On hydrological model complexity, its geometrical interpretations and prediction uncertainty

Abstract

[1] Knowledge of hydrological model complexity can aid selection of an optimal prediction model out of a set of available models. Optimal model selection is formalized as selection of the least complex model out of a subset of models that have lower empirical risk. This may be considered equivalent to minimizing an upper bound on prediction error, defined here as the mathematical expectation of empirical risk. In this paper, we derive an upper bound that is free from assumptions on data and underlying process distribution as well as on independence of model predictions over time. We demonstrate that hydrological model complexity, as defined in the presented theoretical framework, plays an important role in determining the upper bound. The model complexity also acts as a stabilizer to a hydrological model selection problem if it is deemed ill-posed. We provide an algorithm for computing complexity of any arbitrary hydrological model. We also demonstrate that hydrological model complexity has a geometric interpretation as the size of model output space. The presented theory is applied to quantify complexities of two hydrological model structures: SAC-SMA and SIXPAR. It detects that SAC-SMA is indeed more complex than SIXPAR. We also develop an algorithm to estimate the upper bound on prediction error, which is applied on five different rainfall-runoff model structures that vary in complexity. We show that a model selection problem is stabilized by regularizing it with model complexity. Complexity regularized model selection yields models that are robust in predicting future but yet unseen data.

Socio-hydrological modelling: a review asking "why, what and how?"

“…However, the extensive investigations into dealing with uncertainty (particularly the recent focus on prediction in ungauged basins Wagener and Montanari, 2011) can only be of benefit to studies which widen system boundaries. The trade-offs between model complexity and "empirical risk" (Arkesteijn and Pande, 2013) in modelling, ways to deal with large numbers of parameters and limited data (Welsh et al, 2013), as well as statistical techniques to cope with uncertainties (Wang and Huang, 2014) have all been well investigated, and knowledge from these areas can certainly be applied to future studies.…”

Socio-hydrologic modeling to understand and mediate the competition for water between agriculture development and environmental health: Murrumbidgee River basin, Australia

“…While high complexity is desirable to simulate a rich class of emergent patterns, such models when calibrated, especially for sparsely gauged basins (in terms either of socio-economic or hydrological data), may not reliably predict the dynamics driven by future yet unseen exogenous forcing. See for example Sivapalan et al (2003), Jakeman and Letcher (2003), Fenicia et al (2008), Pande et al (2012), Pande (2013), and Arkesteijn and Pande (2013) for extensive analyses of the relationships between model complexity, model structure deficiency, prediction uncertainty. Furthermore, the differences in the shapes of the curves between observations and predictions, especially in the case of irrigation area, points to model improvements that can still be made: for example, the assumption that attractiveness level is a function of irrigation potential may have to be improved with the hindsight of additional data.…”

Toward creating simpler hydrological models: A LASSO subset selection approach

“…In practice, when dealing with constrained models with large numbers of observations it seems unlikely that least squares results would be very different, although there may be numerical problems with the resulting quadratic programming exercise, as noted in Section 3. More recently, Arkesteijn and Pande (2013) found it helpful to use a least absolute deviations fit measure in a formal analysis of complexity in hydrological models.…”

“…A review is beyond the scope of this paper but selected works include Dooge (1997), Schoups et al (2008), Sivapalan et al (2003), Sivakumar (2008), Fenicia et al (2008), Hill (2006), Tonkin and Doherty (2005), Hunt et al (2007), Arkesteijn and Pande (2013), and Diodato et al (2014).…”