Stability of nonlinear filters in nonmixing case

The nonlinear filtering equation is said to be stable if it ``forgets'' the initial condition. It is known that the filter might be unstable even if the signal is an ergodic Markov chain. In general, the filtering stability requires stronger signal ergodicity provided by the, so called, mixing condition. The latter is formulated in terms of the transition probability density of the signal. The most restrictive requirement of the mixing condition is the uniform positiveness of this density. We show that it might be relaxed regardless of an observation process structure.Comment: Published at http://dx.doi.org/10.1214/105051604000000873 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Dimension-free Wasserstein contraction of nonlinear filters

For a class of partially observed diffusions, sufficient conditions are given for the map from initial condition of the signal to filtering distribution to be contractive with respect to Wasserstein distances, with rate which has no dependence on the dimension of the state-space and is stable under tensor products of the model. The main assumptions are that the signal has affine drift and constant diffusion coefficient, and that the likelihood functions are log-concave. Contraction estimates are obtained from an $h$-process representation of the transition probabilities of the signal reweighted so as to condition on the observations

Effect of nonlinear filters on detrended fluctuation analysis

We investigate how various linear and nonlinear transformations affect the scaling properties of a signal, using the detrended fluctuation analysis (DFA). Specifically, we study the effect of three types of transforms: linear, nonlinear polynomial and logarithmic filters. We compare the scaling properties of signals before and after the transform. We find that linear filters do not change the correlation properties, while the effect of nonlinear polynomial and logarithmic filters strongly depends on (a) the strength of correlations in the original signal, (b) the power of the polynomial filter and (c) the offset in the logarithmic filter. We further investigate the correlation properties of three analytic functions: exponential, logarithmic, and power-law. While these three functions have in general different correlation properties, we find that there is a broad range of variable values, common for all three functions, where they exhibit identical scaling behavior. We further note that the scaling behavior of a class of other functions can be reduced to these three typical cases. We systematically test the performance of the DFA method in accurately estimating long-range power-law correlations in the output signals for different parameter values in the three types of filters, and the three analytic functions we consider.Comment: 12 pages, 7 figure

Computing the output distribution and selection probabilities of a stack filter from the DNF of its positive Boolean function

Many nonlinear filters used in practise are stack filters. An algorithm is presented which calculates the output distribution of an arbitrary stack filter S from the disjunctive normal form (DNF) of its underlying positive Boolean function. The so called selection probabilities can be computed along the way.Comment: This is the version published in Journal of Mathematical Imaging and Vision, online first, 1 august 201

On the exchange of intersection and supremum of sigma-fields in filtering theory

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page

Adaptive Signal Processing Strategy for a Wind Farm System Fault Accommodation

In order to improve the availability of offshore wind farms, thus avoiding unplanned operation and maintenance costs, which can be high for offshore installations, the accommodation of faults in their earlier occurrence is fundamental. This paper addresses the design of an active fault tolerant control scheme that is applied to a wind park benchmark of nine wind turbines, based on their nonlinear models, as well as the wind and interactions between the wind turbines in the wind farm. Note that, due to the structure of the system and its control strategy, it can be considered as a fault tolerant cooperative control problem of an autonomous plant. The controller accommodation scheme provides the on-line estimate of the fault signals generated by nonlinear filters exploiting the nonlinear geometric approach to obtain estimates decoupled from both model uncertainty and the interactions among the turbines. This paper proposes also a data-driven approach to provide these disturbance terms in analytical forms, which are subsequently used for designing the nonlinear filters for fault estimation. This feature of the work, followed by the simpler solution relying on a data-driven approach, can represent the key point when on-line implementations are considered for a viable application of the proposed scheme

A new method for the design of energy transfer filters

This paper is concerned with the development of a new method for the design of Energy Transfer Filters (ETFs). ETFs are a new class of nonlinear filters recently proposed by the authors, which employ nonlinear effects to transfer signal energy from one frequency band to a different frequency location. The new method uses the powerful Orthogonal Least Squares (OLS) algorithm to solve the Least Squares problem associated with the design and compared with previous methods achieves much better filtering performance