Seite - 162 - in Short-Term Load Forecasting by Artificial Intelligent Technologies

Energies2018,11, 2038 useasquarederror loss function,whereasaclassificationproblemmayuse logarithmic loss. Indeed, anydifferentiable loss functioncanbeused. Althoughboostingmethods reducesbiasmore thanbagging, theyaremore likely tooverfita trainingdataset. Toovercomethis task, several regularizationtechniquescanbeapplied. • Treeconstraints: thereareseveralways to introduceconstraintswhenconstructingregression trees. Forexample, the followingtreeconstraintscanbeconsideredasregularizationparameters: The number of gradient boosting iterationsN: increasingN reduces the error on the trainingdataset,butmayleadtooverfitting.AnoptimalvalueofN isoftenselectedby monitoringpredictionerroronaseparatevalidationdataset. Treedepth: the size of the trees or number of terminal nodes in trees,which controls themaximumallowed level of interaction betweenvariables in themodel. Theweak learnersneedtohaveskillsbut theyshouldremainweak, thusshorter treesarepreferred. Ingeneral,valuesof treedepthbetween4and8workwellandvaluesgreater than10are unlikely toberequired, see [35]. Theminimumnumberofobservationper split: theminimumnumberofobservations neededbeforeasplit canbeconsidered. Ithelps toreducepredictionvarianceat leaves. • Shrinkageor learningrate: inregularizationbyshrinkage,eachupdate is scaledbythevalueof the learningrateparameter“eta” in (0,1]. Shrinkagereduces the influenceofeach individual tree andleavesspace for future trees to improve themodel.As it is stated in [28], small learningrates provide improvements inmodel’sgeneralizationabilityovergradientboostingwithoutshrinking (eta=1),but thecomputational time increases. Besides, thenumberof iterationsandlearningrate are tightlyrelated: forasmaller learningrate“eta”,agreaterN is required. • Random sampling: to reduce the correlation between the trees in the sequence, at each step, a subsample of the trainingdata is selectedwithout replacement to fit the base learner. Thismodificationprevent overfitting and itwasfirst introduced in [36],which is also called stochasticgradientboosting. Friedmanobservedanimprovement ingradientboosting’saccuracy with samplingsof aroundonehalf of the trainingdatasets. Analternative to rowsampling is columnsampling,which indeedpreventsover-fittingmoreefficiently, see [37]. • Penalize tree complexity: complexityof a tree canbedefinedasacombinationof thenumber of leaves and the L2 normof the leaf scores. This regularization not only avoids overfitting, italso tends toselect simpleandpredictivemodels. Followingthisapproach, ref. [37]describes ascalable treeboostingsystemcalledXGBoost. In thatpaper, theobjective tobeminimized is a combinationof the loss functionand the complexityof the tree. In contrast to theprevious ensemblemethods, XGBoost requires aminimal amount of computational resources to solve real-worldproblems. InXGBoost, themodel is trainedinanadditivemannerandit considersaregularizedobjective that includesa loss functionandpenalizes thecomplexityof themodel. Following[37], ifwedenote by yˆ(t)i , thepredictionof the i-th instanceof theresponseat the t-th iteration,weneedtofindthe tree structure ft thatminimizes the followingobjective: L(t) = n ∑ i= 1 l ( yi, yˆ (t−1) i + ft(xi) ) +Ω(ft) (2) In the first termof (2), l is a differentiable convex loss function thatmeasures the difference betweentheobservedresponseyi andtheresultingprediction yˆi. Thesecondtermof (2)penalizes the complexityof themodel,as follows: Ω(f) = γT+ 1 2 λ‖w‖2 (3) 162