Page - 237 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 4.2. FunctionalModelKWF 4.2.1. StationaryCase Weconsider a stochasticprocessZ= (Zi)i∈Z assumed for themoment, tobe stationary,with values ina functional spaceH (forexampleH= L2([0,1])).WehaveasampleofncurvesZ1, . . . ,Zn andthegoal is to forecastZn+1. The forecastingmethodisdivided in twosteps. First,findamongthe blocksof thepast those thataremostsimilar to the lastobservedblock. Thenbuildaweightvector wn,i, i=1,. . . ,n−1andobtain thedesiredforecastbyaveragingthe futureblockscorrespondingto the indices2,. . . ,n respectively. First step. Totake intoaccount in thedissimilarity the infinitedimensionof theobjects tobecompared, the KWFmodel representseachsegmentZi, i=1,. . . ,n, by itsdevelopmentonawaveletbasis truncated toa scale J> j0. Thus, eachobservationZi isdescribedbya truncatedversionof itsdevelopment obtainedbythediscretewavelet transform(DWT): Zi,J(t)= 2j0−1 ∑ k=0 c(i)j0,kφj0,k(t)+ J ∑ j=j0+1 2j−1 ∑ k=0 d(i)j,kψj,k(t), t∈ [0,1]. Thefirsttermoftheequationisasmoothapproximationtotheresolution j0oftheglobalbehaviour of the trajectory. It containsnon-stationarycomponentsassociatedwith lowfrequenciesora trend. The second termcontains the informationof the local structureof the function. For twoobserved segmentsZi(t)andZi′(t),weuse thedissimilarityDdefinedas follows: D(Zi,Zi′)= J ∑ j=j0+1 2−j 2j−1 ∑ k=0 (d(i)j,k−d (i′) j,k ) 2. (4) SincetheZprocess isassumedtobestationaryhere, theapproximationcoefficientsdonotcontain useful informationfor the forecast since theyprovide localaverages.Asaresult, theyarenot taken intoaccount in theproposeddistance. Inotherwords, thedissimilarityDmakes itpossible tofind similarpatternsbetweencurveseven if theyhavedifferentapproximations. Secondstep. DenoteΞi={c(i)J,k : k=0,1, . . . ,2J−1} thesetof scalingcoefficientsof the i-thsegmentZi at the finerresolution J. Thepredictionofscalingcoefficients (at thescale J ) Ξ̂n+1 ofZn+1 isgivenby: Ξ̂n+1= ∑n−1m=1Khn(D(Zn,J,Zm,J))Ξm+1 1/n+∑n−1m=1Khn(D(Zn,J,Zm,J)) , whereK is aprobabilitykernel. Finally,wecanapply the inverse transformof theDWTto Ξ̂n+1 to obtain the forecastof theZn+1 curve in the timedomain. Ifwenote wn,m= Khn(D(Zn,J,Zm,J)) ∑n−1m=1Khn(D(Zn,J,Zm,J)) , (5) theseweightsallowtorewrite thepredictorasabarycentreof futuresegmentsof thepast: Ẑn+1(t)= n−1 ∑ m=1 wn,mZm+1(t). (6) 237