FLUX: Progressive State Estimation Based on Zakai-type Distributed Ordinary Differential Equations

We propose a homotopy continuation method called FLUX for approximating complicated probability density functions. It is based on progressive processing for smoothly morphing a given density into the desired one. Distributed ordinary differential equations (DODEs) with an artificial time γ∈[0,1] are derived for describing the evolution from the initial density to the desired final density. For a finite-dimensional parametrization, the DODEs are converted to a system of ordinary differential equations (SODEs), which are solved for γ∈[0,1] and return the desired result for γ=1. This includes parametric representations such as Gaussians or Gaussian mixtures and nonparametric setups such as sample sets. In the latter case, we obtain a particle flow between the two densities along the artificial time. FLUX is applied to state estimation in stochastic nonlinear dynamic systems by gradual inclusion of measurement information. The proposed approximation method (1) is fast, (2) can be applied to arbitrary nonlinear systems and is not limited to additive noise, (3) allows for target densities that are only known at certain points, (4) does not require optimization, (5) does not require the solution of partial differential equations, and (6) works with standard procedures for solving SODEs. This manuscript is limited to the one-dimensional case and a fixed number of parameters during the progression. Future extensions will include consideration of higher dimensions and on the fly adaption of the number of parameters

Closed-Form Information-Theoretic Roughness Measures for Mixture Densities

We calculate the smoothest mixture density under a variety of prescribed specifications. This includes constraints on certain moments, specifications on density values and/or its derivatives, and prescribed probability masses in certain regions. As a roughness measure, we use Fisher Information (FI) in the space of mixtures $\cal M$. For mixtures, FI cannot be calculated in closed form. We define the space $\cal R$ of root mixtures (RMs) living on the Hilbert sphere. A transformation of FI to $\cal R$ admits a closed-form solution and yields the desired result in $\cal M$. This naturally leads to a tandem processing with two density representations maintained simultaneously in $\cal R$ and $\cal M$. FI is calculated in RM space $\cal R$ while the constraints are evaluated in mixture space $\cal M$

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