Mechanism-Based Emulation of Dynamic Simulation Models – Concept and Application in Hydrology

1. Mechanism-Based Emulation of Dynamic Simulation Models –Concept and Application in Hydrology Peter Reichert Eawag Dübendorf and ETH ZürichSwitzerland

2. Contents • Motivation • Concept of Emulators • General Concept • Gaussian Process Emulator • Dynamic Emulator • Implementation • Application • Discussion and Outlook Motivation Concept Implementation Application Discussion

3. Motivation Motivation Concept Implementation Application Discussion Motivation

4. Motivation Motivation Concept Implementation Application Discussion Problem • Many important systems analytical techniques, such as optimization, sensitivity analysis, and statistical inference (e.g. Bayesian inference using MCMC) require a large number of model evaluations. • Many environmental simulation models are computationally demanding.  Model-based analysis of environmental systems is often limited by computational requirements.

5. Motivation Motivation Concept Implementation Application Discussion Solution Strategies • Improve the efficiency of the implementation of environmental simulation models. • Improve the efficiency of the implementaton of systems analytical techniques. • Replace the simulation model by a simplified statistical description, an emulator. • Obviously, all three strategies must be followed. • This talk is about recent progress with strategy 3: The construction and use of emulators of dynamic environmental simulation models.

6. Concept Motivation Concept Implementation Application Discussion Concept

7. Concept Emulator: An emulator is a statistical approximation of a deterministic simulation model It can be used for interpolating model results between simulation results gained at carefully chosen design points in model input space. Replacing the simulation model by the emulator can tremendously increase the efficiency of analyses(but it also adds additional uncertainty). The emulator provides a deterministic interpolation result as well as a probability distribution representing our knowledge of the uncertainty of emulation. Motivation Concept Implementation Application Discussion

8. Concept Gaussian Process Emulators: Emulators have quite successfully been constructed by setting-up a Gaussian process prior with a mean consisting of a linear combination of basis functions and then conditioning this prior on the design data. Motivation Concept Implementation Application Discussion O‘Hagan 2006

9. Concept Gaussian Process Emulators: Limitations: • Dense output in the time domain leads to numerical difficulties (large size and poor conditioning of matrices to be inverted). • The knowledge about the mechanisms built into the simulation program is not used.It can be expected that we could built a better emulator when using this knowledge. This is of particular importance if the design set is small. Motivation Concept Implementation Application Discussion This raises the question how to build an emulator of a dynamic model that resolves both of these issues.

10. Concept Emulators for Dynamic Models: Three Options: Motivation Concept Implementation Application Discussion • Application of Gaussian processes with time dimension as an additional input.Can lead to very large and poorly conditioned matrices to invert and numerical problems. • For Markovian or state-space models: Emulate transfer function from one state to the next instead of the complete dynamic response. • Use a simple dynamic model as a prior and model innovations as Gaussian processes in the other input dimensions. These Gaussian processes correct for the bias in the simple model.

11. Concept Emulators for Dynamic Models: All emulators proposed so far (to my knowledge) do not consider our knowledge about the mechanisms implemented in the simulation model (with the exception of an problem-specific choice of basis functions). Approach proposed in this talk: Motivation Concept Implementation Application Discussion • Use a simplified, linear state-space model to describe the approximate dynamics of the simulation model. • Formulate the innovations as Gaussian processes of parameters (and potentially other input). • Derive the emulator (posterior) by Kalman smoothing.

12. Implementation Motivation Concept Implementation Application Discussion Implementation

13. Construction of Emulators Construction of Emulators: We can distinguish five steps of emulator development: Motivation Concept Implementation Application Discussion • Choice of Design Data • Choice of a Simplified Probabilistic Model • Coupling of Replicated Simplified Models • Conditioning the Simplified Model on the Design Data • Calculation of Expected Value and Uncertainty

14. Construction of Emulators 1. Choice of Design Data: Often parameter values are chosen by latin hypercube sampling from reasonable domains of model parameters. However, adaptive sampling schemes could be used that increase the density of sampling points in regions of high variability of results. The design data set consists of these parameter values and the corresponding simulation results: Motivation Concept Implementation Application Discussion

15. Construction of Emulators 2. Choice of a Simplified Probabilistic Model: The emulator is based on a simplified probabilistic model M‘ of the simulation model M. This model expresses our prior beliefs of the behaviour of the deterministic simulation model. Ist likelihood function is given by: Motivation Concept Implementation Application Discussion

16. Construction of Emulators 3. Coupling of Replicated Simplified Models: The augmented model consists of n replicates of the simplified model for different parameter values: Motivation Concept Implementation Application Discussion These models are stochastically coupled. Probabilities represent here beliefs in a Bayesian sense. We construct a model with n = nD+1 replicates of the simplified model. These correspond to models for the nD design parameter sets and for the emulation parameter set.

17. Construction of Emulators 4. Conditioning the Simplified Model on the Design Data: Motivation Concept Implementation Application Discussion We calculate the distribution of the last set of components conditional on results for the first nD sets of components: The emulator is gained by integrating out additional parameters:

18. Construction of Emulators 5. Calculation of Expected Value and Uncertainty: Motivation Concept Implementation Application Discussion The expected value provides the deterministic emulator: The variance-covariance matrix of the emulator is a quantification of emulation uncertainty.

19. Gaussian Process Emulator 1. Choice of Design Data: Often parameter values are chosen by latin hypercube sampling from reasonable domains of model parameters. However, adaptive sampling schemes could be used that increase the density of sampling points in regions of high variability of results. The design data set consists of these parameter values and the corresponding simulation results: Motivation Concept Implementation Application Discussion

20. Gaussian Process Emulator 2. Choice of a Simplified Probabilistic Model: Motivation Concept Implementation Application Discussion The simplified probabilistic model consists of a deterministic model plus a multivariate normal error term with mean zero: The simplified model can contain additional parameters. Often a linear combination of suitably chosen basis function is used:

21. Gaussian Process Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion The augmented model consists of independent replications of the deterministic simplified model and error terms that are stochastically coupled:

22. Gaussian Process Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion A simple stochastic coupling is obtained by:

23. Gaussian Process Emulator 4. Conditioning the Simplified Model on the Design Data: Motivation Concept Implementation Application Discussion The augmented model is then multivariate normal. For this reason, we can apply the standard result for conditioning a multivariate normal distribution on some of ist components:

24. Gaussian Process Emulator 4. Conditioning the Simplified Model on the Design Data: Motivation Concept Implementation Application Discussion This leads to the emulator as a multivariate normal distribution: with

25. Gaussian Process Emulator 5. Calculation of Expected Value and Uncertainty: Motivation Concept Implementation Application Discussion O‘Hagan 2006

26. Dynamic Emulator Dynamic models (and their emulators) have a structured output: Motivation Concept Implementation Application Discussion

27. Dynamic Emulator 1. Choice of Design Data: Often parameter values are chosen by latin hypercube sampling from reasonable domains of model parameters. However, adaptive sampling schemes could be used that increase the density of sampling points in regions of high variability of results. The design data set consists of these parameter values and the corresponding simulation results: Motivation Concept Implementation Application Discussion

28. Dynamic Emulator 2. Choice of a Simplified Probabilistic Model: Motivation Concept Implementation Application Discussion Concept: Use of state-space model – emulation of „observed“ output only. Reasons: • This accounts for the typical „hidden Markov“ structure of environmental simulation models. • It allows us to implement an emulator with a simplied (lower dimensional) state space.

29. Dynamic Emulator 2. Choice of a Simplified Probabilistic Model: Motivation Concept Implementation Application Discussion

30. Dynamic Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion Augmented Model (1):

31. Dynamic Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion Augmented Model (2):

32. Dynamic Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion Augmented Model (3): Stochastic coupling

33. Dynamic Emulator 4. Conditioning the Simplified Model on the Design Data: Motivation Concept Implementation Application Discussion Kalman (forward) filtering (Künsch, 2001):

34. Dynamic Emulator 4. Conditioning the Simplified Model on the Design Data: Motivation Concept Implementation Application Discussion Kalman (backward) smoothing (Künsch, 2001):

35. Dynamic Emulator 5. Calculation of Expected Value and Uncertainty: Motivation Concept Implementation Application Discussion Calculation of expected value and variance-covariance matrix of last set of components:

36. Implementation Due to the dependence on(which depends on the design data as well as on the new parameter values), the smoothing step is very inefficient. By using the general matrix identity we are able to separate-out the inversion of the large sub-matrix that depends only on the design data. This makes the procedure much more efficient as we do not have to perform large matrix inversions when using the emulator at new parameter values. Motivation Concept Implementation Application Discussion

37. Application Motivation Concept Implementation Application Discussion Application

38. Hydrological Model Simple Hydrological Watershed Model (1): Motivation Concept Implementation Application Discussion Kuczera et al. 2006

39. 3 4 6 5 1 2 7 8 Hydrological Model Simple Hydrological Watershed Model (2): Motivation Concept Implementation Application Discussion 8 model parameters 3 initial conditions 1 standard dev. of obs. err. Kuczera et al. 2006

40. 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 averaged discharge. Motivation Concept Implementation Application Discussion

41. Linearization Linearization of model nonlinearities: Motivation Concept Implementation Application Discussion

42. Linearization Derivation of simplified, linear state-space model: Motivation Concept Implementation Application Discussion

43. Results Preliminary results with a simpler model look promising. They demonstrate that the concept works. Unfortunately, the results for the hydrological model are not yet available. Motivation Concept Implementation Application Discussion

44. Discussion Motivation Concept Implementation Application Discussion Discussion

45. Discussion • We developed a general technique of constructing emulators for dynamic simulation models. • In addition to solving technical problems of Gaussian process emulation of dynamic models, this technique easily allows us to rely on mechanisms incorporated in the simulation model. It can be expected that this improves the emulation process. This is of particular importance if the design set is small. • There is need for more research: • Gaining more experience with our approach. • Extending the approach to the estimation of additional parameters of the simplified model. • Learning about advantages and disadvantages of the different approaches to dynamic emulation. Motivation Concept Implementation Application Discussion

46. Acknowledgements • Collaboration for this paper:Gentry White, Susie Bayarri, Bruce Pitman, Tom Santner during my stay at SAMSI, NC, USA • Hydrological example and data:George Kuczera. • More Interactions at SAMSI:Jim Berger, Fei Liu, Rui Paulo, Robert Wolpert, John Paul Gosling, Tony O‘Hagan, and many more. Motivation Concept Implementation Application Discussion