Page - 390 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 Note thatwhenK is compactlysupported,which is thecaseof theEpanechnikovfunctionandallof its relatives, itbecomes: ∫ Ω K(‖x−γi(t)‖)dx= ∫ Rq K(‖x‖)dx providedthatΩcontains theset: {x∈Rq, inf i=1...N,t∈[0,1] ‖x−γi(t)‖≤A} where the interval [−A,A]contains thesupportofK. Thecaseofkernelswithunboundedsupport, like Gaussian functions, may be dealt with providedΩ = Rq. In the application considered, only compactly-supported kernels are used,mainly to allow fastmachine implementation of the densitycomputation. Using the polar coordinates (ρ,θ) and the rotation invariance of the integrand, the relationbecomes: Vol ( S q−1 )∫ R+ K(ρ)ρq−1dρ=1 whichyieldsanormalizingconstantof2/π for theEpanechnikovfunction indimensiontwo, instead of theusual3/4 in therealcase.Whenthenormalizationcondition is fulfilled, theexpressionof the densitysimplifies to: d: x →N−1 N ∑ i=1 ∫ 1 0 K(‖x−γi(t)‖)dt. (5) Thenormalizingconstant is thesameas in (2). As an example, one day of traffic over France is considered and pictured in Figure 1 with the corresponding density map, computed on an evenly-spaced grid with a normalized Epanechnikovkernel. (a) (b) Figure1. (a)TrafficoverFrance; (b)Associateddensity. Unfortunately,densitycomputedthiswaysuffersasevereflawfor theATMapplication: it isnot relatedonly to theshapeof trajectories,butalso to the timebehavior. Formally, it isdefinedontheset Imm([0,1],Rq)ofsmooth immersions from [0,1] toRqwhile thespaceofprimary interestwillbe the quotientbysmoothdiffeomorphismsof the interval [0,1], Imm([0,1],Rq)/Diff([0,1]). Invarianceof thedensityunder theactionofDiff([0,1]) isobtainedas in [10]byaddinga termrelatedtovelocity in the integrals. Thenewdefinitionofdbecomes: d˜: x → ∑ N i=1 ∫1 0 K(‖x−γi(t)‖)‖γ′i(t)‖dt ∑Ni=1 ∫ Ω ∫1 0 K(‖x−γi(t)‖)‖γ′i(t)‖dtdx . (6) 390