Page - 395 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 andasecondonecomingfromthe lengthofγj in thedenominator. This last termisobtainedfromthe usualfirstordervariationformulaofacurve length:∫ [0,1] ∥∥∥γ′j(t)+ ′(t)∥∥∥dt=∫ [0,1] ∥∥∥γ′j(t)∥∥∥dt+∫ [0,1] 〈 γ′j(t) ‖γ′j(t)‖ , ′(t) 〉 dt+o(‖ ‖2). Usinganintegrationbyparts, thefirstorder termcanbewrittenas: ∫ [0,1] 〈 γ′j(t) ‖γ′j(t)‖ , ′(t) 〉 dt= − ∫ [0,1] 〈( γ′′j (t) ‖γ′j(t)‖ ) N , (t) 〉 dt (16) with: ( γ′′j (t) ‖γ′j(t)‖ ) N = γ′′j (t) ‖γ′j(t)‖ − γ′j(t) ‖γ′j(t)‖ 〈 γ′j(t) ‖γ′j(t)‖ , γ′′j (t) ‖γ′j(t)‖ 〉 thenormalcomponentof: γ′′j (t) ‖γ′j(t)‖ . Pleasenote thatwhendealingwithplanarcurves (i.e.,withvalues inR2), it isκj(t)Nj(t)withκj (resp.Nj) thecurvature (resp. theunitnormalvector)ofγj. The integral in (15) canbeexpanded inasimilar fashion. Usingasabove thenotation ()N for normalcomponents, thefirstorder termisobtainedas: ∫ [0,1] 〈( γj(t)−x ‖γj(t)−x‖ ) N , (t) 〉 K′ (‖γj(t)−x‖)‖γ′j(t)‖dt − ∫ [0,1] 〈( γ′′j (t) ‖γ′j(t)‖ ) N , (t) 〉 K (‖γj(t)−x‖)dt. (17) Fromtheexpressions in (16)and(17), thefirstordervariationof theentropyis: 1 ∑Ni=1 li (∫ [0,1] 〈∫ Ω ( γj(t)−x ‖γj(t)−x‖ ) N K′ (‖γj(t)−x‖) log(d˜(x))dx, (t) 〉 ‖γ′j(t)‖dt − ∫ [0,1] (∫ Ω K (‖γj(t)−x‖) log(d˜(x))dx) 〈( γ′′j (t) ‖γ′j(t)‖ ) N , (t) 〉 dt + (∫ Ω d˜(x) log(d˜(x))dx )∫ [0,1] 〈( γ′′j (t) ‖γ′j(t)‖ ) N , (t) 〉 dt ) . (18) 395