Directional Statistics with the Spherical Normal Distribution Supplementary Material

Transcription

1 APPENDIX

2 Directional Statistics with the Spherical Normal Distribution Supplementary Material Søren Hauberg Department of Mathematics and Computer Science Technical University of Denmar Kgs. Lyngby, Denmar The isotropic spherical normal distribution has density SNx µ, = 1 Z 1 Log µx Log µ x, µ = 1, > 0. 1 In the main paper, we claim that the normalization constant is Z even 1 = A D D π D D/ 1 erf π + A D D/ D 1 =0 1 D / 1 [ ] a D π D i Re erf Z odd 1 = A D 1 D 3/ { [ Im erf D 3/ π =0 D i ] + Im D 1 [ erf D π D i ]} b when D is even and odd. Here erf is the imaginary error function, while Re[ ] and Im[ ] taes the real and imaginary parts of a complex number, respectively. In this section, we provide the derivation of this constant. By definition, we have Z 1 = S Log µx Log µ x dx. 3 D 1 We ress this integral in the tangent space of the sphere at µ, i.e. we perform the substitution The appropriate Jacobian is and the integral becomes Z 1 = v = Log µ x. 4 D sin v detj = 5 v v <π D v sin v dv. 6 v This project has received funding from the European Research Council ERC under the European Union s Horizon 00 research and innovation programme grant agreement n SH was supported by a research grant from VILLUM FONDEN.

3 We then write this in hyper-spherical coordinates 1 D Z 1 =... r sinr r D φ 1=0 φ =0 φ D 3 =0 φ D =0 r sinφ 1 D 3 sinφ D 4... sinφ D 3 dφ D dφ D 3... dφ dφ 1 dr. Since the radius r and the angles φ are never mixed in the integrand, we can split this into the product of two integrals Z 1 = r sin D rdr 8 sinφ 1 D 3 sinφ D 4 sinφ D 3 dφ D dφ D 3 dφ dφ 1. φ D 3 =0 φ D =0 φ 1=0 φ =0 The second term is merely the surface area of the D dimensional unit sphere A D = D 1/ Γ D 1 /, 9 where Γ is the usual Gamma function. The integral 9 then reduces to Z 1 = A D r sin D rdr. 10 To evaluate this ression we need the trigonometric power formulas [] sin n x = 1 n n + 1n n 1 n n n 1 1 cosn x, 11 =0 n sin n+1 x = 1n n n 1 sinn + 1 x. 1 =0 We are now ready evaluate the normalization constant. First we consider the case were D is even. Z even 1 = A D r sin D rdr 13 = A D D π r D dr D/ 1 + A D 1 D / 1 D/ =0 1 D r cosd rdr. To evaluate this we need two simple integrals, which we evaluate using Maple, r π π dr = erf 15 r cosa rdr = 4 a π ai π + ai erf + erf, 16 where i denotes the complex unit. Inserting these ressions into Eq. 14 and simplifying ressions gives the desired result a. Note that in the simple special case D =, the spherical normal is a distribution over the unit circle. Here, the normalization constant reduce to Z D= 1 = erf 7 14 π Using the convention of

4 We now consider the case where D is odd. Ain to the previous derivation, we insert Eq. 1 into Eq. 10 and get Z odd 1 = A D r sin D rdr 18 D 3/ = A D 1 D 3/ =0 1 D To evaluate this, we need to evaluate the integral r sina rdr = i 4 r sind rdr. 19 a ia π ia π + ia erf + erf erf, which we have evaluated using Maple. Inserting this ression into Eq. 19 and simplifying gives the result in Eq. b. In the important special-case D = 3, the normalization constant reduce to Z D=3 1 = iπ3 / 1 { } i π i π + i erf + erf erf. 1 This concludes the derivation. A. Quality of the approximation We approximate the inverse normalization constant 1 /Z 1 with a straight line in order to derive an ression for the anisotropic normalization constant. Figure 1 center show the inverse normalization constant for varying values of. The right panel of the figure show the difference between the inverse normalization and a fitted straight line. From this we draw two conclusions: 1 the inverse normalization is indeed not a straight line; a straight line is, however, a good approximation. While using one globally fitted straight line gives a fairly accurate estimate of the normalization constant, we find that accuracy can be slightly improved by fitting the line locally. We mae this local fit through 1 /Z 1 and 1 /Z 1 + α 1. By extensive numerical optimization we have found that α 1 = minimizes the worst-case approximation error of the integral. At times it may be easier to interpret a variance parameter rather than a concentration parameter. The variance of the spherical normal distribution is defined as [1] Var[x] = arccos x µ SNx µ, Λdx. S D 1 When the distribution is isotropic, this ression can be evaluated for S similarly to the proof of proposition 1 to give Var[x] = π i π 1 i π 1 / erf 5 / i π 1 i + π 1 / erf + i 1 1 /erf π / i The left panel of Fig. 1 show how the variance change as a function of. Notice that the curve is roughly shaped as 1 /.. 0 3

5 Fig. 1. Left: the variance as a function of the concentration. Center: the inverse normalization constant for the isotropic distribution. Right: The deviation between the inverse normalization constant and a single linear approximation. REFERENCES [1] X. Pennec. Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements. Journal of Mathematical Imaging and Vision JMIV, 51:17 154, 006. [] E. W. Weisstein. Trigonometric power formulas. From MathWorld A Wolfram Web Resource. mathworld.wolfram.com/ TrigonometricPowerFormulas.html.