Seite - 241 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 Following[33]weuse thewaveletextendedR2 baseddistance (WER)which isconstructedon topof thewavelet coherence. Ifx(t)andz(t)are twosignals, thewavelet coherencebetweenthemis definedas Rx,z(a,b)= |S(Cx,z(a,b))| |S(Cx,x(a,b))|1/2|S(Cz,z(a,b))|1/2, whereCx,z(a,b))=Cx(a,b))C∗z(a,b)) is thecross-wavelet transform,andS isasmoothoperator. Then, thewavelet coherencecanbeseenasa linearcorrelationcoefficientcomputedin thewaveletdomain andso localizedboth in timeandscale.Notice that smoothing isamandatorystep inorder toavoida trivial constantRx,z(a,b)=1 forall a,b. Thewavelet coherence is thenatwodimensionalmapthatquantifies foreachtime-scale location the strengthof the associationbetween the twosignals. Toproduce a singlemeasureof thismap, somekindofaggregationmustbedone. Followingtheconstructionof theextendeddetermination coefficientR2,Ref. [33]propose touse thewaveletextendedR2whichcanbecomputedusing WER2x,z= ∑Jj=1 ( ∑Nk=1 |S(Cx,z(j,k))| )2 ∑Jj=1 ( ∑Nk=1 |S(Cx,x(j,k))|∑Nk=1 |S(Cz,z(j,k))| ). Notice thatWER2x,z isasimilaritymeasureanditcaneasilybe transformedintoadissimilarity oneby D(x,z)= √ JN(1−WER2x,z), where thecomputationsaredoneover thegrids{1,. . . ,N} for the locationsband{aj, j=1,. . . J} for thescales a. Theboundaryscales (smallestandgreatest)aregenerally takenasapowerof twowhichdepend respectively on theminimumdetail resolution and the size of the time grid. The other values correspondusually toa linear interpolationoverabase2 logarithmicscale. While themeasure is justifiedby thepower of thewavelet analysis, in practice this distance impliesheavycomputations involvingcomplexnumbersandsorequiresofa largermemoryspace. This isoneof the tworeasonthat renders itsuseontheoriginaldataset intractable. Thesecondreason is relatedto thesizeof thedissimilaritymatrix that results fromitsapplicationsandthatgrowswith thesquareof thenumberof timeseries. Indeed, suchamatrixobtainedfromtheSCis largely tractable for amoderatenumberof super customersof about somehundreds, but it isnot if appliedon the wholedatasetofsometensofmillionsof individualcustomers. Thetradeoffbetweencomputation timeandprecision is resolvedbyafirst clusteringstepthatdramaticallyreduces thenumberof time seriesusingtheRCfeatures;andasecondstepthat introduces thefinerbutcomputationallyheavier dissimilaritymeasureontheSCaggregates. SinceK′ (thenumberofSC) is sufficiently small, adissimilaritymatrixbetween theSCcanbe computed in a reasonable amount of time. Thismatrix is then used as the input of the classical HierarchicalAgglomerativeClustering (HAC)algorithm,usedherewith theWard link. Itsoutput corresponds to thedesiredhierarchyof (super-)customers. Otherwise, onemayuse other clustering algorithms that use a dissimilaritymatrix as input (for instancePartitioningAroundMediods,PAM)togetanoptimalpartitioningforafixednumber ofclusters. Thesecondrowof theschemeinFigure8represents this secondstepclustering. 6.Upscaling Wediscuss in this section the ideaswedevelop toupscale theproblem. Ourfinal target is to workover twentymillion time-series. For this,werunmanyindependentclustering tasks inparallel, beforemerging the results to obtain anapproximationof thedirect clustering. Wegiveproposed solutions thatwere tested inorder to improve thecodeperformance. Someofour ideasproved to beuseful formoderate sample sizes (say tens of thousands) but turned to be counter-productive 241