Peter Reichert Eawag Dübendorf, ETH Zürich and SAMSI

Analyzing input and structural uncertainty of deterministic models with stochastic, time-dependent parameters Peter Reichert Eawag Dübendorf, ETH Zürich and SAMSI

Contents • Motivation • Approach • Implementation[Tomassini, Reichert, Künsch and Borsuk, 2007, subm.] • Results for a simple epidemiological model[Cintron-Arias, Reichert, Lloyd, Banks, 2007, initiated] • Results for a simple hydrologic model[Reichert and Mieleitner, 2007, in progress] • Discussion Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Motivation Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Motivation

Motivation Problem: • In most applications, there is a bias in results of deterministic models. • This bias is typically caused by input and model structure errors, and not by correlated measurement errors. • Bias modelling describes this bias statistically, but does not directly support the identification of its causes and the formulation of improved model structures. Objective: • Search for improved model structures or stochastic influence factors to model input or parameters that reduce/remove the bias. Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Approach Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Approach

Approach The proposed approach consists of the following steps: • Replace selected (constant) parameters of the deterministic model by a stochastic process in time. • Estimate the states of this process jointly with the model parameters. • Analyze the degree of bias reduction achieved by doing this for different parameters. • Analyze the identified time dependence of the parameters for correlation with external influence factors and internal model variables. • Improve the deterministic model if such dependences were found. • Explain the remaining stochasticity (if any) of the model by including the appropriate stochastic parameter(s) into the model description. Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Implementation Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Implementation

Model Deterministc model: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Consideration of observation error:

Model Model with parameter i time dependent: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Time Dependent Parameter The time dependent parameter is modelled by a mean-reverting Ornstein Uhlenbeck process: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion This has the advantage that we can use the analytical solution: or, after reparameterization:

Inference We combine the estimation of • constant model parameters, , with • state estimation of the time-dependent parameter(s), , and with • the estimation of (constant) parameters of the Ornstein-Uhlenbeck process(es) of the time dependent parameter(s), . Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Inference Gibbs sampling for the three different types of parameters. Conditional distributions: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion simulation model (expensive) Ornstein-Uhlenbeck process (cheap) Ornstein-Uhlenbeck process (cheap) simulation model (expensive)

Inference Metropolis-Hastings sampling for each type of parameter: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion 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, , 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 .

Estimation of Hyperparametersby Cross - Validation Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Due to identifiability problems we select the two hyperparameters (s,t) by cross-validation:

Estimation of Hyperparametersby Cross - Validation For a state-space model of the form Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion we can estimate the pseudo-likelihood from the sample:

Results for Epidemiological Model Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Results for a Simple Epidemiological Model • Model • Model Application • Preliminary Results

1 2 3 4 5 6 Model A Simple Epidemiological Model: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion S: susceptible I: infected N: „total population“ N-S-I: resistant 6 model parameters 2 initial conditions 1 standard dev. of meas. err. Dushoff et al. 2004

Model Application Influenza Outbreaks in the USA Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Model predictions and data Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Residual analysis – constant parameters Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Residual analysis – beta0 time varying Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Time dependent parameter: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Next Steps • More Fourier terms, more general periodicity. • Cross validation for choice of s and t. • Interpretation of remaining stochasticity. • Inclusion of identified forcing in model equations. • Inclusion of stochastic parameter in model equations. • Different data set: measles. Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary for Hydrologic Model Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Results for a Simple Hydrologic Model • Model • Model Application • Preliminary Results

Model A Simple Hydrologic Watershed Model (1): Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Kuczera et al. 2006

3 A 4 5 1 B 2 C 6 7 Model A Simple Hydrologic Watershed Model (2): Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion 7 model parameters 3 initial conditions 1 standard dev. of meas. err. 3 „modification parameters“ Kuczera et al. 2006

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 Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

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, fQ) equal to unity. Estimation of time-dependent parameters: Ornstein-Uhlenbeck process applied to log of the parameter.Hyper-parameters: t = 5d, s fixed, only estimation of initial value and mean (0 for log frain, fpet, fQ). Constant parameters as above. Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Posterior marginals: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Max. post. simulation with constant parameters: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Residuals of max. post. sim. with const. pars.: Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Residual analysis, max. post., constant parameters Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Residual analysis, max. post., q_gw_max time-dependent

Preliminary Results Residual analysis, max. post., s_F time-dependent Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Residual analysis, max. post., f_rain time-dependent

Preliminary Results Residual analysis, max. post., f_pet time-dependent Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Residual analysis, max. post., f_Q time-dependent

Preliminary Results Residual analysis, max. post., k_et time-dependent Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Residual analysis, max. post., q_latmax time-dependent

Preliminary Results Residual analysis, max. post., k_s time-dependent Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion NS parameter NS parameter 0.90 f_rain 0.79 f_pet 0.88 s_F 0.69 q_gwmax 0.87 k_s 0.67 q_latmax 0.83 f_Q 0.67 k_et const. parameter: 0.64

Preliminary Results Time-dependent parameters Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Preliminary Results Scatter plot of parameter vs model variables Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Next Steps • Search for correlations between time dependent parameters and external influence factors and system variables. • Improve formulation of deterministic model and redo the analysis. • Cross validation for choice of s and t. • Inclusion of stochastic parameter in model equations to explain remaining stochasticity. Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Discussion Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion Discussion

Discussion Discussion • 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? • How to improve efficiency? • How to combine statistical with contextual knowledge? • ? Motivation Approach Implementation Results for epidem. Model Results for hydrol. model Discussion

Thank you for your attention

Peter Reichert Eawag Dübendorf, ETH Zürich and SAMSI