Page - 32 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 1009 3.2.2. ForecastingAccuracyIndexes Three forecastingaccuracyevaluation indexesareusedtocompare the forecastingperformances foreachmodel: (1) theMAPEmentionedinEquation(5); (2) therootmeansquareerror (RMSE);and (3) themeanabsoluteerror (MAE).The latter twoindexescouldbecalculatedbyEquations (18)and (19), respectively: RMSE= √ ∑Ni=1(ai− fi)2 N s (18) MAE= 1 N N ∑ i=1 |ai− fi| (19) whereN is the totalnumberofdata; ai is theactualelectric loadvalueatpoint i; fi is the forecasted electric loadvalueatpoint i. 3.2.3. ForecastingAccuracySignificanceTests TodemonstratethesignificantsuperiorityoftheproposedSSVRCCSmodel intermsofforecasting accuracy, somefamousstatistical testsare implemented. BasedonDieboldandMariano’s [50]and Derracetal. [51] researchsuggestions, theWilcoxonsigned-ranktest [52]andFriedmantest [53]are simultaneouslyapplied in thispaper. TheWilcoxonsigned-ranktest isusedtocompare thesignificantdifferences in termsofcentral tendencybetween twodata setwith the same size. Let di represent the i-thpair difference of the i-th forecastingerrors fromanytwoforecastingmodels, thedifferencesarerankedaccordingto their absolutevalues. Let r+ represent the sumof ranks that thefirstmodel larger than the secondone; r− represent the sumof ranks that the secondmodel larger than the first one. In case of dj = 0, then,excludethe j-thpairandreducesamplesize. ThestatisticWof theWilcoxonsigned-ranktest is shownasEquation(20): W=min { r+,r− } (20) IfWmeets thecriterionof theWilcoxondistributionunderNdegreesof freedom, then, thenull hypothesisofequalperformanceof these twocomparedmodelscannotbeaccepted. Italso implies that theproposedmodel is significantly superior to theothermodel. Of course, if the comparison size is larger thanthecritical size, thesamplingdistributionofWwouldapproximate to thenormal distribution insteadofWilcoxondistribution,andtheassociatedp-valuewouldalsobeprovided. Ontheotherhand,dueto thenon-parametric statistical test in theANOVAanalysisprocedure, the Friedman test is devoted to compare the significant differences among two ormoremodels. ThestatisticFof theFriedmantest is shownasEquation(21): F= 12N k(k+1) [ k ∑ j=1 R2j− k(k+1)2 4 ] (21) whereN isthetotalnumberofforecastingresults;k isthenumberofcomparedmodels;Rj istheaverage ranksumobtainedineachforecastingvalue foreachcomparedmodelasshowninEquation(22), Rj= 1 N N ∑ i=1 rji (22) where rji is theranksumfrom1(thesmallest forecastingerror) tok (theworst forecastingerror) for ith forecastingresult, for jthcomparedmodel. Similarly, if theassociatedp-valueofFmeets thecriterionofnotacceptance, thenullhypothesis, equalperformanceamongall comparedmodels, couldalsonotbeheld. 32