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Directional Statistics

Directional statistics is a subfield of statistics where we consider directional quantities, for example, angles, orientations, phase, etc.

To illustrate the need for a special type of statistics in these scenarios consider the following example. If we want to compute the average of two angles, say 20° and 340°, taking the usual mean yields $\frac{20^\circ + 340^\circ}{2} = 180^\circ$. However, the correct result is 0°, the circular mean, which is the exact opposite direction!

Circular Statistics

The most fundamental domain in directional statistics is the unit circle. It is of interest when considering periodic quantities such as angles or phase.

Fig. 1: Wrapped normal distribution.

Popular probability distributions on the unit circle include the von Mises distribution $c \cdot \exp(\kappa \cos(x-\mu)) $ and the wrapped normal distribution $\sum_{k=-\infty}^\infty \mathcal{N}(x + 2 k \pi; \mu, \sigma) $. I have investigated some of their properties in [1] and [2].

Circular Estimation

I am interested in developing recursive filters for estimation of circular quantities, similar to the well-known Kalman filter as well as its nonlinear extensions such as the unscented Kalman filter (UKF).

  • Nonlinear prediction [3]
  • Nonlinear measurement update [4], [5]
  • Generalization to non-additive noise [6]
  • Fourier-based approach [7], [8]
  • Discrete approach [9]

Deterministic sampling. Fig. 2: Deterministic Sampling.

Deterministic Sampling

An important tool for nonlinear filtering is deterministic sampling. The idea is to choose a number of samples that are representative of a continuous probability density. Working with a set of samples is usually much easier than working with a continuous density. In contrast to stochastic sampling, usually a much smaller number of samples is sufficient.

I have published the following deterministic sampling approaches on the unit circle.

  • based on first trigonometric moment [3]
  • based on second trigonometric moment [10], [11]
  • based on a binary tree [11]
  • based on quantization [12]

Applications

Circular statistics has many applications in fields ranging from signal processing to robotics, from geology to medicine, and from neuroscience to computational biology.

During my work on circular statistics, I have published results about application of the developed techniques in the following circumstances.

  • Sensor scheduling for bearings-only measurements [13]
  • Constrained object tracking [14]
  • Model-predictive control on the unit circle [15]
  • Heart phase estimation [16]
  • Optical Belt Sorting [17]

Spherical Statistics

Spherical statistics considers quantities on the unit sphere rather than the unit circle. More generally, the unit hypersphere, i.e., the sphere in $n$ dimensions, can be considered.

Fig. 3: Bingham distribution.

Bingham Distribution

The Bingham distribution is given by $c \cdot \exp(\underline{x}^T \mathbf{M} \mathbf{Z} \mathbf{M}^T \underline{x})$, where $\underline{x}$ is located on the unit hypersphere, $\mathbf{M}$ is an orthogonal matrix, and $\mathbf{Z}$ is a diagonal matrix. This distribution is antipodally symmetric, that is, $x$ and $-x$ always have the same probability. Thus, it can be conveniently applied to quaternions, which can be used to represent rotations in the three-dimensional space $\mathbb{R}^3$.

The following publications contain my work on the Bingham distribution.

  • Recursive filter for the 2D case [18]
  • Recursive filter for the 4D case (for quaternions) [19]
  • Normalization constant [20]
  • Deterministic sampling [21]

Fig. 4: von Mises-Fisher distribution.

von Mises-Fisher Distribution

Another distribution on the unit sphere is the von Mises-Fisher distribution, a generalization of the von Mises distribution on the unit circle. It is given by $c \cdot \exp(\kappa \underline{\mu}^T \underline{x})$, where $\kappa > 0$ and $\underline{\mu}$ as well as $\underline{x}$ are unit vectors.

I have published these results on the von Mises-Fisher distribution.

  • Stochastic sampling [22]
  • Deterministic sampling and nonlinear filtering [23]
  • Parameter estimation [24]

Fig. 5: Bivariate wrapped normal distribution.

Spherical Harmonics

Similar to approximations based on Fourier series on the unit circle, it is possible to approximate probability densities on the unit sphere using so-called spherical harmonics. This way, arbitrary multimodal spherical distributions can be considered.

  • Recursive spherical filter based on spherical harmonics [25]

Toroidal Statistics

While estimation of a single angle is possible with circular statistics, estimating multiple angles at the same time requires toroidal statistics. The key questions is how correlations between these angles can be modeled appropriately. For this purpose, circular-circular correlation coefficients can be used.

In my work, I have investigated the bivariate wrapped normal distribution, a generalization of the wrapped normal distribution on the unit circle to higher dimensions, as well as the bivariate von Mises distribution.

  • Recursive filtering using the bivariate wrapped normal distribution [26], [27]
  • Parameter estimation for the bivariate wrapped normal distribution [28]
  • Fusion using the bivariate von Mises distribution (with matrix parameter) [29]
  • Deterministic sampling on the torus [30]
  • Fourier-based hypertoroidal filtering [31]

Fig. 6: Partially wrapped normal distribution.

Circuar-linear Statistics

Whereas toroidal statistics considers multiple periodic quantities, circular-linear statistics deals with combinations of periodic and nonperiodic quantities, for example orientation and position. Here, the key questions is how to model the correlation between circular and linear quantities. Among the most important applications of circular-linear statistics are problems on SE(2) and SE(3), the groups of rigid body motions in 2D and 3D, respectively.

  • Partially wrapped normal distribution [32]
  • SE(2) Bingham [33], [34], [35]
  • SE(2) using discretization of circle [36]

1. a Gerhard Kurz, Uwe D. Hanebeck, 2015. Trigonometric Moment Matching and Minimization of the Kullback–Leibler Divergence. IEEE Transactions on Aerospace and Electronic Systems, 51, pp.3480-3484.
2. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2014. Efficient Evaluation of the Probability Density Function of a Wrapped Normal Distribution. Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2014), Bonn, Germany.
3. a, b Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2013. Recursive Nonlinear Filtering for Angular Data Based on Circular Distributions. Proceedings of the 2013 American Control Conference (ACC 2013), Washington D. C., USA.
4. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2014. Nonlinear Measurement Update for Estimation of Angular Systems Based on Circular Distributions. Proceedings of the 2014 American Control Conference (ACC 2014), Portland, Oregon, USA.
5. a Igor Gilitschenski, Gerhard Kurz, Uwe D. Hanebeck, 2015. Non-Identity Measurement Models for Orientation Estimation Based on Directional Statistics. Proceedings of the 18th International Conference on Information Fusion (Fusion 2015), Washington D. C., USA.
6. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2016. Recursive Bayesian Filtering in Circular State Spaces. IEEE Aerospace and Electronic Systems Magazine, 31, pp.70-87.
7. a Florian Pfaff, Gerhard Kurz, Uwe D. Hanebeck, 2015. Multimodal Circular Filtering Using Fourier Series. Proceedings of the 18th International Conference on Information Fusion (Fusion 2015), Washington D. C., USA.
8. a Florian Pfaff, Gerhard Kurz, Uwe D. Hanebeck, 2016. Nonlinear Prediction for Circular Filtering Using Fourier Series. Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany.
9. a Gerhard Kurz, Florian Pfaff, Uwe D. Hanebeck, 2016. Discrete Recursive Bayesian Filtering on Intervals and the Unit Circle. Proceedings of the 2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016), Baden-Baden, Germany.
10. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2014. Deterministic Approximation of Circular Densities with Symmetric Dirac Mixtures Based on Two Circular Moments. Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain.
11. a, b Gerhard Kurz, Igor Gilitschenski, Roland Y. Siegwart, Uwe D. Hanebeck, 2016. Methods for Deterministic Approximation of Circular Densities. Journal of Advances in Information Fusion, 11, pp.138–156.
12. a Igor Gilitschenski, Gerhard Kurz, Uwe D. Hanebeck, Roland Siegwart, 2016. Optimal Quantization of Circular Distributions. Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany.
13. a Igor Gilitschenski, Gerhard Kurz, Uwe D. Hanebeck, 2013. Bearings-Only Sensor Scheduling Using Circular Statistics. Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey.
14. a Gerhard Kurz, Florian Faion, Uwe D. Hanebeck, 2013. Constrained Object Tracking on Compact One-dimensional Manifolds Based on Directional Statistics. Proceedings of the Fourth IEEE GRSS International Conference on Indoor Positioning and Indoor Navigation (IPIN 2013), Montbeliard, France.
15. a Gerhard Kurz, Maxim Dolgov, Uwe D. Hanebeck, 2015. Nonlinear Stochastic Model Predictive Control in the Circular Domain. Proceedings of the 2015 American Control Conference (ACC 2015), Chicago, Illinois, USA.
16. a Gerhard Kurz, Uwe D. Hanebeck, 2015. Heart Phase Estimation Using Directional Statistics for Robotic Beating Heart Surgery. Proceedings of the 18th International Conference on Information Fusion (Fusion 2015), Washington D. C., USA.
17. a Florian Pfaff, Gerhard Kurz, Christoph Pieper, Georg Maier, Benjamin Noack, Harald Kruggel-Emden, Robin Gruna, Uwe D. Hanebeck, Siegmar Wirtz, Viktor Scherer, Thomas Laengle, Juergen Beyerer, 2017. Improving Multitarget Tracking Using Orientation Estimates for Sorting Bulk Materials. Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2017), Daegu, Korea.
18. a Gerhard Kurz, Igor Gilitschenski, Simon J. Julier, Uwe D. Hanebeck, 2013. Recursive Estimation of Orientation Based on the Bingham Distribution. Proceedings of the 16th International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey.
19. a Gerhard Kurz, Igor Gilitschenski, Simon Julier, Uwe D. Hanebeck, 2014. Recursive Bingham Filter for Directional Estimation Involving 180 Degree Symmetry. Journal of Advances in Information Fusion, 9, pp.90–105.
20. a Igor Gilitschenski, Gerhard Kurz, Simon J. Julier, Uwe D. Hanebeck, 2014. Efficient Bingham Filtering based on Saddlepoint Approximations. Proceedings of the 2014 IEEE International Conference on Multisensor Fusion and Information Integration (MFI 2014), Beijing, China.
21. a Igor Gilitschenski, Gerhard Kurz, Simon J. Julier, Uwe D. Hanebeck, 2016. Unscented Orientation Estimation Based on the Bingham Distribution. IEEE Transactions on Automatic Control, 61, pp.172-177.
22. a Gerhard Kurz, Uwe D. Hanebeck, 2015. Stochastic Sampling of the Hyperspherical von Mises–Fisher Distribution Without Rejection Methods. Proceedings of the IEEE ISIF Workshop on Sensor Data Fusion: Trends, Solutions, Applications (SDF 2015), Bonn, Germany.
23. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2016. Unscented von Mises-Fisher Filtering. IEEE Signal Processing Letters, 23, pp.463-467.
24. a Gerhard Kurz, Florian Pfaff, Uwe D. Hanebeck, 2016. Kullback-Leibler Divergence and Moment Matching for Hyperspherical Probability Distributions. Proceedings of the 19th International Conference on Information Fusion (Fusion 2016), Heidelberg, Germany.
25. a Florian Pfaff, Gerhard Kurz, Uwe D. Hanebeck, 2017. Filtering on the Unit Sphere Using Spherical Harmonics. Proceedings of the 2017 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2017), Daegu, Korea.
26. a Gerhard Kurz, Igor Gilitschenski, Maxim Dolgov, Uwe D. Hanebeck, 2014. Bivariate Angular Estimation Under Consideration of Dependencies Using Directional Statistics. Proceedings of the 53rd IEEE Conference on Decision and Control (CDC 2014), Los Angeles, California, USA.
27. a Gerhard Kurz, Florian Pfaff, Uwe D. Hanebeck, 2017. Nonlinear Toroidal Filtering Based on Bivariate Wrapped Normal Distributions. Proceedings of the 20th International Conference on Information Fusion (Fusion 2017), Xi'an, China.
28. a Gerhard Kurz, Uwe D. Hanebeck, 2015. Parameter Estimation for the Bivariate Wrapped Normal Distribution. Proceedings of the 54th IEEE Conference on Decision and Control (CDC 2015), Osaka, Japan.
29. a Gerhard Kurz, Uwe D. Hanebeck, 2015. Toroidal Information Fusion Based on the Bivariate von Mises Distribution. Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2015), San Diego, California, USA.
30. a Gerhard Kurz, Uwe D. Hanebeck, 2017. Deterministic Sampling on the Torus for Bivariate Circular Estimation. IEEE Transactions on Aerospace and Electronic Systems, 53, pp.530–534.
31. a Florian Pfaff, Gerhard Kurz, Uwe D. Hanebeck, 2016. Multivariate Angular Filtering Using Fourier Series. Journal of Advances in Information Fusion, 11, pp.206–226.
32. a Gerhard Kurz, Igor Gilitschenski, Uwe D. Hanebeck, 2014. The Partially Wrapped Normal Distribution for SE(2) Estimation. Proceedings of the 2014 IEEE International Conference on Multisensor Fusion and Information Integration (MFI 2014), Beijing, China.
33. a Igor Gilitschenski, Gerhard Kurz, Simon J. Julier, Uwe D. Hanebeck, 2014. A New Probability Distribution for Simultaneous Representation of Uncertain Position and Orientation. Proceedings of the 17th International Conference on Information Fusion (Fusion 2014), Salamanca, Spain.
34. a Igor Gilitschenski, Gerhard Kurz, Uwe D. Hanebeck, 2015. A Stochastic Filter for Planar Rigid-Body Motions. Proceedings of the 2015 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2015), San Diego, California, USA.
35. a Kailai Li, Gerhard Kurz, Lukas Bernreiter, Uwe D. Hanebeck, 2018. Nonlinear Progressive Filtering for SE(2) Estimation. Proceedings of the 21st International Conference on Information Fusion (Fusion 2018), Cambridge, United Kingdom.
36. a Gerhard Kurz, Florian Pfaff, Uwe D. Hanebeck, March 2018. Application of Discrete Recursive Bayesian Estimation on Intervals and the Unit Circle to Filtering on SE(2). IEEE Transactions on Industrial Informatics, 14, pp.1197-1206.

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