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-0641572. Data provided by Environmental Modeling Center, NCEP/NWS/NOAA.
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 1959 31 December 1998 (40 years) 29 February (leap year) excluded 5 March 1997 filled in
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
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
Sample deciles in original space (15-day window), SAV 300 Temperature ( K) 290 280 270 0 100 200 300 Day Maximum 90 80 70 60 50 40 30 20 10 Minimum
Sample mean (15-day), Fourier series (2 nd order), SAV 298 296 Mean (degree K) 294 292 290 288 286 284 0 50 100 150 200 250 300 350 Day
5.5 Sample std. dev. (15-day), Fourier series (2 nd order), SAV 5 Standard Deviation (degree K) 4.5 4 3.5 3 2.5 2 1.5 1 0 50 100 150 200 250 300 350 Day
Sample deciles in standard space (15-day window), SAV 4 3 2 Standardized Temperature 1 0-1 -2-3 -4-5 -6 0 100 200 300 Day Maximum 90 80 70 60 50 40 30 20 10 Minimum
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.
Prior Marginal D. F.: Analyses 0. Given: standardized climatic sample for day k w k n : n 1,...,M M = 600 12 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
Empirical and Parametric Distribution Functions on 1 Jan (k = 1) (M = 600), SAV MAD: 277 α = 971 β = 3.5186 η L = -4.0396 η U = 2.3533 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Feb (k = 32) (M = 600), SAV MAD: 201 α = 467 β = 3.0885 η L = -3.4939 η U = 2.3624 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Mar (k = 60) (M = 600), SAV MAD: 236 α = 139 β = 3.0467 η L = -3.6945 η U = 2.1512 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Apr (k = 91) (M = 600), SAV MAD: 234 α = 974 β = 3.3631 η L = -3.7663 η U = 2.2657 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 May (k = 121) (M = 600), SAV MAD: 387 α = 703 β = 3.3851 η L = -3.4261 η U = 2.2486 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Jun (k = 152) (M = 600), SAV MAD: 199 α = 259 β = 5.2807 η L = -5.7415 η U = 2.1794 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Jul (k = 182) (M = 600), SAV MAD: 289 α = 362 β = 4.2803 η L = -4.5610 η U = 2.9318 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Aug (k = 213) (M = 600), SAV MAD: 185 α = 738 β = 4.2969 η L = -4.1051 η U = 3.2253 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Sep (k = 244) (M = 600), SAV MAD: 115 α = 015 β = 4.7956 η L = -5.0532 η U = 2.7160 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Oct (k = 274) (M = 600), SAV MAD: 249 α = 507 β = 3.7870 η L = -4.0384 η U = 2.1044 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Nov (k = 305) (M = 600), SAV MAD: 242 α = 238 β = 3.2029 η L = -3.8365 η U = 2.1703 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Empirical and Parametric Distribution Functions on 1 Dec (k = 335) (M = 600), SAV MAD: 262 α = 528 β = 2.8569 η L = -3.3537 η U = 2.2668 P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Prior Marginal D. F. : Conclusions Small diversity of empirical D. F. Large similarity of parametric D. F. Near-stationarity of parameters,, L, U
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 ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Parametric Distribution Functions on First Day of Each Month (M = 600), SAV Distribution P(W ' k w ' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
Time-series distribution parameters: First of Each Month (M = 600), SAV Distribution 6 5 Alpha Beta EtaL EtaU 4 3 Parameter Value 2 1-0 -1-2 -3-4 -5-6 0 100 200 300 400 Day
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
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 = 5.4736 η L = -5.8833 η U = 3.3777 P(W ' w' ) -6-5 -4-3 -2-1 0 1 2 3 4 Standardized Temperature w'
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
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 = 5.4736 η Lk = 256.18 η Uk = 302.89 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 254.23 η Uk = 301.90 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 258.07 η Uk = 302.60 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 269.31 η Uk = 301 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 277.76 η Uk = 304 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 286.01 η Uk = 301.80 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 291.76 η Uk = 301.61 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 294.14 η Uk = 301.30 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 291.45 η Uk = 301.57 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 285 η Uk = 303.28 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 275 η Uk = 302.83 P(W k w ) 260 270 280 290 300 Temperature w ( K)
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 = 5.4736 η Lk = 267 η Uk = 303.57 P(W k w ) 260 270 280 290 300 Temperature w ( K)
Stationary Prior Marginal D.F.: Conclusions Sampling Window: 5 15 days (200 600 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)
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
Autocorrelation Autocorrelation in normal in normal space space (15-day), Fourier series series (8th order), (8 th order), SAV SAV 5 5 Correlation 5 5 0 50 100 150 200 250 300 350 Day
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