Analyzing input in addition to structural uncertainty of a hydrological model with stochast

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Analyzing input in addition to structural uncertainty of a hydrological model with stochast

Keener, Lamar, Co-Publisher has reference to this Academic Journal, PHwiki organized this Journal Analyzing input in addition to structural uncertainty of a hydrological model with stochastic, time-dependent parameters Peter Reichert Eawag Dübendorf in addition to ETH Zürich, Switzerl in addition to Contents Motivation Approach Implementation Application Discussion Motivation Approach Implementation Application Discussion Motivation Motivation Motivation Approach Implementation Application Discussion

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Motivation Motivation Approach Implementation Application Discussion Environmental modelling is often based on deterministic models that describe substance in addition to organism mass balances in environmental compartments. Statistical inference with such models is often based on the assumption that the data is independently in addition to identically distributed around the predictions of the deterministic model at „true“ parameter values. The concept underlying this approach is that the deterministic model describes the „true“ system behaviour in addition to the probability distributions centered at the model predictions the measurement process. Motivation Motivation Approach Implementation Application Discussion Empirical evidence often demonstrates the invalidity of these statistical assumptions: Residuals are often heteroscedastic in addition to autocorrelated. The residual error is usually (much) larger than the measurement error. This leads to incorrect results of statistical inference. In particular, parameter in addition to model output uncertainty are usually underestimated. These obviously wrong results lead to ab in addition to oning of the statistical approach in addition to to the development of conceptually poorer techniques in applied sciences. We are interested in a statistically satisfying approach to this problem. Motivation Suggested solution (Kennedy in addition to O‘Hagan, 2001, in addition to many earlier, more case-specific approaches): Extend the model by a discrepancy or bias term. Replace: by: where yM = deterministic model, x = model inputs, q = model parameters, Ey = observation error, B = bias or model discrepancy, YM = r in addition to om variable representing model results. Motivation Approach Implementation Application Discussion The bias term is usually as long as mulated as a non-parametric statistical description of the model deficits (typically as a Gaussian stochastic process).

Motivation Advantage of this approach: Statistical description of model discrepancy improves uncertainty analysis. Disadvantage: Lack of underst in addition to ing of the cause of the discrepancy makes it still difficult to extrapolate. Motivation Approach Implementation Application Discussion We are interested in a technique that supports identification of the causes in addition to reduction of these discrepancies. Motivation Errors in deterministic model structure. Errors in model input. Inadequateness of a deterministic description of systems that contain intrinsic non-deterministic behaviour due to influence factors not considered in the model, model simplifications (e.g. aggregation, adaptation, etc.), chaotic behaviour not represented by the model. Motivation Approach Implementation Application Discussion There are three generic causes of failure of the description of nature with a deterministic model plus measurement error: Motivation Pathway as long as improving models: Reduce errors in deterministic model structure to improve average behaviour. Add adequate stochasticity to the model structure to account as long as r in addition to om influences. Motivation Approach Implementation Application Discussion This requires the combination of statistical analyses with scientific judgment. This talk is about support of this process by statistical techniques. Because of these deficits we cannot expect a deterministic model to describe nature appropriately.

Approach Approach Motivation Approach Implementation Application Discussion Approach Questions: How to make a deterministic, continuous-time model stochastic How to distinguish between deterministic in addition to stochastic model deficits Motivation Approach Implementation Application Discussion Replacement of differential equations (representing conservation laws) by stochastic differential equations can violate conservation laws in addition to does not address the cause of stochasticity directly. It seems to be conceptually more satisfying to replace model parameters (such as rate coefficients, etc.) by sto- chastic processes, as stochastic external influence factors usually affect rates in addition to fluxes rather than states directly. The model consists then of an extended set of stochastic differential equations of which some have zero noise. Approach Motivation Approach Implementation Application Discussion

Approach Note that the basic idea of this approach is very old. The original as long as mulation was, however, limited to linear or weakly nonlinear, discrete-time systems with slowly varying driving as long as ces (e.g. Beck 1987). The bias term approach is a special case of our approach that consists of an additive output parameter. Motivation Approach Implementation Application Discussion Our suggestion is to extend this original approach to continuous-time in addition to nonlinear models; allow as long as rapidly varying external as long as ces; embed the procedure into an extended concept of statistical „bias-modelling“ techniques. Implementation Implementation Motivation Approach Implementation Application Discussion Model Deterministc model: Consideration of observation error: Motivation Approach Implementation Application Discussion

Model Model with parameter i time-dependent: Motivation Approach Implementation Application Discussion Time Dependent Parameter This has the advantage that we can use the analytical solution: The time dependent parameter is modelled by a mean-reverting Ornstein Uhlenbeck process: or, after reparameterization: Motivation Approach Implementation Application Discussion Inference We combine the estimation of constant model parameters, , with state estimation of the time-dependent parameter(s), , in addition to with the estimation of (some of the) (constant) parameters of the Ornstein-Uhlenbeck process of the time dependent parameter(s), . Motivation Approach Implementation Application Discussion

Inference Gibbs sampling as long as the three different types of parameters. Conditional distributions: Ornstein-Uhlenbeck process (cheap) simulation model (expensive) simulation model (expensive) Ornstein-Uhlenbeck process (cheap) Motivation Approach Implementation Application Discussion Tomassini et al. 2007 Inference Metropolis-Hastings sampling as long as each type of parameter: Multivariate normal jump distributions as long as the parameters qM in addition to qP. This requires one simulation to be per as long as med per suggested new value of qM. The discretized Ornstein-Uhlenbeck parameter, , is split into subintervals as long as which OU-process realizations conditional on initial in addition to end points are sampled. This requires the number of subintervals simulations per complete new time series of . Motivation Approach Implementation Application Discussion Tomassini et al. 2007 Application Motivation Approach Implementation Application Discussion Application

Hydrological Model Simple Hydrological Watershed Model (1): Kuczera et al. 2006 Motivation Approach Implementation Application Discussion Hydrological Model Simple Hydrological Watershed Model (2): Kuczera et al. 2006 8 model parameters 3 initial conditions 1 st in addition to ard dev. of obs. err. 3 „modification parameters“ Motivation Approach Implementation Application Discussion Hydrological Model Simple Hydrological Watershed Model (3): Motivation Approach Implementation Application Discussion

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Model Application Data set of Abercrombie watershed, New South Wales, Australia (2770 km2), kindly provided by George Kuczera (Kuczera et al. 2006). Box-Cox trans as long as mation applied to model in addition to data to decrease heteroscedasticity of residuals. Step function input to account as long as input data in the as long as m of daily sums of precipitation in addition to potential evapotranspiration. Daily averaged output to account as long as output data in the as long as m of daily averaged discharge. Motivation Approach Implementation Application Discussion Analysis with Constant Parameters Estimation of 11 model parameters: 8 rate parameters 3 initial conditions 1 measurement st in addition to ard deviation Priors: Independent lognormal distributions as long as all parameters with the exception of the measurement st in addition to ard deviation (1/s). Modification factors (frain, fpet, fQ) kept equal to unity. Motivation Approach Implementation Application Discussion Results as long as Constant Parameters Motivation Approach Implementation Application Discussion

Results as long as Constant Parameters Motivation Approach Implementation Application Discussion Results as long as Constant Parameters Motivation Approach Implementation Application Discussion Results as long as Constant Parameters Motivation Approach Implementation Application Discussion Residuals are heteroscedastic in addition to autocorrelated. The st in addition to ard deviation of the residuals is larger than the measurement error (increasing from 0.24 m3/s at a discharge of zero to 30 m3/s at 100 m3/s). Model predictions are overconfident. In addition: ground water level trend seems unrealistic. The results show the typical deficiencies of deterministic models:

Discussion There is need as long as future research in the following areas: Explore alternative ways of learning from the identified parameter time series. Different as long as mulation of time-dependent parameters ( as long as some applications smoother behaviour). Include multiple time-dependent parameters into the analysis. Use a more specific model to represent input uncertainty. Improve efficiency (linearization, emulation). Learn from more applications. Motivation Approach Implementation Application Discussion Acknowledgements Collaboration as long as this paper: Johanna Mieleitner Development of the technique: Hans-Rudolf Künsch, Rol in addition to Brun, Christoph Buser , Lorenzo Tomassini, Mark Borsuk. Hydrological example in addition to data: George Kuczera. Interactions at SAMSI: Susie Bayarri, Tom Santner, Gentry White, Ariel Cintron, Fei Liu, Rui Paulo, Robert Wolpert, John Paul Gosling, Tony O‘Hagan, Bruce Pitman, Jim Berger, in addition to many more. Motivation Approach Implementation Application Discussion

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