Page - 396 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 As expected, onlymoves normal to the trajectorywill change at first order the value of the criterion: thedisplacementof thecurveγjwill thusbeperformedat t in thenormalbundle toγj and isgiven,upto the (∑Ni=1 li) −1 term,by: ∫ Ω ( γj(t)−x ‖γj(t)−x‖ ) N K′ (‖γj(t)−x‖) log(d˜(x))dx‖γ′j(t)‖ − (∫ Ω K (‖γj(t)−x‖) log(d˜(x))dx) ( γ′′j (t) ‖γ′j(t)‖ ) N + (∫ Ω d˜(x) log(d˜(x))dx )( γ′′j (t) ‖γ′j(t)‖ ) N . (19) Thefirst termintheexpressionwill favormoves towardsareasofhighdensity,while thesecond andthirdonesaremovingalongnormalvectorandwill straightenthe trajectory. This lastpointcanbe enlightenedbyconsideringthecaseofasingleplanarcurvewithfixedendpoints. Proposition3. Let a,bbefixedpoints inR2 andKbeakernel as in (7). Thesegment [a,b] is a criticalpoint for the entropyassociatedwith the curve systeminR2 consistingof single smoothpathswith endpoints a,b. Proof. Let the segment [a,b] be parametrized as γ: t ∈ [0,1] → a+ tvwith v the vector (b− a). Startingwith theexpression(19), it is clear that thesecondandthirdtermsoccurring in the formula will vanishas thesecondderivativeofγ is zero. Letube theunitnormalvector toγ. Anypoint x inR2 canbewrittenasx= a+θv+ξu, θ,ξ∈R, so thatγ(t)−x=(t−θ)v−ξuand‖γ(t)−x‖=√ (t−θ)2‖b−a‖2+ξ2. Thechangeofvariablesx→ (θ,ξ)has Jacobian‖v‖= ‖b−a‖. Forafixed t∈ [0,1], itbecomes: ∫ R2 ( γ(t)−x ‖γ(t)−x‖ ) N K′(‖γ(t)−x‖) log(d˜(x))dx‖γ′(t)‖= ‖b−a‖2 ∫ R ∫ R −ξ√ (t−θ)2‖b−a‖2+ξ2K ′ (√ (t−θ)2‖b−a‖2+ξ2 ) log(d˜(θ,ξ))dξdθ. (20) Thedensity d˜ for theγcurve isexpressed inξ,θ coordinatesas: ∫ [0,1] K (√ (t−θ)2‖b−a‖2+ξ2 ) dt andisanevenfunction inξ. Thesameis true forK′(‖γ(t)−x‖). Finally, themapping: ξ → −ξ√ (t−θ)2‖b−a‖2+ξ2 is odd for afixed θ, so that thewhole integrand is oddas a functionof ξ. By theFubini theorem, integratingfirst inξwill thereforeyieldavanishing integral,provingtheassertion. The result still holds inRq, the only different aspect being that x is now expanded as x = a+θv+∑ q−1 i=1 ξiui withui, i = 1,. . . ,q−1 anorthonormal basis of the orthogonal complement of Rv inRq. Rewritingγ(t)−x=(t−θ)v−∑q−1i=1 ξiui and‖γ(t)−x‖= √ (t−θ)2‖b−a‖2+∑q−1i=1 ξ2i , thesameparityargumentcanbeappliedonanyof thecomponentsξi, i=1,. . . ,q−1, showingthat the integral isvanishing. Theeffectofcurvestraighteningispresentwhenminimizingtheentropyofawholecurvesystem, but is counterbalancedbythegatheringeffect.Dependingonthechoiceof thekernelbandwidth,one or theothereffect isdominant: straightening ispreeminent for lowvalues,beingtheonlyremaining 396