1. Probabilistic Weather Forecasting Using Bayesian Model Averaging
J. McLean Sloughter
Adviser: Tilmann Gneiting
GSR: Susan Joslyn
Committee members: Adrian Raftery & Cliff Mass
8 May, 2009
This work was supported by MURI, JEFS, & NSF grants

2. Background
Motivation
Ensemble forecasting
Bayesian model averaging
Dissertation outline
BMA for vector wind
Data
Decomposing the problem
Bias-correction
Error distributions
Model
Results
Future directions
References
Acknowledgements

3. Why probabilistic forecasting?
Situations where certain ranges or thresholds are of interest
Situations where knowing not just the most likely outcome, but possible extremes are important
Situations involving a cost / loss analysis, where probabilities of different outcomes need to be known
Examples:
Wind energy
Military
Sailing
Airports
Winter road maintenance

4. Ensemble Forecasting
48-hour forecasts for maximum wind speeds on 7 August 2003

5. Ensemble Forecasting
Single forecast model is run multiple times with different initial conditions
Forecasts created on a 12-km grid, and bilinearly interpolated to locations of interest
Ensemble mean tends to outperform individual members
Spread-skill relationship: spread of forecasts tends to be correlated with magnitude of error

6. Ensemble Forecasting
Would like the ensemble to look like draws from the same distribution as the observed values
Ensemble only captures one source of variability – uncertainty in initial conditions
Ensemble distribution is underdispersed relative to observed values
Ensemble members agree with one another more than they agree with observations

7. Bayesian model averaging (BMA)
Weighted average of multiple component models
One component per ensemble member
Each component a distribution of observed value conditioned on an ensemble member forecast
Model fit based on training data – past sets of forecasts / observations
Use a sliding window of training data
Weights determined by how well each member fits the training data

8. Bayesian model averaging
where
is the deterministic forecast from member k,
is the weight associated with member k, and
is the estimated density function for y given member k

9. Background
Motivation
Ensemble forecasting
Bayesian model averaging
Dissertation outline
BMA for vector wind
Data
Decomposing the problem
Bias-correction
Error distributions
Model
Results
Future directions
References
Acknowledgements

10. Dissertation Outline Precipitation forecasting
Sloughter et al., 2007, MWR
Extends BMA to a specific case of skewed and non-continuous distributions
Wind speed forecasting
Sloughter et al., 2009, JASA
Extends methods of Sloughter et al. (2007) to other forms of skewed and non-continuous distributions
Examines robustness of BMA to details of model selection
Vector wind forecasting
This talk
Extends BMA to multivariate distributions

11. Background
Motivation
Ensemble forecasting
Bayesian model averaging
Dissertation outline
BMA for vector wind
Data
Decomposing the problem
Bias-correction
Error distributions
Model
Results
Future directions
References
Acknowledgements

12. BMA for vector wind
Methods exist for using Bayesian Model Averaging to create probabilistic forecasts for weather quantities that can be expressed as a mixture of normals (Raftery et al., 2005), such as temperature and pressure.
Expanded to be applied to non-continuous and skewed quantities such as precipitation and wind speed in Sloughter et al. 2007, Sloughter et al
A method is needed for modeling multivariate quantites such as wind vectors.

13. Knot
A knot is a measure of speed used in nautical, meteorological, and aviation settings
1.852 kilometers per hour
1.151 miles per hour
0.514 meters per second
Sailors would throw out the chip log (a board designed to stay stationary in water) tied to a rope with knots spaced 7 fathoms (42 feet) apart
They would then count how many knots were fed out in 30 seconds

14. Knot
4-6 knots is a light breeze – leaves move, breeze can be felt on one’s face
11-16 knots is a moderate breeze – dust and paper will be blown about, whitecaps will form on the water
20-21 knots is generally the threshold for issuing a small craft advisory
34-40 knots is a gale – small branches break from trees, walking becomes difficult

15. Data
This work uses wind data from the Pacific Northwest for the full year 2003, plus November and December 2002 (results for 2003 data, 2002 used only for training)
“Instantaneous” vector wind measurements
Measured in knots
Each forecast consists of 8 ensemble members
Data were available for 343 days, missing for 83 days
A total of observations, averaging 111 observations per day
All work that follows is based on 30-day training periods, with 2-day-ahead forecasting

16. Data Data from Surface Airway Observation stations
Airports in BC, Washington, Oregon, Idaho, and California

17. Decomposing the problem
Wind has two dimensions, east/west direction and north/south direction
BMA uses a mixture distribution with one component per ensemble member
Consider each mixture component a bivariate distribution parameterized in terms of a mean vector and a covariance matrix
Assume that the mean of the distribution is some function of the forecast vector, and that the covariance matrix does not depend upon the forecast (exploratory plots support these assumptions)

18. Decomposing the problem
h(fk) is the mean (a bias-corrected forecast)
BV(0, Q) is the distribution of the forecast error
Model the distribution of the errors rather than the observed values
Has the advantage of having constant parameters across forecast values
Can then be decomposed into two separate problems:
bias-correcting the forecast
modeling the error distribution

19. Bias-correction For simplicity, consider affine bias corrections
Two potential forms of bivariate bias correction
Additive bias-correction
Full affine bias-correction
Where Y is the observed wind vector, fk is the kth vector forecast, ak is an additive bias vector, and Bk is a transformation matrix

20. Bias-correction Out-of-sample results using 30-day training period
Bivariate root mean squared error (in knots) for one ensemble member
Out-of-sample results using 30-day training period
Similar results hold for other ensemble members
Affine bias-correction shows a marked improvement

21. Error distributions
Now deal with the error field (observations minus bias-corrected forecasts)
Exploratory work suggests that the distributions are ellipsoidal, but have heavier tails than normal distributions
Transform the error vector (rkcosqk, rksinqk)T by raising the magnitude of the vector to the 4/5 power while preserving the angle
Model this as a bivariate normal distribution

22. Model Thus, our final model is:
Where the gk are the distributions on y implied by the distributions of the transformed error vectors
Model parameters are estimated globally using all observation locations

23. Model
Bias-correction fit via linear regression (separate bias correction for each mixture component)
Weights and covariance matrix estimated via maximum likelihood using the EM algorithm
Use latent variables zkst which are indicators that forecast k was the best forecast at station s at time t

24. Model
E step:
M step:

25. Model
M step (continued)

26. Results We simulate a large number of forecasts from our distribution
Can evaluate the forecast of either the wind vector or derived quantities (marginal speed or direction) from the empirical distribution of our forecasts
Essentially creating a new, larger ensemble of forecasts that should be better-calibrated than the original ensemble

27. Example
To illustrate what the BMA distribution is doing, consider the case of forecasting at Omak, Washington on February 4th, 2003

29. Results
Our goal is to maximize sharpness subject to calibration (the Gneiting principle)
By calibration, we mean that we want our probability distribution function to be correct – if we forecast an event as happening with probability .9, we want it to happen 90% of the time
By sharpness, we mean that we want predictive intervals to be as narrow as possible

30. Results
For univariate quantities, the verification rank histogram is a tool that can be used to assess the calibration of an ensemble forecast
Find the rank of each forecast relative to the ensemble members
If the ensemble is properly calibrated, the observation and forecasts should be interchangeable
If so, each potential rank of the forecast should have equal probability
Thus, a histogram of the ranks should look flat

31. Results
For multivariate quantities, there is an analogous multivariate rank histogram (MVRH), again based on the assumption of exchangeability
Define if and only if in every dimension
For each member of the combined set of the observation and the forecasts, find the pre-rank
The multivariate rank is the rank of the observation pre-rank, with any ties resolved at random
If we have a set of 8 forecasts and 1 observation, there are 9 possible rankings of the observation relative to the forecasts

32. Results
MVRH for the raw ensemble (left) and BMA forecast distribution (right)
Raw ensemble is under-dispersed
BMA forecast distribution is much better-calibrated

33. Results
The energy score (ES) is a scoring rule for multivariate probabilistic forecasts that takes into account both calibration and sharpness
In the univariate case, it reduces to the continuous ranked probability score (CRPS)
P is the predictive distribution, x the observed wind vector, X and X’ independent random variables with distribution P

34. Results There may still be interest in a point forecast as well
We can use the spatial median as a point forecast
We can assess the quality of a multivariate point forecast using the multivariate mean absolute error (MMAE)

35. Results
BMA outperforms climatology and the raw ensemble both in terms of the probabilistic forecast and the deterministic forecast

36. Results – marginal speed and direction
Again consider verification rank histograms to assess calibration
Both speed (top) and direction (bottom) are much improved by BMA

37. Results – marginal speed and direction
CRPS is the scalar equivalent of the energy score
DCRPS is the angular equivalent
Scalar point forecasts can be assessed by the MAE, and angular point forecasts by the mean directional error (MDE)
Can also look at coverage and width of 77.8% prediction intervals for scalar forecasts – coverage assesses calibration, width assesses sharpness

38. Results – marginal speed and direction
Wind speed:
Wind direction:

39. Results – marginal speed and direction
We can see that for both speed and direction, BMA improves the quality of both the probabilistic and deterministic forecasts
BMA produces marginal distributions that are better-calibrated than the raw ensemble and sharper than climatology

40. Background
Motivation
Ensemble forecasting
Bayesian model averaging
Dissertation outline
BMA for vector wind
Data
Decomposing the problem
Bias-correction
Error distributions
Model
Results
Future directions
References
Acknowledgements

41. Future Directions
Develop a BMA method to explicitly model marginal instantaneous wind speed and compare to the performance of the forecasts from this model (current BMA for marginal wind speed is for maximum wind speeds, not instantaneous)
Incorporate spatial information, either through explicitly modeling some spatial structure to our parameters or by estimating parameters locally rather than globally
Investigate using an exponential forgetting for training data rather than a sliding window, which could allow for faster computation through the use of updating formulae for parameter estimates
Extend multivariate methods to jointly forecast multiple weather quantities simultaneously

42. References
Raftery, A.E., Gneiting, T., Balabdaoui, F. and Polakowski, M. (2005). Using Bayesian Model Averaging to calibrate forecast ensembles. Monthly Weather Review, 133,
Sloughter, J. M., Raftery, A. E., Gneiting, T. and Fraley, C. (2007). Probabilistic quantitative precipitation forecasting using Bayesian model averaging. Monthly Weather Review, 135,
Sloughter, J. M., Gneiting, T., and Raftery, A.E. (2009). Probabilistic wind speed forecasting using ensembles and Bayesian model averaging. Journal of the American Statistical Association, accepted.
Mass, C., Joslyn, S., Pyle, P., Tewson, P., Gneiting, T., Raftery, A., Baars, J., Sloughter, J. M., Jones, D., and Fraley, C. (2009). PROBCAST: A web-based portal to mesoscale probabilistic forecasts. Bulletin of the American Meteorological Society, in press.

43. Acknowledgements Committee: Statistics folks:
Tilmann Gneiting - adviser
Adrian Raftery, Cliff Mass - committee members
Susan Joslyn - GSR
Statistics folks:
Veronica Berrocal, Chris Fraley, Thordis Thorarinsdottir, Will Kleiber, Larissa Stanberry, Matt Johnson, Robert Yuen, Michael Polakowski, Nicholas Johnson
Atmospheric Sciences folks:
Jeff Baars, Eric Grimit, Jeff Thomason, Tony Eckel
APL folks:
Patrick Tewson, John Pyle, David Jones, Janet Olsonbaker, Scott Sandgathe
Psychology folks:
Limor Nadav-Greenberg, Buzz Hunt, Queena Chen, Jared Le Clerc, Rebecca Nichols, Sonia Savelli