Error Analysis for the Linear Feedback Particle Filter

Taghvaei, Amirhossein; Mehta, Prashant G.

doi:10.23919/acc.2018.8430867

Cited by 6 publications

(6 citation statements)

References 31 publications

(66 reference statements)

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“…For detailed analysis of P-EnKF algorithm, see Tugaut 2018, Bishop and, and extension to nonlinear setting [Del Moral et al 2017, de Wiljes et al 2018]. Analysis of S-EnKF and D-EnKF appears in [Taghvaei and Mehta 2018b] and [Taghvaei and Mehta 2018a] respectively.…”

Section: Particle System and Error Analysismentioning

confidence: 99%

Optimality vs Stability Trade-off in Ensemble Kalman Filters

Taghvaei¹,

Mehta²,

Georgiou³

2020

Preprint

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This paper is concerned with optimality and stability analysis of a family of ensemble Kalman filter (EnKF) algorithms. EnKF is commonly used as an alternative to the Kalman filter for highdimensional problems, where storing the covariance matrix is computationally expensive. The algorithm consists of an ensemble of interacting particles driven by a feedback control law. The control law is designed such that, in the linear Gaussian setting and asymptotic limit of infinitely many particles, the mean and covariance of the particles follow the exact mean and covariance of the Kalman filter. The problem of finding a control law that is exact does not have a unique solution, reminiscent of the problem of finding a transport map between two distributions. A unique control law can be identified by introducing control cost functions, that are motivated by the optimal transportation problem or Schrödinger bridge problem. The objective of this paper is to study the relationship between optimality and long-term stability of a family of exact control laws. Remarkably, the control law that is optimal in the optimal transportation sense leads to an EnKF algorithm that is not stable.

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Section: Particle System and Error Analysismentioning

confidence: 99%

Optimality vs Stability Trade-off in Ensemble Kalman Filters

Taghvaei¹,

Mehta²,

Georgiou³

2020

Preprint

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“…Let G be a measurable vector function from history of state-space R t×n (t is time and n is dimension of state) to R s , where s < ∞. For any 1 ≤ k ≤ T, we define g * k (x 1:k ) as (10). Then,…”

Section: Lemmamentioning

confidence: 99%

“…where For the first part of the error decomposition, [8] uses the result of [9] to show that the distribution of the difference between generic PF estimate and the conditional mean is asymptotically normal in scalar cases as the number of particles gets large. Recently, [10] conducted error analysis specifically on the linear feedback PF, which, as a special variant of PF, includes a feedback control for particles. However, whether a similar result holds for the multivariate case remains unclear and the theoretical foundation for the second part of the error term also remain to be explored.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis for the Particle Filter: Methods and Theoretical Support

Liu¹,

Wei²,

Spall³

2019

Preprint

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The particle filter is a popular Bayesian filtering algorithm for use in cases where the state-space model is nonlinear and/or the random terms (initial state or noises) are non-Gaussian distributed. We study the behavior of the error in the particle filter algorithm as the number of particles gets large. After a decomposition of the error into two terms, we show that the difference between the estimator and the conditional mean is asymptotically normal when the resampling is done at every step in the filtering process. Two nonlinear/non-Gaussian examples are tested to verify this conclusion.

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“…Analysis of the deterministic linear FPF is easier because the update formuala is identical to the Kalman filter update formula. For the linear Gaussian setting, it is shown that (i) the empirical distribution converges to the mean-field limit for any finite time; (ii) and even for a finite number of particles, the long term error converges to zero [25]. The convergence and long term stability results are shown for the nonlinear setting as well, where it is assumed that drift function is Lipschitz and the system is fully observed with small measurement noise [5].…”

Section: Introductionmentioning

confidence: 96%

“…(i) EnKBF with perturbed observation [ In our previous conference publication [25], we presented the analysis of the deterministic linear FPF. The objective of this paper is to extend the analysis to the stochastic linear FPF.…”

Section: Introductionmentioning

confidence: 99%

Error Analysis of the Stochastic Linear Feedback Particle Filter

Taghvaei

Mehta

2018

2018 IEEE Conference on Decision and Control (CDC)

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This paper is concerned with the convergence and long-term stability analysis of the feedback particle filter (FPF) algorithm. The FPF is an interacting system of N particles where the interaction is designed such that the empirical distribution of the particles approximates the posterior distribution. It is known that in the mean-field limit (N = ∞), the distribution of the particles is equal to the posterior distribution. However little is known about the convergence to the mean-field limit. In this paper, we consider the FPF algorithm for the linear Gaussian setting. In this setting, the algorithm is similar to the ensemble Kalman-Bucy filter algorithm. Although these algorithms have been numerically evaluated and widely used in applications, their convergence and long-term stability analysis remains an active area of research. In this paper, we show that, (i) the mean-field limit is well-defined with a unique strong solution; (ii) the mean-field process is stable with respect to the initial condition; (iii) we provide conditions such that the finite-N system is long term stable and we obtain some mean-squared error estimates that are uniform in time.

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Error Analysis for the Linear Feedback Particle Filter

Cited by 6 publications

References 31 publications

Optimality vs Stability Trade-off in Ensemble Kalman Filters

Optimality vs Stability Trade-off in Ensemble Kalman Filters

Error Analysis for the Particle Filter: Methods and Theoretical Support

Error Analysis of the Stochastic Linear Feedback Particle Filter

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