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

Questions?jakub.nowotarski@pwr.edu.pl

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