On optimal global rate of convergence of some nonparametricidentification procedures

A study of orthogonal series procedures for function fitting, motivated by an interest in nonparametric identification of linear dynamic systems, is reported. It is shown that the procedures attain the optimal rate of convergence for the Fourier and the Walsh orthonormal systems. These rates cannot be exceeded by any method of estimation. This is the first analytical result giving a clear answer to the question of which orthonormal system is the best one for the purpose of system identification     

Nonparametric algorithm for input signals identification in static distributed-parameter systems

In this correspondence, a nonparametric algorithm for identification of input signals in linear, static distributed-parameter systems is proposed and investigated. Integral mean-square convergence of the algorithm is proved for an infinite number of point measurements of the system state. The algorithm is a generalized version of the one recently proposed by Rutkowski [10] for nonparametric function fitting, and in a common area, the presented results are complementary.     

Repeated least squares with inversion and its application in identifying linear distributed-parameter systems

In the paper an approach to a certain class of the nonlinear parameter estimation problem is proposed, which is, in particular, applicable to distributed-parameter systems described by elliptic partial differential equations. The approach exploits the special structure of nonlinear dependence, which allows the least-squares algorithm to be applied twice, together with the inversion of a nonlinear characteristic. One can roughly say that the class of considered systems can be described by a feedforward neural net with two hidden layers and monotonic activation functions. In the language of neural nets, the estimation problem can be interpreted as a partial inversion of the net, that is finding part of its inputs from a learning sequence. Simulations confirm that the approach is useful and much simpler than a direct iteration minimization of the sum of squares.     

Moving average restoration of bandlimited signals from noisyobservations

The purpose of this paper is to describe the extension of the Whittaker-Shannon sampling theorem to reconstruction of bandlimited functions in the presence of zero mean, uncorrelated noise. It is shown that the classical Whittaker-Shannon sampling scheme is not consistent in the case of noisy measurements, and new reconstruction algorithms based on the moving average smoothing are proposed. The weak and strong consistency of the algorithms is established, and the rate of convergence is investigated. The theory is verified in the computer simulations     

Optimal experiment design for identification of linear distributed-parameter systems: Frequency domain approach

In this paper optimality conditions for experiment design are derived. The experiment is planned for identification of linear time invariant distributed-parameter systems. The determinant of the averaged information matrix is expressed in the terms of input spectral density matrix and spatial density of measurements, and then used as a measure of estimation accuracy. Presented results are applied to find optimal sensors positions and input signals in several examples.     

Linear systems identification from random threshold binary data

A new identification problem of estimating parameters of linear dynamic systems from random threshold binary observations of its output and input is stated. The only available data are collected as a result of checking whether a signal reached a randomly specified threshold at a randomly chosen instant of time. The proposed estimation algorithm is based on the celebrated von Neumann theorem, which was earlier used mainly for generating random numbers. Strong consistency of parameters estimate from low-cost output binary observations is proved, assuming deterministic input signal of a finite duration. Possibilities of relaxing the assumption used in the theoretical part of the paper are considered by means of simulations     

Postfiltering versus prefiltering for signal recovery from noisy samples

We consider the extension of the Whittaker-Shannon (WS) reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited. Observing that in this situation the classical sampling expansion yields inconsistent reconstruction, we introduce a class of signal recovery methods with a smooth correction of the interpolation series. Two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing. We assess the accuracy of the methods by the global L/sub 2/ error. Both band-limited and non-band-limited signals are considered. A general class of correlated noise processes is taken into account. The weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed. The influence of noise memory and its moment structure on the accuracy is thoroughly examined.     

Testing (non-)existence of input-output relationships by estimating fractal dimensions

Our aim is to propose tests for (non-)existence of nonlinear relationships between signals, which, after passing a test, can be interpreted as input and output signals of a certain system, if its characteristic is sufficiently smooth. The proposed tests are based on the theoretical results on equality of fractal dimensions of these signals as well as on estimation of fractal dimensions from observations. They are applicable when at least one of these signals has the fractal dimensions strictly larger than one, i.e., it is rough enough. The tests are then verified on simulated data. Their applicability is illustrated by two sets of real data, namely, observations of two financial time series and samples of displacement-force signals in a magneto-hydrological damper.     

On restoring band-limited signals

The problem of reconstruction of band-limited signals from discrete and noisy data is studied. The reconstruction schemes employing cardinal expansions are proposed and their asymptotical properties are examined. In particular, the conditions for the convergence of the mean integrated squared error are found and the rate of convergence is evaluated. The main difference between the proposed reconstruction scheme and the classical one is in treating the sampling rate and the reconstruction rate differently. This distinction is necessary to ensure consistency of the reconstruction scheme in the presence of noise     

Exponential weighting algorithms for reconstruction of bandlimitedsignals

The problem of reconstruction of bandlimited signals from discrete, noisy observations is considered. Whittaker-Shannon interpolation series-based estimates involving the weighting factors are proposed. It is shown that they converge as the sampling rate increases to infinity. The rate of convergence of the double exponential weighting algorithm is established. These results are corroborated in computer simulations