1. Bayesian analysis of a conceptual transpiration model with a comparison of canopy conductance sub-models Sudeep Samanta Department of Forest Ecology and Management University of Wisconsin - Madison

2. Motivation Mechanistic or process based conceptual models: basis for extrapolation, scientific understanding. Requires: model testing, parameter estimation, model discrimination through comparison. Problem: Most models are deterministic.

3. Objectives Estimate model parameter and prediction uncertainties with a simple probabilistic error term. Develop a methodology for comparing models that accounts for both model fit and complexity.

4. Bayesian approach Probability as the mathematical expression of the degree of belief. Direct quantification of uncertainty. Allows for incorporating prior knowledge. Computational advantages: Conceptually simple, Computationally feasible.

5. Bayesian approach Unknown parameters β and σ 2 are random quantities. Prior distributions describe knowledge about the parameters prior to the experiment. Posterior distributions expresses information gained from data. Bayes’ Rule:

6. Uncertainty analysis of a mechanistic transpiration model

7. Half hourly measurements made at the ChEAS site, WI: Average transpiration from eight sugar maple trees, Measurements within canopy - Hay Creek, WI, Measurements above canopy - Willow Creek, WI. Transpiration Data 05/05/200109/11/2001

8. Transpiration Model Canopy conductance: Penman – Monteith Equation: Probabilistic Model:

9. Prior and posterior distributions Noninformative prior distribution of parameters: β uniformly distributed within specified limits, σ 2 uniform on log(σ), joint prior density: Joint posterior distribution used for MCMC:

10. Histograms of marginal posterior distributions Symmetric, well defined.

11. Residuals Residuals are not obviously biased or heteroscedastic. Small systematic errors might be present.

12. Posterior interval for transpiration rate Overall, the posterior density regions are consistent with the observations. Relationship between observations and predictions inconsistent from day to day.

13. Conclusions drawn from uncertainty analysis Possible to obtain estimate of error variability. Bayesian approach is useful: to estimate parameter values, to estimate uncertainties associated with the parameters and predictions. These uncertainties may be considerable and should be taken into account when drawing inferences from data. Systematic errors → consider: Other transpiration models, Collection of other data might improve the modeling.

14. Comparison of canopy conductance sub-models

15. Model comparison metric Deviance Information Criterion (DIC): A penalized criterion that combines Bayesian measures of model complexity and fit. Explicitly accounts for prior and posterior distributions. Can be used for comparing models of arbitrary structure. Can be used in situations where all possible models cannot be specified ahead of time. [Spiegelhalter et al., 2002]

16. Measure of fit in DIC “Bayesian deviance”, in this case:

17. Measure of complexity in DIC Measures of fit and complexity combined to define DIC as: Effective number of parameters, p D : where = posterior mean of the deviance, and = deviance at the posterior estimates of the parameters. Model comparison metric: DIC

18. Canopy conductance: Where: Canopy conductance sub-model g Smax, highest conductance per unit leaf area (mol m -2 s -1 ), L max, maximum leaf area index, D, vapor pressure deficit within canopy (kPa), Q p, average PAR photon flux density (  mol m -2 s -1 ), T c, canopy air temperature (˚C), Ψ s, soil water potential (MPa), lf doy, fitted transpiration (mm s -1 ).

19. Constraint functions Estimated parameters: g Smax, δ, δ h, A, Ψ 0, T o, T lo, T hi, and lf scl.

20. Marginal posterior distributions for MC7 Distributions for Ψ 0, T o, T lo, and T hi are skewed, Very wide 95% posterior intervals.

21. Results of comparison using DIC Increased model complexity did not always improve DIC.

22. Conclusions drawn from model comparison DIC is useful as a metric to identify an appropriate conceptual model for a given set of data. DIC helps in the development of more refined models. Some of the parameters in an overly complex model may be poorly identified based on the information available in the data. Future data collection efforts may be directed by: data requirements of proposed models, need for better parameter identification.

23. Future research Usability and effectiveness of this methodology with other models and data. Use of informative priors. Use of other error models. Use of the methodology for models with more than one observed output or spatially distributed output.