Page - 351 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 396 3.NonParametricDensityEstimationontheSiegelSpace LetΩbea space, endowedwithaσ-algebraandaprobabilitymeasure p. LetXbea random variableΩ→Dn. TheRiemannianmeasureofDn is calledvolandthemeasureonDn inducedbyX isnotedμX.WeassumethatμXhasadensity,noted f ,withrespect tovol, andthat thesupportofX is acompactsetnotedSupp. Let (x1,...,xk)∈DknbeasetofdrawsofX. TheDiracmeasureat apoint a∈Dn is denoted δa. Letμk = 1k∑ki=1δxi denotes the empirical measure of the set of draws. This section presents four non-parametric techniques of estimation of the density f from the set of draws (x1,...,xk). The estimated density at x in Dn is noted fˆk(x) = fˆ(x,x1,...,xk). Therelevanceofadensityestimationtechniquedependsonseveralaspects. Whenthespaceallowsit, theestimationtechniqueshouldequallyconsidereachdirectionandlocation. This leads toan isotropyandahomogeneitycondition. In thekernelmethod, for instance,akernel density functionKxi is placed at each observation xi. Firstly, in order to treat directions equally, the functionKxi shouldbe invariantunder the isotropygroupofxi; Secondly, foranotherobservation xj, functionsKxi andKxj shouldbesimilaruptothe isometries thatsendxionxj. Theseconsiderations stronglydependonthegeometryof thespace: if thespace isnothomogeneousandthe isotropygroup is empty, these indifferenceprincipleshavenomeaning. Since theSiegel space is symmetric, it is homogeneousandhasanonempty isotropygroup. Thus, thedensityestimationtechniqueshouldbe chosenaccordingly. The convergence of the different estimation techniques iswidely studied. Resultswere first obtainedintheEuclideancase,andaregraduallyextendedto theprobabilitydensitiesonmanifold (see [1,2,15,25]). The last relevantaspect is computational. Eachestimationtechniquehas itsowncomputational frameworkthatpresentsprosandconsgiventhedifferentapplications. For instance, theestimation byorthogonalseriesneedsaninitialpre-processing,butprovidesa fastevaluationof theestimated density incompactmanifolds. 3.1.Histograms The histogram is the simplest density estimation method. Given a partition of the space Dn= ∪iAi, theestimateddensity isgivenby fˆ(x∈Ai)= 1vol(Ai) k ∑ j=1 1Ai(xj), where1Ai standsfor the indicator functionofAi. Followingtheconsiderationsof theprevioussections, the elements of the partition shouldfirstly be as isotropic as possible, and secondly as similar as possible toeachother. Regardingtheproblemofhistograms, thecaseof theSiegel space is similar to thecaseof thehyperbolic space. Thereexistvariousuniformpolygonal tilingsontheSiegel space that couldbeusedtocomputehistograms.However, thereareratioλ∈R forwhich there isnohomothety. Thus, it is not alwayspossible to adapt the size of thebins to agiven set of drawsof the random variable.Modifyingthesizeof thebinscanrequireachangeof thestructureof the tiling. This iswhy thestudyofhistogramshasnotbeendeepened. 3.2.OrthogonalSeries Theestimationof thedensity f canbemadeoutof theestimationof thescalarproductbetween f andasetof“orthonormal” functions { ej } . Themost standardchoice for { ej } is theeigenfunctions of theLaplacian.WhenthevariableX takes itsvalues inRn, thisestimationtechniquebecomesthe characteristic functionmethod.Whentheunderlyingspace iscompact, thespectrumof theLaplacian operator iscountable,whilewhenthespace isnon-compact, thespectrumisuncountable. In thefirst case, theestimationof thedensity f ismade through theestimationof a sum,while in the second case ismadethroughtheestimationofan integral. Inpractice, thesecondsituationpresentsa larger 351