Sequential Bayesian inference for static parameters in dynamic state space models

A method for sequential Bayesian inference of the static parameters of a dynamic state space model is proposed. The method is based on the observation that many dynamic state space models have a relatively small number of static parameters (or hyper-parameters), so that in principle the posterior can be computed and stored on a discrete grid of practical size which can be tracked dynamically. Further to this, this approach is able to use any existing methodology which computes the filtering and prediction distributions of the state process. Kalman filter and its extensions to non-linear/non-Gaussian situations have been used in this paper. This is illustrated using several applications: linear Gaussian model, Binomial model, stochastic volatility model and the extremely non-linear univariate non-stationary growth model. Performance has been compared to both existing on-line method and off-line methods

Power System Dynamic State Estimation: Motivations, Definitions, Methodologies, and Future Work

This paper summarizes the technical activities of the Task Force on Power System Dynamic State and Parameter Estimation. This Task Force was established by the IEEE Working Group on State Estimation Algorithms to investigate the added benefits of dynamic state and parameter estimation for the enhancement of the reliability, security, and resilience of electric power systems. The motivations and engineering values of dynamic state estimation (DSE) are discussed in detail. Then, a set of potential applications that will rely on DSE is presented and discussed. Furthermore, a unified framework is proposed to clarify the important concepts related to DSE, forecasting-aided state estimation, tracking state estimation, and static state estimation. An overview of the current progress in DSE and dynamic parameter estimation is provided. The paper also provides future research needs and directions for the power engineering community

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

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

Kalman Filter Applications for Traffic Management

An on­line calibration approach for dynamic traffic assignment systems has been developed. The approach is general and flexible and makes no assumptions on the type of the DTA system, the models or the data that it can handle. Therefore, it is applicable to a wide variety of tools including simulation­based and analytical, as well as microscopic and macroscopic models. The objective of the on­line calibration approach is to introduce a systematic procedure that will use the available data to steer the model parameters to values closer to the realized ones. The output of the on­line calibration is therefore a set of parameter values that --when used as input for traffic estimation and prediction-- minimizes the discrepancy between the simulated (estimated and predicted) and the observed traffic conditions. The scope of the on­line calibration is neither to duplicate nor to substitute for the off­line calibration process. Instead, the two processes are complementary and synergistic in nature. The on­line calibration problem is formulated as a state­space model. State­space models have been extensively studied and efficient algorithms have been developed, such as the Kalman Filter for linear models. Because of the non­linear nature of the on­line calibration formulation, modified Kalman Filter methodologies have been presented. The most straightforward extension is the Extended Kalman Filter (EKF), in which optimal quantities are approximated via first order Taylor series expansion (linearization) of the appropriate equations. The Limiting EKF is a variation of the EKF that eliminates the need to perform the most computationally intensive steps of the algorithm on­line. The use of the Limiting EKF provides dramatic improvements in terms of computational performance. The Unscented Kalman Filter (UKF) is an alternative filter that uses a deterministic sampling approach. The computational complexity of the UKF is of the same order as that of the EKF. Empirical results suggest that joint on­line calibration of demand and supply parameters can improve estimation and prediction accuracy of a DTA system. While the results obtained from this real network application are promising, they should be validated in further empirical studies. In particular, the scalability of the approach to larger, more complex networks needs to be investigated. The results also suggest that --in this application-- the EKF has more desirable properties than the UKF (which may be expected to have superior performance over the EKF), while the UKF seems to perform better in terms of speeds than in terms of counts. Other researchers have also encountered situations where the UKF does not outperform the EKF, e.g. LaViola, J. J., Jr. (2003) and van Rhijn et al. (2005). The Limiting EKF provides accuracy comparable to that of the best algorithm (EKF), while providing order(s) of magnitude improvement in computational performance. Furthermore, the LimEKF algorithm is that it requires a single function evaluation irrespective of the dimension of the state vector (while the computational complexity of the EKF and UKF algorithms increases proportionally with the state dimension). This property makes this an attractive algorithm for large­scale applications

Implicit Maximum a Posteriori Filtering via Adaptive Optimization

Bayesian filtering approximates the true underlying behavior of a time-varying system by inverting an explicit generative model to convert noisy measurements into state estimates. This process typically requires either storage, inversion, and multiplication of large matrices or Monte Carlo estimation, neither of which are practical in high-dimensional state spaces such as the weight spaces of artificial neural networks. Here, we frame the standard Bayesian filtering problem as optimization over a time-varying objective. Instead of maintaining matrices for the filtering equations or simulating particles, we specify an optimizer that defines the Bayesian filter implicitly. In the linear-Gaussian setting, we show that every Kalman filter has an equivalent formulation using K steps of gradient descent. In the nonlinear setting, our experiments demonstrate that our framework results in filters that are effective, robust, and scalable to high-dimensional systems, comparing well against the standard toolbox of Bayesian filtering solutions. We suggest that it is easier to fine-tune an optimizer than it is to specify the correct filtering equations, making our framework an attractive option for high-dimensional filtering problems.Comment: Under review at ICLR 202