Deterministic Sampling of Multivariate Densities based on Projected Cumulative Distributions

We want to approximate general multivariate probability density functions by deterministic sample sets. For optimal sampling, the closeness to the given continuous density has to be assessed. This is a difficult challenge in multivariate settings. Simple solutions are restricted to the one-dimensional case. In this paper, we propose to employ one-dimensional density projections. These are the Radon transforms of the densities. For every projection, we compute their cumulative distribution function. These Projected Cumulative Distributions (PCDs) are compared for all possible projections (or a discrete set thereof). This leads to a tractable distance measure in multivariate space. The proposed approximation method is efficient as calculating the distance measure mainly entails sorting in one dimension. It is also surprisingly simple to implement.Comment: 21 pages, 10 figure

Deterministic Von Mises–Fisher Sampling on the Sphere Using Fibonacci Lattices

We propose a von Mises–Fisher sampling scheme using Fibonacci lattices to generate high-quality deterministic samples on the sphere. Key idea is an orthogonal inverse transform to map uniform low-discrepancy or quasi-random samples, ideally from Fibonacci lattices, to the sphere. The proposed new sampling method can be applied in assumed density Riemannian particle filters and controllers. Compared to random sampling, it produces well-separated, locally homogeneous samples, yielding superior convergence in numerical applications. The advantage over UKF-like sampling schemes is the free choice of the number of samples

Directional statistics and filtering using libDirectional

In this paper, we present libDirectional, a MATLAB library for directional statistics and directional estimation. It supports a variety of commonly used distributions on the unit circle, such as the von Mises, wrapped normal, and wrapped Cauchy distributions. Furthermore, various distributions on higher-dimensional manifolds such as the unit hypersphere and the hypertorus are available. Based on these distributions, several recursive filtering algorithms in libDirectional allow estimation on these manifolds. The functionality is implemented in a clear, well-documented, and object-oriented structure that is both easy to use and easy to extend