Page - 362 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 5.Conclusions Threenonparametricdensityestimationtechniqueshavebeenconsidered. Themainadvantage ofhistogramsin theEuclideancontext is their simplicityofuse. Thismakeshistogramsaninteresting tooldespitethefactthattheydonotpresentoptimalconvergencerates.OntheSiegelspace,histograms lose their simplicityadvantage. Theywere thusnotdeeplystudied. Theorthogonal seriesdensity estimation also presents technical disadvantages on the Siegel space. Indeed, the series become integrals,whichmakethenumerical computationof theestimatormoredifficult than in theEuclidean case. Ontheotherhand, theuseof thekerneldensityestimatordoesnotpresentmajordifferences with theEuclideancase. Theconvergencerateobtained in [1] canbeextendedtocompactlysupported randomvariablesonnoncompactRiemannianmanifolds. Furthermore, thecorrective termwhose computation is required to use Euclidean kernels on Riemannianmanifolds turns out to have a reasonably simple expression. Our future efforts will concentrate on the use of kernel density estimation on the Siegel space in radar signal processing. As the experimental section suggests, westronglybelievethat theestimationof thedensitiesof theΩkwillprovideaninterestingdescription of thedifferentbackgrounds. Thisnon-parametricmethodofdensityestimationshouldbecompared withparametricones,as“MaximumEntropyDensity” (Gibbsdensity)onhomogenesousmanifold asproposed in [37]basedontheworksof Jean-MarieSouriau. Asproposed in [38], amedian-shift approachmightalsobe investigated. Acknowledgments:Theauthorswould like to thankSalemSaid,MichalZidorandDmitryGourevitchfor the help theyprovidedin theunderstandingofsymmetric spacesandtheSiegel space. Author Contributions: Emmanuel Chevallier carried out the mathematical development. Thibault Forget has set up the Radar clutter segmentation. Frédéric Barbaresco has introduced Poincaré/SiegelHalf space and Poincaré/Siegel Disk parameterization for Radar Doppler and Space-Time Adaptive Processing based onMetric spacesdeduced fromInformationGeometry. Thisparameterizationhasbeen re-used in thispaper. JesusAngulowas thePh.D. supervisorofEmmanuelChevallierandparticipates in thesupervisionofmaster thesis ofThibault Forget, both thesis are at theoriginof this study. All authorshave readandapproved the finalmanuscript. Conflictsof Interest:Theauthorsdeclarenoconflictof interest. Appendix DemonstrationofTheorem1 LemmaA1. Let (M,g)beaRiemannianmanifold, letCbea compact subset ofMand letUbea relatively compact opensubset ofMcontainingC. Then, there is a compactRiemannianmanifold (M′,g′) such thatU is anopensubset ofM′, the inclusion i :U ↪→M′ is adiffeomorphismonto its imageandg′= gonU. Proof. WecanassumethatM isnotcompact. Let f :M→RbeasmoothfunctiononMwhichtends to+∞at infinity. SinceU is compact, f−1(]−∞,a[)containsU for a largeenough. BySardTheorem, thereexistsavalue a∈Rsuchthat f−1(a)containsnocriticalpointof f andsuchthat f−1(]−∞,a[) containsU. It followsthatN= f−1(]−∞,a]) isasubmanifoldwithboundaryofM. Since f tends to +∞at infinity,N is compactaswellas itsboundary∂N= f−1({a}). CallM′ the double ofN. It is a compactmanifoldwhich containsN such that the inclusion i :N ↪→M′ isadiffeomorphismonto its image(see [39],Theorem5.9andDefinition5.10 ). Choose any metric g0 on M′. Consider two open subsetsW1 andW2 in M′ and two smooth functions f1, f2 :M′→ [0,1] suchthat U⊂W1⊂W1⊂W2⊂W2⊂ intN, the interiorofN, f1(x)=1 onW1,vanishesoutsideofW2, and f2(x)=1 362