Page - 420 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 7.DiscussionandConcludingRemarks Incorporatinganisotropy inmodels fordata innon-linearspacesvia the framebundleaspursued inthispaper leadstoasub-Riemannianstructureandmetric.Adirect implicationis thatmostprobable paths toobserveddata in thesenseofsequencesofstochastic stepsofadrivingsemi-martingaleare not related togeodesics in theclassical sense. Instead,abestestimateof thesequenceofstepswt∈Rd that leads toanobservationx= ϕu(wt)|t=1 isanMPPinthesenseofDefinition1.Asshowninthe paper, thesepathsaregenerallynotgeodesicsorpolynomialswithrespect to theconnectiononthe manifold. Inparticular, ifMhasaRiemannianstructure, theMPPsaregenerallyneitherRiemannian geodesicsnorRiemannianpolynomials. Below,wediscuss thestatistical implicationsof this result. 7.1. StatisticalEstimators MetricdistancesandRiemanniangeodesicshavebeen the traditionalvehicle for representing observeddata innon-linearspaces.Most fundamentally, thesampleFrechétmean xˆ=argminx∈M N ∑ i=1 dgR (x,xi) 2 (19) of observeddata x1, . . . ,xN ∈ M relies crucially on theRiemanniandistance dgR connected to the metric gR. ManyPCAconstructs (e.g., PrincipalGeodesicsAnalysis [6]) use theRiemannianExp. andLogmapstomapbetweenlinear tangentspacesandthemanifold. Thesemapsaredefinedfrom theRiemannianmetric andRiemanniangeodesics. Distributionsmodelledas in the randomorbit model [32]orBayesianmodels [15,33]againrelyongeodesicswithrandominitial conditions. Usingthe framebundlesub-RiemannianmetricgFM,wecandefineanestimatoranalogous to the RiemannianFrechétmeanestimator.Assumingthecovariance isaprioriknown, theestimator xˆ=argminu∈s(M) N ∑ i=1 dFM ( u,π−1(xi) )2 (20) actscorrespondinglytotheFrechétmeanestimator (19).Here s∈Γ(FM) isa (local) sectionofFM that tox∈Mconnects theknowncovariancerepresentedby s(x)∈FM. ThedistancesdFM ( u,π−1(xi) ) , u= s(x)are realizedbyMPPs fromthemeancandidate x to thefibersπ−1(xi). TheFrechétmean problemis thus liftedto the framebundlewith theanisotropicweighting incorporated in themetric gFM. Thismetric isnot relatedtogR, except for itsdependenceontheconnectionC thatcanbedefined as theLevi–CivitaconnectionofgR. The fundamental roleof thedistancedgR andgRgeodesics in (19) is thusremoved. Because covariance is an integral part of themodel, sample covariance canalsobe estimated directlyalongwith thesamplemean. In [3], theestimator uˆ=argminu∈FM N ∑ i=1 dFM ( u,π−1(xi) )2−N log(detgRu) (21) is suggested. The normalizing term−N log(detgRu) is derived such that the estimator exactly corresponds to themaximumlikelihoodestimatorofmeanandcovariance forEuclideanGaussian distributions. Thedeterminant isdefinedvia gR, and the termacts toprevent thecovariance from approachinginfinity.Maximumlikelihoodestimatorsofmeanandcovariancefornormallydistributed Euclideandatahaveuniquesolutions in thesamplemeanandsamplecovariancematrix, respectively. Uniqueness of the Frechétmean (19) is only ensured for sufficiently concentrated data. For the estimator (21), existenceanduniquenesspropertiesarenot immediate,andmoreworkisneededin order tofindnecessaryandsufficientconditions. 420