DOCSLIB.ORG

Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatial Fading Correlation for Multiple-Antenna Systems Yibeltal F

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatial Fading Correlation for Multiple-Antenna Systems Yibeltal F IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 1 Spherical Harmonic Expansion of Fisher-Bingham Distribution and 3D Spatial Fading Correlation for Multiple-Antenna Systems Yibeltal F. Alem, Student Member, IEEE, Zubair Khalid, Member, IEEE and Rodney A. Kennedy, Fellow, IEEE Abstract—This paper considers the 3D spatial fading correla- closed-form expressions for evaluating the SFC function have tion (SFC) resulting from an angle-of-arrival (AoA) distribution been developed in the existing literature (e.g., [10], [16]–[25]). that can be modelled by a mixture of Fisher-Bingham distribu- The elliptic (directional) nature of the Fisher-Bingham tions on the sphere. By deriving a closed-form expression for the spherical harmonic transform for the component Fisher-Bingham distribution offers flexibility in modelling the distribution of distributions, with arbitrary parameter values, we obtain a normalized power or AoA of the multipath components for closed-form expression of the 3D-SFC for the mixture case. The most practical scenarios. Exploiting this fact, a 3D spatial 3D-SFC expression is general and can be used in arbitrary multi- correlation model has been developed in [9], where the dis- antenna array geometries and is demonstrated for the cases of tribution of AoA of the multipath components is modelled by a 2D uniform circular array in the horizontal plane and a 3D regular dodecahedral array. In computational aspects, we use a positive linear sum of Fisher-Bingham distributions, each recursions to compute the spherical harmonic coefficients and with different parameters. Such modelling has been shown give pragmatic guidelines on the truncation size in the series to be useful in a sense that it fits well with the multi-input representations to yield machine precision accuracy results. The multi-output (MIMO) field data and allows the evaluation results are further corroborated through numerical experiments of the SFC as a function of the angular spread, ovalness to demonstrate that the closed-form expressions yield the same results as significantly more computationally expensive numerical parameter, azimuth, elevation, and prior contribution of each integration methods. cluster. Although the SFC function presented in [9] is general in a sense that it is valid for any arbitrary antenna array Index Terms—Fisher-Bingham distribution, spherical har- monic expansion, spatial correlation, MIMO, angle of ar- geometry, it has not been expressed in closed-form and has rival (AoA). been only evaluated using numerical integration techniques, which can be computationally intensive to produce sufficiently accurate results. If the spherical harmonic expansion of the I. INTRODUCTION Fisher-Bingham distribution is given in a closed-form, the The Fisher-Bingham distribution, also known as the Kent SFC function can be computed analytically using the spherical distribution, belongs to the family of spherical distributions harmonic expansion of the distribution of AoA of the multipath in directional statistics [1]. It has been used in applications components [21]. To the best of our knowledge, the spherical in a wide range of disciplines for modelling and analysing harmonic expansion of Fisher-Bingham distribution has not directional data. These applications include sound source lo- been derived in the existing literature. calization [2], [3], joint set identification [4], modelling protein In the current work, we present the spherical harmonic structures [5], 3D beamforming [6], classification of remote expansion of Fisher-Bingham distribution and a closed-form sensing data [7], modelling the distribution of AoA in wireless expression that enables the analytic computation of the spher- communication [8], [9], to name a few. ical harmonic coefficients. We also address the computational arXiv:1501.04395v1 [cs.IT] 19 Jan 2015 In this work, we focus our attention to the use of Fisher- considerations required to be taken into account in the evalua- Bingham distribution for modelling the distribution of angle tion of the proposed closed-form. Using the proposed spherical of arrival (AoA) (also referred as the distribution of scatterers) harmonic expansion of the Fisher-Bingham distribution, we in wireless communication and computation of spatial fading also formulate the SFC function experienced between two correlation (SFC) experienced between elements of multiple- arbitrary points in 3D-space for the case when the distribution antenna array systems. Modelling the distribution of scatterers of AoA of the multipath components is modelled by a mixture and characterising the spatial correlation of fading channels (positive linear sum) of Fisher-Bingham distributions. The is a key factor in evaluating the performance of wireless SFC presented here is general in a sense that it is expressed as communication systems with multiple antenna elements [10]– a function of arbitrary points in 3D-space and therefore can be [15]. It has been an active area of research for the past two used to compute spatial correlation for any 2D and 3D antenna decades or so and a number of spatial correlation models and array geometries. Through numerical analysis, we also validate The authors are with the Research School of Engineering, College the correctness of the proposed spherical harmonic expansion of Engineering and Computer Science, The Australian National Univer- of Fisher-Bingham distribution and the SFC function. In this sity, Canberra, ACT 2601, Australia. Emails: fyilbeltal.alem, zubair.khalid, paper, our main objective is to employ the proposed spherical [email protected]. This work was supported by the Aus- tralian Research Council’s Discovery Projects funding scheme (Project no. harmonic expansion of the Fisher-Bingham distribution for DP150101011). computing the spatial correlation. However, we expect that IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY SUBMISSION 2 ` the proposed spherical harmonic expansion can be useful in where dm;m0 (·) denotes the Wigner-d function of degree ` and various applications where the Fisher-Bingham distribution is orders m and m0 [26]. used to model and analyse directional data (e.g., [2]–[7]). By completeness of spherical harmonics, any finite energy The remainder of the paper is structured as follows. We function f(xb) on the 2-sphere can be expanded as review the mathematical background related to signals defined 1 ` on the 2-sphere and spherical harmonics in Section II. In X X m m f(xb) = (f)` Y` (xb); (4) Section III, we define the Fisher-Bingham distribution and `=0 m=−` present its application in modelling 3D spatial correlation. where (f)m is the spherical harmonic coefficient given by We present the spherical harmonic expansion of the Fisher- ` Z Bingham distribution and analytical formula for computing m m (f)` = f(xb)Y` (xb) ds(xb): (5) the spherical harmonic coefficients in Section IV, where we 2 S also address computational issues. We derive a closed-form The signal f is said to be band-limited in the spectral domain expression for the 3D SFC between two arbitrary points in 3D- m at degree L if (f)` = 0 for ` > L. For a real-valued function space when the AoA of an incident signal follows the Fisher- f(x), we also note the relation Bingham probability density function (pdf) in Section V. b −m m m In Section VI, we carry out a numerical validation of the (f)` = (−1) (f)` ; (6) proposed results and provide examples of the SFC for uniform which stems from the conjugate symmetry property of spher- circular array (2D) and regular dodecahedron array (3D) ical harmonics [26]. antenna elements. Finally, the concluding remarks are made in Section VII. C. Rotation on the Sphere II. MATHEMATICAL PRELIMINARIES The rotation group SO(3) is characterized by Euler angles A. Signals on the 2-Sphere ('; #; !) 2 SO(3), where ' 2 [0; 2π), # 2 [0; π] and ! 2 [0; 2π). We define a rotation operator on the sphere We consider complex valued square integrable functions 2 D('; #; !) that rotates a function on the sphere, according defined on the 2-sphere, S . The set of such functions forms to 0zyz0 convention, in the sequence of ! rotation around the L2( 2) a Hilbert space, denoted by S , that is is equipped with z-axis, # rotation around the y-axis and ' rotation around f g the inner product defined for two functions and defined z-axis. A rotation of a function f(θ; φ) on sphere is given by on S2 as [26] −1 Z (D('; #; !)f)(xb) , f(R xb); (7) hf; gi , f(x)g(x) ds(x); (1) 2 b b b S where R is a 3 × 3 real orthogonal unitary matrix, referred which induces a norm kfk hf; fi1=2. Here, (·) as rotation matrix, that corresponds to the rotation operator , D('; #; !) and is given by denotes the complex conjugate operation and xb , [sin θ cos θ; sin θ sin φ, cos θ]T 2 2 ⊂ 3 represents a point S R R = R (')R (#)R (!); on the 2-sphere, where [·]T represents the vector transpose, z y z (8) θ 2 [0; π] and φ 2 [0; 2π) denote the co-latitude and longitude where 2cos ' − sin ' 03 respectively and ds(xb) = sin θ dθ dφ is the surface measure on the 2-sphere. The functions with finite energy (induced norm) Rz(') = 4sin ' cos ' 05 ; are referred as signals on the sphere. 0 0 1 2 cos # 0 sin #3 B. Spherical Harmonics Ry(#) = 4 0 1 0 5 : Spherical harmonics serve as orthonormal basis functions − sin # 0 cos # for the representation of functions on the sphere and are Here R (') and R (#) characterize individual rotations by defined for integer degree ` ≥ 0 and integer order jmj ≤ ` as z y ' along z-axis and # along y-axis, respectively. m m m m imφ Y` (xb) ≡ Y` (θ; φ) , N` P` (cos θ)e ; III. PROBLEM FORMULATION with s 2` + 1 (` − m)! A. Fisher-Bingham Distribution on Sphere N m ; (2) ` , 4π (` + m)! Definition 1 (Fisher-Bingham Five-Parameter (FB5) Distribu- tion).
Load more

Recommended publications
  • Contributions to Directional Statistics Based Clustering Methods
    University of California Santa Barbara Contributions to Directional Statistics Based Clustering Methods A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Statistics & Applied Probability by Brian Dru Wainwright Committee in charge: Professor S. Rao Jammalamadaka, Chair Professor Alexander Petersen Professor John Hsu September 2019 The Dissertation of Brian Dru Wainwright is approved. Professor Alexander Petersen Professor John Hsu Professor S. Rao Jammalamadaka, Committee Chair June 2019 Contributions to Directional Statistics Based Clustering Methods Copyright © 2019 by Brian Dru Wainwright iii Dedicated to my loving wife, Carmen Rhodes, without whom none of this would have been possible, and to my sons, Max, Gus, and Judah, who have yet to know a time when their dad didn’t need to work from early till late. And finally, to my mother, Judith Moyer, without your tireless love and support from the beginning, I quite literally wouldn’t be here today. iv Acknowledgements I would like to offer my humble and grateful acknowledgement to all of the wonderful col- laborators I have had the fortune to work with during my graduate education. Much of the impetus for the ideas presented in this dissertation were derived from our work together. In particular, I would like to thank Professor György Terdik, University of Debrecen, Faculty of Informatics, Department of Information Technology. I would also like to thank Professor Saumyadipta Pyne, School of Public Health, University of Pittsburgh, and Mr. Hasnat Ali, L.V. Prasad Eye Institute, Hyderabad, India. I would like to extend a special thank you to Dr Alexander Petersen, who has held the dual role of serving on my Doctoral supervisory commit- tee as well as wearing the hat of collaborator.
    [Show full text]
  • A Guide on Probability Distributions
    powered project A guide on probability distributions R-forge distributions Core Team University Year 2008-2009 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic ditribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 132 13 Misc 134 Conclusion 135 Bibliography 135 A Mathematical tools 138 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗).
    [Show full text]
  • Recent Advances in Directional Statistics
    Recent advances in directional statistics Arthur Pewsey1;3 and Eduardo García-Portugués2 Abstract Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio- temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed. Keywords: Classification; Clustering; Dimension reduction; Distributional
    [Show full text]
  • Multivariate Statistical Functions in R
    Multivariate statistical functions in R Michail T. Tsagris [email protected] College of engineering and technology, American university of the middle east, Egaila, Kuwait Version 6.1 Athens, Nottingham and Abu Halifa (Kuwait) 31 October 2014 Contents 1 Mean vectors 1 1.1 Hotelling’s one-sample T2 test ............................. 1 1.2 Hotelling’s two-sample T2 test ............................ 2 1.3 Two two-sample tests without assuming equality of the covariance matrices . 4 1.4 MANOVA without assuming equality of the covariance matrices . 6 2 Covariance matrices 9 2.1 One sample covariance test .............................. 9 2.2 Multi-sample covariance matrices .......................... 10 2.2.1 Log-likelihood ratio test ............................ 10 2.2.2 Box’s M test ................................... 11 3 Regression, correlation and discriminant analysis 13 3.1 Correlation ........................................ 13 3.1.1 Correlation coefficient confidence intervals and hypothesis testing us- ing Fisher’s transformation .......................... 13 3.1.2 Non-parametric bootstrap hypothesis testing for a zero correlation co- efficient ..................................... 14 3.1.3 Hypothesis testing for two correlation coefficients . 15 3.2 Regression ........................................ 15 3.2.1 Classical multivariate regression ....................... 15 3.2.2 k-NN regression ................................ 17 3.2.3 Kernel regression ................................ 20 3.2.4 Choosing the bandwidth in kernel regression in a very simple way . 23 3.2.5 Principal components regression ....................... 24 3.2.6 Choosing the number of components in principal component regression 26 3.2.7 The spatial median and spatial median regression . 27 3.2.8 Multivariate ridge regression ......................... 29 3.3 Discriminant analysis .................................. 31 3.3.1 Fisher’s linear discriminant function ....................
    [Show full text]
  • Handbook on Probability Distributions
    R powered R-forge project Handbook on probability distributions R-forge distributions Core Team University Year 2009-2010 LATEXpowered Mac OS' TeXShop edited Contents Introduction 4 I Discrete distributions 6 1 Classic discrete distribution 7 2 Not so-common discrete distribution 27 II Continuous distributions 34 3 Finite support distribution 35 4 The Gaussian family 47 5 Exponential distribution and its extensions 56 6 Chi-squared's ditribution and related extensions 75 7 Student and related distributions 84 8 Pareto family 88 9 Logistic distribution and related extensions 108 10 Extrem Value Theory distributions 111 3 4 CONTENTS III Multivariate and generalized distributions 116 11 Generalization of common distributions 117 12 Multivariate distributions 133 13 Misc 135 Conclusion 137 Bibliography 137 A Mathematical tools 141 Introduction This guide is intended to provide a quite exhaustive (at least as I can) view on probability distri- butions. It is constructed in chapters of distribution family with a section for each distribution. Each section focuses on the tryptic: definition - estimation - application. Ultimate bibles for probability distributions are Wimmer & Altmann (1999) which lists 750 univariate discrete distributions and Johnson et al. (1994) which details continuous distributions. In the appendix, we recall the basics of probability distributions as well as \common" mathe- matical functions, cf. section A.2. And for all distribution, we use the following notations • X a random variable following a given distribution, • x a realization of this random variable, • f the density function (if it exists), • F the (cumulative) distribution function, • P (X = k) the mass probability function in k, • M the moment generating function (if it exists), • G the probability generating function (if it exists), • φ the characteristic function (if it exists), Finally all graphics are done the open source statistical software R and its numerous packages available on the Comprehensive R Archive Network (CRAN∗).
    [Show full text]
  • A New Unified Approach for the Simulation of a Wide Class of Directional Distributions
    This is a repository copy of A New Unified Approach for the Simulation of a Wide Class of Directional Distributions. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/123206/ Version: Accepted Version Article: Kent, JT orcid.org/0000-0002-1861-8349, Ganeiber, AM and Mardia, KV orcid.org/0000-0003-0090-6235 (2018) A New Unified Approach for the Simulation of a Wide Class of Directional Distributions. Journal of Computational and Graphical Statistics, 27 (2). pp. 291-301. ISSN 1061-8600 https://doi.org/10.1080/10618600.2017.1390468 © 2018 American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America. This is an author produced version of a paper published in Journal of Computational and Graphical Statistics. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A new unified approach for the simulation of a wide class of directional distributions John T.
    [Show full text]
  • A Logistic Normal Mixture Model for Compositions with Essential Zeros
    A Logistic Normal Mixture Model for Compositions with Essential Zeros Item Type text; Electronic Dissertation Authors Bear, John Stanley Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 28/09/2021 15:07:32 Link to Item http://hdl.handle.net/10150/621758 A Logistic Normal Mixture Model for Compositions with Essential Zeros by John S. Bear A Dissertation Submitted to the Faculty of the Graduate Interdisciplinary Program in Statistics In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy In the Graduate College The University of Arizona 2 0 1 6 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the disserta- tion prepared by John Bear, titled A Logistic Normal Mixture Model for Compositions with Essential Zeros and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Date: August 18, 2016 Dean Billheimer Date: August 18, 2016 Joe Watkins Date: August 18, 2016 Bonnie LaFleur Date: August 18, 2016 Keisuke Hirano Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement.
    [Show full text]
  • High-Precision Computation of Uniform Asymptotic Expansions for Special Functions Guillermo Navas-Palencia
    UNIVERSITAT POLITÈCNICA DE CATALUNYA Department of Computer Science High-precision Computation of Uniform Asymptotic Expansions for Special Functions Guillermo Navas-Palencia Supervisor: Argimiro Arratia Quesada A dissertation submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in Computing May 7, 2019 i Abstract In this dissertation, we investigate new methods to obtain uniform asymptotic ex- pansions for the numerical evaluation of special functions to high-precision. We shall first present the theoretical and computational fundamental aspects required for the development and ultimately implementation of such methods. Applying some of these methods, we obtain efficient new convergent and uniform expan- sions for numerically evaluating the confluent hypergeometric functions 1F1(a; b; z) and U(a, b, z), and the Lerch transcendent F(z, s, a) at high-precision. In addition, we also investigate a new scheme of computation for the generalized exponential integral En(z), obtaining one of the fastest and most robust implementations in double-precision arithmetic. In this work, we aim to combine new developments in asymptotic analysis with fast and effective open-source implementations. These implementations are com- parable and often faster than current open-source and commercial state-of-the-art software for the evaluation of special functions. ii Acknowledgements First, I would like to express my gratitude to my supervisor Argimiro Arratia for his support, encourage and guidance throughout this work, and for letting me choose my research path with full freedom. I also thank his assistance with admin- istrative matters, especially in periods abroad. I am grateful to Javier Segura and Amparo Gil from Universidad de Cantabria for inviting me for a research stay and for their inspirational work in special func- tions, and the Ministerio de Economía, Industria y Competitividad for the financial support, project APCOM (TIN2014-57226-P), during the stay.
    [Show full text]
  • Package 'Directional'
    Package ‘Directional’ May 26, 2021 Type Package Title A Collection of R Functions for Directional Data Analysis Version 5.0 URL Date 2021-05-26 Author Michail Tsagris, Giorgos Athineou, Anamul Sajib, Eli Amson, Micah J. Waldstein Maintainer Michail Tsagris <[email protected]> Description A collection of functions for directional data (including massive data, with millions of ob- servations) analysis. Hypothesis testing, discriminant and regression analysis, MLE of distribu- tions and more are included. The standard textbook for such data is the ``Directional Statis- tics'' by Mardia, K. V. and Jupp, P. E. (2000). Other references include a) Phillip J. Paine, Si- mon P. Preston Michail Tsagris and Andrew T. A. Wood (2018). An elliptically symmetric angu- lar Gaussian distribution. Statistics and Computing 28(3): 689-697. <doi:10.1007/s11222-017- 9756-4>. b) Tsagris M. and Alenazi A. (2019). Comparison of discriminant analysis meth- ods on the sphere. Communications in Statistics: Case Studies, Data Analysis and Applica- tions 5(4):467--491. <doi:10.1080/23737484.2019.1684854>. c) P. J. Paine, S. P. Pre- ston, M. Tsagris and Andrew T. A. Wood (2020). Spherical regression models with general co- variates and anisotropic errors. Statistics and Computing 30(1): 153--165. <doi:10.1007/s11222- 019-09872-2>. License GPL-2 Imports bigstatsr, parallel, doParallel, foreach, RANN, Rfast, Rfast2, rgl RoxygenNote 6.1.1 NeedsCompilation no Repository CRAN Date/Publication 2021-05-26 15:20:02 UTC R topics documented: Directional-package . .4 A test for testing the equality of the concentration parameters for ciruclar data . .5 1 2 R topics documented: Angular central Gaussian random values simulation .
    [Show full text]
  • Spherical-Homoscedastic Distributions: the Equivalency of Spherical and Normal Distributions in Classification
    Journal of Machine Learning Research 8 (2007) 1583-1623 Submitted 4/06; Revised 1/07; Published 7/07 Spherical-Homoscedastic Distributions: The Equivalency of Spherical and Normal Distributions in Classification Onur C. Hamsici [email protected] Aleix M. Martinez [email protected] Department of Electrical and Computer Engineering The Ohio State University Columbus, OH 43210, USA Editor: Greg Ridgeway Abstract Many feature representations, as in genomics, describe directional data where all feature vectors share a common norm. In other cases, as in computer vision, a norm or variance normalization step, where all feature vectors are normalized to a common length, is generally used. These repre- sentations and pre-processing step map the original data from Rp to the surface of a hypersphere p 1 S − . Such representations should then be modeled using spherical distributions. However, the difficulty associated with such spherical representations has prompted researchers to model their spherical data using Gaussian distributions instead—as if the data were represented in Rp rather p 1 than S − . This opens the question to whether the classification results calculated with the Gaus- sian approximation are the same as those obtained when using the original spherical distributions. In this paper, we show that in some particular cases (which we named spherical-homoscedastic) the answer to this question is positive. In the more general case however, the answer is negative. For this reason, we further investigate the additional error added by the Gaussian modeling. We conclude that the more the data deviates from spherical-homoscedastic, the less advisable it is to employ the Gaussian approximation.
    [Show full text]
  • Clustering on the Unit Hypersphere Using Von Mises-Fisher Distributions
    JournalofMachineLearningResearch6(2005)1345–1382 Submitted 9/04; Revised 4/05; Published 9/05 Clustering on the Unit Hypersphere using von Mises-Fisher Distributions Arindam Banerjee [email protected] Department of Electrical and Computer Engineering University of Texas at Austin Austin, TX 78712, USA Inderjit S. Dhillon [email protected] Department of Computer Sciences University of Texas at Austin Austin, TX 78712, USA Joydeep Ghosh [email protected] Department of Electrical and Computer Engineering University of Texas at Austin Austin, TX 78712, USA Suvrit Sra [email protected] Department of Computer Sciences University of Texas at Austin Austin, TX 78712, USA Editor: Greg Ridgeway Abstract Several large scale data mining applications, such as text categorization and gene expression anal- ysis, involve high-dimensional data that is also inherently directional in nature. Often such data is L2 normalized so that it lies on the surface of a unit hypersphere. Popular models such as (mixtures of) multi-variate Gaussians are inadequate for characterizing such data. This paper proposes a gen- erative mixture-model approach to clustering directional data based on the von Mises-Fisher (vMF) distribution, which arises naturally for data distributed on the unit hypersphere. In particular, we derive and analyze two variants of the Expectation Maximization (EM) framework for estimating the mean and concentration parameters of this mixture. Numerical estimation of the concentra- tion parameters is non-trivial in high dimensions since it involves functional inversion of ratios of Bessel functions. We also formulate two clustering algorithms corresponding to the variants of EM that we derive. Our approach provides a theoretical basis for the use of cosine similarity that has been widely employed by the information retrieval community, and obtains the spherical kmeans algorithm (kmeans with cosine similarity) as a special case of both variants.
    [Show full text]
  • A Toolkit for Inference and Learning in Dynamic Bayesian Networks Martin Paluszewski*, Thomas Hamelryck
    Paluszewski and Hamelryck BMC Bioinformatics 2010, 11:126 http://www.biomedcentral.com/1471-2105/11/126 SOFTWARE Open Access Mocapy++ - A toolkit for inference and learning in dynamic Bayesian networks Martin Paluszewski*, Thomas Hamelryck Abstract Background: Mocapy++ is a toolkit for parameter learning and inference in dynamic Bayesian networks (DBNs). It supports a wide range of DBN architectures and probability distributions, including distributions from directional statistics (the statistics of angles, directions and orientations). Results: The program package is freely available under the GNU General Public Licence (GPL) from SourceForge http://sourceforge.net/projects/mocapy. The package contains the source for building the Mocapy++ library, several usage examples and the user manual. Conclusions: Mocapy++ is especially suitable for constructing probabilistic models of biomolecular structure, due to its support for directional statistics. In particular, it supports the Kent distribution on the sphere and the bivariate von Mises distribution on the torus. These distributions have proven useful to formulate probabilistic models of protein and RNA structure in atomic detail. Background implementation of Mocapy (T. Hamelryck, University of A Bayesian network (BN) represents a set of variables Copenhagen,2004,unpublished).Today,Mocapyhas and their joint probability distribution using a directed been re-implemented in C++ but the name is kept for acyclic graph [1,2]. A dynamic Bayesian network (DBN) historical reasons. Mocapy supports a large range of is a BN that represents sequences, such as time-series architectures and probability distributions, and has pro- from speech data or biological sequences [3]. One of the ven its value in several published applications [9-13].
    [Show full text]