1. More difficult data sets
Interactions of N and P & Wedge shaped scatter plots

2. How to deal with N and P ?
Assume that N and P act as potentially limiting nutrients.
Biomass of algae limited by the lower of N or P, depending on the relative requirements
EQR likely to be partly dependent on biomass, so also likely to be lowered by an excess of a non-limited nutrient.

3. I will use an artificial data set
200 values of P
random log normal distribution with a mean of 35 ug/L
200 values of N
random log normal distribution with a mean of 350 ug/L
200 EQR values derived by fitting 2 relationships
EQRP ~ a log(P) + c + error
EQRN ~ a log(N) + d + error
Where error is random distribution with a mean of 0 and a specified standard deviation
Assume N:P requirement of 15:1 (molar ratio) or 6.6 (mass ratio)
if N/P > 6.6, EQR = EQRP as P limited
If N/P < 6.6, EQR = EQRN as N limited

4. Relationships between EQR and each nutrient independently
N limited
P limited
Good/Moderate boundary values would be 58 ug/L P and 383 ug/L for N

5. Relationships between combined EQR and each nutrient
Fitting linear models to these data shows how the effect of the sites where either P or N is not influencing status increases the predicted GM nutrient boundary
Solution is to split the data set into P or N limited sites, based on N:P ratio
What ratio should be used?
Alternative is to fit quantile models.
Good/Moderate boundary values become 98 ug/L P and 5204 ug/L for N

6. Fit linear quantiles (using 08_TKit_fit_quantile_reg)
Quantile regression
chops up the data into small categories of nutrient concentration
Looks at the distribution of EQR values in each category
Fits regression lines to different quantiles
Note that the uncertainty will depend on the density of data points at particular points along the nutrient gradient

7. Fit linear quantiles (using 08_TKit_fit_quantile_reg)
Fitting quantile regression lines demonstrates how in this example a lower linear quantile would result in estimating the correct boundary
Inverted wedge shaped scatter for both N and P is indicative of nutrient limited systems

8. Lake data often have scatter plots that resembles an inverted wedge
Very shallow high alkalinity lakes, from N, CB and EC gigs.
(DataTemplate_L-CB2L_EC1.csv)

9. Check for wedge shaped relationships using quantile regression
Screen shot from “Visualise data” section of Shiny
Converging lines for TP
Less evidence of converging lines for TN
This may indicated N limited sites, but the evidence is less strong.

10. Plot the data with N:P as the “Group” variable
High alkalinity very shallow lakes (NGIG, CBGIG, ECGIG
Split the data into separate data sets
N:P molar <10 (N limited)
N:P molar (Co-limited)
N:P molar >20 (P limited)
(need to define linear ranges for each)

11. Regression model outputs from Shiny
SPLIT DATA (N and P limited sites)
All DATA

12. What about the lakes with N:P 10:20 range?
Might try fitting linear models to this category
Alternatively consider using
P boundaries derived from P limited lakes
N boundaries derived from N limited lakes
But combine with a rule that requires both boundary values to be exceeded. (Best of either)
(Sensible logic if assume limiting nutrient approach)

13. Example 2 Lake Phytoplankton LT, type BT3 – L-CB1 High alkalinity, clear water, shallow lakes
Output from 08_TKit_fit_quantile_reg_V2.R
Wedge shape data, rather sparse data set so uncertainty of quantiles is likely to be high.

14. Significance of quantiles for P
Toolkit provides graphs which compare the intercept and slope parameters with those of a OLS regression
Only the lowest and highest quantiles are significantly different from the OLS regression

15. Fitting additive quantile models non-linear
Note the large error bars in the area of the good/moderate boundaries, except for the 50th quantile
Need more data to pursue this

16. Add data from LV Type L5, clear water shallow high alkalinity
Additional data from LV has a similar scatter
Lower quantiles now start to differ significantly

17. Additive smooths Error bars close to the GM boundary are now smaller
> Bound.A #list the estimated boundaries for the Additive quantile model
tau HG.est GM.est
NaN
Note. LT & LV lakes all have a high N:P ratio, so N limitation unlikely to be explanation of inverted wedge

18. LV high alkalinity lakes, Type L1 & L5 – Clear, Types L2 & L6 - Humic
Humic lakes are clearly shifted to higher TP relative to clear
Fit a linear model to the clear water lakes (remove outliers)

19. LV high alkalinity lakes, Type L1 & L5 – Clear, Types L2 & L6 - Humic
Model EQR ~ log(TP) + Hum + Error
Very little difference between Humic (blue) and Clear (red) water lakes

20. LV high alkalinity lakes, Type L1 & L5 – Clear,
For clear water lakes a quantile plot provides more realistic P boundaries

21. LV high alkalinity lakes, Type L1 & L5 – Clear,
Potential GM boundary values
25th quantile 44 ug/L
50th quantile 88 ug/L
But compare with other high alkalinity lakes from CBGIG, potentially combine all these data sets

22. All high alkalinity lakes, NGIG, CBGIG, LV & LT
Outstanding question
What other factors influence EQR in high alkalinity lakes ?

23. LV data with colour as a variable
Multivariate model: EQR ~ aP + bColour + c(P*Colour) + error
Including N was not significant
Including Depth type (shallow or very shallow) was not significant
(All data were transformed and centred prior to analysis)
Good Moderate Boundary for EQR = 0.61
R2 of model = 0.40
Color 30 mgPt/L = 72 ugP/L
Color 60 mgPt/L = 99 ugP/L
Color 90 mgPt/L = 138 ugP/L
Note.
At High/Good boundary, humic lakes more sensitive than clear water lakes