# BAYESIAN THEORY of ENSEMBLE FORECASTING by cometjunkie45

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```									BAYESIAN THEORY of ENSEMBLE FORECASTING
Roman Krzysztofowicz University of Virginia USA Presented at the HEPEX Meeting on Uncertainty Post-Processing for Streamflow Forecasts Delft, The Netherlands 23 – 25 June 2008
Acknowledgments: National Science Foundation, Grant No. ATM–0641572 Henry D. Herr, Coire J. Maranzano

BAYESIAN THEORY
(Krzysztofowicz, 1999)
• Derived from : Total Probability Law Bayes Theorem • • • • Applicable to any deterministic hydrologic model (=> informativeness) Fuses optimally information Quantifies total uncertainty (=> calibration) Outputs probabilistic forecast

• Basis for: system decomposition mathematical models computation methods 1. Bayesian Forecasting System (BFS) Analytic  Numerical For headwater basin 2. Ensemble BFS (EBFS) Ensemble / Monte-Carlo simulation For any basin

DECOMPOSITION OF UNCERTAINTY
OPERATIONAL UNCERTAINTY (exterior to forecasting theory)
Missing data Processing errors
• • •

INPUT UNCERTAINTY (dominant uncertainty)
Future precipitation

HYDROLOGIC UNCERTAINTY (all other uncertainties)
Models Parameters Initial conditions Deterministic inputs
• • •

TOTAL: INPUT and HYDROLOGIC

BAYESIAN THEORY
(Krzysztofowicz, 1999)

Decomposition Theorem
Total Uncertainty Ensemble RSF 



Hydrologic Uncertainty



Input Uncertainty Ensemble QPF

[Bayesian HUP]

Calibration Theorem In a Bayesian Forecasting System (BFS) with any deterministic hydrologic model, if the input distribution is well calibrated, then the output distribution is well calibrated. Inferred Requirements on Input 1. Ensemble QPF must be well calibrated. 2. Ensemble size must be large enough.

ENSEMBLE (NAIVE) SYSTEM

Precipitation Amounts {(w1,…,wM)} Deterministic Hydrologic Model Model River Stages {(s1,…,sN)}

How to implement Bayesian Theory?

ENSEMBLE BAYESIAN SYSTEM
Precipitation Ensemble Processor PEP (Monte Carlo) Precipitation Amounts {(w1,…,wM)} Deterministic Hydrologic Model Model River Stages {(s1,…,sN)} Hydrologic Uncertainty Processor HUP Real-Time Processing Off-line Simulation

Integrator INT (Monte Carlo) Actual River Stages {(h1,…,hN)}

NOTATION
EQPF Period
0 1 2 3

n

h0

h1 s1

h2 s2

h3 s3

Actual Model

v
n

– indicator of precip. occurrence during EQPF period:

v  1 yes,
– index of times – actual river stage

v 0

no

hn sn

– model river stage induced by actual precipitation during EQPF period no precip. uncertainty

HYDROLOGIC UNCERTAINTY PROCESSOR
Hypothesized Observed Precipitation River Event Stage V=1 H0 = h0 Forecasted River Stage H1 HN

V=0

H0 = h0

H1

HN

Process: Two-branch, Non-stationary, Markov Two families of joint conditional densities:
 v hN |s N , h 0  v  0, 1

Bayesian Formulation Meta-Gaussian Model: • Marginal distributions: Any form • Dependence structure: Non-linear Heteroscedastic

HYDROLOGIC UNCERTAINTY PROCESSOR
Prior 1-step Transition Density
r nv h n |h n−1  n  1, . . . , N v  0, 1

•

stochastic model of river stage process

Likelihood Function
f nv s n |h n , h n−1 , h 0 

n  1, . . . , N

v  0, 1

•

stochastic representation of deterministic model

Posterior 1-step Transition Density
 nv h n |s n , h n−1 , h 0   f nv  r nv

n  1, . . . , N

v  0, 1

•

Joint Conditional Density
 v h N |s N , h 0     nv h n |s n , h n−1 , h 0 
n1 N

v  0, 1

HUP: BAYESIAN REVISION
Conditional Expected Density
 nv s n |h n−1 , h 0    f nv s n |h n , h n−1 , h 0  r nv h n |h n−1  dh n
− 

n  1, . . . , N

v  0, 1

Posterior 1–step Transition Density
f nv s n |h n , h n−1 , h 0  r nv h n |h n−1   nv h n |s n , h n−1 , h 0    nv s n |h n−1 , h 0 

n  1, . . . , N

v  0, 1

SAMPLES
EQPF Period

v  1
h1 s1

or

v  0
h2 s2 h3 s3

h0

Actual Model

Prior Distribution
v;h 0 , h 1 , h 2 , h 3 

historical record (long)

Likelihood Function

simulation experiment (short)

v; s 1 , s 2 , s 3 , h 0 , h 1 , h 2 , h 3 

Hydrologic model re-initialized (as in real-time) Real-time inputs, except precipitation Actual precip. during EQPF period No precip. uncertainty

Marginal Distribution Functions of H1
(a)
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 10 12 14 16 18 20
Empirical Log-Weibull α11 = 1.59 β11 = 3.02 γ11 = 3.45

V1

Γ11(h1)

(b)
1.0 0.9 0.8 0.7

Actual River Stage h1 [ft]

Γ10(h1)

V0

0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 10 12 14
Empirical Log-logistic α10 = 2.66 β10 = 3.02 γ10 = 3.45

16

18

20

Actual River Stage h1 [ft]

Prior Dependence Structure:
n  1, 24 h
(a)
3 2 1 c11 = 0.702 3 2 1 c21 = 0.810 3 2 1

Hn |Hn−1 ,

V1

n  2, 48 h

n  3, 72 h
c31 = 0.789

w1

w2

0 -1 -2 -3

0 -1 -2 -3

w3
-3 -2 -1 0 1 2 3

0 -1 -2 -3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

w0

w1

w2

(b)
20 18 ρ11 = 0.685 20 18 ρ21 = 0.796 20 18 ρ31 = 0.774

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h3 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h0 [ft]

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

Prior Dependence Structure:
n  1, 24 h
(a)
3 2 1 c10 = 0.948 3 2 1 c20 = 0.797 3 2 1

Hn |Hn−1 ,

V0

n  2, 48 h

n  3, 72 h
c30 = 0.813

w1

w2

0 -1 -2 -3

0 -1 -2 -3

w3
-3 -2 -1 0 1 2 3

0 -1 -2 -3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

w0

w1

w2

(b)
20 18 ρ10 = 0.944 20 18 ρ20 = 0.783 20 18 ρ30 = 0.800

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h3 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h0 [ft]

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

Marginal Distribution Functions of H2 and S 2
(a)
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4 6 8 10 12 14
Empirical Log-Weibull

v=0 v=1

Γ2v(h2)

16

18

20

(b)
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 2 4

Actual River Stage h2 [ft]

v=0 v=1

Λ2v(s2)

Empirical Log-Weibull

6

8

10

12

14

16

18

20

Model River Stage s2 [ft]

Likelihood Dependence Structure:
n  1, 24 h
(a)
3 2 1 a11 = 0.946 3 2 1 a21 = 0.838 3 2 1 a31 = 0.682

S n |Hn ,

V1

n  2, 48 h

n  3, 72 h

x1

x2

0 -1 -2 -3

0 -1 -2 -3

x3
-3 -2 -1 0 1 2 3

0 -1 -2 -3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

w1

w2

w3

(b)
20 18 ρ11 = 0.941 20 18 ρ21 = 0.826 20 18 ρ31 = 0.665

Model River Stage s1 [ft]

Model River Stage s2 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Model River Stage s3 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

Actual River Stage h3 [ft]

Likelihood Dependence Structure:
n  1, 24 h
(a)
3 2 1 a10 = 0.998 3 2 1 a20 = 0.901 3 2 1 a30 = 0.811

S n |Hn ,

V0

n  2, 48 h

n  3, 72 h

x1

x2

0 -1 -2 -3

0 -1 -2 -3

x3
-3 -2 -1 0 1 2 3

0 -1 -2 -3

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

w1

w2

w3

(b)
20 18 ρ10 = 0.998 20 18 ρ20 = 0.893 20 18 ρ30 = 0.797

Model River Stage s1 [ft]

Model River Stage s2 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Model River Stage s3 [ft]

16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 18 20

Actual River Stage h1 [ft]

Actual River Stage h2 [ft]

Actual River Stage h3 [ft]

Prior and Posterior 1-Step Transition Densities
n1
24 h
Density

V  1
(a)
1.00 r11 (h1 | H0 = 8) φ11 (h1 | S1 = 6, H0 = 8) φ11 (h1 | S1 =15, H0 = 8)

V  0
(a)
1.00 r10 (h1 | H0 = 8) φ10 (h1 | S1 = 6, H0 = 8) φ10 (h1 | S1 =15, H0 = 8) 0.75

0.75

0.50

Density
4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00 2

0.00

n2
48 h

Actual River Stage h1 [ft]

2

4

6

8

10

12

14

16

18

20

(b)

Actual River Stage h1 [ft]

1.00 r21 (h2 | H1 = 6) φ21 (h2 | S2 = 9, H1 = 6, H0 = 8) r21 (h2 | H1 = 15) φ21 (h2 | S2 = 10, H1 = 15, H0 = 8)

(b)

1.00 r20 (h2 | H1 = 6) r20 (h2 | H1 = 15)

0.75

0.75

Density

0.50

Density
4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00 2

0.00

n3
72 h

Actual River Stage h2 [ft]

2

4

6

8

10

12

14

16

18

20

(c)

Actual River Stage h2 [ft]

1.00 r31 (h3 | H2 = 9) r31 (h3 | H2 = 10)

(c)

1.00 r30 (h3 | H2 = 9) r30 (h3 | H2 = 10)

0.75

0.75

Density

0.50

Density
4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00 2

0.00

Actual River Stage h3 [ft]

2

4

6

8

10

12

14

16

18

20

Actual River Stage h3 [ft]

Posterior 1-Step Transition Densities
n1
24 h
Density

V  1
(a)
1.00

V  0
(a)
1.00 H0 = 4, 6, 8, 10, 12, 14 S 1 = 15 0.75

H0 = 4, 6, 8, 10, 12, 14 S 1 = 15

0.75

0.50

Density
2 4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00

n2
48 h

Actual River Stage h1 [ft]

0.00 2

4

6

8

10

12

14

16

18

20

(b)

Actual River Stage h1 [ft]

1.00

H0 = 8 H1 = 4, 6, 8, 10, 12, 14

(b)

1.00

H0 = 8 H1 = 4, 6, 8, 10, 12, 14

0.75

S 2 = 10

0.75

S 2 = 10

Density

0.50

Density
2 4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00

n3
72 h

Actual River Stage h2 [ft]

0.00 2

4

6

8

10

12

14

16

18

20

(c)

Actual River Stage h2 [ft]

1.00

H0 = 8 H2 = 4, 6, 8, 10, 12, 14

(c)

1.00

H0 = 8 H2 = 4, 6, 8, 10, 12, 14

0.75

S3 = 5

0.75

S3 = 5

Density

0.50

Density
2 4 6 8 10 12 14 16 18 20

0.50

0.25

0.25

0.00

Actual River Stage h3 [ft]

0.00 2

4

6

8

10

12

14

16

18

20

Actual River Stage h3 [ft]

ENSEMBLE BAYESIAN SYSTEM
Precipitation Ensemble Processor PEP (Monte Carlo) Precipitation Amounts {(w1,…,wM)} Deterministic Hydrologic Model Model River Stages {(s1,…,sN)} Hydrologic Uncertainty Processor HUP Real-Time Processing Off-line Simulation

Integrator INT (Monte Carlo) Actual River Stages {(h1,…,hN)}

INTEGRATOR (Monte Carlo)
• Input: one realization
Model River Stages Indicator of Precip. Initial River Stage Given Generate
h1 h2 h3
• • •

s 1 , s 2 , s 3 , . . . , s N 
v

h0

From Posterior Distribution
 1v |s 1 , h 0   2v |s 2 , h 1 , h 0 

s1 s 2, h 1 s 3, h 2
• • •

 3v |s 3 , h 2 , h 0 
• • •

• Output: multiple realizations
Actual River Stages
h 1 , h 2 , h 3 , . . . , h N 

Bayesian Ensemble Forecast of River Stages

20 18

M = 100

Stage, hn [feet]

16 14 12 10 8 6 4 0

ν = 0.81
v=0 v=1

6

12

18

24

30

36

42

48

54

60

66

72

time, n x 6 h

Predictive n-Step Transition Distributions
Empirical Distributions Ensemble Size M = 100
1.0 0.9 0.8 0.7 1.0 0.9 1.0 0.9 1.0 0.9

n=1

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n=2

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n=3

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n=4

Ψ n(hn)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

1.0 0.9 0.8 0.7

1.0 0.9

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 5 10 15 20 25 0.0 0 5 10 15 20 25

1.0 0.9

n=5

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n=6

n=7

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15

n=8

Ψ n(hn)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25

20

25

1.0 0.9 0.8 0.7

1.0 0.9

1.0 0.9

1.0 0.9

n=9

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n = 10

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n = 11

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

n = 12

Ψ n(hn)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15 20 25

0

5

10

15

20

25

0

5

10

15

20

25

0

5

10

15

20

25

hn

hn

hn

hn

ERROR IN DISTRIBUTION Due to Ensemble Size
1.0 0.9 0.8 0.7

Ψ 30(h3)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10

v=0 M = 100 Analytical Ensemble 1 Ensemble 2 Ensemble 3 max MAD = 0.1815 15 20 25

h3 [feet]
1.0 0.9 0.8 0.7

Ψ 31(h3)

0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10

v=1 M = 100 Analytical Ensemble 1 Ensemble 2 Ensemble 3 max MAD = 0.2010 15 20 25

h3 [feet]

̂ MAD Between Ensemble nv h n  and Analytical nv h n 
0.14 0.12 0.14

0.12 0.10 0.08 0.06 0.04 0.02

v=1 n= 1

0.10 0.08 0.06 0.04 0.02 0.00 0 2000 4000 6000

8000

10000

0.00

0

2000

4000

6000

8000

10000

M0
0.14 0.12 0.14

M1

0.12 0.10 0.08 0.06 0.04 0.02

v=1 n= 2

0.10 0.08 0.06 0.04 0.02 0.00 0 2000 4000 6000

8000

10000

0.00

0

2000

4000

6000

8000

10000

M0
0.14 0.12 0.14

M1

0.12 0.10 0.08 0.06 0.04 0.02

v=1 n= 3

0.10 0.08 0.06 0.04 0.02 0.00 0 2000 4000 6000

8000

10000

0.00

0

2000

4000

6000

8000

10000

M0

M1

̂ MAD Between Ensemble Θ nv h n |h n −1  and Analytical Θ nv h n |h n −1 
0.24 0.22 0.20 0.18 0.24 0.22 0.20 0.18

v=0 n= 2 Average MAD d = 0.1 h1 = 6 ft

v=1 n= 2 d = 0.5 h1 = 8 ft

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 2000 4000 6000

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

8000

10000

0

2000

4000

6000

8000

10000

M0

M1

0.24 0.22 0.20 0.18

v=0 n= 3 Average MAD d = 0.3 h2 = 6 ft

0.24 0.22 0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00

v=1 n= 3 d = 0.5 h1 = 8 ft

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 2000 4000 6000

8000

10000

0

2000

4000

6000

8000

10000

M0

M1

BAYESIAN THEORY: SUMMARY
Ensemble Bayesian Forecasting System (EBFS) 0. Deterministic Hydrologic Model 1. Precipitation Ensemble Processor • calibrated ensemble QPF 2. Hydrologic Uncertainty Processor • all hydrologic uncertainties • correct theoretic structure (Bayesian) • model fitting data (meta-Gaussian) 3. Integrator • Monte Carlo re-generation (ensemble size) Forecast Products • calibrated distributions

REFERENCES
Krzysztofowicz, R., Bayesian Theory of Probabilistic Forecasting via Deterministic Hydrologic Model, Water Resources Research, 35(9), 2739–2750, 1999. Krzysztofowicz, R. and Herr, H.D., Hydrologic Uncertainty Processor for Probabilistic River Stage Forecasting: Precipitation-Dependent Model, Journal of Hydrology, 249(1–4), 46–68, 2001. Maranzano, C.J. and Krzysztofowicz, R., Identification of Likelihood and Prior Dependence Structures for Hydrologic Uncertainty Processor, Journal of Hydrology, 290(1–2), 1–21, 2004. Krzysztofowicz, R. and Maranzano, C.J., Hydrologic Uncertainty Processor for Probabilistic Stage Transition Forecasting, Journal of Hydrology, 293(1–4), 57–73, 2004. Krzysztofowicz, R. and Maranzano, C.J., Bayesian System for Probabilistic Stage Transition Forecasting, Journal of Hydrology, 299(1–2), 15–44, 2004. Herr, H.D. and Krzysztofowicz, R., Bayesian Ensemble Forecast of River Stages and Ensemble Size Requirements, Research Paper RK–0801, July 2008.

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