Page - 235 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1893 Thewavelet transformcanbeefficientlycomputedusingthenotionofmutiresolutionanalysisof H (MRA), introducedbyMallat,whoalsodesignedafamilyof fastalgorithms(see [27]).UsingMRA, thefirst termat the right hand side of (1) describe a smooth approximationof the function z at a resolution level j0while thesecondtermis theapproximationerror. It isexpressedas theaggregation of thedetailsat scales j≥ j0. Ifonewants to focusonthefinerdetails thenonly the informationat the scales{j : j≥ j0} is tobe looked. Figure5 is themultiresolutionanalysisofadaily loadcurve. Theoriginal curve is representedon thetopleftmostpanel. Thebottomrightmostpanelcontainstheapproximationpartatthecoarsestscale j0=0, that is, aconstant level function. Thesetofdetailsareplottedbyscalewhichcanbeconnected to frequencies.With this, thedetail functionsclearlyshowthedifferentpatternsrangingbetween low andhighfrequencies. Thestructureof thesignal is centredonthehighest scales (lowest frequencies), while the lowestscale (highest frequencies)keepthenoiseof thesignal. ofadaily loadcurve. Figure5. Multiresolutionanalysis Fromapracticalpointofview, letus suppose for simplicity that each function isobservedon afinetimesamplinggridofsizeN=2J (ifnot,onemayinterpolatedata to thenextpowerof two). In this contextweuseahighlyefficientpyramidal algorithm([28]) toobtain the coefficientsof the DiscreteWaveletTransform(DWT). Denotebyz= {z(tl) : l= 0,. . . ,Ni−1} thefinitedimensional sampleofthefunctionz. Fortheparticular levelofgranularitygivenbythesizeNofthesamplinggrid, onerewrites (1)using the truncation imposedbythe2Jpointsandthecoarserapproximation level j0=0,as: z˜J(t)= c0φ0,0(t)+ J−1 ∑ j=0 2j−1 ∑ k=0 dj,kψj,k(t). (2) Hence, foragivenwaveletψandacoarseresolution j0=0,onemaydefinetheDWToperator: Wψ :RN→RN, z → ( d0, . . . ,dJ−1,c0 f ) 235