The Kalman-Levy filter

The 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 taken Gaussian. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and L\'evy laws. The main tool needed to solve this ``Kalman-L\'evy'' 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 $\mu \leq 2$). We present the general solution and discuss its properties on pedagogical examples. The standard Kalman-Gaussian filter is recovered for the case $\mu = 2$. The optimal Kalman-L\'evy filter is found to deviate substantially fro the standard Kalman-Gaussian filter as $\mu$ deviates from 2. As $\mu$ decreases, novel observations are assimilated with less and less weight as a small exponent $\mu$ 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 matrice equation for the Kalman-L\'evy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.Comment: 41 pages, 9 figures, correction of errors in the general multivariate cas

Ensemble transform Kalman-Bucy filters

Two recent works have adapted the Kalman-Bucy filter into an ensemble setting. In the first formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-BGR09, the ensemble of perturbations is updated by the solution of an ordinary differential equation (ODE) in pseudo-time, while the mean is updated as in the standard KF. In the second formulation, BR10, the full ensemble is updated in the analysis step as the solution of single set of ODEs in pseudo-time. Neither requires matrix inversions except for the frequently diagonal observation error covariance. We analyze the behavior of the ODEs involved in these formulations. We demonstrate that they stiffen for large magnitudes of the ratio of background to observational error covariance, and that using the integration scheme proposed in both BGR09 and BR10 can lead to failure. An integration scheme that is both stable and is not computationally expensive is proposed. We develop transform-based alternatives for these Bucy-type approaches so that the integrations are computed in ensemble space where the variables are weights (of dimension equal to the ensemble size) rather than model variables. Finally, the performance of our ensemble transform Kalman-Bucy implementations is evaluated using three models: the 3-variable Lorenz 1963 model, the 40-variable Lorenz 1996 model, and a medium complexity atmospheric general circulation model (AGCM) known as SPEEDY. The results from all three models are encouraging and warrant further exploration of these assimilation techniques

Brief Communication: Breeding vectors in the phase space reconstructed from time series data

Bred vectors characterize the nonlinear instability of dynamical systems and so far have been computed only for systems with known evolution equations. In this article, bred vectors are computed from a single time series data using time-delay embedding, with a new technique, nearest-neighbor breeding. Since the dynamical properties of the standard and nearest-neighbor breeding are shown to be similar, this provides a new and novel way to model and predict sudden transitions in systems represented by time series data alone

Density imaging of heterochromatin in live cells using orientation-independent-DIC microscopy

© The Author(s), 2017. This article is distributed under the terms of the Creative Commons Attribution License. The definitive version was published in Molecular Biology of the Cell 28 (2017): 3349-3359, doi:10.1091/mbc.E17-06-0359.In eukaryotic cells, highly condensed inactive/silenced chromatin has long been called “heterochromatin.” However, recent research suggests that such regions are in fact not fully transcriptionally silent and that there exists only a moderate access barrier to heterochromatin. To further investigate this issue, it is critical to elucidate the physical properties of heterochromatin such as its total density in live cells. Here, using orientation-independent differential interference contrast (OI-DIC) microscopy, which is capable of mapping optical path differences, we investigated the density of the total materials in pericentric foci, a representative heterochromatin model, in live mouse NIH3T3 cells. We demonstrated that the total density of heterochromatin (208 mg/ml) was only 1.53-fold higher than that of the surrounding euchromatic regions (136 mg/ml) while the DNA density of heterochromatin was 5.5- to 7.5-fold higher. We observed similar minor differences in density in typical facultative heterochromatin, the inactive human X chromosomes. This surprisingly small difference may be due to that nonnucleosomal materials (proteins/RNAs) (∼120 mg/ml) are dominant in both chromatin regions. Monte Carlo simulation suggested that nonnucleosomal materials contribute to creating a moderate access barrier to heterochromatin, allowing minimal protein access to functional regions. Our OI-DIC imaging offers new insight into the live cellular environments.This work was supported by MEXT and Japan Society for the Promotion of Science (JSPS) grants (Nos. 23115005 and 16H04746, respectively), as well as a Japan Science and Technology Agency (JST) CREST grant (No. JPMJCR15G2). R.I. and T.N. are JSPS Fellows. R.I. was supported by the SOKENDAI Short-Stay Study Abroad Program in fiscal year 2016

Oscillatory Finite-Time Singularities in Finance, Population and Rupture

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.Comment: Latex document of 59 pages including 20 eps figure

Development of Data Assimilation Systems for the Ionosphere, Thermosphere, and Mesosphere

The past decade saw the development of several data assimilation systems for the ionosphere, thermosphere, and mesosphere (ITM). To fully realize the capabilities of ITM data assimilation systems for both scientific investigations and operations, several critical advances are needed. This white paper outlines some of the outstanding challenges facing ITM data assimilation that need to be addressed in the coming decade in order to achieve robust, high-quality, ITM data assimilation systems. Benefits to both the scientific and operational communities of advancing ITM data assimilation capabilities are also provided. These include, but are not limited to, providing the framework for investigating ITM predictability, scientific investigations into day-to-day ITM variability driven by the lower atmosphere and geomagnetic storms, as well as advancing space weather forecasting capabilities

Ensemble clustering in deterministic ensemble Kalman filters

Ensemble clustering (EC) can arise in data assimilation with ensemble square root filters (EnSRFs) using non-linear models: an M-member ensemble splits into a single outlier and a cluster of M−1 members. The stochastic Ensemble Kalman Filter does not present this problem. Modifications to the EnSRFs by a periodic resampling of the ensemble through random rotations have been proposed to address it. We introduce a metric to quantify the presence of EC and present evidence to dispel the notion that EC leads to filter failure. Starting from a univariate model, we show that EC is not a permanent but transient phenomenon; it occurs intermittently in non-linear models. We perform a series of data assimilation experiments using a standard EnSRF and a modified EnSRF by a resampling though random rotations. The modified EnSRF thus alleviates issues associated with EC at the cost of traceability of individual ensemble trajectories and cannot use some of algorithms that enhance performance of standard EnSRF. In the non-linear regimes of low-dimensional models, the analysis root mean square error of the standard EnSRF slowly grows with ensemble size if the size is larger than the dimension of the model state. However, we do not observe this problem in a more complex model that uses an ensemble size much smaller than the dimension of the model state, along with inflation and localisation. Overall, we find that transient EC does not handicap the performance of the standard EnSRF