see section 2.2.3): (a) Cyano, (b) CyanoMax (corresponding to Table 2c and d, respectively). For more information, see Table 2. Fig. B3. ...
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... Cyano, (b) CyanoMax (corresponding to Table 2c and d, respectively). For more information, see Table 2. Fig. B3. ...

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... At present, global lake management is facing the dual stress of climate warming and eutrophication (Carter and Schindler, 2012;Moe et al., 2016). The severe human disturbance caused a large amount of internal and external organic matter to enter the freshwater lakes, leading to the intensification of the ecosystem eutrophication and subsequent formation of algae blooms. ...
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... MyLake has been successfully applied to various projects as a standalone simulation tool. For example, assessing ice regime (Livingstone and Adrian, 2009), lake thermodynamics (Woolway et al., 2017), greenhouse gas emissions (Kiuru et al., 2018), light dynamics (Pilla and Couture, 2021) and predicting cyanobacterial blooms (Moe et al., 2016). ...
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... Thus, water quality degradation affects the ecosystem, the water supply and recreation activities (Ansari et al., 2010;Wurtsbaugh et al., 2019). Furthermore, hydroclimatic variability associated with climate change, especially temperature, precipitation and evaporation, are important drivers that can act synergistically with the anthropic influence ruling changes in the environmental status of water bodies (O'Neil et al., 2012;Moe et al., 2016;Moorhouse et al., 2018;Nazari-Sharabian et al., 2018;Huo et al., 2019). ...
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... Our case study site is the western basin of Vansjø, a shallow mesotrophic/eutrophic lake in southeastern Norway. A number of BN models have previously been applied in the lake (Couture et al., 2014(Couture et al., , 2018Moe et al., 2016Moe et al., , 2019Barton et al., 2008), but these were all discrete metamodels, i.e. the BN nodes summarised process-based model simulations, statistical relationships, expert opinion, and/or data distributions, and the studies were focused on the longer-term impacts of climate, land use, and land management change. Here, the aim was to provide medium-term forecasts to support lake management by developing a model able to predict, in the spring of a given year, water quality for the coming growing season (May-October), including the probability of lying within WFD ecological status classes for total phosphorus (TP), chl a, and cyanobacteria. ...
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... Thus, water quality degradation affects the ecosystem, the water supply and recreation activities (Ansari et al., 2010;Wurtstbaugh et al., 2019). Furthermore, hydroclimatic variability associated with climate change, especially temperature, precipitation and evaporation, are important drivers that can act synergistically with the anthropic influence ruling changes in the environmental status of water bodies (O'Neil et al., 2012;Moe et al., 2016;Moorhouse et al., 2018, Nazari-Sharabian et al., 2018Huo et al., 2019). ...
... However, this theory has also met with strong opposition. erefore, there is a need to select appropriate fund performance evaluation models to further develop the fund evaluation system [7][8][9][10][11][12][13][14]. ...
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Based on Bayesian method, this paper constructs a model for estimating fund performance evaluation, and uses machine learning algorithm to construct a sampler that can sample on the basis of conditional distribution. Sampling is used for stress test, so as to give the closeness of all possible test results and data results. The results show that performance evaluation is affected by many factors, and the resistance to risk plays an important role in the whole performance evaluation. At the same time, the Bayesian model in machine learning can quickly and accurately approach the statistical results, which is of great significance for predicting performance evaluation.
... Here, we develop a GBN to forecast seasonally water quality in the western basin of lake Vansjø, a shallow 105 mesotrophic/eutrophic lake in southeast Norway. A number of BN models have previously been applied in the lake (Barton et al., 2008;Couture et al., 2018;Couture et al., 2014;Moe et al., 2019;Moe et al., 2016), but these were all discrete metamodels, i.e. the underlying network nodes were 'response surfaces' summarising a combination of process-based model simulations, expert opinion or data distributions, and the studies were focused on the longer-term impacts of climate, land use and land management change. Here, the aim was to provide medium-term forecasts to support lake management, by 110 developing a model able to predict, in spring, water quality for the coming growing season (May -October), including the probability of lying within WFD ecological status classes for TP, chl-a and cyanobacterial. ...
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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.
... These findings and principles are widely applicable, and further accumulation of high-frequency monitoring data at different sites and for different variables would enable the selection of inference methods and development of efficient priors to expand. Bayesian modelling can be applied to quantify uncertainty of classifying not only the physicochemical status but also the biological status and, consequently, the overall ecological status (Moe et al. 2016;Loga et al. 2018). It is important to discuss further how to apply Bayesian methods to the overall procedure of WFD status classification and which statistical models to use. ...
Article
River water quality monitoring at limited temporal resolution can lead to imprecise and inaccurate classification of physicochemical status due to sampling error. Bayesian inference allows for the quantification of this uncertainty, which can assist decision-making. However, implicit assumptions of Bayesian methods can cause further uncertainty in the uncertainty quantification, so-called second-order uncertainty. In this study, and for the first time, we rigorously assessed this second-order uncertainty for inference of common water quality statistics (mean and 95th percentile) based on sub-sampling high-frequency (hourly) total reactive phosphorus (TRP) concentration data from three watersheds. The statistics were inferred with the low-resolution sub-samples using the Bayesian lognormal distribution and bootstrap, frequentist t test, and face-value approach and were compared with those of the high-frequency data as benchmarks. The t test exhibited a high risk of bias in estimating the water quality statistics of interest and corresponding physicochemical status (up to 99% of sub-samples). The Bayesian lognormal model provided a good fit to the high-frequency TRP concentration data and the least biased classification of physicochemical status (< 5% of sub-samples). Our results suggest wide applicability of Bayesian inference for water quality status classification, a new approach for regulatory practice that provides uncertainty information about water quality monitoring and regulatory classification with reduced bias compared to frequentist approaches. Furthermore, the study elucidates sizeable second-order uncertainty due to the choice of statistical model, which could be quantified based on the high-frequency data.
... Bayesian networks are commonly used in environmental modelling (Aguilera et al., 2011;Barton et al., 2012;Landuyt et al., 2013). For example, there are numerous BN models developed for assessment and management of water quality (Barton et al., 2014;Borsuk et al., 2004Borsuk et al., , 2012Moe et al., 2016Moe et al., , 2019. BN modelling has been applied more rarely within ecotoxicology and ecological risk assessment, but recent publications have demonstrated the applicability of BN modelling in these fields Landis et al., 2017Landis et al., , 2019Lehikoinen et al., 2015). ...
Conventional environmental risk assessment of chemicals is based on a calculated risk quotient, representing the ratio of exposure to effects of the chemical, in combination with assessment factors to account for uncertainty. Probabilistic risk assessment approaches can offer more transparency, by using probability distributions for exposure and/or effects to account for variability and uncertainty. In this study, a probabilistic approach using Bayesian network modelling is explored as an alternative to traditional risk calculation. Bayesian networks can serve as meta-models that link information from several sources and offer a transparent way of incorporating the required characterization of uncertainty for environmental risk assessment. To this end, a Bayesian network has been developed and parameterised for the pesticides azoxystrobin, metribuzin, and imidacloprid. We illustrate the development from deterministic (traditional) risk calculation, via intermediate versions, to fully probabilistic risk characterisation using azoxystrobin as an example. We also demonstrate seasonal risk calculation for the three pesticides. This article is protected by copyright. All rights reserved.