Electricity Consumption Prediction with Functional Linear Regression - PowerPoint PPT Presentation

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Electricity Consumption Prediction with Functional Linear Regression Jarom´ ır Antoch, Luboˇ s Prchal, Maria Rosaria De Rosa and Pascal Sarda MODELLING SMART GRIDS 2015

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Goals Functional linear regression model linking observations of a functional response variable with measurements of an explanatory functional variable is considered. Our aim is to analyze effect of a functional variable on a functional response by means of functional linear regression models when slope function is estimated with tensor product splines. Model is applied to real data comprising electricity consumption of Sardinia 2000 – 2005. Computational issues are addressed.

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Data Model serves to analyze real data set concerning electricity consumption of Sardinia. Data set consists of 52 584 values of electricity consumption collected every hour within January 1, 2000 – December 31, 2005. 1800 1600 Consumption 1400 1200 1000 800 0 10000 20000 30000 40000 50000 Hours

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Data ORIGINAL DATA 1800 1600 1400 HOURLY CONSUMPTION 1200 1000 800 600 400 200 0 1 2 3 4 5 TIME 4 x 10

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Official data – Sardinia Energia richiesta Energia richiesta in Sardegna GWh 12.611,6 Deficit (-) Superi (+) della produzione rispetto alla richiesta GWh +419,9 % 3,3 1973 = +14 2005 = +419,9 14.000 12.000 10.000 8.000 6.000 4.000 2.000 0 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Deficit Superi Energia elettrica prodotta Energia elettrica richiesta Consumi: complessivi 12.036,7 GWh; per abitante 7.286 kWh

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Official data – Italy Energia richiesta Energia richiesta in Italia GWh 330.443,0 Deficit (-) Superi (+) della produzione rispetto alla richiesta GWh -49.154,5 % 14,9 1973 = -879 2005 = -49.154,5 350.000 300.000 250.000 200.000 150.000 100.000 50.000 0 1973 1975 1977 1979 1981 1983 1985 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Deficit Superi Energia elettrica prodotta Energia elettrica richiesta Consumi: complessivi 309.816,8 GWh; per abitante 5.286 kWh

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Basic trends 1800 1600 Consumption 1400 1200 1000 800 0 10000 20000 30000 40000 50000 Hours

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Basic trends 1800 1600 Consumption 1400 1200 1000 800 0 10000 20000 30000 40000 50000 Hours 1000 1200 1400 1600 1800 Total electricity consumption 800 0 10000 20000 30000 40000 50000 Hours

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Consumptions for one day COMPLETE DATA 1800 1600 1400 1200 CONSUMPTION 1000 800 600 400 200 0 0 5 10 15 20 25 HOUR

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Mean consumption for individual days MEAN HOURLY CONSUMPTIONS FOR DIFFERENT DAYS 1800 1700 1600 1500 MEAN COMSUMPTION 1400 1300 1200 MONDAY 1100 TUESDAY WEDNESDAY 1000 THURSDAY FRIDAY 900 SATURDAY SUNDAY 800 0 5 10 15 20 25

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Mean consumption for individual months MEANS FOR DIFFERENT MONTHS 1800 GENNAIO FEBBRAIO 1700 MARZO APRILE MAGGIO 1600 GIUGNO LUGLIO AGOSTO 1500 SETTEMBRE MEAN CONSUMPTIONS OTTOBRE NOVEMBRE 1400 DICEMBRE 1300 1200 1100 1000 900

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Mean consumption : Individual months over years MEAN CONSUMPTIONS MEAN CONSUMPTIONS MEAN CONSUMPTIONS GENNAIO −− HOURLY MEANS OVER YEARS FEBBRAIO −− HOURLY MEANS OVER YEARS MARZO −− HOURLY MEANS OVER YEARS 1800 1800 1800 1600 1600 1600 2000 2000 2000 1400 1400 1400 2001 2001 2001 1200 1200 1200 2002 2002 2002 1000 1000 1000 2003 2003 2003 800 800 800 0 10 20 0 10 20 0 10 20 2004 2004 2004 MEAN CONSUMPTIONS MEAN CONSUMPTIONS MEAN CONSUMPTIONS HOUR HOUR HOUR 2005 2005 2005 APRILE −− HOURLY MEANS OVER YEARS MAGGIO −− HOURLY MEANS OVER YEARS GIUGNO −− HOURLY MEANS OVER YEARS 1800 1800 1800 1600 1600 1600 2000 2000 2000 1400 1400 1400 2001 2001 2001 1200 1200 1200 2002 2002 2002 1000 1000 1000 2003 2003 2003 800 800 800 0 10 20 0 10 20 0 10 20 2004 2004 2004 MEAN CONSUMPTIONS MEAN CONSUMPTIONS MEAN CONSUMPTIONS HOUR HOUR HOUR 2005 2005 2005 LUGLIO −− HOURLY MEANS OVER YEARS AGOSTO −− HOURLY MEANS OVER YEARS SETTEMBRE −− HOURLY MEANS OVER YEARS 1800 1800 1800 1600 1600 1600 2000 2000 2000 1400 1400 1400 2001 2001 2001 1200 1200 1200 2002 2002 2002 1000 1000 1000 2003 2003 2003 800 800 800 0 10 20 0 10 20 0 10 20 2004 2004 2004 MEAN CONSUMPTIONS MEAN CONSUMPTIONS MEAN CONSUMPTIONS HOUR HOUR HOUR 2005 2005 2005 OTTOBRE −− HOURLY MEANS OVER YEARS NOVEMBRE −− HOURLY MEANS OVER YEARS DICEMBRE −− HOURLY MEANS OVER YEARS 1800 1800 1800 1600 1600 1600 2000 2000 2000 1400 1400 1400 2001 2001 2001 1200 1200 1200 2002 2002 2002 1000 1000 1000 2003 2003 2003 800 800 800 0 10 20 0 10 20 0 10 20 2004 2004 2004 HOUR HOUR HOUR 2005 2005 2005

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Consumption for one week 1600 1500 Consumption 1400 1300 1200 1100 0 40 80 120 160 Hours 1800 1600 Consumption 1400 1200 1000 800 0 40 80 120 160 Hours

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Main tasks Main interest is predicting oncoming weekend and/or weekdays consumption curve if present weekdays consumption is known and functional predictor is curve of present weekdays consumption. Model � Y i ( t ) = α ( t ) + X i ( s ) β ( s , t ) ds + ε i ( t ) , t ∈ I 2 , i = 1 , . . . , n I 1 Data Functional predictors X i ’s represent weekdays curves Y i ’s represent a weekend curves or a weekday curve in which case Y i = X i +1 Recall that model corresponds to ARH(1) Complete data series has been cut into 307 weeks Weekdays (Mo to Fri) and weekends (Sa to Su) separated (reason, leading to two sets of discretized electricity consumption curves, is fundamental difference between weekdays and weekend consumption).

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Assumptions Data are observations of identically distributed random functional � � variables X i ( s ) , Y i ( t ) , s ∈ I 1 , t ∈ I 2 , i = 1 , . . . , n , defined on same probability space and taking values in some functional spaces. We consider separable real Hilbert spaces L 2 ( I 1 ) and L 2 ( I 2 ) of square integrable functions defined on compact intervals I 1 ⊂ R and I 2 ⊂ R , equipped with standard inner products. We focus on functional linear relation � Y i ( t ) = α ( t ) + X i ( s ) β ( s , t ) ds + ε i ( t ) , t ∈ I 2 , i = 1 , . . . , n I 1 α ( t ) ∈ L 2 ( I 2 ) and β ( s , t ) ∈ L 2 ( I 1 × I 2 ) are unknown functional parameters ε 1 ( t ) , . . . , ε n ( t ) are i.i.d. centered random variables ∈ L 2 ( I 2 ) ε i ( t ) and X i ( s ) are uncorrelated

J. ANTOCH et al. FUNCTIONAL LINEAR REGRESSION . . . ELECTRICITY CONSUMPTION SEPTEMBER 10th, 2015 Assumptions (cont.) For generic interval I set L 2 ( I ) is equipped with usual inner product � I φ ( t ) ψ ( t ) dt , φ, ψ ∈ L 2 ( I ) and associated norm � φ, ψ � = � φ � = � φ, φ � 1 / 2 . We often omit arguments of functional variables and parameters and write X i , Y i , ε i and β instead of X i ( s ) , Y i ( t ) , ε i ( t ) and β ( s , t ) Recall the model � Y i ( t ) = α ( t ) + X i ( s ) β ( s , t ) ds + ε i ( t ) t ∈ I 2 , i = 1 , . . . , n (1) I 1 X i ’s represent a weekdays curves Y i ’s represent a weekend curves, or a weekday curve, in which case Y i = X i +1 α ( t ) and β ( s , t ) are unknown functional parameters Model (1) corresponds to an ARH(1)