Optimal Filtering of a Gaussian Signal in the Presence of Levy Noise

Ahn, Hyungsok; Feldman, Raisa E.

doi:10.21236/ada336874

Cited by 4 publications

(9 citation statements)

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“…In this section we look at three different methods for filtering an infinite variance observation process in a linear framework. We work in the onedimensional case using (4.17) and compare our results with those of [1] and [7]. The example we will look at will be that of infinite variance α-stable noisy obervations of a mean reverting Brownian motion, i.e.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…An exception to this is the paper [7], but that uses a very different approach to the one that we will present, and it seems that the only concrete examples that fit naturally into that framework are the α-stable Lévy processes. We also mention [1] that deals with a very similar problem to ours. We compare the approach of that paper with ours below.…”

Section: Introductionmentioning

confidence: 96%

“…In contrast to our linear filter, [1] obtain a non-linear optimal filter for the best measurable estimate. Having the system evolve with Gaussian noise is also a vital aspect of their set-up, indeed the authors describe this as a "significant limitation" in the closing paragraph of their paper.…”

Section: Introductionmentioning

confidence: 99%

“…In section 4 we describe the limiting processes, and obtain our main results. Some numerical simulations are presented in section 5, where we make practical comparisons of our filter with those of [1] and [7]. A more detailed account of the results presented here can be found in [6].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

Applebaum

Blackwood

2015

Journal of Applied Probability

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We extend the Kalman-Bucy filter to the case where both the system and observation processes are driven by finite dimensional Lévy processes, but whereas the process driving the system dynamics is square-integrable, that driving the observations is not; however it remains integrable. The main result is that the components of the observation nose that have infinite variance make no contribution to the filtering equations. The key technique used is approximation by processes having bounded jumps.

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Section: Numerical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

Applebaum

Blackwood

2015

Journal of Applied Probability

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“…The filter in [4] is nonlinear and recursive, and thus may greatly limit its application in practice. Ahn and Feldman [1] proposed to minimize the difference between the true state and the filtered observation in the L µ -norm. However, as pointed in [9], this method does not really address the Kalman filtering problem which consists of combining forecasts and observations.…”

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confidence: 99%

State estimation under non-Gaussian Lévy noise: A modified Kalman filtering method

Sun

Duan

et al. 2015

Banach Center Publ.

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The Kalman filter is extensively used for state estimation for linear systems under Gaussian noise. When non-Gaussian Lévy noise is present, the conventional Kalman filter may fail to be effective due to the fact that the non-Gaussian Lévy noise may have infinite variance.A modified Kalman filter for linear systems with non-Gaussian Lévy noise is devised. It works effectively with reasonable computational cost. Simulation results are presented to illustrate this non-Gaussian filtering method.

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The Kalman–Lévy filter

Sornette

Ide

2001

Physica D: Nonlinear Phenomena

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Physica D, in press.Acknowledgments: We acknowledge stimulating discussions with Dan Cayan and Larry Riddle at Scripps and Michael Ghil and Andrew Robertson at UCLA and thank R. Mantegna for exchanges on how to generate noise with Lévy distributions.We thank the two referees for spotting analytical errors in the submitted version and for their insightful remarks that helped improve the presentation. All errors remain ours. This work was partially supported by ONR N00014-99-1-0020 (KI). AbstractThe Kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error. The Kalman filter has two main restrictions: (i) the dynamical system is assumed linear and (ii) forecasting errors and observational noises are projected onto Gaussian distributions. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and Lévy laws. The main tool needed to solve this "Kalman-Lévy" filter is the "tail-covariance" matrix which generalizes the covariance matrix in the case where it is mathematically ill-defined (i.e. for power law tail exponents µ ≤ 2).We present the general solution and discuss its properties on pedagogical examples. The standard Kalman-Gaussian filter is recovered for the case µ = 2. The optimal Kalman-Lévy filter is found to deviate substantially from the standard Kalman-Gaussian filter as µ deviates from 2. As µ decreases, novel observations are assimilated with less and less weight as a small exponent µ implies large errors with significant probabilities. In terms of implementation, the price-to-pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the Gaussian case is transformed into a nonlinear matrix equation for the Kalman-Lévy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.

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Optimal Filtering of a Gaussian Signal in the Presence of Levy Noise

Cited by 4 publications

References 0 publications

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

The Kalman-Bucy filter for integrable Lévy processes with infinite second moment

State estimation under non-Gaussian Lévy noise: A modified Kalman filtering method

The Kalman–Lévy filter

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