Page - 394 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 as the cost ofmoving the smoothed density around γ1 to the uniformdistribution on the curve, thenmoving γ1 to γ2, keepingpointswith equal scaled arclength in correspondence, andfinally, movingtheuniformdistributiononγ2 to thesmootheddensity. Havingthedensityathand, theentropyof thesystemofcurvesγ1, . . . ,γN isdefinedtheusual wayas: E(γ1, . . . ,γN)=− ∫ Ω d˜(x) log ( d˜(x) ) dx. The entropy isdependenton theparticular choiceof thekernelK. Asmentionedbefore, it is acommonpractice in thefieldofnon-parametricstatistics to introduceatuningparameterν>0 in thekernel, calledbandwidth, so that it is expressedasa scaledversionK= fν of agiven function f :R+→R+. Thevalueofν is themost influentialparameter intheestimationofthedensityandmust beselectedcarefully. Forcurveclusteringapplications, it isdefinedbythedesired interaction length: ifν tends tozero, thecurveswillbehaveas independentobjects,whileontheotherendof thescale, veryhighbandwidthwill tendtoremovethe influenceof thecurves themselves. For themoment,no automatedmeansoffindinganoptimalνwasused,althoughitwillbepartofa futurework. 2.4.Minimizing theEntropy In order to fulfill the initial requirement of finding bundles of curve segments as straight as possible, one seeks after the system of curves minimizing the entropy E(γ1, . . . ,γN), orequivalentlymaximizing: ∫ Ω d˜(x) log ( d˜(x) ) dx. The reasonwhy this criterion gives the expected behaviorwill becomemore apparent after derivationof itsgradientat theendof thispart.Nevertheless,whenconsideringasingle trajectory, it is intuitivethat themostconcentrateddensitydistributionisobtainedwithastraightsegmentconnecting theendpoints: thispointwillbemaderigorous later. Letting beaperturbationof thecurveγj, suchthat (0)= (1)=0, thefirstorderexpansion of−E(γ1, . . . ,γN)willbecomputed inorder togetamaximizingdisplacementfield, analogous to a gradient ascent (the choice has beenmade tomaximize the opposite of the entropy, so that the algorithmwillbeagradientascentone) in thefinitedimensional setting. Thenotation: ∂F ∂γj willbeused in thesequel todenote thederivativeofa functionFof thecurveγj in thesense that fora perturbation : F(γj+ )=F(γj)+ ∂F ∂γj ( )+o(‖ ‖2). Firstofall,pleasenote thatsince d˜has integraloneover thedomainΩ: ∫ Ω ∂d˜(x) ∂γj ( )dx=0 so that: − ∂ ∂γj E(γ1, . . . ,γN)( )= ∫ Ω ∂d˜(x) ∂γj ( ) log ( d˜(x) ) dx. (14) Starting fromtheexpressionof d˜given inEquation(7), thefirstorderexpansionof d˜withrespect to theperturbation ofγj isobtainedasasumofa termcomingfromthenumerator:∫ 1 0 K (‖x−γj(t)‖)‖γ′j(t)‖dt. (15) 394