Parameter-dependent convergence bounds and complexity measure for a class of conceptual hydrological models

We provide analytical bounds on convergence rates for a class of hydrologic models and consequently derive a complexity measure based on the Vapnik–Chervonenkis (VC) generalization theory. The class of hydrologic models is a spatially explicit interconnected set of linear reservoirs with the aim of representing globally nonlinear hydrologic behavior by locally linear models. Here, by convergence rate, we mean convergence of the empirical risk to the expected risk. The derived measure of complexity measures a model's propensity to overfit data. We explore how data finiteness can affect model selection for this class of hydrologic model and provide theoretical results on how model performance on a finite sample converges to its expected performance as data size approaches infinity. These bounds can then be used for model selection, as the bounds provide a tradeoff between model complexity and model performance on finite data. The convergence bounds for the considered hydrologic models depend on the magnitude of their parameters, which are the recession parameters of constituting linear reservoirs. Further, the complexity of hydrologic models not only varies with the magnitude of their parameters but also depends on the network structure of the models (in terms of the spatial heterogeneity of parameters and the nature of hydrologic connectivity).