In state estimation problems, the Kalman filter (KF) algorithm considers the noise in the measurements and the systems facilitating convergence to the true state. This paper presents the Bayesian derivation of the discrete-time KF algorithm for a simple example known as the random walk model. However, if the KF coefficients are not well-tuned, it can significantly impact the estimation accuracy and may lead to algorithmic inconsistency. The Kalman gain is a quantitative measure which plays a crucial role in achieving the optimum convergence and stability. In this study, we evaluate the importance of the Kalman gain in the KF algorithm across several choices of the error covariance within the context of the random walk model. Furthermore, we demonstrate that the optimal Kalman gain is determined by minimizing the mean squared error (MSE), producing an unbiased and efficient estimate. This adaptive adjustment enables the KF to tune parameters easily. The theoretical and numerical investigations were carried out using the random walk plus noise model.

The Role of Kalman Gain and Noise Covariance Selection on the Convergence in State Estimation

This paper reviews the growing field of Bayesian prediction. Bayes point and interval prediction are defined and exemplified and situated in statistical prediction more generally. Then, four general approaches to Bayes prediction are defined and we turn to predictor selection. This can be done predictively or non-predictively and predictors can be based on single models or multiple models. We call these latter cases unitary predictors and model average predictors, respectively. Then we turn to the most recent aspect of prediction to emerge, namely prediction in the context of large observational data sets and discuss three further classes of techniques. We conclude with a summary and statement of several current open problems.

pdf/a-cheat-sheet-for-bayesian-prediction-3r8q734a.pdf

A Cheat Sheet for Bayesian Prediction

We apply recent work on category theoretical probability to the idea of Bayesian filtering, making use of the concept of a strongly representable Markov category. We show that there is an adjunction between 'dynamical' and 'epistemic' models of a hidden Markov process. Concepts such as Bayesian filtering and conjugate priors arise as natural consequences of this adjunction. Along the way we define a notion of unifilar machine, which is a kind of stochastic Moore machine in which the output is chosen stochastically, but the update function is deterministic given the output. Unifilar machines are useful as models of the behaviour of stochastic systems; we show that in the Kleisli category of the distribution monad there is a terminal unifilar machine, and its elements are controlled stochastic processes, mapping sequences of the input alphabet probabilistically to sequences of the output alphabet. 

/pdf/unifilar-machines-and-the-adjoint-structure-of-bayesian-3qpm52g1.pdf

Unifilar Machines and the Adjoint Structure of Bayesian Models

We elucidate the mathematical structure of Bayesian filtering, and Bayesian inference more broadly, by applying recent work on category theoretical probability, specifically the concept of a strongly representable Markov category. We show that filtering, along with related concepts such as conjugate priors, arise from an adjunction: the process of taking a hidden Markov process is right adjoint to a forgetful functor. This has an interesting consequence. In practice, filtering is usually implemented using parametrised families of distributions. The Kalman filter is a particularly important example, which uses Gaussians. Rather than calculating a new posterior each time, the implementation only needs to udpate the parameters. This structure arises naturally from our adjunction; the correctness of such a model is witnessed by a map from the model into the system being modelled. Conjugate priors arise from this construction as a special case. In showing this we define a notion of unifilar machine, which has its origins in the literature on epsilon-machines. Unifilar machines are useful as models of the"observable behaviour"of stochastic systems; we show additionally that in the Kleisli category of the distribution monad there is a terminal unifilar machine, and its elements are controlled stochastic processes, mapping sequences of the input alphabet probabilistically to sequences of the output alphabet.

Unifilar Machines and the Adjoint Structure of Bayesian Filtering

Battery health assessments are essential for roadside energy storage systems that facilitate electric transportation. This paper uses the samples from the charging and discharging data of the base station and the power station under different working conditions at different working hours and at different temperatures to demonstrate the decay of the battery health of a roadside energy storage system under different cycles. In this paper, for the first time, the predicted state-of-health values are obtained by extracting the characteristic quantities affecting the battery health based on three indicators: the internal resistance, the rate of change of voltage, and the change of temperature. Data on state of health are clustered by K-Means, GMM, K-Means++ and divided into high, medium, and low levels. Using a comparison of the three methods, GMM clustering appears to be the best at reflecting the charging and discharging capacity of the battery.

Battery State-of-Health Evaluation for Roadside Energy Storage Systems in Electric Transportation

The Kalman filter is popularly known to be an optimal recursive implementation of the Bayesian prediction and correction in the sense that it minimises the estimated error covariance. The filter has been originally derived in this error minimising framework and there is extensive literature on the same. The Kalman filter has also been derived under other frameworks, like the maximum likelihood approach, etc., which all converge to the true posterior. In this paper we present a purely Bayesian filtering approach to the Kalman filter. We first build an analogy to the principles of Bayesian estimation and then present a step-by-step derivation for the Kalman filter following the Bayesian principles. From this derivation, we show that the Kalman filter gives a tractable solution to the Bayesian filtering process by computing the underlying probability densities exactly. This derivation is known to some in the research community but no formal article in the literature presents it in detail. This paper fills this gap and will be a good read for Bayesian enthusiasts. The filter is simulated in the proposed framework on a simple 4-D linear Gaussian model.

Derivation of the Kalman filter in a Bayesian filtering perspective

The particle filter is known to be a powerful tool for estimating hidden Markov processes in nonlinear and nonGaussian state space models. The filter involves generating new particles from old ones, from regions of high importance in the state space using a proposal distribution and then weighing them using the incoming observation. However a poor choice of the proposal distribution may migrate the new particles into regions that do not contribute to the posterior and hence lead to one particle accumulating all the weight (termed particle degeneracy). This degeneracy is overcome using the resampling step that eliminates those particles with low weights and replaces them by those with large weights. However this resampling step is a computationally demanding operation. In the literature, the methods that speed up the particle filter, like the Gaussian particle filter, trade tracking accuracy with speed while methods that sample particles from high importance regions, like the auxiliary particle filters and lookahead particle filters, trade speed with accuracy. In this paper we propose a simple lookahead sampling scheme. Here the particles that fall into high importance regions are predetermined (seen ahead) and then propagated in copies to make up for those that should be discarded. This strategy avoids the resampling step and consequently leads to high speed and accuracy. Using two nonlinear models, we show the tracking efficiency of the proposed method.

A sorted weighting lookahead sampling scheme for accurate and fast particle filtering

The particle filter is an important approximation method for online state estimation in nonlinear nonGaussian scenarios. The resampling step in the particle filter is critical because it eliminates the wasteful use of particles that do not contribute to the posterior (degeneracy). The fully stochastic resamplers, despite being unbiased in approximating the posterior density, involve exhaustive and sequential communication within the particles and thus are computationally expensive. The alternate partial deterministic resamplers overcome this problem by reducing the communication within particles but this leads to approximation bias. This paper proposes a fast resampling procedure that gives an accurate approximation of the posterior and tracks as accurately as the conventional resamplers.

A Fast and Unbiased Minimalistic Resampling Approach for the Particle Filter

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Derivation of the Kalman filter in a Bayesian filtering perspective

A sorted weighting lookahead sampling scheme for accurate and fast particle filtering

A Fast and Unbiased Minimalistic Resampling Approach for the Particle Filter