Electricity Demand Probabilistic Forecasting
With Quantile Regression Averaging
Bidong Liu, Jakub Nowotarski, Tao Hong, Rafa l Weron
Department of Operations Research, Wroc law University of Technology, PolandBig Data Energy Analytics Laboratory, University of North Carolina at Charlotte
Riverside, 24.06.2015
Based on:Bidong Liu, Jakub Nowotarski, Tao Hong and Rafal Weron, Probabilistic Load Forecasting via Quantile Regression Averaging
on Sister Forecasts, IEEE Transactions on Smart Grid, forthcomingThis work was supported by funds from NCN (Poland) through grant no. 2013/11/N/HS4/03649
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 1 / 21
Motivation: probabilistic forecasts
Stochastic nature of forecasting
Assessment of future uncertainty
Ability to plan different strategies for the range of possibleoutcomes indicated by the probabilistic forecast
Variability of the electricity demand becoming a challenge to theutility industry
→ useful in practice (risk management and decision-making)
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 2 / 21
Motivation: combining forecasts
Similar to portfolio diversification and management
Availability of various models/experts’ predictions
No single best forecasting method
Generally believed to improve forecast accuracy
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 3 / 21
Motivation: load forecasting
Interval/density forecast, combining not so popular in loadforecasting
Combine point predictions for probabilistic forecasting →opportunity to leverage existing research
Use methodology proved to work well (J. Nowotarski and R.Weron (2014), T. Hong, B.Liu, and P. Wang (2015))
Relative simplicity of the two key components
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 4 / 21
Background: Point forecast averaging
f1
f2
fN
…Weights
estimationfC
Individualforecasts
Combinedforecast
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 5 / 21
Background: Interval forecast averaging
For point forecasts: fc =∑M
i=1 wi fi(e.g. a linear regression model)
For interval forecasts the above formula may not hold
A linear combination of α-th quantiles is not an α-th quantile ofa linear combination of random variables
qαc 6=M∑i=1
wiqαi
→ A possibility for development of new approaches
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 6 / 21
Background: quantile regression
10 15 20 25 30−50
0
50
100
150
200
250
300
X
Y
Linear regression
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 7 / 21
Background: quantile regression
10 15 20 25 30−50
0
50
100
150
200
250
300
X
Y
Linear regressionQuantile regression, α=0.95, α=0.05
Interval forecast
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 7 / 21
Proposed model: Quantile Regression Averaging
f1
f2
fN
…Quantile
regressionfC
Ind
ivid
ual
po
int
fore
cast
s
Combinedintervalforecast(2 quantiles)
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 8 / 21
Methodology: sister models and sister forecasts
Motivation: variable selection is core in regression model for loadforecasting
Sister models – constructed by different subsets of variableswith overlapping components
Here: 2 or 3 years for calibration and 4 ways of partitioningtraining and validation periods
Sister forecasts are generated from sister models
The family of sister recency effect models:
yt = β0 + β1Mt + β2Wt + β3Ht + β4WtHt + f (Tt) +
+∑d
f (Tt,d) +∑lag
f (Tt−lag ),
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 9 / 21
Methodology: the data (GEFCom2014)
2 or 3 years for calibration of sister (individual) models
1 year for validation of sister (individual) models (variableselection)
1 year for validation of probabilistic forecasts (best modelsselection)
1 year for testing probabilistic forecasts
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 10 / 21
Methodology: benchmarks
Two naive benchmarks
Scenario generation from historical weather data, no recencyeffect (Vanilla)Quantiles interpolated from 8 individual forecasts (Direct)
Benchmarks from individual models
8 individual models (Ind) with residuals’ distributionBest Individual (BI) individual model according to MAE
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 11 / 21
Methodology: evaluation of forecasts
Pinball loss function for 99 percentiles
Pt =
{(1− q)(yq
t − yt), yt < yqt
q(yt − yqt ), yt ≥ yq
t
Winkler score for 50% and 90% two-sided day-ahead predictionintervals:
Wt =
δt for pt ∈ [Lt|t−1,Ut|t−1],
δt + 2α
(Lt|t−1 − pt) for pt < Lt|t−1,
δt + 2α
(pt − Ut|t−1) for pt > Ut|t−1,
where δt = Ut|t−1 − Lt|t−1 is the interval’s width
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 12 / 21
Results: validation period
7 QRA models, 8+1 individual models
4 lengths of calibration period
One year of validation to pick up best (model, length) pairs
→ QRA models are dominantly better than the benchmarkmodels
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 13 / 21
Results: test period
Model class Pinball Winkler (50%) Winkler (90%)QRA(8,183) 2.85 25.04 55.85Ind(1,91) 3.22 26.35 56.38BI(-,365) 3.00 26.38 57.17Direct 3.19 26.62 94.27Vanilla 8.00 70.51 150.0
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 14 / 21
Discussion
Resolution – log-transform caused intervals to be wider in peakhours
Practicality
Sister forecasts easy to generateNo need of independent expert forecastsSimple way to leverage from point to probabilistic forecasts
Extensions
Sister forecasts eg. for machine learning methodsQRA for expert forecasts
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 15 / 21
Summary
QRA – a new technique the load forecasting literature
Practical value (1) – input to QRA from point forecasts
Practical value (2) – the sister forecasts are easy to generate
Publicly available data (GEFCom2014)
Accurate – confirmed by the pinball loss function andWinkler scores
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 16 / 21
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 17 / 21
Methodology: sister models and sister forecasts
yt = β0 +
calendar effects︷ ︸︸ ︷β1Mt + β2Wt + β3Ht + β4WtHt +
temp. dependence︷ ︸︸ ︷f (Tt) +
+∑d
f (Tt,d) +∑lag
f (Tt−lag )︸ ︷︷ ︸recency effect
,
where:
f (Tt) = β5Tt + β6T2t + β7T
3t + β8TtMt + β9T
2t Mt+
+ β10T3t Mt + β11TtHt + β12T
2t Ht + β13T
3t Ht
Tt,d =1
24
24d∑lag=24d−23
Tt−lag
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 18 / 21
Extension: Factor Quantile Regression Averaging
f1
f2
fN
…Quantile
regression fC
Ind
ivid
ual
po
int
fore
cast
s
Combinedintervalforecast(2 quantiles)
F1
FK
…
K factors extracted from a panel of point forecasts,K<N
PCA
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 19 / 21
Price forecasting results: case study 1J. Nowotarski and R. Weron (2014, Computational Statistics)
1 6 12 18 240
5
10
15
20Conditional coverage LR
AR
1 6 12 18 240
5
10
15
20Unconditional coverage LR
1 6 12 18 240
5
10
15
20
SNAR
1 6 12 18 240
5
10
15
20
1 6 12 18 240
5
10
15
20
QRA
Hour
1 6 12 18 240
5
10
15
20
50% PI 90% PI
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 20 / 21
Price forecasting results: case study 2K. Maciejowska, J. Nowotarski and R. Weron (2015, IJF)
6 12 18 24 30 36 42 48−5%
0%5%
10%15%20%25%
Rel
ativ
e W
inkl
ersc
ore,
50%
PI
1 − W
hQRA/W
hARX 1 − W
hFQRA/W
hARX
6 12 18 24 30 36 42 48−5%
0%5%
10%15%20%25%
Load period (h)
Rel
ativ
e W
inkl
ersc
ore,
90%
PI
B. Liu, J. Nowotarski, T. Hong & R. Weron Electricity Demand Probabilistic Forecasting Riverside, 24.06.2015 21 / 21