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Is a group of forecasts better than one?. Caio A. S. Coelho. Department of Meteorology, University of Reading. Plan of talk. Introduction Combination in meteorology Bayesian combination Conclusion and future directions. The Pioneer of combination. Pierre-Simon Laplace (1749-1827). a. - PowerPoint PPT Presentation
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Caio A. S. Coelho
Is a group of forecastsbetter than one?
Department of Meteorology, University of Reading
• Introduction• Combination in meteorology• Bayesian combination• Conclusion and future
directions
Plan of talk
Pierre-Simon Laplace(1749-1827)
C)yy(yy MSLSLS
The Pioneer of combination
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)1757,ichcovBos(SituationofMethod:y
squareLeast:y
MS
LS
))x(,x(CCwhere
Laplace (1818): Combination of two estimators
)xmin( 2
|)xmin(|
slope residual
+ numerical = methods
Have combined forecasts better skill thanindividual forecasts?
In modern times…
Forecasts from
•••
Forecast combination literature
• Trenkler and Gotu (2000): ~600 publications
(1970-2000)• Widely applied in Economics and
Meteorology
• Overlap of methods in these areas
• Combined forecasts are better than individual forecasts
Some issues
• What is the best method for combining?
• Is it worth combining unbiased forecasts with biased forecasts?
DEMETER coupled model forecasts
Nino-3.4 index (Y)
Period: 1987-99
9 members
Jul -> Dec
5 months lead
DEMETER web page: http://www.ecmwf.int/research/demeter
ECMWFMeteo-France (MF)Max Planck Institut (MPI)),(N~Y 2
tt
December Nino-3.4 raw forecasts
]ˆ96.1ˆ,ˆ96.1ˆ[:.I.P%95 tttt
Xt
tt
sˆ
Xˆ
),(N~Y 2tt
Forecast combination in meteorology
:wandw
:F
:M
:X
)t(Xww)t(F
io
M
1iiio
Linear combination of M ensemble mean forecasts X
constants
models
ensemble mean
combined forecast
• Bias-removed multimodel ensemble mean forecast (Uem)
• Regression-improved multimodel ensemble mean forecast(Rem)
• Regression-improved multimodel forecast (Rall)
Combination methods used in meteorology
Kharin and Zwiers (2002)
M
1iiio )t(Xww)t(F
How to estimate wo and wi ?
Kharin and Zwiers combined forecasts
2iii
n
1i
2i
2mRe
2mRe
)mReY(
n
1s
)s,m(ReN~Y
2iii
n
1i
2i
2Rall
2Rall
)RallY(
n
1s
)s,Rall(N~Y
]XXX[X
)s,Uem(N~Y
321
2X
Y: Observed December Nino-3.4 index
X: Ensemble mean forecast of Y for December
Thomas Bayes (1701-1761)
The process of belief revision on any event Y consists in updating the probability of Y
when new information X becomes available
The Bayesian approach
)xX(p
)Y(p)Y|xX(p)xX|Y(p
Example: Ensemble mean (X=x=27C)
Likelihood:p(X=x|Y)
Prior:p(Y)
Posterior:p(Y|X=x)
)C,Y(N~Y b
1TT
111T
obba
)SGCG(CGL
C)LGI()CGSG(D
)]YY(GX[LYY
)S],YY[G(N~Y|X o
Prior:
Likelihood:
Posterior:
1YYXYSSG
YGXGYo T
YYXX GGSSS
)D,Y(N~X|Y a
calibration
Bayesian multimodel forecast (B)
bias
July and December Reynolds OI V2 SST (1950-2001)
Nino-3.4 index observational data
Nino-3.4 indexmean values:Jul: 27.1CDec: 26.5Cr: 0.87
R2 =0.76
tyeartheforindex4.3NinoJuly
tyeartheforindex4.3NinoDecemberY
50.1C14.14),(N~|Y
t
t
1o
o2
t0t1ott
),(N~Y:iorPr 2otot t1oot
All combined forecasts
Uem
Rall
Rem
B
Why are Rall and B so similar?
Combined forecasts in Bayesian notationForecast
Likelihood Prior
Uem
Rem
Rall
B
)S,YY(N~Y|X o
)S],YY[G(N~Y|X o
)S],YY[G(N~Y|X o
)S],YY[G(N~Y|X o
)S,Y(N~Y YY
)S,Y(N~Y YY
),(N~Y 2otot
)C(Uniform
B priorRem and
Rall prior
)C,Y(N~Y b
)S],YY[G(N~Y|X oBayesian
combinationPrior:Likelihoo
d:
Skill and uncertainty measuresForecast MSE
(C)2
Skill Score(%)
Uncert.(C)2
Climatology 1.98 0 1.20
ECMWF (raw) 1.64 18 0.49
ECMWF (bias-removed) 0.18 91 0.49
ECMWF (regression-improved)
0.22 89 0.47
MF (raw) 0.27 86 0.39
MF (bias-removed) 0.22 89 0.39
MF (regression-improved) 0.20 90 0.46
MPI (raw) 15.24 -669 0.49
MPI (bias-removed) 1.52 23 0.49
MPI (regression-improved) 1.06 46 0.86
Uem 0.25 88 1.79
Rem 0.46 77 0.56
Rall 0.12 94 0.38
B 0.11 94 0.23Skill Score = [1- MSE/MSE(climatology)]*100%
Conclusions and future directions
• Forecast can be combined in several different ways
• Combined forecasts have better skill than individual forecasts
• Rall and B had best skill for Nino-3.4 example
• Inclusion of very biased forecast has not deteriorated combination
• Extend method for South America rainfall forecasts