Page - 398 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 For the tangentialpart, thestartingpoint is therelation: ∂η (K(‖φ(0,η)−x‖)T(η))=liK′(‖φ(0,η)−x‖) 〈 φ(0,η)−x ‖φ(0,η)−x‖,T(η) 〉 T(η) +K(‖φ(0,η)−x‖)∂ηT(η). (24) where thesubscriptT stands for tangential component. Itbecomes: li ∑Nj=1 lj ∫ 1 0 〈( γi(η)−x ‖γi(η)−x‖ ) T ,(∂tφ(0,η))T 〉 K′(‖γi(η)−x‖)dηdx= li ∑Nj=1 lj ∫ 1 0 〈 ∂η (K(‖φ(0,η)−x‖)T(η)) ,(φ(0,η))T 〉 dηdx − li ∑Nj=1 lj ∫ 1 0 〈 K(‖φ(0,η)−x‖)∂ηT(η),(φ(0,η))T 〉 dηdx. (25) Withan integrationbyparts, thefirst integral in theright-handsidebecomes: − li ∑Nj=1 lj ∫ 1 0 〈 K(‖φ(0,η)−x‖)T(η),∂η (∂tφ(0,η))T 〉 dηdx= − li ∑Nj=1 lj ∫ 1 0 K(‖φ(0,η)−x‖)∂tlφ(0)dηdx. (26) Gatheringterms, theexpression(18) is recovered.Asexpected,onlythenormalcomponentsenter therelation,but ithas tobenotedthat the tangential componentof∂tφ(0,η) isnotarbitraryandcan bededucedfrom(22). Thegradientwithrespect to the i-thcurve isobtainedfromtheexpressionof theentropyvariationandcanbewritten in its simplest formas: li ∑Nj=1 lj ∫ Rq ∫ 1 0 γi(η)−x ‖γi(η)−x‖K ′(‖γi(η)−x‖)dη log d˜(x)dx. (27) where d˜ is the estimatedspatialdensity. Onemustkeep inmind the constrainton ∂tφ(0,η) that is hiddenwithin theapparentsimplicityof theexpression. 3.Numerical Implementation Thetwoformulations(19)and(27)of thegradientmaybeused. Thefirstoneismorecomplicated, butdoesnot requireanyadditional constraint tobe taken intoaccount. Thesecondonecannotbe appliedreadilyas thetangentialcomponentmustcomplywithRelation(22). Inbothcases, it isneeded toevaluateaspatial integral,whichmayyieldtoprohibitivecomputational time,especially inhigh dimensions. In theair trafficapplication,onlyplanar3Dcurvesareconsidered,greatlysimplifyingthe problem.Nevertheless, theperformanceof thealgorithmsisstill aconcern,andthechoicemadewas toreplace thespatial integralbyadiscretesumoveranevenly-spacedgrid. Fromnow, it isassumed thatall curvesareplanar, so that theambientspace for thespatialdensity d˜ isR2.Goingbackto the expressionof d˜givenby(7), afirst step is toreplace the integralover tbyadiscretesum. Inpractice, curves aredescribedbya sequence of sampledpointsγi(tij)where the sampling times tij will be assumedtobe identical forall curves. Thisassumption isnotsatisfied in theair trafficapplication, so thatapre-processingstepmustbe takenbefore theactualentropyminimizationstage. Itwillnotbe describedhere, asanystandard interpolationprocedurecanbeappliedwithnegligibledifferences onthefinal result. Toobtain theresultspresentedhere,acubicsplinesmootherwasused. Since the samplingtimesareassumedtobethesameforall trajectories, thedoublesubscriptwillbedropped, so that thesamplesoneach trajectorywillbedenotedasγij=γi(tj). It is furtherassumedthat the 398