Abstract In this study we propose an approach to predicting the probability that river waters will exceed a water quality standard. The study used a two-part generalised linear modelling approach within a Bayesian framework. Binomial regression was used to model the probability that a water quality standard would be exceeded and included two factors – the difference between sampling sites and difference between years of sampling. Using a Bayesian approach meant that information could be drawn from all observations from all sites, across all years, and that all results would come with a measure of uncertainty. Furthermore, although some known factors could not be included in the binomial regression, they could be included using Bayes’ rule to enhance and inform the results. This approach was applied to assessing the probability of nitrate concentrations in English river waters exceeding the current nitrate water quality standard of 11.3 mg N/l. The study showed that the Bayesian approach decreased the measures of uncertainty in the predicted outcomes was reduced by an average of 60% and increased the effective sample size by 64%.The best-fit model had a root mean square error (RMSE) of 7.9% which equated to an error of ±1 sample above the water quality standard for the median site. When interaction of factors could be included, then RMSE decreased to 3.8%. It was not possible to include a diurnal cycle, owing to a paucity of sub-daily sampling, but there was a significant seasonal cycle and so outputs could be adjusted by means of Bayes’ rule to predict water quality standard exceedance each month. Comparison with the current method of classification shows no significant difference between five out of the six lowest classifications with only the highest classification being correlated with the estimated exceedence rate. With respect to nitrate in English river waters, the average exceedance rate was 8.3% but was declining at a statistically-significant rate of 0.09%/yr2.

中文翻译：

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违反水质标准的概率——河水硝酸盐浓度的概率模型

摘要 在这项研究中，我们提出了一种预测河水超过水质标准的概率的方法。该研究使用了贝叶斯框架内的两部分广义线性建模方法。二项式回归用于模拟超过水质标准的概率，包括两个因素——采样地点之间的差异和采样年份之间的差异。使用贝叶斯方法意味着可以从所有地点、所有年份的所有观测中获取信息，并且所有结果都带有一定的不确定性。此外，虽然一些已知因素无法包含在二项式回归中，但可以使用贝叶斯规则将它们包含在内以增强和告知结果。该方法用于评估英国河水中硝酸盐浓度超过当前硝酸盐水质标准 11.3 mg N/l 的可能性。研究表明，贝叶斯方法降低了预测结果中的不确定性度量，平均减少了 60%，有效样本量增加了 64%。最佳拟合模型的均方根误差 (RMSE) 为 7.9 %，相当于高于中位点水质标准±1个样本的误差。当可以包括因素的相互作用时，RMSE 下降到 3.8%。由于缺乏次日采样，不可能包括昼夜循环，但存在显着的季节性循环，因此可以通过贝叶斯规则调整输出，以预测每个月的水质超标情况。与目前的分类方法相比，六个最低分类中的五个之间没有显着差异，只有最高分类与估计的超标率相关。就英国河水中的硝酸盐而言，平均超标率为 8.3%，但以 0.09%/yr2 的统计显着率下降。