Use of time-dependent parameters for improvement and uncertainty estimation of dynamic models

1. Use of time-dependent parameters for improvement and uncertainty estimation of dynamic models Peter Reichert Eawag Dübendorf and ETH Zürich

2. Contents • Motivation • References • Approach • Concept • Implementation • Preliminary Results for a Simple Hydrologic Model • Problems / Challenges Motivation References Approach Preliminary Results Problems/Challenges

3. Motivation Motivation References Approach Preliminary Results Problems/Challenges Motivation

4. Motivation Fundamental Objectives: • Improve understanding of mechanisms governing the behaviour of the system described by the model. • Estimate realistic uncertainty bounds / decrease the width of uncertainty bounds of model predictions. Technical Objectives: • Improve the formulation of the deterministic model component. • Make the stochastic component of the model more realistic. Motivation References Approach Preliminary Results Problems/Challenges

5. Motivation Achieve these objectives by: • Improving the input error model. • Allowing model parameters to vary (e.g. in time) to address model structure error. • Improving the output error model (by addressing bias explicitly). In particular: • Search for statistical model components that cannot be rejected by the data. • Try to „explain“ the bias by input and/or model structure error („trace“ the causes of the bias). Motivation References Approach Preliminary Results Problems/Challenges

6. References Motivation References Approach Preliminary Results Problems/Challenges References

7. References The idea of using time-dependent parameters for model structure deficit evaluation is very old (e.g. review by Beck, 1987). Our work applies this idea to continuous time models and provides algorithms to apply it to nonlinear dynamic systems. Motivation References Approach Preliminary Results Problems/Challenges This talk is based on: • Brun, PhD dissertation, 2002: First trials with filtering algorithm. • Buser, Masters thesis, 2003: Smoothing, MCMC algorithm. • Tomassini, Reichert, Künsch, Buser, Borsuk, 2007:Estimation of process parameters, cross-validation.

8. Approach Motivation References Approach Preliminary Results Problems/Challenges Approach

9. Notation (according to Bayarri et al. 2005) Input: Field data = reality plus measurement error: Motivation References Approach Preliminary Results Problems/Challenges Output: Field data = reality plus measurement error: An ideal model describes reality: A realistic model approximates an ideal model: All terms together: inputerror effect of model structure error meas.error

10. Problem Motivation References Approach Preliminary Results Problems/Challenges inputerror effect of model structure error meas.error Problem: The bias term describes the effect, but not the cause of the model structure error. This leads to a satisfying statistical description of the past, but is hard to extra-polate into the future. For uncertainty reduction and extrapolation it would be better to reduce the bias by improving the mechanistic description of the system. In particular, trends must be described by the model, not by the bias term. How can statistical procedures support this?

11. Concept Motivation References Approach Preliminary Results Problems/Challenges inputerror effect of model structure error meas.error Concept: • Allow model parameters to vary. Add parameters where appropriate (input, output). • Try to reduce the bias by finding an adequate behaviour of these parameters. • Explore dependency of parameter variability on external or model variables. If successful (from a statistical and physical point of view), modify the model structure to reflect this dependency. • Redo the analysis with improved model structure and reduced bias.

12. Use for dynamic models Motivation References Approach Preliminary Results Problems/Challenges inputerror effect of model structure error meas.error Formulation for time dependent models: xt : Correction accounting for input error. t : Model-internal correction of model structure error. yt : Model-external correction for remaining effect of model structure error.In the ideal case, this error could be neglected as it would be accounted for by the internal correction.

13. Approach Motivation References Approach Preliminary Results Problems/Challenges • Fit model with constant parameters, identify presence of bias. If bias exists: • Identify, separately or jointly, time-dependent • input variation (xt ) • parameter variation (t ) • output variation (yt ) • Identify dependences of time-dependent parameters on external or model variables. • Improve the model structure by deterministic or sto-chastic elements (according to statistical and physi-cal considerations), try to avoid output error (yt ). • Use the extended model for understanding and prediction.

14. Implementation The time dependent parameter is modelled by a mean-reverting Ornstein Uhlenbeck process: Motivation References Approach Preliminary Results Problems/Challenges This has the advantage that we can use the analytical solution: or, after reparameterization:

15. Implementation We combine the estimation of • constant model parameters, , with • state estimation of the time-dependent parameter(s), t, and with • the estimation of (constant) parameters of the Ornstein-Uhlenbeck process(es) of the time dependent parameter(s), =(,, ,to). Motivation References Approach Preliminary Results Problems/Challenges

16. Conceptual Framework Motivation References Approach Preliminary Results Problems/Challenges Original deterministic model Model extended by input- and output-parameter and measurement error

17. Simplified Framework Simplifications: • Omit representation of given measured input, xF. • Add parameter to input to represent input uncertainty by parameter uncertainty. • Add parameter to output to represent output uncertainty by parameter uncertainty. Motivation References Approach Preliminary Results Problems/Challenges

18. Numerical Implementation (1) Gibbs sampling for the three different types of parameters. Conditional distributions: Motivation References Approach Preliminary Results Problems/Challenges simulation model (expensive) Ornstein-Uhlenbeck process (cheap) Ornstein-Uhlenbeck process (cheap) simulation model (expensive)

19. Numerical Implementation (2) Metropolis-Hastings sampling for each type of parameter: Motivation References Approach Preliminary Results Problems/Challenges Multivariate normal jump distributions for the parameters q and x. This requires one simulation to be performed per suggested new value of q. The discretized Ornstein-Uhlenbeck parameter, ft, is split into subintervals for which OU-process realizations conditional on initial and end points are sampled. This requires the number of subintervals simulations per complete new time series of ft.

20. Estimation of Hyperparametersby Cross - Validation Motivation References Approach Preliminary Results Problems/Challenges Due to identifiability problems we selected the hyperparameters, x, in a previous application (Tomassini et al., 2006) alternatively by cross-validation:

21. Preliminary Results Motivation References Approach Preliminary Results Problems/Challenges Preliminary Results for a Simple Hydrologic Model • Model • Model Application • Preliminary Results(based on Markov chains of insufficient length)

22. Model A Simple Hydrologic Watershed Model (1): Motivation References Approach Preliminary Results Problems/Challenges Kuczera et al. 2006

23. 3 A 4 5 1 B 2 C 6 7 Model A Simple Hydrologic Watershed Model (2): Motivation References Approach Preliminary Results Problems/Challenges 7 model parameters 3 initial conditions 1 standard dev. of meas. err. 3 „modification parameters“ Kuczera et al. 2006

24. Model A Simple Hydrologic Watershed Model (3): Motivation References Approach Preliminary Results Problems/Challenges Kuczera et al. 2006

25. Model Application Model application: • Data set of Abercrombie watershed, New South Wales, Australia (2770 km2), kindly provided by George Kuczera (Kuczera et al. 2006). • Box-Cox transformation applied to model and data to decrease heteroscedasticity of residuals. • Step function input to account for input data in the form of daily sums of precipitation and potential evapotranspiration. • Daily averaged output to account for output data in the form of daily average discharge. Motivation References Approach Preliminary Results Problems/Challenges

26. Model Application Prior distribution: Estimation of constant parameters: Independent uniform distributions for the loga-rithms of all parameters (7+3+1=11), keeping correction factors (frain, fpet) equal to unity and bias (bQ) equal to zero. Estimation of time-dependent parameters: Ornstein-Uhlenbeck process applied to log of the parameter (with the exception of bQ). Hyper-parameters: t = 5d, s fixed, only estimation of initial value and mean (0 for frain, fpet, bQ). Constant parameters as above. Motivation References Approach Preliminary Results Problems/Challenges

27. Preliminary Results (MC of insufficient length) Posterior marginals: Motivation References Approach Preliminary Results Problems/Challenges

28. Preliminary Results (MC of insufficient length) Max. post. simulation with constant parameters: Motivation References Approach Preliminary Results Problems/Challenges

29. Preliminary Results (MC of insufficient length) Residuals of max. post. sim. with const. pars.: Motivation References Approach Preliminary Results Problems/Challenges

30. Preliminary Results (MC of insufficient length) Residual analysis, max. post., constant parameters Motivation References Approach Preliminary Results Problems/Challenges Residual analysis, max. post., q_gw_max time-dependent

31. Preliminary Results (MC of insufficient length) Residual analysis, max. post., s_F time-dependent Motivation References Approach Preliminary Results Problems/Challenges Residual analysis, max. post., f_rain time-dependent

32. Preliminary Results (MC of insufficient length) Time-dependent parameters Motivation References Approach Preliminary Results Problems/Challenges

33. Problems / Challenges Motivation References Approach Preliminary Results Problems/Challenges Problems / Challenges (= Working Group Opportunities)

34. Problems / Challenges Problems / Challenges • Other formulations of time-dependent parameters? • Dependence on other factors than time. • How to estimate hyperparameters? (Reduction in correlation time always improves the fit.) • How to avoid modelling physical processes with the bias term? • Learn from more applications. • Compare results with methodology by Bayarri et al. (2005). Combine/extend the two methodologies? • ? Motivation References Approach Preliminary Results Problems/Challenges

35. Problems / Challenges Motivation References Approach Preliminary Results Problems/Challenges Problems / Challenges (= Working Group Opportunities) Discussion slides from talk at Oct. 16.

36. Problems / Challenges Research Questions / Options for Projects (1) • Compare results when making different model parameters stochastic and time-dependent. (Ongoing with a postdoc in Switzerland extending earlier work with continuous-time stochastic parameters.) • Develop a better statistical description of rainfall uncertainty.(Option for a collaboration with climate/weather working groups.) • Explore alternative options for making parameters time-dependent.(Suggestions so far: storm-dependent parameters, time-dependent parameter as an Ornstein-Uhlenbeck process.) Motivation References Approach Preliminary Results Problems/Challenges

37. Problems / Challenges Research Questions / Options for Projects (2) • Investigate how to learn from state estimation of stochastic hydrological models.(Can the pattern of state adaptations lead to insights of model structure deficits or input errors?) • Develop uncertainty estimates when using multi-objective optimization.(How to use information on Pareto set for uncertainty estimation of parameters and results?) • Analyse differences in results of suggested approaches when using different models.(Is there a generic behaviour of different techniques when they are applied to different models/data sets?) Motivation References Approach Preliminary Results Problems/Challenges

38. Problems / Challenges Research Questions / Options for Projects (3) • Improve the efficientcy of posterior maximisation and posterior sampling.(Efficiency becomes important when having complex watershed models in mind. Efficient global optimizers and sampling from multi-modal posterior distributions becomes then important.) • More questions will come up during discussions. Motivation References Approach Preliminary Results Problems/Challenges

39. Thank you for your attention