Page - 355 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 Sinceφ fromEquation(10) isan isometryuptoascalingfactor, ifY∈pandCφ(exp(Y))C−1= exp0(u∈T0Dn), then dlog∗(vol) dLeb0 (u)= dlog∗(vol′) dY (Y), whereLeb0 referstotheLebesguemeasureonthetangentspaceT0DnasinEquation(6).GivenZ∈Dn, HZ fromEquation(4)verifiesCφ(exp(Adk(HZ)))C−1=Z forsomek inK. Thus, θ0(Z)= dlog∗(vol′) dY (Adk(HZ))= ∏ λ∈Λ+ sinh(λ(HZ)) λ(HZ) . Wehavethen θ0(Z)=∏ i<j sinh(τi−τj) τi−τj ∏i≤j sinh(τi+τj) τi+τj , where the (τi)aredescribed inSection2.2.GivenZ1,Z2∈Dn, θZ1(Z2)= θ0(g −1 Z1 .Z2), where g−1Z1 is defined in Equation (3). It is thus possible to use the density estimator defined in Equation(7). Indeed, 1 θZ1(Z2) K ( d(Z1,Z2) r ) =∏ i<j τi−τj sinh(τi−τj)∏i≤j τi+τj sinh(τi+τj) K ( (2∑τ2i ) 1/2 r ) , (19) wherethe(τi)arethediagonalelementsofHg−1Z1 .Z2 . Recall thatwhenn=1, theSiegeldiskcorresponds to thePoincarédisk. Thus,weretrieve theexpressionof thekernel for thehyperbolic space, 1 θZ1(Z2) K ( d(Z1,Z2) r ) = 2τ sinh(2τ) K ( (2τ2)1/2 r ) . (20) 4.ApplicationtoRadarProcessing 4.1. RadarData Inspace timeadaptativeradarprocessing(STAP), thesignal is formedbyasuccessionofmatrices X representing therealizationofa temporalandspatialprocess. LetB+n,mbe thesetofpositivedefinite blockTeoplitzmatricescomposedofn×nblocksofm×mmatrices (PDBT).Forastationarysignal, theautocorrelationmatrixR isPDBT(see [5,6,14]).Authorsof [5,6,14]proposedageneralizationof VerblunskycoefficientsanddefinedaparametrizationofPDBTmatrices, B+n,m → Sym+×Dm−1n R → (P0,Ω1,...,Ωm−1), (21) inwhich themetric inducedbytheKählerpotential is theproductmetricofanaffine invariantmetric on Sym+ andm−1 times themetric of the Siegel disk, up to a scaling factor. When the signal is notGaussian, reflection/Verblunskycoefficients inPoincaréorSiegelDisksshouldbenormalizedas described in [28]byanormalizedBurgalgorithm. Amongother references,positivedefiniteblock Teoplitzmatriceshavebeenstudied in thecontextofSTAP-radarprocessing in [4–6]. 4.2.MarginalDensities ofReflectionCoefficients In this section,weshowdensityestimationresultsof themarginalparametersΩk. For thesakeof visualization,only theSiegeldiskD1 is considered. Recall thatD1 coincideswith thePoincarédisk. 355