Recent advances in electricity price forecasting (EPF)

Transcription

1 Recent advances in electricity price forecasting (EPF) Rafał Weron Department of Operations Research Wrocław University of Science and Technology, Poland Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 1 / 34

2 Agenda 1 Beyond point forecasts probabilistic forecasts 2 Combining forecasts Point forecasts Probabilistic forecasts 3 Variable selection and shrinkage LASSO Elastic nets 4 Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 2 / 34

3 1. Beyond point forecasts Probabilistic forecasting A new book on EPF... forthcoming in 218 Chapters: 1. The Art of Forecasting 2. Markets for Electricity 3. Forecasting for Beginners 4. Evaluating Models and Forecasts 5. Forecasting for Experts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 3 / 34

4 1. Beyond point forecasts Probabilistic forecasting Point probabilistic path forecasting Electricity price in EUR/MWh Electricity price in EUR/MWh Electricity price in EUR/MWh D D Time Electricity price in EUR/MWh D D Time D D Time D D Time Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 4 / 34

5 1. Beyond point forecasts Probabilistic forecasting A (very) recent review of probabilistic forecasting Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 5 / 34

6 1. Beyond point forecasts Literature review Papers, cites <2 Journal articles Citations ( 5) <2 Neural network only Neural network & time series Time series only Other methods Probabilistic EPF Number of Scopus-indexed articles and citations Number of Scopus-indexed articles Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 6 / 34

7 1. Beyond point forecasts GEFCom214 GEFCom214 (Hong, Pinson, Fan et al., 216, IJF) Incremental data sets released on weekly basis Price Track: 287 contestants Submit 99 quantiles for 24h load periods of the next day Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 7 / 34

8 1. Beyond point forecasts GEFCom214 Price Track: Top winning teams (1st and) 2nd place for QRA! 1 Pierre Gaillard, Yannig Goude, Raphaël Nedellec (EDF R&D, F) 2 Katarzyna Maciejowska, Jakub Nowotarski (Wrocław UT, PL) 3 Grzegorz Dudek (Czȩstochowa UT, PL) 4 Zico Kolter, Romain Juban, Henrik Ohlsson, Mehdi Maasoumy (C3 Energy, USA) 5 Frank Lemke (KnowledgeMiner Software, D) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 8 / 34

9 Agenda 1 Beyond point forecasts probabilistic forecasts 2 Combining forecasts Point forecasts Probabilistic forecasts 3 Variable selection and shrinkage LASSO Elastic nets 4 Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 9 / 34

10 Individual forecasts 2. Combining forecasts Point forecast averaging: The idea f 1 f 2 f N Weights estimation f C Combined forecast Dates back to the 196s and the works of Bates, Crane, Crotty & Granger AI world : committee machines, ensemble averaging, expert aggregation Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 1 / 34

11 2. Combining forecasts QRA Combining probabilistic forecasts is more tricky Gneiting & Ranjan (213): a linearly combined probabilistic forecast is more dispersed than the least dispersed of the component distributions Helps if the component distributions tend to be underdispersed Lichtendahl et al. (213): averaging quantiles is better (sharper) 1 Averaging probabilities 1 Averaging quantiles 1 Comparison CDF.6.4 SCARX Q HP5e1 CDF.6.4 SCARX Q HP5e1 CDF SCARX Q S6 F-Ave 2 Q.2 SCARX Q S6 Q-Ave 2 Q.2 F-Ave 2 Q Q-Ave 2 Q System price (EUR/MWh) System price (EUR/MWh) System price (EUR/MWh) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 11 / 34

12 2. Combining forecasts QRA Alternative: Quantile Regression Averaging (QRA) (Submitted on , 21:26 ;-) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 12 / 34

13 Individual point forecasts 2. Combining forecasts QRA Quantile Regression Averaging: The idea y 1,t y 2,t Quantile regression: min β q t q 1 yt <X t β q y t X t β q y t L, y t U y m,t X t = 1, y 1,t,, y m,t β q - vector of parameters Combined interval forecast (e.g. for q=.5 &.95) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 13 / 34

14 2. Combining forecasts QRA Quantile regression 3 Linear regression Y X Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 14 / 34

15 2. Combining forecasts QRA Quantile regression 3 25 Linear regression Quantile regression, α=.95, α= Y 1 5 Interval forecast X Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 14 / 34

16 2. Combining forecasts QRA How does the score function look like? For vector X t = [1, ŷ 1,t,..., ŷ m,t ] of point forecasts, i.e. explanatory variables, weights β q are estimated by minimizing: min β q {t:y t X t β q } q y t X t β q + {t:y t<x t β q } (1 q) y t X t β q q=5% q=25% q=5% Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 15 / 34

17 2. Combining forecasts QRA Case Study Case study Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 16 / 34

18 2. Combining forecasts QRA Case Study QRA at work 15 NP Price [EUR/MWh] Forecast (weeks 1 26) Aug 8, 212 Jul 3, 213 Dec 31, 213 Hours [Aug 8, 212 Dec 31, 213] Nord Pool hourly prices ( ) Seven months for calibration of individual models Four weeks for calibration of quantile regression 26 weeks for evaluation of interval forecasts Six individual point forecasting models AR, TAR, SNAR, MRJD, NAR, FM Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 17 / 34

19 2. Combining forecasts QRA Case Study Evaluation of forecasts 5% and 9% two-sided day-ahead prediction intervals Two benchmark models: AR and SNAR Christoffersen s (1998, IER) test for unconditional and conditional coverage 1 y t [ŷt L, ŷt U ] The focus on the sequence: I t = y t [ŷt L, ŷt U ] Conditional Coverage test Unconditional Coverage test (UC + independece) Asymptotically χ 2 (2) Asymptotically χ 2 (1) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 18 / 34

20 2. Combining forecasts QRA Case Study Results: Unconditional coverage PI AR SNAR QRA Unconditional coverage 5% % Mean width (STD of interval width) 5% 4.55 (1.34) 2.76 (.61) 2.23 (.81) 9% (3.31) 9.33 (2.45) 6.78 (2.2) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 19 / 34

21 2. Combining forecasts QRA Case Study Results: Christoffersen s test 2 Conditional coverage LR 2 Unconditional coverage LR AR SNAR QRA Hour 5% PI 9% PI Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 2 / 34

22 Individual point forecasts 2. Combining forecasts FQRA FQRA: When the number of predictors is large (Maciejowska, Nowotarski & Weron, 216, IJF) y 1,t y 2,t PCA f 1,t Quantile regression: y t L, y t U X t = 1, f 1,t,, f k,t y m,t f k,t k <m factors extracted from a panel of point forecasts Combined interval forecast (e.g. for q=.5 &.95) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 21 / 34

23 Agenda 1 Beyond point forecasts probabilistic forecasts 2 Combining forecasts Point forecasts Probabilistic forecasts 3 Variable selection and shrinkage LASSO Elastic nets 4 Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 22 / 34

24 3. Variable selection and shrinkage Stepwise regression Automated variable selection Consider a general regression: p ŷ i = β j x i,j + ε i j=1 How to select predictors x i,j? How to estimate β j s? Single-step elimination of insignificant predictors In EPF: Gianfreda & Grossi (212) Stepwise regression Forward stepwise selection Backward stepwise elimination In EPF: Karakatsani & Bunn (28), Misiorek (28), Bessec et al. (216), Keles et al. (216) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 23 / 34

25 3. Variable selection and shrinkage Shrinkage (regularization) What is shrinkage (regularization)? Minimize the residual sum of squares (RSS) + a penalty function of the betas: ˆβ = argmin β j { N ( p ) 2 n } y i β j x i,j + λ β j q i=1 } j=1 {{ } j=1 }{{ } RSS penalty Ridge regression (q = 2) Introduced by: Hoerl & Kennard (197, Technometrics) In EPF: Barnes & Balda (213) Least Absolute Shrinkage & Selection Operator (LASSO; q = 1) Introduced by: Tibshirani (1996, JRSSB) In EPF: Ludwig et al. (215), Ziel et al. (215), Gaillard et al. (216), Ziel (216), Ziel and Weron (216) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 24 / 34

26 3. Variable selection and shrinkage Shrinkage (regularization) How does it work? Lasso Ridge regression Blue areas constraint regions, i.e., β 1 + β 2 t and β β2 2 t Red ellipses contours of the least squares error function Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 25 / 34

27 3. Variable selection and shrinkage Shrinkage (regularization) Elastic net RSS penalized by a mixed quadratic and linear shrinkage factor ˆβ EN = argmin β j RSS + λ 1 α 2 n n βj 2 + α β j j=1 j=1 Ridge regression, α= Elastic net, α=.75 Lasso, α=1 β 1 β 1 β 1 β 2 β 2 β 2 Introduced by: Zou & Hastie (215, JRSSB) In EPF: Uniejewski, Nowotarski & Weron (216, Energies) Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 26 / 34

28 3. Variable selection and shrinkage Shrinkage (regularization) How ˆβ s change when λ increases?.5 Ridge regression.5 Elastic net.5 Lasso λ λ λ Left: Ridge regression with λ (, 2), linear scale Center: Elastic net with α =.5 and λ (, 1), log-scale Right: Lasso with λ (, 1), log-scale Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 27 / 34

29 3. Variable selection and shrinkage Results Results: WMAE errors (Uniejewski et al., 216, Energies) Full ARX model, 17 variables: 72 hourly prices from 3 previous days min, max & average price of 3 previous days 2 load forecasts, one lagged (1, 7 days) weekly seasonality (daily dummies, multiplied by loads or by prices) ARX-type AR-type AR - ARX GEFCom Nord Pool GEFCom N2EX (UK) GEFCom Naive (.975) (.778) Naive (.975) (.778) (.975) (.975) (.31) (.31) Expert benchmarks ARX1 (.639) (.614) AR1 (.639) (.614) (.71) (.71) (.253) (.253) ARX1h (.639) (.616) AR1h (.639) (.616) (.74) (.74) (.253) (.253) ARX1hm (.617) (.516) AR1hm (.617) (.516) (.657) (.657) (.247) (.247) marx1 (.621) (.61) mar1 (.621) (.61) (.696) (.696) (.253) (.253) marx1h (.622) (.62) mar1h (.622) (.62) (.699) (.699) (.254) (.254) marx1hm (.598) (.518) mar1hm (.598) (.518) (.644) (.644) (.246) (.246) ARX2 (.575) (.546) AR2 (.575) (.546) (.7) (.7) (.253) (.253) ARX2h (.575) (.546) AR2h (.575) (.546) (.74) (.74) (.253) (.253) ARX2hm (.565) (.485) AR2hm (.565) (.485) (.656) (.656) (.249) (.249) Full ARX model farx (.57) (.78) far (.57) (.78) (.62) (.62) (.334) (.334) Selection and shrinkage methods ssarx (.577) (.537) ssar (.577) (.537) (.644) (.644) (.27) (.27) ssarx1 (.548) (.57) ssar1 (.548) (.57) (.641) (.641) (.261) (.261) fsarx (.52) (.52) fsar (.592) (.272) (.52) (.52) (.592) (.272) bsarx (.52) (.599) bsar (.582) (.31) (.52) (.599) (.582) (.31) RidgeX (.544) (.479) Ridge (.653) (.26) (.544) (.479) (.653) (.26) LassoX (.516) (.53) Lasso (.69) (.253) (.516) (.53) (.69) (.253) EN75X (.517) (.489) EN75 (.61) (.253) (.517) (.489) (.61) (.253) EN5X (.518) (.496) EN5 (.611) (.253) (.518) (.496) (.611) (.253) EN25X (.522) (.53) EN25 (.613) (.253) (.522) (.53) (.613) (.253) Comparisons Expert - Best Expert - Best Best farx - Best far far - Best Best Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 28 / 34

30 3. Variable selection and shrinkage Results Variable significance across hours Table 5: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso Table 6: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample (15) test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) (Ziel & Weron, 216, RePEc) HQC DoW,p,nl model, see Eqn. for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued in Table 6. in Table 7. Day (d 3) Day (d 2) Day (d 1) h φ h,1,1, φ h,1,2, φ h,1,3, φ h,1,4, φ h,1,5, φ h,1,6, φ h,1,7, φ h,1,8, φ h,1,9, φ h,1,1, φ h,1,11, φ h,1,12, φ h,1,13, φ h,1,14, φ h,1,15, φ h,1,16, φ h,1,17, φ h,1,18, φ h,1,19, φ h,1,2, φ h,1,21, φ h,1,22, φ h,1,23, φ h,1,24, φ h,2,1, φ h,2,2, φ h,2,3, φ h,2,4, φ h,2,5, φ h,2,6, φ h,2,7, φ h,2,8, φ h,2,9, φ h,2,1, φ h,2,11, φ h,2,12, φ h,2,13, φ h,2,14, φ h,2,15, φ h,2,16, φ h,2,17, φ h,2,18, φ h,2,19, φ h,2,2, φ h,2,21, φ h,2,22, φ h,2,23, φ h,2,24, φ h,3,1, φ h,3,2, φ h,3,3, φ h,3,4, φ h,3,5, φ h,3,6, φ h,3,7, φ h,3,8, φ h,3,9, φ h,3,1, φ h,3,11, φ h,3,12, φ h,3,13, φ h,3,14, φ h,3,15, φ h,3,16, φ h,3,17, φ h,3,18, φ h,3,19, φ h,3,2, φ h,3,21, φ h,3,22, φ h,3,23, φ h,3,24, Day (d 6) Day (d 5) Day (d 4) h φ h,4,1, φ h,4,2, φ h,4,3, φ h,4,4, φ h,4,5, φ h,4,6, φ h,4,7, φ h,4,8, φ h,4,9, φ h,4,1, φ h,4,11, φ h,4,12, φ h,4,13, φ h,4,14, φ h,4,15, φ h,4,16, φ h,4,17, φ h,4,18, φ h,4,19, φ h,4,2, φ h,4,21, φ h,4,22, φ h,4,23, φ h,4,24, φ h,5,1, φ h,5,2, φ h,5,3, φ h,5,4, φ h,5,5, φ h,5,6, φ h,5,7, φ h,5,8, φ h,5,9, φ h,5,1, φ h,5,11, φ h,5,12, φ h,5,13, φ h,5,14, φ h,5,15, φ h,5,16, φ h,5,17, φ h,5,18, φ h,5,19, φ h,5,2, φ h,5,21, φ h,5,22, φ h,5,23, φ h,5,24, φ h,6,1, φ h,6,2, φ h,6,3, φ h,6,4, φ h,6,5, φ h,6,6, φ h,6,7, φ h,6,8, φ h,6,9, φ h,6,1, φ h,6,11, φ h,6,12, φ h,6,13, φ h,6,14, φ h,6,15, φ h,6,16, φ h,6,17, φ h,6,18, φ h,6,19, φ h,6,2, φ h,6,21, φ h,6,22, φ h,6,23, φ h,6,24, Rafał Weron (Wrocław, PL) 3 Recent advances in EPF , 31 ISEA17, Cairns 29 / 34

31 3. Variable selection and shrinkage Results Variable significance across hours cont. (Ziel & Weron, 216, RePEc) Table 7: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables. Continued in Table 8. Day (d 8) Day (d 7) h φ h,7,1, φ h,7,2, φ h,7,3, φ h,7,4, φ h,7,5, Table 8: Mean occurrence (in %) of the multivariate lasso model parameters across all 12 datasets and the full out-ofsample test period. Columns represent the hours and rows the parameters of the 24lasso HQC DoW,p,nl model, see Eqn. (15) φ h,7,6, φ h,7,7, φ h,7,8, φ h,7,9, for details. A heat map is used to indicate more ( green) and less ( red) commonly-selected variables φ h,7,1, φ h,7,11, h φ h,7,12, φ h,1,min, φ h,7,13, φ h,2,min, φ h,7,14, φ h,3,min, φ h,7,15, φ h,4,min, φ h,7,16, φ h,5,min, φ h,7,17, φ h,6,min, φ h,7,18, φ h,7,min, φ h,7,19, φ h,8,min, φ h,7,2, φ h,1,max, φ h,7,21, φ h,2,max, φ h,7,22, φ h,3,max, φ h,7,23, φ h,4,max, φ h,7,24, φ h,5,max, φ h,8,1, φ h,6,max, φ h,8,2, φ h,7,max, φ h,8,3, φ h,8,max, φ h,8,4, φ h,,, φ h,8,5, φ h,,, φ h,8,6, φ h,,, φ h,8,7, φ h,,, φ h,8,8, φ h,,, φ h,8,9, φ h,,, φ h,8,1, φ h,,, φ h,8,11, φ h,1,h, φ h,8,12, φ h,1,h, φ h,8,13, φ h,1,h, φ h,8,14, φ h,1,h, φ h,8,15, φ h,1,h, φ h,8,16, φ h,1,h, φ h,8,17, φ h,1,h, φ h,8,18, φ h,1,24, φ h,8,19, φ h,1,24, φ h,8,2, φ h,1,24, φ h,8,21, φ h,1,24, φ h,8,22, φ h,1,24, φ h,8,23, φ h,1,24, φ h,8,24, φ h,1,24, Daily minimums Daily maximums DoW dummies Periodic on Y,h Periodic on Y, Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 3 / 34

32 Agenda 1 Beyond point forecasts probabilistic forecasts 2 Combining forecasts Point forecasts Probabilistic forecasts 3 Variable selection and shrinkage LASSO Elastic nets 4 Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 31 / 34

33 4. Guidelines for evaluating forecasts Guidelines for evaluating probabilistic forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 32 / 34

34 4. Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances 1 in EPF , 11 ISEA17, Cairns 33 / 34 Maximizing sharpness subject to reliability (Gneiting & Katzfuss, 214; Nowotarski & Weron, 217) Reliability refers to statistical consistency (x% PI covers x% of obs.) Table 1: Sharpness A comparison of refers evaluation tometrics howfor tightly probabilistic theforecasting. PI covers Statistics theand observations tests in italics are discussed in the text, but not illustrated in the empirical study in Section 5. Interval forecasts Density forecasts Statistics Tests Statistics Tests Reliability / calibration / unbiasedness Unconditional coverage [46, 74] Conditional coverage [46] (CC = UC + Independence) Sharpness (and reliability) Pinball loss [84, 85] Winkler (interval) score [86] Kupiec [74] Christoffersen [46] (Lagged [78]) Ljung-Box Christoffersen [79] Duration-based tests [8, 81] Dynamic Quantile (DQ) [82] VQR [83] Diebold-Mariano [87, 88] Model confidence set [89] Forecast encompassing [9] Probability Integral Transform (PIT) [14, 75] Visual tests [14, 16] Tests for uniformity [76, 77] Berkowitz CC statistic [48] Berkowitz [48] Continuous Ranked Probability Score (CRPS) [15, 91] Logarithmic score [92] Diebold-Mariano [87, 88] Model confidence set [89] Forecast encompassing [9]

35 Take-home messages 1 Beyond point forecasts probabilistic forecasts 2 Combining forecasts Point forecasts Probabilistic forecasts 3 Variable selection and shrinkage LASSO Elastic nets 4 Guidelines for evaluating forecasts Rafał Weron (Wrocław, PL) Recent advances in EPF , ISEA17, Cairns 34 / 34