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`Bottom-up' and `Top-down' Modelling Dominant Mode Analysis: Data-Based Mechanistic Modelling and methodology Global Carbon Cycle Examples Conclusions Dominant Mode Analysis: Simplicity Out of Complexity Peter Young Centre for Research on Environmental Systems and Statistics Lancaster University and Centre for Resource and Environmental Studies Australian National University Prepared for Complexity Seminar, May 26th 2004 Two Illustrative examples

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Data-Based Mechanistic (DBM) Modelling: An Inductive Approach to Modelling Stochastic, Dynamic Systems

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Data-Based Mechanistic (DBM) Modelling The general approach of DBM modelling is as follows: 1. Define the objectives of the modelling exercise. 2. Prior to the acquisition of data (or in situations of insufficient data), develop a simulation model that satisfies the defined objectives. Then, evaluate the sensitivity of the model to uncertainty and develop a reduced order, dominant mode model that captures the most important aspects of its dynamic behaviour. 3. When sufficient data become available: identify and estimate from these data a parsimonious, dominant mode, stochastic model that again satisfies the defined objectives; reflects the information content in the data; and can be interpreted in physically meaningful terms. 4. Attempt to reconcile the models obtained in 1. and 2. Data-Based Mechanistic (DBM) modelling has been developed at Lancaster over the past twenty years but is derived from research dating back to the early 1970 s. Note: not all stages in this modelling process will be required; this will depend on the nature of the problem; the available date; and the modelling objectives.

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The DBM Approach to Modelling When Data are Available The most parametrically efficient (parsimonious, identifiable) model structure is first inferred statistically from the available time series data in an inductive manner, based on a generic class of dynamic models. After this initial inductive modelling stage is complete, the model is interpreted in a physically meaningful, mechanistic manner based on the nature of the system under study and the physical, chemical, biological or socio-economic laws that are most likely to control its behaviour. This inductive approach can be contrasted with the alternative hypothetico-deductive modelling approach (including simple grey-box modelling), where the physically meaningful model structure is based on prior assumptions; and the parameters that characterise this simplified structure are estimated from data only after this structure has been specified by the modeller. By delaying the mechanistic interpretation of the model in this manner, the DBM modeller avoids the temptation to attach too much importance to prior, subjective judgement when formulating the model equations.

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Methods of Recursive Time Series Analysis The methodological tools used in DBM modelling are largely based on recursive methods (Gauss, 1826: see Young, 1984) of time series analysis that perform various tasks: In the case of time-invariant parameter (stationary) models, the statistical identification of the Transfer Function model (an ODE model in the continuous- time case) structure and the optimal estimation of the constant parameters that characterise this structure. In the case of nonstationary and nonlinear models, the estimation of Time Variable (TVP) or State-Dependent (SDP) Parameters that, respectively, characterise the model in these cases. On-line forecasting and data-assimilation (Bayesian recursive estimation). In addition, MCS-based tools for uncertainty and sensitivity analysis are used to evaluate the implications of uncertainty in: (i) derived physically meaningful parameters (e.g. time constants, residence times, decay rates, partition percentages, etc. ); and (ii) model predictions.

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Dominant Mode Analysis and Modelling

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Example 1: Uncertainty and Dominant Mode Analysis of the Enting-Lassey Global Carbon Cycle Model Research carried out by Stuart Parkinson and Peter Young (1996) ( based on a Simulink version of a E-L simulation model developed by Stuart Parkinson )

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Globally Averaged Carbon Cycle Climate Data

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The Enting-Lassey Global Carbon Cycle Model

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Comparison of Uncertainty Bounds for Deterministic Model Stochastic Uncertainty Analysis Using MCS

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Comparison of Reduced Order and 23rd Order EL Model Impulse Responses

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Comparison of Reduced Order and 23rd Order EL Model Responses to Measured Fossil Fuel Input

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Parallel Pathway Decomposition of Reduced Order Model: Physical Interpretation

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MCS Analysis of Reduced Order Model Showing Uncertainty in the Residence Time Estimates

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Example 2: Dominant Mode Analysis of the Lenton Global Carbon Cycle (GCC) Simulation Model Research currently being carried out by Andrew Jervis, Peter Watkins and Peter Young ( based on a Simulink version of a GCC simulation model developed by Tim Lenton, CEH, Bush, Edinburgh )

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Lenton Global Carbon Cycle (GCC) Simulation: Top Level of Complete Hierarchical Model

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Lenton GCC Model, Second Level: Sub-Model Inside Green Block of Complete Model

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Lenton GCC Model, Third Level: Model Inside Green Block of Sub-Model

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Second Order DBM Model Estimated from the Response of the High Order Simulation Model to an 1000 Unit Impulsive Emissions Input

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Second Order DBM Model Mimics the Behaviour of the High Order Simulation Model

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Second Order DBM Model Based on the Impulse Response Estimation Mimics the Response of the High Order Simulation Model to the Historic Emissions Input

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Block Diagram and Physical Interpretation of Second Order DBM Reduced Order Model

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DBM Modelling from the Global Data Based on recent research by Jervis and Young

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Globally Averaged Carbon Cycle Climate Data

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Parameterisation of the Identified Non-Parametric SDP Model The initial non-parametric modelling phase suggests the following SDP differential equation model form: where is a very small, coloured noise process and is a pure time delay. A variety of parameterisations of the SDP were investigated but linear and sigmoidal relation- ships in were found to be most suitable. Note the pure time delay: this suggests that higher order dynamic processes are possibly operative but their aggregative effect is to introduce the pure time delay (a common effect in real systems).

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Parametric Model with External Temperature as an Input: Estimation and Validation Note the expanded scale on error plot

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Wigley (1993, 2000) Model Fitting Results

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A Controlled Future? Hypothetical but showing how SDP models can provide a vehicle for assessing future carbon dioxide emissions policy

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Achieving Stabilization Concentration Profiles One approach to a more discerning definition of emission scenarios is to consider the inverse problem (e.g. Wigley, 2000): i.e. compute the emission scenarios that are able to achieve a range of stabilization concentration profiles for atmospheric, such as those utilized in the studies carried out by the IPCC (Schimel, et al 1996). An alternative and more flexible approach that we have used is to exploit automatic control theory and so generate an emissions scenario (control input) that achieves some required objective, as defined by a specified optimal criterion function. Here, the control system is designed to adjust the emissions input so that one of the required stabilization concentration profiles suggested by Wigley is followed as closely as possible.

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Optimal PIP Control of Emissions for Future Atmospheric Carbon Dioxide and Global Temperature Stabilization: Simulink Diagram Note:most conservative linear model

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PIP Emissions Policy for Future Atmospheric Carbon Dioxide and Global Temperature Stabilization using the Linear DBM Model

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Comparison with Wigleys Stabilization Results We see that the objectives have been realized by 2200, with an emissions scenario stabilizing at a level of 7.4 Gt y -1, with a 5%- 95% percentile range between 6.1 and 8.5 Gt y -1 ; i.e. between 3.6 to 6 Gt y -1 higher than that computed by Wigley for the same profile: i.e. the same result is achieved with a much smaller and more practically feasible reduction in the emissions. Note also that, unlike Wigley, we are able to provide a estimate of the uncert- ainties involved We believe that this is a very important result. Wigleys specifications for the emissions would be impossible to achieve in practice; whereas those suggested here are reasonably realistic (provided, of course, that binding international agreements are negotiated and maintained - which is an entirely different matter!)

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Large simulation models are products of the scientist's (hopefully inspired) imagination, most often constructed within a deterministic-reductionist (`bottom-up') paradigm using a hypothetico-deductive approach to modelling. The responses to external stimuli of such models (and the systems they seek to simulate) are dictated by the dominant modes of the system behaviour, which are normally few in number. As a result, there is only sufficient information in the data (simulated or real) to identify and estimate the parameters that characterise the dominant modal behaviour: i.e. the large model is not fully identifiable from the data. Data-Based Mechanistic (DBM) `Top-Down' models, identified and estimated in relation to observational data (simulated or real) using an inductive approach, capture the essence of the modally-dominant mechanisms. CONCLUSIONS

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Large simulation models can be reduced to their modally dominant form by DBM model reduction Dominant Mode Analysis (DMA) applied to data obtained from experiments conducted on the simulation model. This can help the modeller to better understand the strengths and limitations of the simulation model and what parts of it may be estimated from real data. Often, the DBM models obtained from either simulated or real data will not have sufficient fine detail to allow for `what-if' studies. However, such detail can be added to the model in the same way that it would be in the synthesis of a simulation model (e.g. adding a wier to a DBM flow-routing model). All of the analyses reported here were carried out using our MatlabToolbox CAPTAIN: http://www.es.lancs.ac.uk/cres/captain/. CONCLUSIONS continued

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