Tracking properties of adaptive signal processing algorithms

Adaptive signal processing algorithms derived from LS (least squares) cost functions are known to converge extremely fast and have excellent capabilities to "track" an unknown parameter vector. This paper treats analytically and experimentally the steady-state operation of RLS (recursive least squares) adaptive filters with exponential windows for stationary and nonstationary inputs. A new formula for the "estimation-noise" has been derived involving second- and fourth-order statistics of the filter input as well as the exponential windowing factor and filter length. Furthermore, it is shown that the adaptation process associated with "lag effects" depends solely on the exponential weighting parameter λ. In addition, the calculation of the excess mean square error due to the lag for an assumed Markov channel provides the necessary information about tradeoffs between speed of adaptation and steady-state error. It is also the basis for comparison to the simple LMS algorithm, in a simple case of channel identification, it is shown that the LMS and RLS adaptive filters have the same tracking behavior. Finally, in the last part, we present new RLS restart procedures applied to transversal structures for mitigating the disastrous results of the third source of noise, namely, finite precision arithmetic.

The paper considers the problem of estimating the parameters in a continuous time regression model with a non-Gaussian noise of pulse type. The vector of unknown parameters is assumed to belong to a compact set. The noise is specified by the Ornstein–Uhlenbeck process driven by the mixture of a Brownian motion and a compound Poisson process. Improved estimates for the unknown regression parameters, based on a special modification of the James–Stein procedure with smaller quadratic risk than the usual least squares estimates, are proposed. The developed estimation scheme is applied for the improved parameter estimation in the discrete time regression with the autoregressive noise depending on unknown nuisance parameters.

This paper considers the estimation problem for linear continuous-time systems with the interrupted observation mechanism which is characterized in terms of the jump Markov process taking on the values of 0 or 1. The minimum variance estimator algorithm is derived. The approach adopted here is that we express the jump process in terms of the initial value and the jump times instead of its instantaneous values and that in order to apply Lainiotis' formula we regard the initial value and jump times as the unknown system parameter vector. The parameter vector is, however, infinite dimensional so that Lainiotis' formula cannot be applied directly. This is because the formula is applicable only to linear systems with a finite dimensional parameter vector. Therefore, the minimum variance estimator algorithm is derived by using some limiting procedure. The resultant optimal algorithm is infinite dimensional, so that feasible approximate algorithms should be developed for practical implementation. A simple illustrating example is also worked out.

We consider a generalized partially linear model E(Y|X, T) = G{X T β + m(T)}, where G is a known function, β is an unknown parameter vector, and m is an unknown function. We introduce a test statistic that allows one to decide between a parametric and a semiparametric model: (a) m is linear (i.e., m(t) = t T γ for a parameter vector γ), and (b) m is a smooth (nonlinear) function. Under linearity (a), we show that the test statistic is asymptotically normal. Moreover, we prove that the bootstrap works asymptotically. Simulations suggest that (in small samples) the bootstrap outperforms the calculation of critical values from the normal approximation. The practical performance of the test is demonstrated in applications to data on East–West German migration and credit scoring.

Stability analysis of high-order Hopfield type neural networks with uncertainty

We consider estimation in a semiparametric generalized linear model for clustered data using estimating equations. Our results apply to the case where the number of observations per cluster is finite, whereas the number of clusters is large. The mean of the outcome variable μ is of the form g(μ) = XTβ + θ(T), where g(·) is a link function, X and T are covariates, β is an unknown parameter vector, and θ(t) is an unknown smooth function. Kernel estimating equations proposed previously in the literature are used to estimate the infinite-dimensional nonparametric function θ(t), and a profile-based estimating equation is used to estimate the finite-dimensional parameter vector β. We show that for clustered data, this conventional profile-kernel method often fails to yield a √n-consistent estimator of β along with appropriate inference unless working independence is assumed or θ(t) is artificially undersmoothed, in which case asymptotic inference is possible. To gain insight into these results, we derive the semiparametric efficient score of β, which is found to have a complicated form, and show that, unlike for independent data, the profile-kernel method does not yield a score function asymptotically equivalent to the semiparametric efficient score of β, even when the true correlation is assumed and θ(t) is undersmoothed. We illustrate the methods with an application to infectious disease data and evaluate their finite-sample performance through a simulation study.

This paper discusses the problem of estimation for two classes of nonlinear models, namely random coefficient autoregressive (RCA) and autoregressive conditional heteroskedasticity (ARCH) models. For the RCA model, first assuming that the nuisance parameters are known we construct an estimator for parameters of interest based on Godambe's asymptotically optimal estimating function. Then, using the conditional least squares (CLS) estimator given by Tjostheim (1986, Stochastic Process. Appl., 21, 251–273) and classical moment estimators for the nuisance parameters, we propose an estimated version of this estimator. These results are extended to the case of vector parameter. Next, we turn to discuss the problem of estimating the ARCH model with unknown parameter vector. We construct an estimator for parameters of interest based on Godambe's optimal estimator allowing that a part of the estimator depends on unknown parameters. Then, substituting the CLS estimators for the unknown parameters, the estimated version is proposed. Comparisons between the CLS and estimated optimal estimator of the RCA model and between the CLS and estimated version of the ARCH model are given via simulation studies.

In this paper, we study the properties of information matrices of the difference operator or the so-called delta (/spl delta/) operator-based algorithms for adaptive signal processing. We show that the conditioning of a transformed information matrix in the /spl delta/ domain is always better than that of the original information matrix in the conventional q domain for ill-conditioned problems. The results obtained in this paper give an explanation of the advantages of using delta operator algorithms for adaptive signal processing that have been developed. The analysis in this paper also helps to clarify the problem about the effect of /spl Delta/ in the delta operator algorithms and justify the use of sampling interval as /spl Delta/ in most cases when the ill conditioning is caused by fast sampling of continuous time systems.

The present chapter deals with the study of an important but least known case in the theory of parametric estimation of controlled MS, the case when the vector of unknown system parameters, or that of unknown perturbations affecting the control object, is an unknown (uncontrollable, nonmeasurable) vector function of time. Certain models of solving the problems of adaptive stabilization and optimization under parametric drift conditions, based on utilizing the formalism of Lyapunov functions, were given in earlier researches under a variety of restrictions, for example on the information about the drift model and the rate of change of the parameters. The dominant difficulties that occur in these problems of forming convergent estimation algorithms, concern the proof of the fact that the Lyapunov function monotonically decreases on the trajectories of the controlled process.

The iterative method, which can be applied to estimation of the unknown vector of discrete parameters in combination with filtering of the unknown vector of continuous signal parameters, is proposed. It is demonstrated that this method makes it possible to construct highly efficient algorithms for joint demodulation and filtering of communications channel parameters in code-division multiple-access and orthogonal frequency-division multiplexing systems.

The inversion of seismic amplitude variation with incident angle and azimuth data for estimating fracture and fluid indicators is a crucial task. Tangential fracture weakness and fluid bulk modulus are sensitive to fracture density and fluid saturation, respectively, and serve as key indicators of fractures and fluids. We first express the PP-wave reflection coefficient and azimuthal elastic impedance (AEI) in terms of these fracture and fluid indicators. We develop a two-stage inversion method that uses azimuthal seismic data to estimate multiple parameters, involving P- and S-wave moduli, density, and fracture and fluid indicators. To enhance the accuracy of the multiparameter inversion, we focus on constructing robust initial models for the unknown parameters to constrain the inversion process. Specifically, we introduce a novel approach that uses prestacked seismic attributes — such as the amplitude-variation-with-angle intercept and isotropic and anisotropic gradients — to derive initial models for fracture and fluid indicators. In particular, for constructing initial models of fluid indicators, we develop an extended AEI approach to transform seismic attributes into fluid indicators. To further refine the multiparameter inversion, we incorporate the Hessian matrix, which is calculated using the first- and second-order derivatives of AEI, to compute perturbations in the unknown parameter vector during the second stage of the inversion. Finally, we apply our method to noisy synthetic and real data, which demonstrates its robustness and ability to generate reliable fracture and fluid indicators that match well with logging data. We conclude that our inversion method is a promising tool for predicting potentially hydrocarbon-bearing fractured reservoirs.

Over the last couple of decades the demand for high precision and enhanced performance of physical systems has been steadily increasing. This demand often results in miniaturization and complex design, thus increasing the need for complex nonlinear control methods. Some of the state of the art nonlinear methods are stymied by the requirement of full state information, model and parameter uncertainties, mathematical complexity, etc. For many scenarios it is nearly impossible to consider all the uncertainties during the design of a feedback controller. Additionally, while designing a modelbased nonlinear control there is no standard mechanism to incorporate performance measures. Some of the mentioned issues can be addressed by using online learning. Animals and humans have the ability to share, explore, act or respond, memorize the outcome and repeat the task to achieve a better outcome when they encounter the same or a similar scenario. This is called learning from interaction. One instance of this approach is reinforcement learning (RL). However, RL methods are hindered by the curse of dimensionality, non-interpretability and non-monotonic convergence of the learning algorithms. This can be attributed to the intrinsic characteristics of RL, as it is a modelfree approach and hence no standard mechanism exists to incorporate a priori model information. In this thesis, learning methods are proposed which explicitly use the available system knowledge. This can be seen as a new class of approaches that bridge model-based and model-free methods. These methods can address some of the hurdles mentioned earlier. For example, i) a prior system information can speed up the learning, ii) new control objectives can be achieved which otherwise would be extremely difficult to attain using only model-based methods, iii) physical meaning can be attributed to the learned controller. The developed approach is as follows: themodel of the given physical system is represented in the port-Hamiltonian (PH) form. For the system dynamics in PH form a passivity-based control (PBC) law is formulated, which often requires the solution to a set of partial differential equations (PDEs). Instead of finding an analytical solution, the PBC control law is parameterized using an unknown parameter vector. Then, by using a variation of the standard actor-critic learning algorithm, the unknown parameters can be learned online. Using the principles of stochastic approximation theory, a proof of convergence for the developed method is shown. The proposedmethods are evaluated for the stabilization and regulation ofmechanical and electro-mechanical systems. The simulation and experimental results show comparable learning curves. In the final part of the thesis a novel integral reinforcement learning approach is developed to solve for the optimal output tracking control problem for a set of linear heterogeneous multi-agent systems. Unlike existing methods, this approach does not need to solve either the output regulator equation or requires a p-copy of the leader’s dynamics in the agent’s control law. A detailed numerical evaluation has been conducted to show the feasibility of the developed method.

Multi-armed bandit is a widely-studied model for sequential decision-making problems. The most studied model in the literature is stochastic bandits wherein the reward of each arm follows an independent distribution. However, there is a wide range of applications where the rewards of different alternatives are correlated to some extent. In this paper, a class of structured bandit problems is studied in which rewards of different arms are functions of the same unknown parameter vector. To minimize the cumulative learning regret, we propose a globally-informative Thompson sampling algorithm to learn and leverage the correlation among arms, which can deal with unknown multi-dimensional parameter and non-monotonic reward functions. Our studies demonstrate that the proposed algorithm achieves significant improvement in the learning speed. In particular, the designed algorithm is used to solve an edge transcoder selection problem in crowdsourced live video streaming systems and shows superior performance as compared to the existing schemes.

The identification problem of systems with logarithmic quantized data is analyzed in this paper. A discrete-time system with an unknown parameter vector and Gaussian noise is considered. The output of the system is quantized by a logarithmic quantizer. The estimator of the unknown parameter vector is obtained using the quantile-based estimation. The convergence almost everywhere of the estimator and the upper bound of the estimation error are also analyzed, respectively.

A robust output feedback control which globally stabilizes a class of SISO minimum-phase nonlinear systems with known and constant relative degree containing a vector of unknown parameters is developed. The constant parameter vector is not restricted to enter linearly in the state equations but it is assumed to belong to a known compact set and an imprecise knowledge of the nonlinearities (e.g., lookup tables) is allowed. The class of nonlinear systems is determined by geometric conditions; an additional assumption, which generalizes the knowledge of the sign of high-frequency gain for linear systems, is also required. The nonlinearities are restricted to depend, in suitable coordinates, on the output only: no growth conditions, such as sector or Lipschitz, are required. The robust output feedback control stabilizes the system for every value of the parameter vector in a known compact set. The order of the compensator is equal to the relative degree minus one. >