Page - 391 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 Assumingagainanormalizedkernelandletting libe the lengthof thecurveγi, theexpressionof thedensitysimplifies to: d˜: x →∑ N i=1 ∫1 0 K(‖x−γi(t)‖)‖γ′i(t)‖dt ∑Ni=1 li . (7) ThenewDiff-invariantdensity ispictured inFigure2alongwith thestandarddensity.While the overall aspectof theplot is similar,onecanobserve that routesaremoreapparent in therightpicture andthat thedensitypeak locatedabove theParisarea isof less importanceandlesssymmetricdue to the fact thatnearairports, aircraftareslowingdown,andthiseffectexaggerates thedensitywith the non-invariantdefinition. (a) (b) Figure2.Density (a) andDiff invariantdensity (b) for12February2013 traffic. Theextensionofthetwo-dimensionaldefinedthatwaytothegeneralcaseofcurvesinanarbitrary spaceRq is straightforward. 2.3. FurtherProperties of theDensity In this section, thecurvesconsideredareasmoothmappingfromtheclosed interval [0,1] toRq, withanon-vanishingderivative in ]0,1[. AllmultivariatekernelsKwillbeassumedsmooth,positive, withaunit integralandof the formx →K(‖x‖).However, it isnot requiredthat theyarecompactly supportedunlessexplicitlystated.All resultsarepresentedfor thewholespaceRq, butapplyalmost verbatimtoanopensubset. Definition1. Let f bea smoothsummablemapping fromR toR. The scaling fν of f isdefined, for eachν>0, to be themapping: fν : x∈R → 1 ν f (x ν ) . It is clear that theL1-normof theoriginalmappingispreservedbythescaling.Givenasummable kernel functionK fromR toR+, it defines amultivariate kernelK onRq thatmaps x toK(‖x‖). Onemayderive fromitaparametrizedfamilyofkernels inRbymappingeachν in ]0,1] to thescaled kernelKν. If theoriginalK isofunit integral, soareallof theKν. Proposition1. Letγ: [0,1]→Rq bea smoothpathwithanon-vanishingderivative in ]0,1[. LetKν,ν>0 beaparametrized familyofunit integral kernels. The familyofBorelmeasuresμν defined for anyBorel setAby: μν(A)= ∫ A ∫ 1 0 Kν(‖x−γ(t)‖)‖γ′(t)‖dtdx is tightandconvergesnarrowly to theLebesguemeasureonγ([0,1]). 391