The Kalman–Lévy filter

Abstract

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.

On the Linearity of Bayesian Interpolators for Non-Gaussian Continuous-Time AR(1) Processes

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Statistical signal extraction using stable processes

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“…The finite length filter discussed above can be extended to the case of multivariate filter and predictor with appropriate modifications of Sornette and Ide (2001). State space representation of model (1) can be used to extract the signal and noise defined in (5) by applying the multivariate filter and predictor.…”

Discrete scale invariance and its logarithmic extension

“…Let us stress that various dynamical mechanisms generate log-periodicity, without relying on a pre-existing discrete hierarchical structure. Thus, DSI may be produced dynamically (see in particular the recent nonlinear dynamical model introduced in [4]) and does not need to be pre-determined by, e.g., a geometrical network. This is because there are many ways to break a symmetry, the subtlety here being to break it only partially.…”