Page - 348 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 Onealreadyencounters theproblemofdensityestimation in thehyperbolicspace forelectrical impedance [15],networks [16]andradarsignals [17]. In [18], ageneralizationof theGaussian lawon thehyperbolicspacewasproposed.Apart from[19],whereauthorsproposeageneralizationof the Gaussian law,probabilitydensityestimationontheSiegel spacehasnotyetbeenaddressed. Thecontributionsof thepaperare the following. Wereviewthemainnonparametricdensity estimationtechniquesontheSiegeldisk.Weprovidesomerathersimpleexplicit expressionsof the kernelsdefinedbyPelletier in [1]. These expressionsmake thekerneldensity estimation themost adaptedmethod.Wepresentvisual resultsofestimateddensities in thesimplecasewhere theSiegel diskreduces to thePoincarédisk. Thepaperbeginswithan introduction to theSiegel space inSection2. Section3reviewsthemain non-parametricdensityestimation techniquesontheSiegel space. Section3.3contains theoriginal resultsof thepaper. Section4presentsanapplicationtoradardataestimation. 2.TheSiegelSpace This sectionpresents facts about theSiegel space. The interested reader canfindmoredetails in [20,21]. ThenecessarybackgroundonLiegroupsandsymmetric spacecanbefoundin[22]. 2.1. TheSiegelUpperHalfSpace TheSiegelupperhalf space isageneralizationof thePoincaréupperhalf space (see [23]) fora descriptionof thehyperbolic space. LetSym(n)bethespaceof real symmetricmatricesofsizen×n andSym+(n) thesetof real symmetricpositivedefinitematricesofsizen×n. TheSiegelupperhalf space isdefinedby Hn={Z=X+ iY|X∈Sym(n),Y∈Sym+(n)} . Hn isequippedwith the followingmetric: ds=2tr(Y−1dZY−1dZ). Thesetof real symplecticmatricesSp(n,R) isdefinedby g∈Sp(n,R)⇔ gtJg= J, where J= ( 0 In −In 0 ) , and In is then×n identitymatrix. Sp(n,R) is a subgroupofSL2n(R), the setof 2n×2n invertible matrices of determinant 1. Let g = ( A B C D ) ∈ Sp(n,R). Themetric ds is invariant under the followingactionofSp(n,R), g.Z=(AZ+B)(CZ+D)−1. Thisaction is transitive, i.e., ∀Z∈Hn,∃g∈Sp(n,R),g.iI=Z. The stabilizerK of iI is the set of elements g of Sp(n,R)whose action leaves iI fixed. K is a subgroupofSp(n,R)calledthe isotropygroup.Wecanverify that K= {( A B −B A ) ,A+ iB∈SU(n) } . 348