Approximate Gaussian conjugacy: parametric recursive filtering under nonlinearity, multimodality, uncertainty, and constraint, and beyond

Abstract

Since the landmark work of R. E. Kalman in the 1960s, considerable efforts have been devoted to time series state space models for a large variety of dynamic estimation problems. In particular, parametric filters that seek analytical estimates based on a closed-form Markov-Bayes recursion, e.g., recursion from a Gaussian or Gaussian mixture (GM) prior to a Gaussian/GM posterior (termed 'Gaussian conjugacy' in this paper), form the backbone for a general time series filter design. Due to challenges arising from nonlinearity, multimodality (including target maneuver), intractable uncertainties (such as unknown inputs and/or non-Gaussian noises) and constraints (including circular quantities), etc., new theories, algorithms, and technologies have been developed continuously to maintain such a conjugacy, or to approximate it as close as possible. They had contributed in large part to the prospective developments of time series parametric filters in the last six decades. In this paper, we review the state of the art in distinctive categories and highlight some insights that may otherwise be easily overlooked. In particular, specific attention is paid to nonlinear systems with an informative observation, multimodal systems including Gaussian mixture posterior and maneuvers, and intractable unknown inputs and constraints, to fill some gaps in existing reviews and surveys. In addition, we provide some new thoughts on alternatives to the first-order Markov transition model and on filter evaluation with regard to computing complexity.

Joint Smoothing and Tracking Based on Continuous-Time Target Trajectory Function Fitting

“…The prevailing, considered the standard, "model-driven" estimation solution is to apply a hidden Markov model (HMM) to link the target state over time, for online recursive computing. The best known methodology in this category is the sequential Bayesian inference, for which a filter consisting of prediction and correction steps is applied iteratively [12]. The HMM can be written in either discrete-time (mainly for convenience) or continuous-time (which is the nature of reality), as given by difference equation (2a) and differential equation (2b), respectively,…”

“…All target models with parameters are no more than statistical simplification to the truth and inevitably suffer from approximation errors and disturbances. Challenges involved in system modeling/identification have been noted in several aspects, e.g., the model must meet practical constraints [12]- [15] and match the sensor revisit rate [12] while noises need to be properly identified [16]- [18]. In particular, the noise u k /u t represents the uncertainty of the state process model, which has to be modeled with respect to the occasionally irregular revisit rate of the sensor (including missed detection, delayed or out of sequence measurements).…”

Detection and Tracking of Moving Pedestrians with a Small Unmanned Aerial Vehicle

“…IMM was researched with an unscented Kalman filter (UKF) in [19]. The effect of the multi-modal approach on high maneuvering was emphasized in [20].…”

Measurement Bias Estimation in the Problem of Target Tracking

“…In contrast, the bias is treated as a random variable with limited fluctuation in different filtering frameworks [12][13][14]. In our previous work [15], the TOA measurement is employed such that a nonlinear state-space model is used to formulate the tracking problem [16]. The measurement bias is assumed to be strictly stationary Gaussian distributed and incorporated into two different state-space models.…”