Approaching the upper boundary of driver-response relationships:identifying factors using a novel framework integrating quantile regression with interpretable machine learning

The identification of factors that may be forcing ecological observations to approach the upper boundary provides insight into potential mechanisms affecting driver-response relationships,and can help inform ecosystem management,but has rarely been explored.In this study,we propose a novel framework integrating quantile regression with interpretable machine learning.In the first stage of the framework,we estimate the upper boundary of a driver-response relationship using quantile regression.Next,we calculate "potentials" of the response variable depending on the driver,which are defined as vertical distances from the estimated upper boundary of the relationship to observations in the driver-response variable scatter plot.Finally,we identify key factors impacting the potential using a machine learning model.We illustrate the necessary steps to implement the framework using the total phosphorus(TP)-Chlorophyll a(CHL) relationship in lakes across the continental US.We found that the nitrogen to phosphorus ratio(N:P),annual average precipitation,total nitrogen(TN),and summer average air temperature were key factors impacting the potential of CHL depending on TP.We further revealed important implications of our findings for lake eutrophication management.The important role of N:P and TN on the potential highlights the co-limitation of phosphorus and nitrogen and indicates the need for dual nutrient criteria.Future wetter and/or warmer climate scenarios can decrease the potential which may reduce the efficacy of lake eutrophication management.The novel framework advances the application of quantile regression to identify factors driving observations to approach the upper boundary of driver-response relationships.