Efficient evaluation of the probability density function of a wrapped normal distribution

The wrapped normal distribution arises when the density of a one-dimensional normal distribution is wrapped around the circle infinitely many times. At first look, evaluation of its probability density function appears tedious as an infinite series is involved. In this paper, we investigate the evaluation of two truncated series representations. As one representation performs well for small uncertainties, whereas the other performs well for large uncertainties, we show that in all cases a small number of summands is sufficient to achieve high accuracy.

[1]  Yannis Stylianou,et al.  Wrapped Gaussian Mixture Models for Modeling and High-Rate Quantization of Phase Data of Speech , 2009, IEEE Transactions on Audio, Speech, and Language Processing.

[2]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[3]  D. Collett,et al.  Discriminating Between the Von Mises and Wrapped Normal Distributions , 1981 .

[4]  E. Breitenberger,et al.  Analogues of the normal distribution on the circle and the sphere , 1963 .

[5]  Ivan Markovic,et al.  Bearing-only tracking with a mixture of von Mises distributions , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[6]  Gerhard Kurz,et al.  Recursive nonlinear filtering for angular data based on circular distributions , 2013, 2013 American Control Conference.

[7]  M. C. Jones,et al.  Discrimination between the von Mises and wrapped normal distributions: just how big does the sample size have to be? , 2005 .

[8]  Karim El Mokhtari,et al.  A circular interacting multi-model filter applied to map matching , 2013, Proceedings of the 16th International Conference on Information Fusion.

[9]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[10]  N. I. Fisher.  Problems with the Current Definitions of the Standard Deviation of Wind Direction , 1987 .

[11]  Serge Reboul,et al.  A multi-temporal multi-sensor circular fusion filter , 2014, Inf. Fusion.

[12]  S. R. Jammalamadaka,et al.  Directional Statistics, I , 2011 .

[13]  S. R. Jammalamadaka,et al.  Topics in Circular Statistics , 2001 .

[14]  D. Owen.  Handbook of Mathematical Functions with Formulas , 1965 .

[15]  Serge Reboul,et al.  A recursive fusion filter for angular data , 2009, 2009 IEEE International Conference on Robotics and Biomimetics (ROBIO).

[16]  Gerhard Kurz,et al.  Bearings-only sensor scheduling using circular statistics , 2013, Proceedings of the 16th International Conference on Information Fusion.

[17]  Gerhard Kurz,et al.  Constrained object tracking on compact one-dimensional manifolds based on directional statistics , 2013, International Conference on Indoor Positioning and Indoor Navigation.

[18]  Paris Smaragdis,et al.  A Wrapped Kalman Filter for Azimuthal Speaker Tracking , 2013, IEEE Signal Processing Letters.

[19]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[20]  Gerhard Kurz,et al.  Nonlinear measurement update for estimation of angular systems based on circular distributions , 2014, 2014 American Control Conference.