Abstract-In this work we propose an approximate Minimum Mean-Square Error (MMSE) filter for linear dynamic systems with Gaussian Mixture noise. The proposed estimator tracks each component of the Gaussian Mixture (GM) posterior with an individual filter and minimizes the trace of the covariance matrix of the bank of filters, as opposed to minimizing the MSE of individual filters in the commonly used Gaussian sum filter (GSF). Hence, the spread of means in the proposed method is smaller than that of GSF which makes it more robust to removing components. Consequently, lower complexity reduction schemes can be used with the proposed filter without losing estimation accuracy and precision. This is supported through simulations on synthetic data as well as experimental data related to an indoor localization system. Additionally, we show that in two limit cases the state estimation provided by our proposed method converges to that of GSF, and we provide simulation results supporting this in other cases.

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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]

In many signal processing applications it is required to estimate the unobservable state of a dynamic system from its noisy measurements. For linear dynamic systems with Gaussian Mixture (GM) noise distributions, Gaussian Sum Filters (GSF) provide the MMSE state estimate by tracking the GM posterior. However, since the number of the clusters of the GM posterior grows exponentially over time, suitable reduction schemes need to be used to maintain the size of the bank in GSF. In this work we propose a low computational complexity reduction scheme which uses an initial state estimation to find the active noise clusters and removes all the others. Since the performance of our proposed method relies on the accuracy of the initial state estimation, we also propose five methods for finding this estimation. We provide simulation results showing that with suitable choice of the initial state estimation (based on the shape of the noise models), our proposed reduction scheme provides better state estimations both in terms of accuracy and precision when compared with other reduction methods.