Page - 401 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 result is shownintherightpartofFigure4,whereencountersno longerexists,butaircraftarebound tosimple trajectories,withamergingandasplittingpoint.Note that since theautomatedplanneracts onvelocity,all aircraftareseparated in timeonthe innerpart. (a) (b) Figure4. (a) Initialflightplans; (b)Finalflightplans. 4.ConclusionsandFutureWork Algorithms coming from thefieldof shape spaces emerge as avaluable tool for applications inATM. In thiswork, the foundationsofapost-processingprocedure thatmaybeappliedafteran automatedflightpathplannerarepresented. Entropyminimizationmakesstraightsegmentbundles emerge,which fulfills theoperational requirements. Computational efficiencyhas tobe improved inorder toreleaseausablebuildingblockfor futureATMsystems.Onewaytoaddress this issue is to computekerneldensityestimatorsusingGPUs,whichexcel in thiskindof task, very similar to texturemanipulations. Furthermore, statisticalproperties, suchas theoptimalchoiceof thebandwidth parameter in thekernelestimation, shouldbeexplored inmoredetail in thenextstepof thiswork. Another important point thatmust be addressed in futureworksdealswith theflight paths that areverysimilar in shape, butareoriented inoppositedirections. As the spatialdensity isnot sensitive to thedirectional information, the entropy-basedprocedurepresented in this paperwill tend to aggregateflightpaths that shouldbe sufficiently separated inorder toprevent hazardous encounters. In [13], anotionofdensitybasedonpositionandvelocity isdeveloped. Thisworkrelies onLiegroupmodelingasaunifyingstate representation that takes intoaccount thedirectionandthe positionof thecurves. Thecurvesystementropyhasbeenextendedto this setting. AuthorContributions: StéphanePuechmorelhas conceived the theoretical aspectsof thiswork, aswell as to the conceptionanddesignof the experiments; FlorenceNicol has contributed to review the theoretical tools. Bothauthorshavecontributedtoanalyze thedataandtowrite thepaper. Bothauthorshavereadandapproved thefinalmanuscript. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. References 1. De Bondt, A.; Leleu, C. 7-Year IFR Flight Movements and Service Units Forecast Update: 2014–2020; EUROCONTROL:Brussels,Belgium,2014. 2. Roussos,G.P.;Dimarogonas,D.V.;Kyriakopoulos,K.J. Distributed3Dnavigationandcollisionavoidance fornonholonomicaircraft-likevehicles. InProceedingsof the2009EuropeanControlConference,Budapest, Hungary,23–26August2009. 401