Page - 416 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 425 Note that the introductionof theRiemannianmetricgR implies thatWij arenowdependenton themanifoldcoordinatesxi. 6.NumericalExperiments Weaimat visualizingmost probable paths for thedrivingprocess andprojections of curves satisfying the MPP Equation (13) in two cases: On 2D surfaces embedded inR3 and on finite dimensional landmarkmanifoldsthatarise fromequippingasubsetof thediffeomorphismgroupwith aright-invariantmetricandlettingtheactiondescendto the landmarksbya leftaction. Thesurface examples are implemented in Python using the Theano [30] framework for symbolic operations, automatic differentiation, andnumerical evaluation. The landmark equations are detailed below and implemented inNumpyusingNumpy’s standardODEintegrators. Thecode for running the experiments isavailableathttp://bitbucket.com/stefansommer/mpps/. 6.1. EmbeddedSurfaces WevisualizenormalMPPsandprojectionsofcurvessatisfyingtheMPPEquation(13)onsurfaces embeddedinR3 inthreecases: ThesphereS2,onanellipsoid,andonahyperbolicsurface. Thesurfaces are chosen in order to have bothpositive andnegative curvature, and to have varyingdegree of symmetry. In all cases, an open subset of the surfaces are represented in a single chart by amap F : R2 → R3. For the sphere and ellipsoid, this gives a representation of the surface, except for thesouthpole. ThemetricandChristoffel symbolsarecalculatedusingthesymbolicdifferentiation featuresofTheano. The integrationareperformedbyasimpleEuler integrator. Figures4–6showfamiliesofcurvessatisfyingtheMPPequations in threecases: (1)Withfixed startingpointx0∈Mandinitialvelocity x˙0∈TMbutvaryinganisotropyrepresentedbychanging frameu in thefiberabove x0; (2)minimizingnormalMPPswithfixedstartingpoint andendpoint x0,x1∈Mbutchanging frameuabovex0; (3)fixedstartingpointx0∈Mandframeubutvarying V∗FMverticalpartof the initialmomentum ξ0∈T∗FM. Thefirst andsecondcases thusshowthe effectofvaryinganisotropy,while the thirdcase illustrates theeffectof the“twist” that thed2 degrees in theverticalmomentumallows.Note thedisplayedanti-developedcurves inR2 that forclassicalC geodesicswouldalwaysbestraight lines. (a) (b) (c) Figure4.CurvessatisfyingtheMPPequations (toprow)andcorrespondinganti-development (bottom row) on three surfaces embedded inR3: (a) An ellipsoid; (b) a sphere; (c) a hyperbolic surface. The familyofcurves isgeneratedbyrotatingbyπ/2radians theanisotropiccovariancerepresented in the initial frameu0 anddisplayedin thegrayellipse. 416