1. Gap filling using a Bayesian-regularized neural network B.H. Braswell University of New Hampshire

2. MacKay DJC (1992) A practical Bayesian framework for backpropagation networks. Neural Computation, 4, 448-472. Bishop C (1995) Neural Networks for Pattern Recognition, New York: Oxford University Press. Nabney I (2002) NETLAB: algorithms for pattern recognition. In: Advances in Pattern Recognition, New York: Springer-Verlag. Proper Credit

3. Two-layer ANN is a nonlinear regression

4. e.g., tanh() usually nonlinear usually linear

5. Neural networks are efficient with respect to number of estimated parameters Polynomial of order M: N p ~ d M Consider a problem with d input variables Neural net with M hidden nodes: N p ~ d∙M

6. ✦ Early stopping ✦ Regularization ✦ Bayesian methods Avoiding the problem of overfitting:

7. ✦ Early stopping ✦ Regularization ✦ Bayesian methods

8. Avoiding the problem of overfitting: ✦ Early stopping ✦ Regularization ✦ Bayesian methods

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15. Hagen SC, Braswell BH, Frolking, Richardson A, Hollinger D, Linder E (2006) Statistical uncertainty of eddy flux based estimates of gross ecosystem carbon exchange at Howland Forest, Maine. Journal of Geophysical Research, 111. Braswell BH, Hagen SC, Frolking SE, Salas WE (2003) A multivariable approach for mapping subpixel land cover distributions using MISR and MODIS: An application in the Brazilian Amazon. Remote Sensing of Environment, 87:243-256. Previous Work

16. ANN Regression for Land Cover Estimation Band1 Band2 Band3 Band4 Forest Fraction Cleared Fraction Secondary Fraction Training data supplied by classified ETM imagery

17. ANN Regression for Land Cover Estimation

18. ANN Estimation of GEE and Resp, with Monte Carlo simulation of Total Prediction uncertainty ClimFlux

19. Weekly GEE from Howland Forest, ME based on NEE ANN Estimation of GEE and Resp, with Monte Carlo simulation of Total Prediction uncertainty

20. Some demonstrations of the MacKay/Bishop ANN regression with 1 input and 1 output

21. Noise=0.10 1.4

22. Noise=0.10 Linear Regression

23. Noise=0.10 ANN Regression

24. Noise=0.02 ANN Regression

25. Noise=0.20 ANN Regression

26. Noise=0.20 ANN Regression

27. Noise=0.10 ANN Regression

28. Noise=0.05 ANN Regression

29. Noise=0.05 ANN Regression

30. Noise=0.05 ANN Regression

31. Issues associated with multidimensional problems ✦ Sufficient sampling of the the input space ✦ Data normalization (column mean zero and standard deviation one) ✦ Processing time ✦ Algorithm parameter choices

32. Our gap-filling algorithm 1.Assemble meteorological and flux data in an Nxd table 2.Create five additional columns for sin() and cos() of time of day and day of year, and potential PAR 3.Standardize all columns 4.First iteration: Identify columns with no gaps; use these to fill all the others, one at a time. 5.Create an additional column, NEE(t-1), flux lagged by one time interval 6.Second iteration: Remove filled points from the NEE time series, refill with all other columns

33. Room for Improvement 1.Don’t extrapolate wildly, revert to time-based filling in areas with low sampling density, especially at the beginning and end of the record 2.Carefully evaluate the sensitivity to internal settings (e.g., alpha, beta, Nnodes) 3.Stepwise analysis for relative importance of driver variables 4.Migrate to C or other faster environment 5.Include uncertainty estimates in the output 6.At least, clean up the code and make it available to others in the project, and/or broader community