Overview

This paper presents a Gaussian Bayesian Network (GBN) for seasonal lake water quality (TP, chl-a, cyanobacteria and colour) forecasting. The GBN was developed and applied to Lake Vansjø in southeast Norway. The GBN was found well-suited for seasonal water quality forecasting and could be parameterised purely using observed data, despite this dataset being small. The forecasting performance of the GBN was assessed using a cross validation scheme, and the performance was also compared to that of a discrete BN (with the same structure) and a naïve forecast model; it was found that the 3 models performed similarly largely due to low interannual variability and high temporal autocorrelation in the study lake.

General comment

Overall, I think this is an interesting study that is very relevant to the HESS special issue. Although the forecasting results with the GBN were considered a mixed success at the study site, I do agree with the authors that the GBN seems to be a sensible and promising approach for water quality forecasting, and I think that by sharing all the code on GitHub the authors have provided a very useful tool for others to use and adapt. I very much enjoyed reading the paper which is both well-written and well-presented, and I believe it can be accepted after some minor revisions.

Below are my comments which I hope the authors will find useful. Lykke til!

Specific comments

1. My main concern is related to the discrete BN and the comparison to the GBN. It is interesting that the discrete BN did a mixed job of representing the relationships, however, I don’t understand why this happens and I think this could be elaborated on further. Specific comments in relation to this:
   
   (i) I’m not sure how the method you used to fit the CPTs works, but considering you have a small dataset and that you are using flat priors, I’m surprised that the fitted CPT in Table 6 seems to suggest that the evidence was strong (i.e., most of parent state combinations results in low-high probabilities of around 99%-1% and 95%-5% or vice versa). Intuitively, I would have thought that the probabilities would still be influenced by the flat prior given the small dataset, but the priors have been completely “outweighed” by the data. To me this suggest that there is something odd about the discretisation of the data and/or the target node states.
   
   (ii) I had a brief look at what I believe is your discretised input data files on Github (.. \BayesianNetwork\Data\DataMatrices\Discretized\), and I think these look a bit strange (although I appreciate these may not be the final version). First of all, the ‘colour_prevSummer’ node seems to have been given 3 states (L, M, H) contrary to what is stated in the manuscript. It also looks like the value for ‘chla’ does not always match the value of ‘chla_prevSummer’ the previous year. The same is the case for ‘colour’ and ‘colour_prevSummer’. I would urge the authors to double check these data files and see if this possibly explain (at least partly) the results of the discrete BBN.
   
   (iii) Finally, I wonder if it would not have been better to use expert opinion to reflect the priors in the discrete network before training, especially as you have a small dataset? To me this would seem sensible, and you already use expert opinion to inform the structure of the network. I also wonder whether you could just have discretised your GBN after it was created (in software like Netica and Genie you can specify continuous distributions and then subsequently discretise these distribution) and how the discretised model would then perform?
2. I’m not sure I fully understand how the leave-one-out cross validation works and I think it would be great if the authors could make this a bit clearer in section 2.7.1. Do you leave one data point (i.e., a year?) out at the time and then fit the GBN to the remaining data and see how well the GBN predicts the target node time-series? Or how well the GBN predicts the data point that was left out? Or something else? I also don’t really understand why the cross validation is stochastic and why it was run a default 20 times.

Minor comments

1. Author name: I believe it should be James E. Sample. Alternatively, change JES to JS in author contributions (L670).
2. L21: change “wasn’t” to “was not”
3. L63-64: maybe worth explaining what polymictic and dimictic lakes are; at least I’m not familiar with these terms.
4. Figure 1: where is the outlet from Vanemfjorden? At Moss River?
5. L127+: Can you explain briefly why Vanemfjorden with its short residence time is more susceptible to eutrophication and cyano blooms than Storefjorden, and why it does not seem to be related to the major input source from River Hobol?
6. L176: Should it be 1998-2013? At least in L179 you seem to suggest NIVA for 2013 as well.
7. L188: specify that it is River Hobol
8. L192: Change “As the aim” to “The aim”. Alternatively combine the two sentences in L192-195 and remove “therefore” on L194.
9. Figure 2: You could consider plotting error bars to give an idea of the variation in the different parameters.
10. L227-229: I’m not sure I understand why these features would have to be included as latent variables. Because they are not measured? From Figure 1, it looks like there are monitoring stations in the eastern lake basin (the same as Storefjorden?), so would you not have water quality data from here?
11. Table 1 and Table 2: I find it slightly confusing what features are included. Are all the features for the 6-month growing season as well as for the previous winter season (Nov-Apr), i.e., the number of features used for all variables are at least 2x13? Looking at Table 2, and if I understand the caption correctly, it looks like cyano has 8 additional features, so 34 in total (not 33).
12. Table 2: Are the features chl-a_prev, cyano chl-a and cyano_prevSummer for the lake?
13. L293-300: I think this would be better presented as a table, where you clearly state what is defined as Low and High in the model. The specific comments related to the water quality parameter in question could then be added in a separate column (e.g., that L and H for TP is in fact lower and upper moderate and so on).
14. L304+: I don’t follow this part of the discretisation process and why you get unbalanced class sizes. Are the variables still transformed in the discrete version and fairly normal?
15. L348+ and Figure 3: Is the relationship between number of calm days and TP negative? To me it looks like the two are positively correlated.
16. L355: Are wind speed (winter_wind) and TP(PS) positively correlated?
17. Figure 3-6: What are the bell-shaped curves and how were they derived?
18. Figure 7: Is TP_prev supposed to be linked to chl-a_prev? If so, should chl-a_prev not have a beta1_TP_prev coefficient?
19. L456: Should it not say: “For parentless nodes…”? Some of your nodes are both parent and child nodes (e.g. lake TP is the parent of lake chl-a but the child of TP_prev)
20. L526: As you say, this bias in cyano is likely due to the box-cox transformation. Rather than the mean, would it not have been better to use the median (or mode)? Also, did you calculate the mean before or after back-transformation?
21. L656: change wasn’t to was not

General comments:

This study presents an application of a Continuous Bayesian Network (CBN) to seasonal (6-month average) algal forecasting in a northern lake.  This is likely the first use of CBN for this purpose.  In general, the model performs similarly to a traditional (discretized) BN and a naïve model (using the mean from the previous year).  It could be a good fit for this special issue, but I do have several concerns, as outlined below.

I’m not really sure that there is a strong contribution, as the CBN does not perform particularly well. Also, the model appears to be based on existing software (an R package), so there isn’t new methods development.   If the objective of the study is to provide a thorough demonstration of CBNs for algal bloom modeling, that could potentially be an important contribution.  In this case, I’d like to see more demonstrations of how the CBN approach (e.g., Figure 7) can be advantageous for studying a system or supporting management.  In my opinion, the current discussion is too focused on skill assessment (e.g., R2), which probably doesn’t do justice to the CBN approach.  Also, probabilistic predictions using various linear covariates can also be obtained through multiple linear regression (frequentist or Bayesian), so why use a CBN?  I think there are potentially good reasons for using a CBN, but they aren’t compellingly demonstrated in the current manuscript.  Also, I’d like to see more discussion of how this effort compares to other CBN (or BN) applications for water quality or environmental sciences, more broadly.

Major comments:

The paper includes a tangential analysis on making predictions at smaller time scales (e.g., Lines 208-215).  I recommend removing this material, as it doesn’t seem relevant to the main focus of this paper (no CBN was used).  Furthermore, this additional analysis doesn’t provide new insights (that aren’t available through existing phytoplankton literature).  It seems a bit “tacked on”.  If you do keep this analysis, the data should be presented (as in Figure 2 for the six-month model).

The variable selection process seems ad hoc (Section 3.1.1), making it somewhat hard to follow and likely difficult to reproduce.  Some of the explanations seem questionable.  For example, the article cites previous literature showing that “windier summers” are relevant, but the CBN uses winds from the previous 6 months (prior to summer), right?  I have two general suggestions.  First use clear and consistent terminology that clarifies which time periods you are talking about (also use consistent notation across the different figures and tables).  Second, drop wind from the 6-month analysis altogether.  Much of the text is a somewhat tortuous explanation (at least for this reader) of reasons to include/exclude wind speed, while in reality, the authors readily acknowledge that wind speed is only relevant at smaller time scales (e.g., Lines 443-445: “wind would likely only have an immediate and relatively short-lived effect…”), not ~6 months in advance.

Detail-oriented comments:

Line 11:  Clarify in the abstract that you are predicting a May-October average (rather than daily predictions).

Line 20:  The term “purely parameterized” is used multiple times throughout this manuscript, but I don’t understand what it means or how it is justified.  As noted above, the parameterization process seems somewhat ad hoc to me. 

Line 23:  Suggest clarifying what is meant by a “naïve forecast” here.

Line 44:  Models for Lake Erie cyanobacteria blooms (including Bayesian models) predict the maximum bloom size months in advance.

Line 56:  Could you explain why “colour” is particularly relevant to water treatment or provide a reference?

Figure 1:  Suggest including arrows to show dominant flow directions.

Table 2:  Clarify what averaging periods were used.

Line 273:  Clarify what normality test was used.

Figure 3, 4, 5:  Clarify why only certain features are shown in each figure.

Table 4:  The “Feature subset” column is confusing.  Use consistent terminology and explain in the caption.

Line 370-371:  Revise for clarity.

Line 422:  Suggest “wind-related” instead of “related” for clarity.

Line 458:  The term “credible” usually refers to the uncertainty in a parameter.  It could be good to present actual parameter uncertainties (e.g., credible intervals).  Also, I don’t think relationships matching the simple bivariate correlations necessarily makes them “credible” in any sense.  For example, see literature on Simpson’s Paradox.

Line 470:  Again, I’m not sure using simple bivariate correlations to evaluate a more sophisticated model makes sense.

Table 6:  To me, making some numbers bold isn’t effective for highlighting unexpected results.  It really depends on which particular pair of numbers is being compared.  Also, I wouldn’t describe some of these relationships as a “physical” response.

Line 569:  This statement seems too strong and/or requires clarification.

Line 644:  This is clearly true (based on the general nature of a GBN), but it wasn’t really explored in this study.  I’m not sure why it is a conclusion.

Line 659:  This seems like a bit of a stretch.  I’m not sure that any “expert” can predict an extreme event ~6 months in advance.  Maybe the authors mean something else, but I can’t imagine what.