Page - 397 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 effect in the limit,whilegatheringdominatesathighbandwidths. For theair trafficapplication,a rule of the thumbis to take2–3-times theseparationnormasaneffectivesupport for thekernel.Usingan adaptivebandwidthmaybeofsomeinterestalso: startingwithmediumtohighvalues favorscurve gathering; then,graduallyreducing itwill straightenthe trajectories. Usingthescaledarclength in theentropygivesanequivalent,butsomewhateasier to interpret result. Startingwith theexpression(7) that takes in thiscase the form: d˜: x →∑ N i=1 li ∫1 0 K(‖x−γi(η)‖)dη ∑Ni=1 li . (21) Let i ∈ {1,. . . ,N} be fixed. An admissible variation of the curve γi is a smooth mapping from ]−a,a[×[0,1] toRq,with a>0satisfyingthe followingproperties: (a) ∀η∈ [0,1],φ(0,η)=γi(η). (b) ∀(t,η)∈]−a,a[×]0,1[,‖∂ηφ(t,η)‖= lφ(t)with lφ(t) the lengthof thecurveη →φ(t,η). (c) ∀t∈]−a,a[,φ(t,0)=γi(0), φ(t,1)=γi(1). Takingthederivativewithrespect to tatzeroofEquation(b)yields:〈 ∂t∂ηφ(0,η),∂ηφ(0,η) 〉 = ∂tlφ(0)li. LettingT(η)betheunit tangentvector toγi atηandnotingthat∂ηφ(0,η)= liT(η), itbecomes:〈 ∂t∂ηφ(0,η),T(η) 〉 = ∂tlφ(t). (22) Thisrelationputsaconstraintonthevariationof thetangentialcomponentof thecurvederivative andshowsthat ithas tobeconstant inη. Proposition4. LetDbe themapping from ]−a,a[×Rq toR+ definedby: D: (t,x) →∑ N j=1,j =i lj ∫1 0 K (‖x−γj(η)‖)dη+∫10 K(‖x−φ(t,η)‖)dη ∑Nj=1 lj . whereη refers collectively to the scaledarclengthparameter for eachcurve. Thepartialderivative∂tD(0,x) is givenby: ∂tD(0,x)= li ∑Nj=1 lj ∫ 1 0 〈 γi(η)−x ‖γi(η)−x‖,∂tφ(0,η) 〉 K′(‖γi(η)−x‖)dη. Theproof is straightforwardand isomitted. FromProposition4, thevariationof the entropy isderived: ∂tE=− ∫ Rq li ∑Nj=1 lj ∫ 1 0 〈 γi(η)−x ‖γi(η)−x‖,∂tφ(0,η) 〉 K′(‖γi(η)−x‖)dηdx. (23) This relation is equivalent to (18): it can be seen by splitting the terms into a normal and atangential component. Thefirstoneyields: − ∫ Rq li ∑Nj=1 lj ∫ 1 0 〈( γi(η)−x ‖γi(η)−x‖ ) N ,(∂tφ(0,η))N 〉 K′(‖γi(η)−x‖)dηdx. 397