Page - 236 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 with dj = {dj,0, . . . ,dj,2j−1}. Since the DWT operator is based on an L2-orthonormal basis decomposition, theenergyofasquare integrablesignal ispreserved: ‖z‖22= c20+ J−1 ∑ j=0 2j−1 ∑ k=0 d2j,k= c 2 0+ J−1 ∑ j=0 ‖dj‖22. (3) Hence, theglobalenergy‖z‖22 ofz isdistributedoversomeenergeticcomponents. Thekeyfact thatwe are going to exploit is how these energies aredistributed andhow they contribute to the globalenergyofasignal. Thenwecangenerateahandynumberof features thataregoingtobeused forclustering. 4.KWF 4.1. FromDiscrete toFunctionalTimeSeries Theoretical developments andpractical applications associatedwith functional data analysis weremainly guidedby the case of independent observations. However, there is awide range of applications inwhichthishypothesis isnot reasonable. Inparticular,whenweconsiderrecordson afinergridof timeassuming that themeasures comefromasamplingofanunderlyingunknown continuous-timesignal. Formally, the problem can be written by considering a continuous stochastic process X=(X(t),t∈R). So the informationcontained ina trajectoryofXobservedon the interval [0,T], T > 0 is also represented by a discrete-time process Z = (Zk(t),k = 0,. . . ,n;t ∈ [0,δ])where Zk(t) = X((δ−1)k+ t) comes from the segmentation of the trajectory X in n blocks of size δ = T/n ([29]). Then, the processZ is a time series of functions. For example, we can forecast Zn+1(t) fromthedataZ1, . . . ,Zn. This isequivalent topredictingthefuturebehaviourof theXprocess over theentire interval [T,T+δ]byhavingobservedXon [0,T]. Pleasenote thatbyconstruction, theZ1, . . . ,Zn areusuallydependent functional randomvariables. Thisframeworkisofparticularinterestinthestudyofelectricityconsumption. Indeed,thediscrete consumptionmeasurementscannaturallybeconsideredasasamplingof theloadcurveofanelectrical system.Theusual segmentsize,δ=1day, takes intoaccount thedailycycleofconsumption. In [21], the authorsproposedapredictionmodel for functional time series in thepresenceof non stationary patterns. Thismodel has been applied to the electricity demand of Electricité de France (EDF).Thegeneralprincipleof the forecastingmodel is tofindinthepast, situationssimilar to the present and linearly combine their futures to build the forecast. The concept of similarity is basedonwavelets and several strategies are implemented to take into account thevariousnon stationary sources. Ref. [30] proposes for the sameproblem touse apredictor of a similar nature but applied toamultivariateprocess. Next, [31]provideanappropriate framework for stationary functionalprocessesusing thewavelet transform. The lattermodel isadaptedandextendedto the caseofnon-stationaryfunctionalprocesses ([32]). Thus,a forecastqualityof thesameorderofmagnitudeasothermodelsusedbyEDFisobtained for the national curve (highly aggregated) even though ourmodel can represent the series in a simpleandparsimoniousway. Thisavoidsexplicitlymodeling the linkbetweenconsumptionand weathercovariates,whichareknowntobe important inmodelingandoftenconsideredessential to take intoaccount.Anotheradvantageof the functionalmodel is itsability toprovidemulti-horizon forecasts simultaneouslybyrelyingonawholeportionof the trajectoryof therecentpast, rather than oncertainpointsasunivariatemodelsdo. 236