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## Mechanism-Based Emulation of Dynamic Simulation Models Concept in addition to Application

Keener, Theresa, Co-Publisher has reference to this Academic Journal, PHwiki organized this Journal Mechanism-Based Emulation of Dynamic Simulation Models Concept in addition to Application in Hydrology Peter Reichert Eawag Dübendorf in addition to ETH Zürich Switzerl in addition to Contents Motivation Concept of Emulators General Concept Gaussian Process Emulator Dynamic Emulator Implementation Application Discussion in addition to Outlook Motivation Concept Implementation Application Discussion Motivation Motivation Motivation Concept Implementation Application Discussion

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Motivation Motivation Concept Implementation Application Discussion Problem Many important systems analytical techniques, such as optimization, sensitivity analysis, in addition to statistical inference (e.g. Bayesian inference using MCMC) require a large number of model evaluations. Many environmental simulation models are computationally dem in addition to ing. Model-based analysis of environmental systems is often limited by computational requirements. 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 in addition to use of emulators of dynamic environmental simulation models. Concept Concept Motivation Concept Implementation Application Discussion

Concept Emulator: An emulator is a statistical approximation of a deterministic simulation model It can be used as long as 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 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 in addition to then conditioning this prior on the design data. Motivation Concept Implementation Application Discussion OHagan 2006 Concept Gaussian Process Emulators: Limitations: Dense output in the time domain leads to numerical difficulties (large size in addition to 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.

Concept Emulators as long as 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 in addition to poorly conditioned matrices to invert in addition to 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 in addition to model innovations as Gaussian processes in the other input dimensions. These Gaussian processes correct as long as the bias in the simple model. Concept Emulators as long as 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 ( in addition to potentially other input). Derive the emulator (posterior) by Kalman smoothing. Implementation Implementation Motivation Concept Implementation Application Discussion

Construction of Emulators Construction of Emulators: We can distinguish five steps of emulator development: 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 in addition to Uncertainty Motivation Concept Implementation Application Discussion 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 in addition to the corresponding simulation results: Motivation Concept Implementation Application Discussion 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

Construction of Emulators 3. Coupling of Replicated Simplified Models: The augmented model consists of n replicates of the simplified model as long as 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 as long as the nD design parameter sets in addition to as long as the emulation parameter set. 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 as long as the first nD sets of components: The emulator is gained by integrating out additional parameters: Construction of Emulators 5. Calculation of Expected Value in addition to 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.

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 in addition to the corresponding simulation results: Motivation Concept Implementation Application Discussion 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: 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 in addition to error terms that are stochastically coupled:

Gaussian Process Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion A simple stochastic coupling is obtained by: 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 st in addition to ard result as long as conditioning a multivariate normal distribution on some of ist components: 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

Gaussian Process Emulator 5. Calculation of Expected Value in addition to Uncertainty: OHagan 2006 Motivation Concept Implementation Application Discussion Dynamic Emulator Motivation Concept Implementation Application Discussion Dynamic models ( in addition to their emulators) have a structured output: 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 in addition to the corresponding simulation results: Motivation Concept Implementation Application Discussion

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 as long as the typical hidden Markov structure of environmental simulation models. It allows us to implement an emulator with a simplied (lower dimensional) state space. Dynamic Emulator 2. Choice of a Simplified Probabilistic Model: Motivation Concept Implementation Application Discussion Dynamic Emulator 3. Coupling of Replicated Simplified Models: Motivation Concept Implementation Application Discussion Augmented Model (1):

Discussion We developed a general technique of constructing emulators as long as 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 as long as more research: Gaining more experience with our approach. Extending the approach to the estimation of additional parameters of the simplified model. Learning about advantages in addition to disadvantages of the different approaches to dynamic emulation. Motivation Concept Implementation Application Discussion Acknowledgements Collaboration as long as this paper: Gentry White, Susie Bayarri, Bruce Pitman, Tom Santner during my stay at SAMSI, NC, USA Hydrological example in addition to data: George Kuczera. More Interactions at SAMSI: Jim Berger, Fei Liu, Rui Paulo, Robert Wolpert, John Paul Gosling, Tony OHagan, in addition to many more. Motivation Concept Implementation Application Discussion

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