## Von Mises-like probability density functions on surfaces

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## Résumé

Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more

angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a

quotient of the hyperbolic plane.

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*Florence Nicol, Stephane Puechmorel*

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Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more

angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a

quotient of the hyperbolic plane.

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Von Mises-like probability density functions on surfaces Florence Nicol1 and Stéphane Puechmorel1 Université Fédérale de Toulouse, Laboratoire ENAC, 7 Avenue Edouard Belin, F-31055 TOULOUSE florence.nicol@enac.fr, stephane.puechmorel@enac.fr Abstract. Directional densities were introduced in the pioneering work of von Mises, with the definition of a rotationally invariant probability distribution on the circle. It was further generalized to more complex objects like the torus or the hyperbolic space. The purpose of the present work is to give a construction of equivalent objects on surfaces with genus larger than or equal to 2, for which an hyperbolic structure exists. Although the directional densities on the torus were introduced by several authors and are closely related to the original von Mises distribution, allowing more than one hole is challenging as one cannot simply add more angular coordinates. The approach taken here is to use a wrapping as in the case of the circular wrapped Gaussian density, but with a summation taken over all the elements of the group that realizes the surface as a quotient of the hyperbolic plane. Keywords: directional densities, hyperbolic geometry, von Mises prob- ability distributions 1 Introduction Estimating probability densities by the means of kernels is a basic procedure in non-parametric statistics. For finite dimensional vector spaces, the choice of the kernel bandwidth is the critical point, while the kernel itself is not as important. When dealing with manifolds, it is no longer the case, since the kernel must be well defined as a density on the manifold itself. A classical case arises when data of interest belong to a unit sphere, that yields the von Mises-Fischer distribution. It relies on the embedding of the unit sphere Sd−1 in Rd to build a kernel that depends on the inner product hx, yi of the radial unit vectors associated to a couple of points (x, y) of Sd−1 . It is obviously invariant by rotation, as applying an isometry will not change the inner product. Since hx, yi is also cos θ, with θ the angle between x and y, it can be seen as a function of the geodetic dis- tance d(x, y) on the sphere. Finally, it has the maximum entropy property, which makes it similar to the normal distribution (in fact, the normal distribution is a limiting case of the von Mises-Fischer, the other being the uniform distribution). Spherical distributions have numerous applications in statistics, as many data can be interpreted as directions in Rd . Attempts was made to generalize them to the d-dimensional torus [6] to obtain multivariate von Mises distributions. Here, the approach taken is somewhat different, since the primary goal was to build a density that is only invariant coordinate-wise. The existence of an angular coordinate system on the d-dimensional torus is the basis of the construction: rotation invariance in each coordinate is gained just by using angle differences in the overall distribution. An interpretation using the geodesic distance on the embedded surface is no longer possible. While not strictly compliant with the geometer’s view of a torus density, the multivariate von Mises-Fischer distri- bution has still a geometrical interpretation. If we restrict our attention to the two-dimensional case, T2 admits a flat structure, with universal covering space R2 and fundamental domain a rectangle. It can be obtain as the quotient of R2 by a group G generated by two translations, respectively parallel to the x and y axis. The most natural density for such an object will be a wrapped R2 heat kernel, namely a sum of the form p(x, y, t) = P g∈G k(x, gy, t) with k the heat kernel on R2 and x the point at which the density is centered. It turns out that, due to the commutativity of the two translations and the particular shape of the R2 heat kernel, it boils down to a product of wrapped normal densities. Recall- ing that the one-dimensional von Mises distribution is an approximation of the wrapped normal and is rotation invariant, one can think of the multivariate von Mises density as a product of two densities invariant by the respective actions of the two translation cyclic groups. The purpose of the present work is to introduce a class of probability dis- tributions on orientable surfaces of genus larger than 1 and endowed with an hyperbolic structure, that may be used as kernels for non-parametric density estimates or to generate random data on such surfaces. By analogy with the multivariate von Mises distribution on the d-dimensional torus, the proposed density will approximate the wrapped heat kernel on the surface in the limit of time parameter going to 0. Furthermore, invariance of the density with re- spect to the action of a primitive element, similar to rotation invariance, will be enforced. The overall procedure will closely mimic the construction of the multivariate von Mises distribution, starting with a representation of the surface as a quotient of the hyperbolic space H2 by a group G of hyperbolic isometries. The heat kernel is obtained readily as a wrapped sum of heat kernels on H2 over the elements of G. Using the property that the centralizer of an hyperbolic element is an infinite cyclic group with generator a primitive element, the wrapped kernel can be written in such a way that an equivalent to a wrapped one dimensional kernel appears. 2 Directional densities : a brief survey Directional densities are roughly speaking probability distributions depending on angular parameters. One of the most commonly used is the von Mises-Fischer on the unit sphere Sd−1 of Rd , that depend on two parameters µ ∈ Sd−1 and κ > 0, respectively called the mean and concentration. Its value at a point x ∈ Sd−1 is given by: p(x; µ, κ) = κd/2−1 (2π) d/2 Id/2−1(κ) exp (κ hµ, xi) , (1) where Ik stands for the modified Bessel function of order k and x, µ are given as unit vectors in Rd . It enjoys many properties, like infinite divisibility [4] and maximal entropy [5]. It has been generalized to other Riemannian manifolds like the d-dimensional torus Td , on which it becomes the multivariate von Mises distribution [6]: p(θ; µ, κ, Λ) ∝ exp hκ, c(θ, µ)i + 1 2 s(θ, µ)T Λ s(θ, µ) , (2) where θ, µ are d-dimensional vectors of angles, κ is a d-dimensional vector of positive real numbers and Λ is a d×d symmetric, positive definite matrix describ- ing the covariance between the angular parameters. The terms c(θ, µ), s(θ, µ) occurring in the expression are given by: c(θ, µ)T = (cos(θ1 − µ1), . . . , cos(θd − µd)) , (3) s(θ, µ)T = (sin(θ1 − µ1), . . . , sin(θd − µd)) . (4) Another generalization is made in [1] and, with a different approach in [3], to the hyperbolic d-dimensional space Hd . Following the later, the starting point is the hyperbolic Brownian motion defined as a diffusion on Hd with infinitesimal generator: x2 d 2 d X i=1 ∂2 ∂x2 i ! − (d − 2)xd 2 ∂ ∂xd , (5) where all the coordinates are given in the half-space model of Hd : Hd = {x1, . . . , xd : xi ∈ R, i = 1, . . . , d − 1, xd ∈ R+ }. In the sequel, only the case d = 2 will be considered, as the primary object of interest are surfaces. It is convenient to represent the half-space model of H2 in C, with z = x1 +ix2. The 2-dimensional hyperboloid embedded in R3 associated with H2 is given by: {(x1, x2, x3): x2 1 + x2 2 − x2 3 = −1}. (6) It admits hyperbolic coordinates: x1 = sinh(r) cos(θ), x2 = sinh(r) sin(θ), x3 = cosh(r) (7) that transforms to the unit disk model as: u = sinh(r) cos(θ) 1 + cosh(r) , v = sinh(r) sin(θ) 1 + cosh(r) (8) where θ and r are the angular and radius coordinates. Finally, using a complex representation z = u+iv and the Möbius mapping z → i(1−z)/(1+z), it comes the expression of the half-plane coordinates: x = sinh(r) sin(θ) cosh(r) + sinh(r) cos(θ) , y = 1 cosh(r) + sinh(r) cos(θ) . (9) The hyperbolic Von Mises distribution is then defined, for a given r > 0, as the density of the first exit on the circle of center i and radius r of the hyperbolic Brownian motion starting at i. Its expression is given in [3] as: pvm(r, θ) = 1 2πP0 −ν(cosh(r)) (cosh(r) + sinh(r) cos(θ)) −ν (10) where P0 −ν is the Legendre function of the first kind with parameters 0, −ν, that acts as a normalizing constant to get a true probability density. The parameter ν is similar to the concentration used in the classical Von Mises distribution. 3 Closed geodesics and wrapping A classical circular density is the wrapped (centered) Gaussian distribution: pwg(θ; σ) = 1 √ 2πσ X k∈Z exp(−(θ + 2kπ)/(2σ2 )) (11) It is clearly a periodic distribution with period 2π, with σ acting as an in- verse concentration parameter. Von Mises densities can approximate the circular wrapped Gaussian density quite well when the concentration is large enough. It worth notice that the wrapped Gaussian can be seen as a circular heat kernel K(θ, t) = pwg(θ, σ2 /2), with the angular parameter θ being interpreted as a dis- tance on the unit circle. The starting point for defining an equivalent of the von Mises distributions on surfaces is the wrapping formula given above. First of all, only compact orientable surfaces of genus g larger that 1 will be considered, as the case of the sphere or the torus is already covered. It is a classical result in hyperbolic geometry that such a surface M can be endowed with an hyperbolic structure that is obtained as the quotient of the hyperbolic plane by a group G of hyperbolic isometries. Any non-trivial element of G is conjugate to an isometry of hyperbolic type and in turn to a scaling acting as: z 7→ q2 z, q2 6= 0, 1. It is quite interesting to note that this is exactly the case considered in [3] for the definition of the Brownian motion with drift in H2 , the drift component being an hyperbolic isometry. The first possible definition of a directional density on an orientable surface of genus g > 1 will be to use kH2 (x, y, t), the heat kernel on H2 , as an analogous of the Gaussian heat kernel, and to consider its wrapping over all possible elements in G. The expression of kH2 is given by [8]: KH2 (x, y, t) = √ 2e− t 4 (4πt)3/2 Z ∞ d(x,y) se− s2 4t p cosh s − cosh d(x, y) ds (12) with d(x, y) the hyperbolic distance between x, y. Please note that KH2 (x, y, t) can be written using the hyperbolic distance only as: KH2 (x, y, t) = kH2 (d(x, y), t) The group G admits an hyperbolic polygon with 4g sides as fundamental region in H2 . For any g ∈ G, its length is defined to be l(g) = infx d(x, gx), or using the conjugacy class of g: l(g) = infx d(x, kgk−1 ) where k runs over G. Elements of G with non zero length are conjugate to hyperbolic elements in SL(2, R) (elliptic and parabolic ones are associated to rotations and translation in H2 so that the length can be made arbitrary small), and are thus conjugate to a scaling x 7→ λ2 x. Furthermore, a conjugacy class represents a free homotopy class of closed curves, that contains a unique minimal geodesic whose length is l(g), where g is a representative element. M can be identified with the quotient H2 /G so that one can define a wrapped heat kernel on M by the formula: KM : (x, y, t) ∈ M2 × R+ 7→ X g∈G KH2 (x, gy, t) (13) KM is clearly invariant by the left action of G and is symmetric since: KM (x, y, t) = X g∈G kH2 (d(x, gy), t) (14) = X g∈G kH2 (d(g−1 x, y), t) = X g∈G kH2 (d(y, g−1 x), t) (15) = KM (y, x, t) (16) Finally, primitive elements in G (i.e. those p ∈ G that cannot be written as a non trivial power of another element) play a central role in the sum defining KM . For p a primitive element, let Gp denote its centralizer in G. The conjugacy classes in G are all of the form gpn g−1 , g ∈ G/Gp with p primitive and n ∈ Z. The wrapped kernel can then be rewritten as: KM (x, y, t) = X p X g∈G/Gp X n∈Z KH2 (gx, pn gy, t) (17) where p runs through the primitive elements of G. It indicates that the kernel KM can be understood as a sum of elementary wrapped kernels associated to primitive elements, namely those k̃p defined by: k̃p(x, y, t) = X n∈Z KH2 (x, pn y, t) (18) with p primitive. Finally, p being hyperbolic, it is conjugate to a scaling, so it is enough to consider kernels of the form: k̃p(x, y, t) = X n∈Z KH2 (x, (λ2 )n y, t) (19) with λ > 1 a real number. To each primitive element p, a simple closed mini- mal geodesic loop is associated, which projects onto the axis of the hyperbolic transformation p. In the Poincaré half-plane model, such a loop unwraps onto the segment of the imaginary axis that lies between i and iλ2 . It is easily seen that the action of the elements pn , n ∈ Z will give rise to a tiling of the positive imaginary axis with segments of the form [λ2n , λ2(n+1) [. This representation al- lows a simple interpretation of the elementary wrapped kernels k̃p, where the wrapping is understood as a winding. 4 von Mises like distributions The wrapped kernels are most natural from the viewpoint of surfaces as quotient spaces, since the group action appears directly within the definition. However, it quite difficult to use them for the purpose of density estimation: even with truncated expansions, it requires quite a huge amount of computation. In the case of circular data, the usual von Mises distribution behaves much like the wrapped Gaussian, but does not involves a summation. The same is true for the generalized multivariate von Mises (2). As mentioned in the introduction, it is invariant under the action of the two generating translations. Using the same principle, a distribution invariant under the action of a primitive element will be used in place of the wrapped sum defining the elementary kernels (19). Since any primitive element is conjugate to a scaling λ2 , it is natural to seek after a distribution on a simple hyperbolic surface that is obtained from the quotient of the hyperbolic half-plane by the cyclic group ξλ generated by λ2 . A fundamental domain for its action is the subset of C defined as: {z = x + iy, x ≥ 0, 1 ≤ y < λ2 } In the quotient, the upper line y = λ2 will be identified with the lower line y = 1, yielding an hyperbolic cylinder. It is convenient to use an exponential coordinate system in order to allow for a simple invariant expression. For a given z = x + iy in the hyperbolic half plane, let x = uev , y = ev . The hyperbolic distance between two elements z1 = (u1, v1), z2 = (u2, v2) is given by: cosh d(z1, z2) = 1 + |z1 − z2|2 2=z1=z2 = 1 + (u1e v1−v2 2 − u2e v2−v1 2 ) + 2 sinh2 v1 − v2 2 = (u1e v1−v2 2 − u2e v2−v1 2 ) + cosh(v1 − v2) If the second term in the sum is considered only, it remains |v1 − v2|, which is similar to the angle difference in the case of the multivariate von Mises density. This fact can be made explicit when considering points (y1, y2) located on the y axis only. In such a case, the hyperbolic distance between them is easily seen to be l = | log(y2/y1)|. Letting L = 2 log(λ), we have the following result: Theorem 1. The wrapped kernel k̃p(y1, y2, t) = k̃p(l, t) is periodic of period L, with Fourier coefficient of order p given by: ap(t) = r π 2 1 L e−t/8 Z ∞ 0 v−1/2 Kip2π L (v)θv(t)dv where K is the modified bessel function and θv is: θv(t) = v √ 2π3t Z ∞ 0 e(π2 −b2 )/2t e−v cosh b sinh(b) sin(πb/t)db Proof. (Sketch) The starting point is the hyperbolic heat kernel representation given in [7]: k(r, t) = e−t/8 √ 2π Z ∞ 0 v−1/2 exp(−v cosh r)θv(t)dv By wrapping it, it appears the periodic function: X n∈Z e−v cosh(l+nL) whose fourier coefficients can be expressed using the modified bessel function Kip2π L (v). The technical part is the use of asymptotics of K to legitimate the summation. The fourier series expansion of the wrapped kernel has quickly decreasing coef- ficients, so that it is well approximated by the first term. Please note that the wrapped distance may be interpreted as an angular distance on the circle after scaling by 2π/l, and due to the previous remark, it gives rise to a standard one dimensional (circular) von Mises kernel depending on the wrapped hyperbolic distance between the two points. Unlike the flat torus case, the hyperbolic translations induced by primitive elements will not commute, so that summation is much more intricate. The way the computation must be performed is currently under study to obtain if possible the most tractable expression. However, it can be organized in such a way to make appear the multivariate von Mises distribution (2) on the angular parameters associated to primitive elements, the correlation matrix Λ in the expression being related to the commutation relations between the corresponding primitive elements. 5 Conclusion and future work The extension of directional statistics to surfaces of genus larger than 1, en- dowed with an hyperbolic structure, can be performed using a special kind of angular parameters. Writing the surface as the quotient of the hyperbolic space by a group G of hyperbolic isometry, one can construct an invariant kernel by summing over the translates of an element. Using the primitive elements of G, the summation can be split in such a way that the innermost sum can be under- stood as a wrapped kernel on an hyperbolic cylinder. This allows to replace it by a standard directional kernel depending on an angular parameter. Proceeding further in the sum, the summation over the elements of the conjugacy classes will yield a multivariate directional density, with correlated components. The practical computation of such a density is still difficult due to the fact that all the possible combinations between primitive elements must be consid- ered. An approximation can be made by neglecting those words in G involving more that a given number of terms: this makes senses as the kernels considered must decay very fast. An important part of the future developments will be ded- icated to the computational aspect as it is one of the key points for being able to use the densities on real data. The second aspect that needs to be addressed is the shape of the kernel itself. Due to the fact that important parameters may be reduced to angles, the initial approach was to use the already available multivariate von Mises. Since it is known that the choice of the kernel is of secondary importance in classical settings, the same may be expected here. However, if one wants to get some extra properties, like maximum entropy, an especially tailored distribution will be needed. The possibility of defining it using the Abel transform [2]of kernels on R is currently investigated. References 1. O. Barndorff-Nielsen. Hyperbolic distributions and distributions on hyperbolae. Scandinavian Journal of Statistics, 5(3):151–157, 1978. 2. R.J. Beerends. An introduction to the abel transform. In Miniconference on Har- monic Analysis, pages 21–33, Canberra AUS, 1987. Centre for Mathematical Anal- ysis, The Australian National University. 3. Jean-Claude Gruet. A note on hyperbolic von mises distributions. Bernoulli, 6(6):1007–1020, 2000. 4. J. T. Kent. The infinite divisibility of the von mises-fisher distribution for all values of the parameter in all dimensions. Proceedings of the London Mathematical Society, s3-35(2):359–384, 1977. 5. K. V. Mardia. Statistics of directional data. Journal of the Royal Statistical Society. Series B (Methodological), 37(3):349–393, 1975. 6. K V. Mardia, Gareth Hughes, Charles C. Taylor, and Harshinder Singh. A multivari- ate von mises distribution with applications to bioinformatics. Canadian Journal of Statistics, 36(1):99–109, 2008. 7. Hiroyuki Matsumoto. Closed form formulae for the heat kernels and the green functions for the laplacians on the symmetric spaces of rank one. Bulletin des Sciences Mathématiques, 125(6):553 – 581, 2001. 8. H. P. McKean. An upper bound to the spectrum of δ on a manifold of negative curvature. J. Differential Geom., 4(3):359–366, 1970.