Page - 400 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 although the second form is probablymore efficient fromanumerical point of view andwill be investigated inasecondstage. Having the density at hand, the gradient of the entropy with respect to the points γij, i = 1,. . . ,N, j = 1,. . . ,m can be easily computed using a straightforward application of the formula (19).Whendealingwithplanarcurves,asimplificationoccurs for thesecondderivative term since forasmoothcurveγj: ( γ′′j (t) ‖γ′j(t)‖ ) N = κ(t)N(t). where κ is the curvature andN theunit normal vector. These quantitiesmaybe computedusing numerical differentiation, but a coarse approximation based on the rotation rate of the vectors γi,j+1−γi,j,γi,j+2−γi,j+1workswell inmanycases. Thecaseof scaledarclengthparametrizationneedssomeextraattention,due to thecondition on the tangential component. The simplest approach is to move the points γij according to an unconstrainedgradient, thentore-sample theobtainedcurvesoas togetadjustedγij thatcorrespond to theabscissaηj, j=1,. . . ,m. Inanumerical implementation, thescalingfactor infrontof thewholeexpressionmaybedropped duetothefact thatallgradient-basedalgorithmswilluseanautomatically-tunedsteplength.Asusual withgradientalgorithms,onemustcarefullyselect thesteptakinginthemaximizingdirectioninorder toavoiddivergence.Asimplefixedstepstrategywasfirstappliedandgivessatisfactoryresultson smalldatasets.Asaferapproach is toadapt thestepsizesoas toensureasufficientdecreaseof the entropy.Duetothepotentiallyhugedimensionof thesearchspace, thisprocedurehas tobesimple enough.Anapproximatequadratic search[12]wasusedin thefinal implementation. Theprocedureappliedtoonedayof trafficoverFranceyields thepictureofFigure3.Asexpected, a route-like network emerges. In such a case, since the traffic comes froman already organized situation, the recovered network is indeed a subset of the route network in the french airspace. Pleasenote that there isa trade-offbetweenthedensityconcentrationandtheminimalcurvatureof therecoveredtrajectories, asalreadymentioned. Thekernelbandwidthwaschosenempirically in the examplepresented,with theaidofvisual interaction. (a) (b) Figure3.Trafficof24February2013: (a) Initial traffic; (b)Bundledtraffic. In thesecondexampleofFigure4, theproblemofautomaticconflict solving isaddressed. In the initial situation,aircraftareconvergingtoasinglepoint,which isunsafe.Air trafficcontrollerswill proceed in sucha case bydiverting aircraft from their initial flightpath so as to avoid eachother, butonlyusingverysimplemaneuvers.Anautomatedtoolwillmakefulluseof theavailableairspace, andtheresultingsetof trajectoriesmayfail tobemanageablebyahuman: in theeventofasystem failure,nobackupcanbeprovidedbycontrollers. Theentropyminimizationprocedurewasaddedto anautomatedconflict solver inorder toendupwithflightpathsstill tractablebyhumans. Thefinal 400