Page - 404 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 estimationandstatistics. Thismetricnaturallyaccounts foranisotropyinasimilarwayastheprecision matrixweights the innerproduct in thenegative log-likelihoodofaEuclideannormaldistribution. Theconnectionbetweentheweighteddistanceandstatisticsofmanifoldvalueddatawaspresented in [2], and theunderlyingsub-Riemannianandfiber-bundlegeometry, togetherwithpropertiesof thegenerateddensities,was furtherexplored in[3]. Thefundamental idea is toperformstatisticson manifoldsbymaximumlikelihood(ML) insteadofparametric constructions thatuse, forexample, approximatinggeodesicsubspaces;bydefiningnatural familiesofprobabilitydistributions (in this caseusingdiffusionprocesses),MLparameterestimatesgiveacoherentwaytostatisticallymodel non-lineardata. Theanisotropicallyweighteddistanceand the resulting familyof extremalpaths arises in this situationwhen thediffusionprocesses havenon-isotropic covariance (i.e.,when the distribution isnotgeneratedfromastandardBrownianmotion). In thispaper,wefocusonthe familyofmostprobablepaths for thesemi-martingales thatdrives the stochasticdevelopment,andin turn themanifoldvaluedanisotropicstochasticprocesses. Suchpaths, as exemplified in Figure 1, extremize the anisotropicallyweighted action functional. Wepresent derivationsofevolutionequations for thepaths fromdifferentviewpoints, andwediscuss theroleof framesasrepresentingeithermetricsorcometrics. In thederivation,weexplicitlysee the influence of theconnectionanditscurvature.Wethenturnto therelationbetweenthesub-Riemannianmetric and the Sasaki–Mokmetric on the frame bundle, andwedevelop a construction that allows the sub-Riemannian metric to be defined as a sum of a rank-deficient generator and an underlying Riemannianmetric. Finally,werelate thepaths togeodesicsandpolynomials inRiemanniangeometry, and we explore computational representations on different manifolds including a specific case: thefinitedimensionalmanifoldsarising in theLargeDeformationDiffeomorphicMetricMapping (LDDMM)[8] landmarkmatchingproblem.Thepaperendswithadiscussionconcerningstatistical implications,openquestions,andconcludingremarks. (a) (b) Figure 1. (a)Amost probable path (MPP) for adrivingEuclideanBrownianmotiononan ellipsoid. Thegrayellipsisover thestartingpoint (reddot) indicates thecovarianceof theanisotropicdiffusion. Aframeut (black/grayvectors) representingthesquarerootcovariance isparallel transportedalong the curve, enabling the anisotropic weighting with the precision matrix in the action functional. With isotropiccovariance,normalMPPsareRiemanniangeodesics. Ingeneral situations, suchas the displayedanisotropiccase, thefamilyofMPPsismuchlarger; (b)Thecorrespondinganti-development inR2 (red line)of theMPP.Comparewith theanti-developmentofaRiemanniangeodesicwithsame initialvelocity (bluedotted line). Theframesut∈GL(R2,TxtM)provide local framecoordinates for eachtime t. Background Generalizing common statistical tools for performing inference on Euclidean space data to manifoldvalueddatahasbeen thesubjectof extensivework (e.g., [9]). Perhapsmost fundamental 404