# American Institute of Mathematical Sciences

June  2017, 14(3): 673-694. doi: 10.3934/mbe.2017038

## Moments of von mises and fisher distributions and applications

 1 University of Alberta, Centre for Mathematical Biology, Edmonton, Alberta, T6G2G1, Canada 2 Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK 3 Cross Cancer Institute, 11560-University Ave NW, Edmonton, Alberta, T6G 1Z2, Canada

* Corresponding author: Thomas Hillen

Received  May 09, 2016 Accepted  November 22, 2016 Published  December 2016

The von Mises and Fisher distributions are spherical analogues to theNormal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth.

Citation: Thomas Hillen, Kevin J. Painter, Amanda C. Swan, Albert D. Murtha. Moments of von mises and fisher distributions and applications. Mathematical Biosciences & Engineering, 2017, 14 (3) : 673-694. doi: 10.3934/mbe.2017038
##### References:
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Stuart-Smith, Distribution of caribou and wolves in relation to linear corridors, The Journal of Wildlife Management, 64 (2000), 154-159. Google Scholar [19] A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. Swanson, M. Pelegrini-Issac, R. Guillevin and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Mang. Res. Med., 54 (2005), 616-624. Google Scholar [20] J. Kent, The Fisher-Bingham Distribution on the Sphere, J. Royal. Stat. Soc., 1982.Google Scholar [21] E. Konukoglu, O. Clatz, P. Bondiau, H. Delignette and N. Ayache, Extrapolation glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Medical Image Analysis, 14 (2010), 111-125. Google Scholar [22] K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, New York, 2000. Google Scholar [23] H. McKenzie, E. Merrill, R. Spiteri and M. Lewis, How linear features alter predator movement and the functional response, Royal Society Interface, 2 (2012), 205-216. Google Scholar [24] P. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, USA, 2006. Google Scholar [25] J. A. Mortimer and A. Carr, Reproduction and migrations of the Ascension Island green turtle (chelonia mydas), Copeia, 1987 (1987), 103-113. Google Scholar [26] P. Mosayebi, D. Cobzas, A. Murtha and M. Jagersand, Tumor invasion margin on the Riemannian space of brain fibers, Medical Image Analysis, 16 (2011), 361-373. Google Scholar [27] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. Google Scholar [28] H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298. doi: 10.1007/BF00277392. Google Scholar [29] K. Painter, Modelling migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543. doi: 10.1007/s00285-008-0217-8. Google Scholar [30] K. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of diffusion tensor imaging DTI data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014. Google Scholar [31] K. Painter and T. Hillen, Navigating the flow: Individual and continuum models for homing in flowing environments, Royal Society Interface, 12 (2015), 20150,647. Google Scholar [32] K. J. Painter, Multiscale models for movement in oriented environments and their application to hilltopping in butterflies, Theor. Ecol., 7 (2014), 53-75. Google Scholar [33] J. Rao, Molecular mechanisms of glioma invasiveness: The role of proteases, Nature Reviews Cancer, 3 (2003), 489-501. Google Scholar [34] A. Swan, T. Hillen, J. Bowman and A. Murtha, A patient-specific anisotropic diffusion model for brain tumor spread, Bull. Math. Biol., to appear (2017).Google Scholar [35] K. Swanson, E. A. Jr and J. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33 (2000), 317-329. Google Scholar [36] K. Swanson, R. Rostomily and E. Alvord Jr, Predicting survival of patients with glioblastoma by combining a mathematical model and pre-operative MR imaging characteristics: A proof of principle, British Journal of Cancer, 98 (2008), 113-119. Google Scholar [37] P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, 1998.Google Scholar [38] C. Xue and H. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Journal for Applied Mathematics, 70 (2009), 133-167. doi: 10.1137/070711505. Google Scholar

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##### References:
 [1] wikipedia. com, https://en.wikipedia.org/wiki/VonMises%E2%80%93Fisher_distribution, last accessed on 11/22/2016.Google Scholar [2] A. Banerjee, I. S. Dhillon, J. Ghosh and S. Sra, Clustering on the unit hypersphere using von Mises-Fisher distributions, Journal of Machine Learning Research, 6 (2005), 1345-1382. Google Scholar [3] E. Batschelet, Circular Statistics in Biology, Academic Press, London, 1981. Google Scholar [4] C. Beaulieu, The basis of anisotropic water diffusion in the nervous system -a technical review, NMR Biomed., 15 (2002), 435-455. Google Scholar [5] R. Bleck, An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates, Ocean Mod., 4 (2002), 55-88. Google Scholar [6] E. A. Codling, N. Bearon and G. J. Thorn, Thorn, Diffusion about the mean drift location in a biased random walk, Ecology, 91 (2010), 3106-3113. Google Scholar [7] E. A. Codling, M. J. Plank and S. Benhamou, Random walk models in biology, J. Roy. Soc. Interface, 5 (2008), 813-834. Google Scholar [8] C. Darwin, Perception in the lower animals, Nature, 7 (1873), 360. Google Scholar [9] C. Engwer, T. Hillen, M. Knappitsch and C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015), 551-582. doi: 10.1007/s00285-014-0822-7. Google Scholar [10] N. I. Fisher, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511564345. Google Scholar [11] A. Giese, L. Kluwe, B. Laube, H. Meissner, M. E. Berens and M. Westphal, Migration of human glioma cells on myelin, Neurosurgery, 38 (1996), 755-764. Google Scholar [12] P. G. Gritsenko, O. Ilina and P. Friedl, Interstitial guidance of cancer invasion, Journal or Pathology, 226 (2012), 185-199. Google Scholar [13] H. Hatzikirou, A. Deutsch, C. Schaller, M. Simaon and K. Swanson, Mathematical modelling of glioblastoma tumour development: A review, Math. Models Meth. Appl. Sci., 15 (2005), 1779-1794. doi: 10.1142/S0218202505000960. Google Scholar [14] T. Hillen, ${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585-616. doi: 10.1007/s00285-006-0017-y. Google Scholar [15] T. Hillen, E. Leonard and H. van Roessel, Partial Differential Equations; Theory and Completely Solved Problems, Wiley, Hoboken, NJ, 2012. Google Scholar [16] T. Hillen and K. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, In: Lewis, M., Maini, P., Petrovskii, S. (Eds.), Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective, Springer, Heidelberg, 2071 (2013), 177-222. doi: 10.1007/978-3-642-35497-7_7. Google Scholar [17] T. Hillen, On the $L^2$-moment closure of transport equations: The general case, Discr.Cont.Dyn.Systems, Series B, 5 (2005), 299-318. doi: 10.3934/dcdsb.2005.5.299. Google Scholar [18] A. R. C. James and A. K. Stuart-Smith, Distribution of caribou and wolves in relation to linear corridors, The Journal of Wildlife Management, 64 (2000), 154-159. Google Scholar [19] A. Jbabdi, E. Mandonnet, H. Duffau, L. Capelle, K. Swanson, M. Pelegrini-Issac, R. Guillevin and H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Mang. Res. Med., 54 (2005), 616-624. Google Scholar [20] J. Kent, The Fisher-Bingham Distribution on the Sphere, J. Royal. Stat. Soc., 1982.Google Scholar [21] E. Konukoglu, O. Clatz, P. Bondiau, H. Delignette and N. Ayache, Extrapolation glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Medical Image Analysis, 14 (2010), 111-125. Google Scholar [22] K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, New York, 2000. Google Scholar [23] H. McKenzie, E. Merrill, R. Spiteri and M. Lewis, How linear features alter predator movement and the functional response, Royal Society Interface, 2 (2012), 205-216. Google Scholar [24] P. Moorcroft and M. A. Lewis, Mechanistic Home Range Analysis, Princeton University Press, USA, 2006. Google Scholar [25] J. A. Mortimer and A. Carr, Reproduction and migrations of the Ascension Island green turtle (chelonia mydas), Copeia, 1987 (1987), 103-113. Google Scholar [26] P. Mosayebi, D. Cobzas, A. Murtha and M. Jagersand, Tumor invasion margin on the Riemannian space of brain fibers, Medical Image Analysis, 16 (2011), 361-373. Google Scholar [27] A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6. Google Scholar [28] H. Othmer, S. Dunbar and W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988), 263-298. doi: 10.1007/BF00277392. Google Scholar [29] K. Painter, Modelling migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009), 511-543. doi: 10.1007/s00285-008-0217-8. Google Scholar [30] K. Painter and T. Hillen, Mathematical modelling of glioma growth: The use of diffusion tensor imaging DTI data to predict the anisotropic pathways of cancer invasion, J. Theor. Biol., 323 (2013), 25-39. doi: 10.1016/j.jtbi.2013.01.014. Google Scholar [31] K. Painter and T. Hillen, Navigating the flow: Individual and continuum models for homing in flowing environments, Royal Society Interface, 12 (2015), 20150,647. Google Scholar [32] K. J. Painter, Multiscale models for movement in oriented environments and their application to hilltopping in butterflies, Theor. Ecol., 7 (2014), 53-75. Google Scholar [33] J. Rao, Molecular mechanisms of glioma invasiveness: The role of proteases, Nature Reviews Cancer, 3 (2003), 489-501. Google Scholar [34] A. Swan, T. Hillen, J. Bowman and A. Murtha, A patient-specific anisotropic diffusion model for brain tumor spread, Bull. Math. Biol., to appear (2017).Google Scholar [35] K. Swanson, E. A. Jr and J. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33 (2000), 317-329. Google Scholar [36] K. Swanson, R. Rostomily and E. Alvord Jr, Predicting survival of patients with glioblastoma by combining a mathematical model and pre-operative MR imaging characteristics: A proof of principle, British Journal of Cancer, 98 (2008), 113-119. Google Scholar [37] P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, 1998.Google Scholar [38] C. Xue and H. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Journal for Applied Mathematics, 70 (2009), 133-167. doi: 10.1137/070711505. Google Scholar
Left: The unimodal von Mises distribution as a function of ${{\bf n}} \in \mathbb{S}^1$ with a peak at $(1,0)^T$. Right: The bimodal von Mises distribution as a function of ${{\bf n}} \in \mathbb{S}^1$ with peaks at ${{\bf n}}=\pm (1,0)^T$
The Fisher distribution on the unit sphere. From left to right we plot the distribution for increasing values of concentration parameter $k$. In each case, the dominant direction is taken as ${\bf u} = (1/2,1/2,\sqrt{2}/2)$
Coefficients $a(k), b(k), c(k), d(k)$ from (33) as functions of $k$
Comparison between simulations of the stochastic velocity-jump model and the fundamental solution of (6), equation (9) in two dimensions. Results shown at times $t=50$ (left) and $t=200$ (right). Red lines indicate contours of the average distribution for the particle position, generated through repeated simulations of the stochastic velocity-jump (VJ) model, while blue lines indicate contours of the fundamental solution (FS), equation (9); contours are shown here for $c({\bf x},t) = 10^{-4}$ and $10^{-2}$. For the stochastic simulations we initialize the individual at $(0,0)$ and compute the random walk path using the von Mises distribution for $q$ (with $\phi = \pi/4$ and $\kappa = 10$), $s = 10$ and $\lambda = 100$. ${\bf a}$ and $D$ for (9) are subsequently determined using (10) and (11)
Comparison between simulations of the stochastic velocity-jump model and the fundamental solution of (6) in three dimensions. All plots are generated at time $t = 50$. In (a)-(b) we plot the three dimensional contour surface at the fixed density $c({\bf x},t)=10^{-5}$ for (a) the fundamental solution given by equation (9) and (b) the particle probability distribution generated from multiple simulations (200000) of the 3D velocity-jump model. In (c-d) we illustrate the quantitative match by plotting contour lines of fixed density $c({\bf x},t)=10^{-4}$ (solid line) and $c({\bf x},t)=10^{-6}$ (dashed line) for projections onto the (c) $x-z$ plane (fixing $y=50$) and (d) $y-z$ plane (fixing $x=460$). For the simulations we initialise the individual at $(0,0,0)$ and compute the random walk path using the Fisher distribution (2) for $q$ (with ${\bf u} = (1,0,0)$) and $\kappa = 10$) along with $s = 10$ and $\lambda = 100$. ${\bf a}$ and $D$ for equation (9) are subsequently determined by equations (25) and (26)
Simulations of green sea turtle navigation towards Ascension island: we refer to [31] for full details. We consider (a) weak navigators and (b) strong navigators, showing in each case (top row) the individual-based model and (bottom row) the macroscopic model. For the IBM turtle positions (white circles) are shown at the indicated times, superimposed on ocean currents as illustrated by its direction (arrows) and magnitude (arrow length/density map), while for the macroscopic model (6) the population density is indicated by the density map, where the scale indicates the number/km$^2$ and white regions represent a density below $10^{-5}$. Simulations start on 1st of January 2014 and turtles swim with mean fixed swimming speed of 40 km/day and make turns once per hour. Ocean currents are obtained from a standard ocean forecasting model (HYCOM, [5])
Population density distributions describing the different responses of a population to linear landscape features such as seismic lines. In each case we solve (6), where ${\bf a}$ and $D$ are calculated using the moments of the bimodal von Mises distribution, i.e. equations (23-24). In (a) we impose a regular array of criss-crossing seismic lines: the population density distribution is either (a1) raised or (a2) decreased along the seismic lines according to whether they tend to (a1) follow along the paths or (a2) move directly off of them. In (b) an identical set of simulations is conducted for a genuine landscape, based on an aerial photograph of a Northern Alberta landscape
Simulations of brain tumour growth using real patient data: see [34] for details. These are test cases to show the effect of changing the concentration parameter. (a) shows the fractional anisotropy for a two-dimensional axial slice of a real patient brain. Yellow indicates high fractional anisotropy, and thus high alignment, while blue indicates isotropic tissue. The initial condition for the brain tumour simulation is indicated by a black dot. (b)-(e) show two artificial tumours generated using real patient DTI data for two different values of $\kappa$, and thus $k$. (b) and (c) use $\kappa=0.5$, and (d) and (e) use $\kappa=15$. (b) and (d) represent a two-dimensional slice of the cell density (where dark blue = low cancer cell density, yellow = high cancer cell density), while (c) and (e) represent an isosurface corresponding to $c(x,t)=0.16$. It has been estimated that this is the cell-denstity that shows up on T2-MRI images, thus the isosurfaces in (c) and (e) can be thought of as the visible tumour boundaries
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