Ewaryst Rafajłowicz

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.

On detecting jumps in time series: Nonparametric setting

Motivated by applications in statistical quality control and signal analysis, we propose a sequential detection procedure which is designed to detect structural changes, in particular jumps, immediately. This is achieved by modifying a median filter by appropriate kernel-based jump-preserving weights (shrinking) and a clipping mechanism. We aim at both robustness and immediate detection of jumps. Whereas the median approach ensures robust smooths when there are no jumps, the modification ensure immediate reaction to jumps. For general clipping location estimators, we show that the procedure can detect jumps of certain heights with no delay, even when applied to Banach space-valued data. For shrinking medians, we provide an asymptotic upper bound for the normed delay. The finite sample properties are studied by simulations which show that our proposal outperforms classical procedures in certain respects.

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

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.

Nonparametric Sequential Change-Point Detection by a Vertically Trimmed Box Method

This paper examines a new method for sequential detection of a sudden and unobservable change in a sequence of independent observations with completely unspecified distribution functions. A nonparametric detection rule is proposed which relies on the concept of a moving vertically trimmed box. As such, it will be coined as the Vertical Box Control Chart (V-Box Chart). Its implementation requires merely to count the number of data points which fall into the box attached to the last available observation. No a priori knowledge of data distributions is required and proper tuning of the box size provides a quick detection technique. This is supported by establishing statistical properties of the method which explain the role of the tuning parameters used in the V-Box Chart. These theoretical results are verified by simulation studies which indicate that the V-Box Chart may provide quick detection with zero delay for jumps of moderate sizes. Its averaged run length to detection is more favorable than the one for the classical EWMA method. By comparison with the classical Shewhart chart, which was optimized for normal errors, our method provides comparable or better performance.

Optimum experimental design for a regression on a hypercube-generalization of Hoel′s result

In the note Hoels result (1965, Ann. Math. Statist., 36, 1097–1106) is generalized to a large family of experimental design optimality criterions. Sufficient conditions for optimality criterion are given, which ensure existence of the optimum experimental design measure which is a product of design measures on lower dimensional domains.

Moving average restoration of bandlimited signals from noisy observations

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.

When product type experimental design is optimal? Brief survey and new results

In the paper the problem of optimum experimental design for estimating parameters of multivariate regression functions is considered. We address the question: under what conditions one can compose the optimal design from partial designs, obtained by considering partial regressions, which depend on reduced number of variables. After reinterpreting and reviewing briefly existing results we provide some new conditions.

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.

SUSAN edge detector reinterpreted, simplified and modified

Our aim is to provide a new interpretation of the SUSAN edge detector and to propose its modified version, which is still simple, robust against errors and provides thinner edges, without further efforts on additional thinning. The new interpretation uses the idea of vertically weighted regression and in its simplest form it leads to interpreting USAN in terms of a box sliding on the surface of an image. This way of interpreting the SUSAN detector suggests hints on tuning its parameters. It also reveals why and to what extent this detector is robust against errors. Guided by this interpretation we also propose simple modifications of the SUSAN algorithm, which provide thinner edges without any additional thinning procedure, keeping other advantages unchanged.