Recap on Data Assimilation

Transcription

1 Concluding Thoughts

2 Recap on Data Assimilation FORECAST ANALYSIS

3 Kalman Filter Forecast Analysis Analytical projection of the ANALYSIS mean and cov from t-1 to the FORECAST mean and cov for t Update FORECAST mean and cov for t to the ANALYSIS mean and cov for t Normal Prior = Model FORECAST mean & cov Normal Likelihood = Data Conjugate Normal posterior

4 Ensemble Kalman Filter Forecast Analysis Sample from ANALYSIS posterior from t-1 Project each sample to t Approx FORECAST with sample mean & cov Update FORECAST mean and cov for t to the ANALYSIS mean and cov for t Normal Prior = Model FORECAST mean & cov Normal Likelihood = Data Conjugate Normal posterior

5 EnKF Process model applied to each ensemble member FORECAST Sample ensemble members from analysis posterior Calculate mean and cov across ensemble model predictions ANALYSIS Normal prior + Normal likelihood

6 Particle Filter Forecast Analysis Project each ensemble member from t t+1 Normal Likelihood = weight Multiply new weight x previous weight Optional: If any ensemble member has too much weight resample ensemble Monte Carlo posterior based on weighted statistics

7 PF Process model applied to each ensemble member FORECAST ANALYSIS Weight = Normal likelihood

8 Ensemble Analysis Allows us to account for multiple sources of uncertainty by sampling from their dist'ns Model state variables / initial conditions Parameter uncertainty Covariates / drivers / scenarios Different models For computational reasons, the number of ensemble members, m, is often not large Some error propagation is better than none! For large m, analogous to model CI / PI

9 Model Averaging Any forecasting/prediction (not just DA) Often have multiple candidate models that are not significantly different (AIC, DIC, etc) Predict using a weighted average of the predictions by each model P X D = k P X M k, D P M k D

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11 Bayesian Model Averaging P X D = P X M k, D P M k D P M k D = model prior P D M k P M k P D M k P M k P D M = P D k, M k k likelihood P M k k prior d k Hoeting et al Bayesian Model Averaging: A Tutorial

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13 Predictions and Decision Making Predicting the behavior of some natural phenomenon Global change, endangered spp, disease spread Fully specified uncertainties Information content inversely proportional to forecast uncertainty Falsely overconfident prediction leads to poor decisions Contingent on explicit scenarios Indicate possibilities, not definitive probabilites What is forecastable? Clark et al 2001 Ecological Forecasting

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15 What is statistical modeling? Confronting models with data Design the statistical analysis to fit the data rather than the data to fit the test

16 Objectives Literacy Read and evaluate advanced stats used in papers Proficiency underlying statistical concepts Software: R, OpenBUGS Exposure to advanced topics Paradigm shift

17 A bit more on motivation... Data are usually complex Violate the assumptions of classical tests This complexity can be addressed with modern techniques

18 Statistical Paradigms Statistical Estimator Method of Estimation Output Data Complexity Prior Info Classical Cost Function Analytical Solution Point Estimate Simple No Maximum Likelihood Probability Theory Numerical Optimization Point Estimate Intermediate No Bayesian Probability Theory Sampling Probability Distribution Complex Yes The unifying principal for this course is statistical estimation based on probability

19 Where to go from here... Ecological Models and Data in R (Bolker 2008) Mostly Likelihood based Lots of great R tricks Well written / easy to follow PDF's of draft version still on Ben's webpage Bayesian Methods for Ecology (McCarthy 2007) Lots of good / simple BUGS examples DANGEROUS for the untrained Devoid of theory, caveats, assumptions, and any discussion of numerical methods

20 Hierarchical Modeling for the Environmental Sciences (Clark and Gelfand 2006) Mostly a collection of case studies Bayesian Data Analysis (Gelman et al 2004) The standard reference for Bayes and Hierarchical Bayes Is a stats book, not biostats, so it's even harder to read than Clark A number of other books listed on the course website

21 Advanced Topics... Spatial: Banerjee et al 2004 Hierarchical modeling and analysis for spatial data Time-series Diggle 1990 Time Series: A biostatistical introduction Data assimilation Lewis et al 2006 Dynamics Data Assimilation

22 Additional Resources lists: BUGS In addition to an endless stream of people asking questions, will have occasional postings about workshops Often useful nuggets of wisdom R-sig-ecology Mostly frequentist, but many useful R tricks for ecological data Equivalent lists for many other subjects: e.g. genetics, phylogenetics, geostatistics, dynamic models

23 It usually takes multiple exposures to this material before it really sinks in