Linear Filtering Problem Research Articles

Multilevel Ensemble Kalman–Bucy Filters

In this article we consider the linear filtering problem in continuous-time. We develop and apply multilevel Monte Carlo (MLMC) strategies for ensemble Kalman-Bucy filters (EnKBFs). These filters can be viewed as approximations of conditional McKean-Vlasov-type diffusion processes. They are also interpreted as the continuous-time analogue of the \textit{ensemble Kalman filter}, which has proven to be successful due to its applicability and computational cost. We prove that an ideal version of our multilevel EnKBF can achieve a mean square error (MSE) of $\mathcal{O}(\epsilon^2), \ \epsilon>0$ with a cost of order $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In order to prove this result we provide a Monte Carlo convergence and approximation bounds associated to time-discretized EnKBFs. This implies a reduction in cost compared to the (single level) EnKBF which requires a cost of $\mathcal{O}(\epsilon^{-3})$ to achieve an MSE of $\mathcal{O}(\epsilon^2)$. We test our theory on a linear problem, which we motivate through high-dimensional examples of order $\sim \mathcal{O}(10^4)$ and $\mathcal{O}(10^5)$.

The conditional law of the Bacry–Muzy and Riemann–Liouville log correlated Gaussian fields and their GMC, via Gaussian Hilbert and fractional Sobolev spaces

We compute E(Xt|(Xs)0≤s≤L) for the standard Bacry–Muzy log-correlated Gaussian field X with covariance log+Tt−s, which corrects the finite-horizon prediction formula in Vargas et al. (Duchon et al., 0000). The problem can be viewed as a linear filtering problem, and we solve the problem by showing that the L2(P) closure of {∫[0,L]ϕ(s)Xsds:ϕ∈S,supp(ϕ)⊆[0,L]} is equal to {X(ϕ):ϕ∈H−12,supp(ϕ)⊆[0,L]}, where X(ϕ) is defined as a continuous linear extension of X acting on S⊂Hs, Hs denotes the fractional Sobolev space of order s and P is the law of the field X on the space of tempered distributions. The explicit formula for the filter is obtained as the solution to a Fredholm integral equation of the first kind with logarithmic kernel. From this we characterize the conditional law of the Gaussian multiplicative chaos (GMC) Mγ generated by X, using that Mγ is measurable with respect to X. We also outline how one can adapt this result for the Riemann–Liouville GMC introduced in Forde et al. (2019), which has a natural application to the Rough Bergomi volatility model in the H→0 limit.11We would like to thank Juhan Aru, Janne Junilla, Vincent Vargas and Lauri Vitasaari for helpful discussions.

Noise Reduction with Optimal Variable Span Linear Filters

In this paper, the problem of noise reduction is addressed as a linear filtering problem in a novel way by using concepts from subspace-based enhancement methods, resulting in variable span linear filters. This is done by forming the filter coefficients as linear combinations of a number of eigenvectors stemming from a joint diagonalization of the covariance matrices of the signal of interest and the noise. The resulting filters are flexible in that it is possible to trade off distortion of the desired signal for improved noise reduction. This tradeoff is controlled by the number of eigenvectors included in forming the filter. Using these concepts, a number of different filter designs are considered, like minimum distortion, Wiener, maximum SNR, and tradeoff filters. Interestingly, all these can be expressed as special cases of variable span filters. We also derive expressions for the speech distortion and noise reduction of the various filter designs. Moreover, we consider an alternative approach, wherein the filter is designed for extracting an estimate of the noise signal, which can then be extracted from the observed signals, which is referred to as the indirect approach. Simulations demonstrate the advantages and properties of the variable span filter designs, and their potential performance gain compared to widely used speech enhancement methods.

Nonlinear portfolio views: an efficient extension to the Black-Litterman approach

Two efficient approximate methods for optimal (Bayesian) blending of views on portfolios of assets with nonlinear payoff profiles are introduced. The idea is based on the observation that the application of the Black-Litterman model, with respect to market views, is equivalent to the measurement update in the linear filtering problem in engineering and statistics. The approaches suggested here are motivated by results from nonlinear filtering theory. In particular it is shown that the simple Gaussian framework can be approximately maintained, despite of nonlinear relations with the respective risk factors, by using the extended Kalman filter update or the unscented transform. Both methods are well suited for high dimensional problems from a computational point of view, and can thus be applied to large portfolios of derivatives.

Measurement Saving Versus Estimate Accuracy in Linear Filtering Problems.

AbstractWith reference to a (discrete time) linear system filtering problem, we consider the problem of on line deciding at which instant actually measure and process the output values. We propose a convenient measurement policy, able to sensibly reduce the measurement cost, while keeping the estimate accuracy at satisfactory levels. By some probabilistic arguments we analyse the upper bounds for the measurement and non measurement time intervals. Finally, with reference to a stable and an unstable system, we investigate the influence of input and output noise variances and of the measurement cost over saving of measurements and over estimate accuracy.

A measurement policy in stochastic linear filtering problems

With reference to a (discrete time) linear system filtering problem, we consider the problem of online deciding at which instant we actually measure and process the output values. We propose a convenient measurement policy, able to sensibly reduce the measurement cost, while keeping the estimate accuracy at satisfactory levels. By some probabilistic arguments we analyse the upper bounds for the measurement and non-measurement time intervals.

On widely linear Wiener and tradeoff filters for noise reduction

Noise reduction is often formulated as a linear filtering problem in the frequency domain. With this formulation, the core issue of noise reduction becomes how to design an optimal frequency-domain filter that can significantly suppress noise without introducing perceptually noticeable speech distortion. While higher-order information can be used, most existing approaches use only second-order statistics to design the noise-reduction filter because they are relatively easier to estimate and are more reliable. When we transform non-stationary speech signals into the frequency domain and work with the short-time discrete Fourier transform coefficients, there are two types of second-order statistics, i.e., the variance and the so-called pseudo-variance due to the noncircularity of the signal. So far, only the variance information has been exploited in designing different noise-reduction filters while the pseudo-variance has been neglected. In this paper, we attempt to shed some light on how to use noncircularity in the context of noise reduction. We will discuss the design of optimal and suboptimal noise reduction filters using both the variance and pseudo-variance and answer the basic question whether noncircularity can be used to improve the noise-reduction performance.

Hermite Polynomials Expansions for Discrete-Time Nonlinear Filtering

A finite-dimensional approximation to general discrete-time non linear filtering problems is provided. It consists in a direct approximation to the recursive Bayes formula, based on a Hermite polynomials expansion of the transition density of the signal process. Tha approximation is in the sense of convergence, in a suitable weighted norm, to the conditional density of a signal process given the observations. The choise of the norm is in turn made so as to guarantee also the convergence of the conditional moments as well as to allow the evaluation of an upper bound for the approximation error.

Sensitivity Limitations for Multivariable Linear Filtering

This paper examines fundamental limitations in performance which apply to linear filtering problems associated with multivariable systems having as many inputs as outputs. The results of this paper quantify unavoidable limitations in the sensitivity of state estimates to process and measurement disturbances, as represented by the maximum singular values of the relevant transfer matrices. These limitations result from interpolation constraints imposed by open right half-plane poles and zeros in the transfer matrices linking process noise and output noise with state estimates. Using the Poisson integral inequality, this paper shows how sensitivity limitations and tradeoffs in multivariable filtering problems are intimately related to the directionality properties of the open right half-plane poles and zeros in these transfer matrices.

A nonlinear filtering algorithm immune to singular errors

The problem of linear discrete filtering is resolved for the case when the observation channel contains piecewise continuous interference having a finite number of the first-kind discontinuities on the whole observation segment and is described, within the continuity intervals, by power polynomials with random coefficients. An illustrative example is included.

Linear time-variant transformations of generalized almost-cyclostationary signals.II. Development and applications

For pt. I see ibid., vol.50, no.12, p.2947-61 (2000). In Part I, the problem of the linear time-variant (LTV) filtering is addressed in the fraction-of-time (FOT) probability framework. The adopted approach, which is an alternative to the classical stochastic one, provides a statistical characterization of the systems in terms of functions that can be estimated by a single time-series. The analysis is carried out with reference to the wide class of the generalized almost-cyclostationary (GACS) signals, which includes, as a special case, the class of the almost-cyclostationary (ACS) signals. Examples of applications and developments of the theory introduced in Part I are presented here in Part II. Specifically, the countability of the set of the output cycle frequencies is studied with reference to linear time-variant systems for both ACS and GACS not containing any ACS component input signals. Thus, the linear almost-periodically time-variant filtering and the product modulation are considered in detail. Moreover, several Doppler channel models are analyzed. In all these examples, it is shown that the FOT probability approach allows one to characterize the system and its output in terms of statistical functions that can be measured by a single time-series. Furthermore, the usefulness of considering the linear filtering problem within the class of the GACS signals is clarified, and several pitfalls arising from continuing to adopt for the observed time-series the ACS model when an increase in the data-record length makes the GACS model more appropriate are pointed out.

Filtering interpretation of acoustic inverse scattering under coherent and incoherent insonification

The reconstruction process of two-dimensional and three-dimensional sound-soft obstacles from a set of scattered field measurements is formulated as a linear filtering problem. Under coherent and incoherent insonification, the impulse response of the equivalent filter completely characterizes the acoustical imaging process. An energy-based criterion is proposed to describe the impulse response features in both cases. Numerical simulations are also presented, showing the better performance of the incoherent insonification.

Optimal linear filtering and extrapolation of realizations of a vector random function observed with errors

A solution to an optimal linear filtering and extrapolation problem is proposed. The investigation object is a realization of multidimensional random function, which is measured with random errors. The solution method does not put any essential constraints on the investigated random fimction, measurment error characteristics, and on the character of relations between them. The solution is based on Pugachov’s canonical decomposition of the investigated random function, and its form is convenient for use with a computer.

Linear filtering with fractional Brownian motion in the signal and observation processes

Integral equations for the mean‐square estimate are obtained for the linear filtering problem, in which the noise generating the signal is a fractional Brownian motion with Hurst index h ∈ (3/4, 1) and the noise in the observation process includes a fractional Brownian motion as well as a Wiener process.

A Design Method of VSS Observer for Continuous-Time Systems with Output Uncertainties.

VSS observers are considered for continuous-time systems, in which the nominal plant model is linear, but there exist bounded plant and output uncertainties. We first present a design method of a VSS observer based on use of some extensions of a conventional approach that uses an algebraic Lyapunov equation. It is then shown that such a method cannot be successfully applied to the system with any linear part, if we desire to use a unique solution to the algebraic Lyapunov equation. Namely, this method is only applicable to a restricted class of systems. In order to develop a VSS observer applicable to a wider class of systems, we next propose a design method based on use of an algebraic Riccati equation, associated with a linear filtering problem. A relationship between the two\design methods is also discussed. Simulation results show that the proposed method is effective for designing a VSS observer for a second-order nonlinear system with band limited output noise, as the output uncertainty.

Stochastic models of environmental pollution

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.

Stochastic models of environmental pollution

In this paper, we consider several stochastic models arising from environmental problems. First, we study pollution in a domain where undesired chemicals are deposited at random times and locations according to Poisson streams. The chemical concentration can be modeled by a linear stochastic partial differential equation (SPDE) which is solved by applying a general result. Various properties, especially the limit behavior of the pollution process, are discussed. Secondly, we consider the pollution problem when a tolerance level is imposed. The chemical concentration can still be modeled by a SPDE which is no longer linear. Its properties are investigated in this paper. When the leakage rate is positive, it is shown that the pollution process has an equilibrium state given by the deterministic model treated in [2]. Finally, the linear filtering problem is considered based on the data of several observation stations.

Lattice algorithms for recursive least squares adaptive second-order Volterra filtering

This paper presents two computationally efficient recursive least-squares (RLS) lattice algorithms for adaptive nonlinear filtering based on a truncated second-order Volterra system model. The lattice formulation transforms the nonlinear filtering problem into an equivalent multichannel, linear filtering problem and then generalizes the lattice solution to the nonlinear filtering problem. One of the algorithms is a direct extension of the conventional RLS lattice adaptive linear filtering algorithm to the nonlinear case. The other algorithm is based on the QR decomposition of the prediction error covariance matrices using orthogonal transformations. Several experiments demonstrating and comparing the properties of the two algorithms in finite and "infinite" precision environments are included in the paper. The results indicate that both the algorithms retain the fast convergence behavior of the RLS Volterra filters and are numerically stable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Estimation of the Quadratic Variation of Nearly Observed Semimartingales with Application to Filtering

Consider a filtering problem in which the available information is a noisy observation of a continuous semimartingale $H_t $. In the case of a high signal-to-noise ratio, it is proved that $H_t $ and its quadratic variation can be jointly estimated by means of a finite-dimensional filter; moreover, for this result, the observation noise and $H_t $ are not required to be independent. This problem can be viewed as a linear filtering problem with randomly time-varying parameters, and our filter is auto-adaptive with respect to changes of the parameters. These results are then applied to the nonlinear filtering of Markov diffusion processes when the observation function is not injective but satisfies a weaker detectability assumption. It appears that filtering such a system involves two timescales. The study is based on time discretization; the main tools are an averaging principle and an application of the asymptotic ordinary differential equation method for the study of stochastic algorithms.

Moment of cepstrum and its applications

It is shown that the moments of a signal sequence and of its corresponding cepstral sequence are related in a way which obviates the need for the direct calculation of the cepstral coefficients. Hence, an ideal calculation for the moments of the cepstrum is possible even if the duration of the cepstrum is infinite. It is also possible to describe all the properties of a signal sequence from its moments. The effectiveness of this method is demonstrated in the problems of homomorphic deconvolution and echo removal. It is also shown that in linear filtering problems, the moments of the minimum-phase sequence are calculated from the corresponding linear-phase sequence without actually calculating the minimum-phase sequence. For a discrete-time real signal sequence with finite duration, complete equivalence is demonstrated between the signal sequence and its moments. This permits a new description and a new set of design specifications based on moments for such systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>