Analytic Minimum Mean-Square Error Bounds in Linear Dynamic Systems With Gaussian Mixture Noise Statistics

Abstract

Using state-space representation, mobile object positioning problems can be described as dynamic systems, with the state representing the unknown location and the observations being the information gathered from the location sensors. For linear dynamic systems with Gaussian noise, the Kalman filter provides the Minimum Mean-Square Error (MMSE) state estimation by tracking the posterior. Hence, by approximating non-Gaussian noise distributions with Gaussian Mixtures (GM), a bank of Kalman filters or Gaussian Sum Filter (GSF), can provide the MMSE state estimation. However, the MMSE itself is not analytically tractable. Moreover, the general analytic bounds proposed in the literature are not tractable for GM noise statistics. Hence, in this work, we evaluate the MMSE of linear dynamic systems with GM noise statistics and propose its analytic lower and upper bounds. We provide two analytic upper bounds which are the Mean-Square Errors (MSE) of implementable filters, and we show that based on the shape of the GM noise distributions, the tighter upper bound can be selected. We also show that for highly multimodal GM noise distributions, the bounds and the MMSE converge. Simulation results support the validity of the proposed bounds and their behavior in limits. Index Terms-Minimum Mean-Square Error estimator, analytic bounds on Minimum Mean-Square Error, Gaussian mixture noise, online Bayesian filtering, Gaussian sum filter arXiv:1506.07603v1 [cs.SY]

Why Mixture?

“…[1,2] In the state estimation realm, the need for a mixture distribution may arise from stochastically switched models [3,4,5], multi-modal data/noise [6,7,8,9] and data association uncertainty [10,11,12]. The most known mixture is the Gaussian mixture [13,7], which consists of a finite number of Gaussian distributions. Recently, it has been further shown that the arithmetic average (AA) fusion which has provided a compelling approach to multi-target density fusion/consensus over sensor networks [14,15,16,17,18,19,20,21] will also result in a mixture distribution.…”

Mobility Management in 5G and Beyond: A Novel Smart Handover With Adaptive Time-to-Trigger and Hysteresis Margin

“…• The estimation of unknown variables using the Kalman filter tends to be more accurate than that based on a single measurement [42]. • The Kalman Filter optimizes the estimation error; specifically, the mean squared error is minimized by it while considering systems with Gaussian noise [43]. • The Kalman filter has a promising ability to estimate (track) the system states (parameters) from noisy measurements [44].…”

Why Mixture?

“…Finite mixtures are flexible and powerful probabilistic modeling tools for both univariate and multivariate data, which have been well acknowledged and widely used for pattern recognition, machine learning, state estimation, etc. [1,2] In the state estimation realm, the need for a mixture distribution may arise from stochastically switched models [3,4,5], multi-modal data/noise [6,7,8,9] and data association uncertainty [10,11,12]. The most known mixture is the Gaussian mixture [13,7], which consists of a finite number of Gaussian distributions.…”

“…Second, the linear fusion of a finite number of mixtures of the same parametric family remains a mixture of the same family. Therefore, the finite mixture has been one of the most important filter structures such as the known Gaussian mixture [13,7], Student's-t mixture [24] and multi-Bernoulli mixture of various forms [25,26,27]. There are many other types of mixture models such as Watson mixture model [28] for axially symmetric data, inverted Beta mixture model [29] for non-symmetric data, von-Mises Fisher mixture model [30] for directional data (such as bearing measurements).…”