The Ensemble Epanechnikov Mixture Filter

TL;DR

This work makes use of the optimal multivariate Epanechnikov mixture kernel density estimate for the sequential filtering scenario through what is term the ensemble Epanechnikov mixture filter (EnEMF).

Abstract

In the high-dimensional setting, Gaussian mixture kernel density estimates become increasingly suboptimal. In this work we aim to show that it is practical to instead use the optimal multivariate Epanechnikov kernel. We make use of this optimal Epanechnikov mixture kernel density estimate for the sequential filtering scenario through what we term the ensemble Epanechnikov mixture filter (EnEMF). We provide a practical implementation of the EnEMF that is as cost efficient as the comparable ensemble Gaussian mixture filter. We show on a static example that the EnEMF is robust to growth in dimension, and also that the EnEMF has a significant reduction in error per particle on the 40-variable Lorenz '96 system.

Figures (7)

• Figure 1: Ensemble mixture model filtering diagram. Clock wise from the left-most rectangle: given a collection of particles from the previous time, (i) propagate to the current time, (ii) build a mixture model representing the prior from the particles through kernel density estimation, (iii) update the mixture model by making use of the measurements to create a mixture model representing the posterior, and (iv) resample to create a new collection of particles. This process is repeated until a desired time is reached or ad infinitum.
• Figure 2: (a) Gaussian and (b) Epanechnikov distribution for $n=2$ spatial dimensions, mean of zero and identity covariance.
• Figure 3: Efficiency of the Gaussian kernel relative to the dimension $n$. Values of the efficiency for non-integer dimensions are plotted for completeness.
• Figure 4: Comparison of Gaussian sum updates for inference on a Gaussian distribution with a non-linear measurement. Three standard deviations of the Gaussian prior are represented by the large dashed blue lines on the the left of the figures. The measurement distribution is represented by the dotted red circles The minimal mean squared error Gaussian approximation to the posterior is represented by solid green lines. The left figure represents an update using the extended Kalman filter, while the right figure represents an update using the Bayesian recursive update.
• Figure 5: A visual description of the approximated posterior Epanechnikov sampling procedure for one Epanechnikov mode. First, in the top left panel the Gaussian component update from the prior to the candidate posterior is performed using the measurement likelihood with samples taken therefrom in the top right panel. Next, in the bottom left panel the samples are projected onto the shell of the prior Epanechnikov component, and finally the projected samples are randomly scaled in the bottom left panel.

Theorems & Definitions (13)

• Lemma 2.1: Radially symmetric integral
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Corollary 2.5.1
• Theorem 4.1
• proof
• Lemma 4.2: Posterior ellipsoid