Mirek Pawlak

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 finite-state vector quantization for noisy channels

Finite-state vector quantization (FSVQ) over a noisy channel is studied. A major drawback of a finite-state decoder is its inability to track the encoder in the presence of channel noise. In order to overcome this problem, we propose a nontracking decoder which directly estimates the code vectors used by a finite-state encoder. The design of channel-matched finite-state vector quantizers for noisy channels, using an iterative scheme resembling the generalized Lloyd algorithm, is also investigated. Simulation results based on encoding a Gauss-Markov source over a memoryless Gaussian channel show that the proposed decoder exhibits graceful degradation of performance with increasing channel noise, as compared with a finite-state decoder. Also, the channel-matched finite-state vector quantizers are shown to outperform channel-optimized vector quantizers having the same vector dimension and rate. However, the nontracking decoder used in the channel-matched finite-state quantizer has a higher computational complexity, compared with a channel-optimized vector-quantizer decoder. Thus, if they are allowed to have the same overall complexity (encoding and decoding), the channel-optimized vector quantizer can use a longer encoding delay and achieve similar or better performance. Finally, an example of using the channel-matched finite-state quantizer as a backward-adaptive quantizer for nonstationary signals is also presented.

Kernel regression estimators for signal recovery

We consider the problem of estimating a class of smooth functions defined everywhere on a real line utilizing nonparametric kernel regression estimators. Such functions have an interpretation as signals and are common in communication theory. Furthermore, they have finite energy, bounded frequency content and often are jammed by noise. We examine the expected L2-error of two types of estimators, one is a classical kernel regression estimator utilizing kernel functions of order p, p [greater-or-equal, slanted] 2 and the other one is motivated by the Whittaker-Shannon sampling expansion. The latter estimator employs a non-integrable kernel function sin(t)/nt, t [epsilon] . The comparison shows that the second technique outperforms the first one as long as the frequency band is finite.

Nonparametric estimation of a class of smooth functions

The problem of recovering an analytic function in the class of bandlimited functions is studied. Estimation techniques derived from the Whittaker-Shannon cardinal expansion are introduced and their statistical properties are established This includes consistency and rate of convergence in the mean square error sense as well as asymptotic normality The estimators are of the kernel convolution type with the non-integrable kernel function sin (t)/πt. Both an ordinary kernel type regression estimate and a version based on binned data are considered.

Nonparametric Identification of Generalized Hammerstein Models

Abstract The popular Hammerstein model consists of a static (zero-memory) non-linearity followed in series by a linear dynamic model. This paper considers an extension of this structure in which the static nonlinearity is replaced by a finite-memory nonlinearity. The result is a large class of fading memory models that includes many interesting special cases. We consider the nonparametric identification problem, building on previous results for nonparametric Hammerstein model identification, emphasizing interactions between (partial) model structure specifications, input sequence characteristics, and modeling error/distribution assumptions.

Kernel Density Estimation with Generalized Binning

We propose kernel density estimators based on prebinned data. We use generalized binning schemes based on the quantiles points of a certain auxiliary distribution function. Therein the uniform distribution corresponds to usual binning. The statistical accuracy of the resulting kernel estimators is studied, i.e. we derive mean squared error results for the closeness of these estimators to both the true function and the kernel estimator based on the original data set. Our results show the influence of the choice of the auxiliary density on the binned kernel estimators and they reveal that non-uniform binning can be worthwhile.

Vector quantisation for finite‐state Markov channels and application to wireless communications

Vector quantisation for joint source-channel (JSC) coding over a finite-state Markov channel (FSMC) is studied. In particular, minimum mean square error (MMSE) decoding of a vector quantised source in the absence of channel state information (CSI) is considered. Based on hidden Markov modelling of the channel output, two decoding strategies are proposed. The first one is a soft-decoder which estimates the source reconstruction vectors directly from a sequence of channel output samples. The second one is a hard-decoder based on joint maximum a posteriori (MAP) probability estimation of channel symbols and channel states. An iterative procedure for designing JSC optimised vector quantisers (VQs) is also proposed. Finally, the design of VQs for wireless channels using a finite-state model is examined. Experimental results are provided for a Gauss-Markov source and flat-fading wireless channels. A comparison with an idealised tandem source-channel coding system is also provided to demonstrate the advantage of the proposed JSC coding approach. Copyright

Reconstruction of bandlimited signals from noisy data by thresholding

The problem of recovering bandlimited signals from discrete and noisy data is studied. A nonlinear signal processing algorithm based on thresholding noisy data prior to reconstruction is proposed for improving the accuracy and robustness of reconstruction. The upper bound and the exact formula for mean integrated squared error of the proposed reconstruction scheme is established. The performance of the proposed reconstruction scheme is compared to that of the classical Whittaker-Shannon interpolation scheme.

Correlation Model for Uniform Scalar Quantizers with Arbitrary Representation Levels

This article derives closed-form expressions for correlation values of quantization errors in a set of uniform quantizers with arbitrary representation levels, operating on a vector of Gaussian stochastic variables. The problem is relevant to subband coding of random vectors using linear filter hunks, a technique that may be useful fur lossy compression of micro-array images. The results obtained here generalize previous results in which the representation levels are constrained to be midpoints of uniform quantization intervals. In particular, a set of expressions for auto-and cross-correlation values of the quantization error vector are derived. Monte-Carlo simulations are used to verify the validity of the given expressions, which indicate a very good match between numerical results and the values predicted by the derived expressions.

Signal Sampling and Recovery Under Long-Range Dependent Noise with Application to Lack-of-Fit Tests

The paper examines the impact of the additive correlated noise on the accuracy of a signal reconstruction algorithm originating from the Whittaker-Shannon sampling interpolation formula. The proposed reconstruction method is a smooth post-filtering correction of the classical Whittaker-Shannon interpolation series. We assess the global accuracy of the proposed reconstruction algorithm for long memory stationary errors being independent on the sampling rate. We also examine a class of long memory noise processes for which the correlation function depends on the sampling rate. Exact rates at which the reconstruction error tends to zero are evaluated. We apply our theory to the problem of designing non-parametric lack-of-fit tests for verifying a parametric assumption on a signal. The theory of the asymptotic behavior of quadratic forms of stationary sequences is utilized in this case