BAYESIAN PROCESSOR OF ENSEMBLE (BPE): PRIOR DISTRIBUTION FUNCTION

Transcription

1 BAYESIAN PROCESSOR OF ENSEMBLE (BPE): PRIOR DISTRIBUTION FUNCTION Parametric Models and Estimation Procedures Tested on Temperature Data By Roman Krzysztofowicz and Nah Youn Lee University of Virginia 4 December 2008 Acknowledgments: Work supported by NSF, Grant No. ATM Data provided by Environmental Modeling Center, NCEP/NWS/NOAA.

2 DATA Predictand: 2m temperature at 12 UTC Forecast time: 00 UTC Location: Savannah, GA Actual Location: 32º 8'N / 81º 13'W Climatic Data: 32.5º N / 80º W Data Climatic Data: 1 January December 1998 (40 years) 29 February (leap year) excluded 5 March 1997 filled in

3 PRIOR DISTRIBUTION FUNCTION k index of the day (k = 1,, 365) l lead time in days (l = 1, 2,, 16) W k predictand (variate) w k realization of W k Prior Marginal D. F. G k (w k ) = P(W k w k ) Prior Markov D. F. H kl (w k w k l ) = P(W k w k W k l = w k l ) Prior D. F. Climatic Probabilistic Forecast Challenges Reference for calibration of ensemble Limit to which calibrated ensemble converges as LT Time series W 1,, W 365 is nonstationary Distributions G k and H kl are non-gaussian

4 STANDARDIZATION Purpose: obtain time series that has stationary mean, variance, marginal D.F. (possibly) Climatic sample for each day k (k = 1,, 365) 15-day sampling window centered on day k w k n : n 1,...,M M = 15 days x 40 years = 600 Sample estimates m k prior (climatic) mean s k prior (climatic) standard deviation Standardized climatic sample w k n w k n m k s k n 1,...,M w k n : n 1,...,M

5 Sample deciles in original space (15-day window), SAV 300 Temperature ( K) Day Maximum Minimum

6 Sample mean (15-day), Fourier series (2 nd order), SAV Mean (degree K) Day

7 5.5 Sample std. dev. (15-day), Fourier series (2 nd order), SAV 5 Standard Deviation (degree K) Day

8 Sample deciles in standard space (15-day window), SAV Standardized Temperature Day Maximum Minimum

9 PRIOR MARGINAL D.F. in Standard Space Two alternative hypotheses (tested empirically) 1. Nonstationary Prior (estimate 12 parametric D.F.) 48 = 4 x 12 parameters + interpolation (=> 365 D.F.) MAD across 12 samples (1 Jan,, 1 Dec): 12 (min), 24 (average), 39 (max) 2. Stationary Prior (estimate 1 parametric D.F.) 4 parameters (=> 365 D.F.) MAD across 12 samples (1 Jan,, 1 Dec): 38 (min), 53 (average), 65 (max) MAD = max Empirical D. F. Parametric D. F.

10 Prior Marginal D. F.: Analyses 0. Given: standardized climatic sample for day k w k n : n 1,...,M M = days k (1 Jan,..., 1Dec) 1. Hypothesize several parametric models for 2. Estimate parameters of each model 3. Choose model that minimizes average (across k): MAD = max Empirical D. F. Parametric D. F. Model: G k G w 1 exp 1 ln ln U L U w 1 Parameters:,, L, U

11 Empirical and Parametric Distribution Functions on 1 Jan (k = 1) (M = 600), SAV MAD: 277 α = 971 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

12 Empirical and Parametric Distribution Functions on 1 Feb (k = 32) (M = 600), SAV MAD: 201 α = 467 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

13 Empirical and Parametric Distribution Functions on 1 Mar (k = 60) (M = 600), SAV MAD: 236 α = 139 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

14 Empirical and Parametric Distribution Functions on 1 Apr (k = 91) (M = 600), SAV MAD: 234 α = 974 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

15 Empirical and Parametric Distribution Functions on 1 May (k = 121) (M = 600), SAV MAD: 387 α = 703 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

16 Empirical and Parametric Distribution Functions on 1 Jun (k = 152) (M = 600), SAV MAD: 199 α = 259 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

17 Empirical and Parametric Distribution Functions on 1 Jul (k = 182) (M = 600), SAV MAD: 289 α = 362 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

18 Empirical and Parametric Distribution Functions on 1 Aug (k = 213) (M = 600), SAV MAD: 185 α = 738 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

19 Empirical and Parametric Distribution Functions on 1 Sep (k = 244) (M = 600), SAV MAD: 115 α = 015 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

20 Empirical and Parametric Distribution Functions on 1 Oct (k = 274) (M = 600), SAV MAD: 249 α = 507 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

21 Empirical and Parametric Distribution Functions on 1 Nov (k = 305) (M = 600), SAV MAD: 242 α = 238 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

22 Empirical and Parametric Distribution Functions on 1 Dec (k = 335) (M = 600), SAV MAD: 262 α = 528 β = η L = η U = P(W ' k w ' ) Standardized Temperature w'

23 Prior Marginal D. F. : Conclusions Small diversity of empirical D. F. Large similarity of parametric D. F. Near-stationarity of parameters,, L, U

24 Empirical Distribution Functions: the two most diverse on 1 Jun (k = 152) and 1 Dec (k = 335) (M = 600), SAV 1 Jun 1 Dec P(W ' k w ' ) Standardized Temperature w'

25 Parametric Distribution Functions on First Day of Each Month (M = 600), SAV Distribution P(W ' k w ' ) Standardized Temperature w'

26 Time-series distribution parameters: First of Each Month (M = 600), SAV Distribution 6 5 Alpha Beta EtaL EtaU 4 3 Parameter Value Day

27 Stationary Prior Marginal D.F. in Standard Space 1. Form a pooled standardized climatic sample from several days 2. Determine bounds from the grand sample (365 days, 40 years) L U min w k n k,n max w k n k,n 3. Estimate stationary D.F. from the pooled sample G w, L w U stationary parameters:,, L, U

28 Stationary Prior Marginal Distribution Functions: Empirical and Parametric, SAV Pooled Sample from 4 Days: 1 Jan, 1 Apr, 1 Jul, 1 Oct (M = 2400) MAD: 139 α = 659 β = η L = η U = P(W ' w' ) Standardized Temperature w'

29 Prior Marginal D.F. for day k in Original Space 0. Given: prior mean and standard deviation m k, s k stationary parameters 1. Construct,, L, U G k w G w m k s k Lk w Uk Lk L s k m k Uk U s k m k

30 Empirical Distribution Function on 1 Jan (k = 1) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 455 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

31 Empirical Distribution Function on 1 Feb (k = 32) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 464 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

32 Empirical Distribution Function on 1 Mar (k = 60) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 588 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

33 Empirical Distribution Function on 1 Apr (k = 91) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 421 α = 659 β = η Lk = η Uk = 301 P(W k w ) Temperature w ( K)

34 Empirical Distribution Function on 1 May (k = 121) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 604 α = 659 β = η Lk = η Uk = 304 P(W k w ) Temperature w ( K)

35 Empirical Distribution Function on 1 Jun (k = 152) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 592 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

36 Empirical Distribution Function on 1 Jul (k = 182) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 652 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

37 Empirical Distribution Function on 1 Aug (k = 213) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 583 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

38 Empirical Distribution Function on 1 Sep (k = 244) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 382 α = 659 β = η Lk = η Uk = P(W k w ) Temperature w ( K)

39 Empirical Distribution Function on 1 Oct (k = 274) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 580 α = 659 β = η Lk = 285 η Uk = P(W k w ) Temperature w ( K)

40 Empirical Distribution Function on 1 Nov (k = 305) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 432 α = 659 β = η Lk = 275 η Uk = P(W k w ) Temperature w ( K)

41 Empirical Distribution Function on 1 Dec (k = 335) (M = 600), SAV Parametric Distribution Function from Pooled Sample from 4 Days MAD: 620 α = 659 β = η Lk = 267 η Uk = P(W k w ) Temperature w ( K)

42 Stationary Prior Marginal D.F.: Conclusions Sampling Window: 5 15 days ( realizations) 15-day => empirical D. F. smoother, tails better delineated Pooled Sample Size M = 2400, 4 days, MAD = 14 M = 3600, 6 days, MAD = 14 M = 7200, 12 days, MAD = 11 Families of Good Parametric D.F. Bounded: (Weibull, Log-Logistic) Bounded Above: (Weibull, Log-Logistic)

43 CLIMATIC AUTOCORRELATION W k ' standardized predictand G' prior marginal D. F. (stationary) V k normalized predictand Q standard normal D. F. Normal Quantile Transform (NQT) V k = Q 1 (G'(W k ')) Autocorrelation coefficient (Pearson s product-moment) c k = Cor(V k 1, V k ) k = 1,, 365 *Note: Standardization does not make c k stationary

44 Autocorrelation Autocorrelation in normal in normal space space (15-day), Fourier series series (8th order), (8 th order), SAV SAV 5 5 Correlation Day

45 Prior Markov D. F. for day k in Original Space k day for which forecast is made (k = 1,, 365) l lead time in days (l = 1, 2,, 14) k l day of the last observation (forecast day) 0. Given: prior marginal D. F. G k, G k l autocorrelation coefficient c k 1. Construct H kl w k w k l Q Q 1 G k w k c k l Q 1 G k l w k l 1 c k 2l 1/2