Moments of von mises and fisher distributions and applications

Thomas Hillen; Kevin J. Painter; Amanda C. Swan; Albert D. Murtha; Thomas Hillen; Kevin J. Painter; Amanda C. Swan; Albert D. Murtha

doi:10.3934/mbe.2017038

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 3: 673-694. doi: 10.3934/mbe.2017038

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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

Received: 09 May 2016 Accepted: 22 November 2016 Published: 01 June 2017
MSC : Primary: 35Q92; Secondary: 62E15, 92B05

The von Mises and Fisher distributions are spherical analogues to the Normal 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.

Keywords:

Citation: Thomas Hillen, Kevin J. Painter, Amanda C. Swan, Albert D. Murtha. Moments of von mises and fisher distributions and applications[J]. Mathematical Biosciences and Engineering, 2017, 14(3): 673-694. doi: 10.3934/mbe.2017038

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Abstract

A correction on

Numerical investigation and improvement of the aerodynamic performance of a modified elliptical-bladed Savonius-style wind turbine. By Sri Kurniati, Sudirman Syam and Arifin Sanusi. AIMS Energy, 2023, Volume 11, Issue 6: 1211–1230. Doi: 10.3934/energy.2023055

The authors would like to make the following corrections to the published paper [1].

On page 1213, we updated the contents of "one symbol statement: ρ" in section 2. The updated contents are as follows:

- ρ is the the density of air,

On page 1215, we updated the contents of "Eq 16" in section 2. The updated contents are as follows:

$\frac{{{\bf{∂}} {\bf{ \pmb{\mathsf{ ϕ}} }}}_{{\bf{ℓ}}}}{{\bf{∂}} \boldsymbol{t}}+{\bf{∇}} \left({{\varnothing }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}\right)-{\bf{∇}} \left({\bf{\Gamma }}_{{\bf{ℓ}}}\overrightarrow{\boldsymbol{V}}{{\varnothing }}_{{\bf{ℓ}}}\right) = {\boldsymbol{R}}_{{\bf{ℓ}}}$

(16)

On page 1216, we updated the contents of "Table 2" in section 2. The updated contents are as follows:

Table 2. The terms in the general transfer equation Eq 16.

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{{\partial {\rm{ \mathsf{ ϕ} }}}_{\ell }}{\partial t}+\nabla \left({\varnothing }_{\ell }\overrightarrow{V}\right)-\nabla \left({\mathrm{\Gamma }}_{\ell }\overrightarrow{V}{\varnothing }_{\ell }\right)={R}_{\ell } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(16)$
$l$	${\varnothing }_{\ell }$	${\mathrm{\Gamma }}_{\ell }$
1	U	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial x}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial x}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial x}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial x}\right)+gx$
2	V	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial y}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial y}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial y}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial y}\right)+gy$
3	W	$vt$	$-\frac{1}{\rho }\frac{\partial \rho }{\partial z}+\frac{\partial }{\partial x}\left(vt\frac{\partial u}{\partial z}\right)+\frac{\partial }{\partial y}\left(vt\frac{\partial v}{\partial z}\right)+\frac{\partial }{\partial z}\left(vt\frac{\partial w}{\partial z}\right)+gz$
4	1	0	0
5	K	$vt/$	$G-\varepsilon$
6	$\varepsilon$	$vt/$	${C}_{1}\frac{\varepsilon }{k}G-{c}_{2}\frac{\varepsilon }{k}\varepsilon$

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Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	[ wikipedia. com, https://en.wikipedia.org/wiki/VonMises%E2%80%93Fisher_distribution, last accessed on 11/22/2016.
[2]	[ A. Banerjee,I. S. Dhillon,J. Ghosh,S. Sra, Clustering on the unit hypersphere using von Mises-Fisher distributions, Journal of Machine Learning Research, 6 (2005): 1345-1382.
[3]	[ E. Batschelet, null, Circular Statistics in Biology, Academic Press, London, 1981.
[4]	[ C. Beaulieu, The basis of anisotropic water diffusion in the nervous system -a technical review, NMR Biomed., 15 (2002): 435-455.
[5]	[ R. Bleck, An oceanic general circulation model framed in hybrid isopycnic-cartesian coordinates, Ocean Mod., 4 (2002): 55-88.
[6]	[ E. A. Codling,N. Bearon,G. J. Thorn, Thorn, Diffusion about the mean drift location in a biased random walk, Ecology, 91 (2010): 3106-3113.
[7]	[ E. A. Codling,M. J. Plank,S. Benhamou, Random walk models in biology, J. Roy. Soc. Interface, 5 (2008): 813-834.
[8]	[ C. Darwin, Perception in the lower animals, Nature, 7 (1873): 360.
[9]	[ C. Engwer,T. Hillen,M. Knappitsch,C. Surulescu, Glioma follow white matter tracts: A multiscale DTI-based model, J. Math. Biol., 71 (2015): 551-582.
[10]	[ N. I. Fisher, null, Statistical Analysis of Circular Data, Cambridge University Press, Cambridge, 1993.
[11]	[ A. Giese,L. Kluwe,B. Laube,H. Meissner,M. E. Berens,M. Westphal, Migration of human glioma cells on myelin, Neurosurgery, 38 (1996): 755-764.
[12]	[ P. G. Gritsenko,O. Ilina,P. Friedl, Interstitial guidance of cancer invasion, Journal or Pathology, 226 (2012): 185-199.
[13]	[ H. Hatzikirou,A. Deutsch,C. Schaller,M. Simaon,K. Swanson, Mathematical modelling of glioblastoma tumour development: A review, Math. Models Meth. Appl. Sci., 15 (2005): 1779-1794.
[14]	[ T. Hillen, ${M}^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006): 585-616.
[15]	[ T. Hillen, E. Leonard and H. van Roessel, Partial Differential Equations; Theory and Completely Solved Problems, Wiley, Hoboken, NJ, 2012.
[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.
[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.
[18]	[ A. R. C. James,A. K. Stuart-Smith, Distribution of caribou and wolves in relation to linear corridors, The Journal of Wildlife Management, 64 (2000): 154-159.
[19]	[ A. Jbabdi,E. Mandonnet,H. Duffau,L. Capelle,K. Swanson,M. Pelegrini-Issac,R. Guillevin,H. Benali, Simulation of anisotropic growth of low-grade gliomas using diffusion tensor imaging, Mang. Res. Med., 54 (2005): 616-624.
[20]	[ J. Kent, The Fisher-Bingham Distribution on the Sphere, J. Royal. Stat. Soc., 1982.
[21]	[ E. Konukoglu,O. Clatz,P. Bondiau,H. Delignette,N. Ayache, Extrapolation glioma invasion margin in brain magnetic resonance images: Suggesting new irradiation margins, Medical Image Analysis, 14 (2010): 111-125.
[22]	[ K. V. Mardia and P. E. Jupp, Directional Statistics, Wiley, New York, 2000.
[23]	[ H. McKenzie,E. Merrill,R. Spiteri,M. Lewis, How linear features alter predator movement and the functional response, Royal Society Interface, 2 (2012): 205-216.
[24]	[ P. Moorcroft,M. A. Lewis, null, Mechanistic Home Range Analysis, Princeton University Press, USA, 2006.
[25]	[ J. A. Mortimer,A. Carr, Reproduction and migrations of the Ascension Island green turtle (chelonia mydas), Copeia, 1987 (1987): 103-113.
[26]	[ P. Mosayebi,D. Cobzas,A. Murtha,M. Jagersand, Tumor invasion margin on the Riemannian space of brain fibers, Medical Image Analysis, 16 (2011): 361-373.
[27]	[ A. Okubo and S. Levin, Diffusion and Ecological Problems: Modern Perspectives, Second edition. Interdisciplinary Applied Mathematics, 14. Springer-Verlag, New York, 2001.
[28]	[ H. Othmer,S. Dunbar,W. Alt, Models of dispersal in biological systems, Journal of Mathematical Biology, 26 (1988): 263-298.
[29]	[ K. Painter, Modelling migration strategies in the extracellular matrix, J. Math. Biol., 58 (2009): 511-543.
[30]	[ K. Painter,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.
[31]	[ K. Painter,T. Hillen, Navigating the flow: Individual and continuum models for homing in flowing environments, Royal Society Interface, 12 (2015): 20150,647.
[32]	[ K. J. Painter, Multiscale models for movement in oriented environments and their application to hilltopping in butterflies, Theor. Ecol., 7 (2014): 53-75.
[33]	[ J. Rao, Molecular mechanisms of glioma invasiveness: The role of proteases, Nature Reviews Cancer, 3 (2003): 489-501.
[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).
[35]	[ K. Swanson,E. A. Jr,J. Murray, A quantitative model for differential motility of gliomas in grey and white matter, Cell Proliferation, 33 (2000): 317-329.
[36]	[ K. Swanson,R. Rostomily,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.
[37]	[ P. Turchin, Quantitative Analysis of Movement, Sinauer Assoc., Sunderland, 1998.
[38]	[ C. Xue,H. Othmer, Multiscale models of taxis-driven patterning in bacterial populations, SIAM Journal for Applied Mathematics, 70 (2009): 133-167.

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