Page - 234 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 thehighestzoneoffluctuationswhichcorresponds to thedays. CWTis thenextremelyredundant but it isuseful forexample, tocharacterize theHolderianregularityof functionsor todetect transient phenomenaorchange-points.Amorecompactwavelet transformcanalsobedefined. Figure4. Wavelet spectrumofaweekofelectrical loaddemand. The Discrete Wavelet Transform is a technique of hierarchical decomposition of the finite energy signals. It allows representing a signal in the time-scale domain, where the scale plays a roleanalogous to thatof the frequency in theFourieranalysis ([27]). It allowstodescribeareal-valued functionthroughtwoobjects: anapproximationofthis functionandasetofdetails. Theapproximation part summarizes theglobal trendof the function,while the localizedchanges (in timeandfrequency) arecaptured in thedetail componentsatdifferent resolutions. Theanalysisof signals is carriedoutby waveletsobtainedasbefore fromsimple transformationsofasinglewell-localized(both in timeand frequency)motherwavelet.Acompactlysupportedwavelet transformprovideswithanorthonormal basisofwaveformsderivedfromscaling(i.e.,dilatingorcompressing)andtranslatingacompactly supportedscaling function φ˜andacompactly supportedmotherwavelet ψ˜. If oneworksover the interval [0,1],periodizedwaveletsareusefuldenotingby φ(t)=∑ l∈Z φ˜(t− l) and ψ(t)=∑ l∈Z φ˜(t− l), for t∈ [0,1], theperiodizedscalingfunctionandwavelet, thatwedilateorstretchandtranslate φj,k(t)=2j/2φ(2jt−k), ψj,k(t)=2j/2φ(2jt−k). Forany j0≥0, thecollection {φj0,k,k=0,1, . . . ,2j0−1;ψj,k, j≥ j0,k=0,1, . . . ,2j−1}, isanorthonormalbasisofH. Then, foranyfunctionz∈H, theorthogonalbasisallowsonetowrite thedevelopment z(t)= 2j0−1 ∑ k=0 cj0,kφj0,k(t)+ ∞ ∑ j=j0 2j−1 ∑ k=0 dj,kψj,k(t), (1) where cj,k anddj,k arecalledrespectively thescaleandthewavelet coefficientsofzat thepositionkof thescale jdefinedas cj,k=< z,φj,k>H dj,k=< z,ψj,k>H . 234