This paper addresses the problem of detecting and estimating hidden periodicity from noisy observations when the noise distribution is asymmetric with heavy tail on one side. The ordinary periodogram is less effective in handling such noise. In this paper, we introduce an alternative periodogram-like function, called the quantile periodogram. The quantile periodogram is constructed from trigonometric regression where a specially designed objective function is used to substitute the squared ℓ2norm that leads to the ordinary periodogram. Simulation results are provided to demonstrate the superior performance of the quantile periodogram in comparison with the ordinary periodogram when the noise is asymmetrically distributed with a heavy tail. The asymptotic distribution of the quantile periodogram is derived under the white noise assumption. Extensions to the multivariate case and the complex case are also discussed.

BACKGROUND: Time-course gene expression analysis has become important in recent developments due to the increasingly available experimental data. The detection of genes that are periodically expressed is an important step which allows us to study the regulatory mechanisms associated with the cell cycle. RESULTS: In this work, we present the Laplace periodogram which employs the least absolute deviation criterion to provide a more robust detection of periodic gene expression in the presence of outliers. The Laplace periodogram is shown to perform comparably to existing methods for the Sacharomyces cerevisiae and Arabidopsis time-course datasets, and to outperform existing methods when outliers are present. CONCLUSION: Time-course gene expression data are often noisy due to the limitations of current technology, and may include outliers. These artifacts corrupt the available data and make the detection of periodicity difficult in many cases. The Laplace periodogram is shown to perform well for both data with and without the presence of outliers, and also for data that are non-uniformly sampled.

Accurate estimation of the frequency of sinusoidal signals from noisy observations is an important problem in signal processing applications such as radar, sonar, and telecommunications. In this paper, we study the problem under the assumption of non-Gaussian noise in general and Laplace noise in particular. We prove that the Laplace maximum likelihood estimator is able to attain the asymptotic Cramer-Rao lower bound under the Laplace assumption which is one half of the Cramer-Rao lower bound in the Gaussian case. This provides the possibility of improving the currently most efficient methods such as nonlinear least-squares and periodogram maximization in non-Gaussian cases. We propose a computational procedure that overcomes the difficulty of local extrema in the likelihood function when computing the maximum likelihood estimator. We also provide some simulation results to validate the proposed approach.

Accurate estimation of sinusoidal frequencies from noisy observations is an important problem in many applications including radar, sonar, and data communications. Among many algorithms is the iterative filtering algorithm (IFA), proposed by Kay, which provides a computationally simple procedure yet capable of accurate frequency estimation especially at low signal-to-noise ratio (SNR). However, the convergence and other numerical/statistical properties of IFA have not been established beyond simulation. This paper makes several important contributions: (a) It shows that the poles of the AR filter must be reduced via a shrinkage parameter to accommodate possibly poor initial values. (b) It shows that the AR estimates in each iteration must be bias-corrected to produce more accurate frequency estimates; a closed-form expression is provided for bias correction. (c) It shows that for a sufficiently large sample size, the resulting algorithm, called new IFA, or NIFA, converges to the desired fixed-point which constitutes a consistent frequency estimator. Numerical examples, including a radar data example, are provided to demonstrate the findings.

This article addresses the problem of estimating a multivariate linear system from its output when the input is an unobservable sequence of random vectors with finite-alphabet distribution. By explicitly utilizing the finite-alphabet property, an estimation method is proposed under the traditional inverse filtering paradigm as a generalization of a univariate method that has been studied previously. Identifiability of multivariate systems by the proposed method is proved mathematically under very mild conditions that can be satisfied even if the input is nonstationary and has both cross-channel and serial statistical dependencies. Statistical super-efficiency in estimating both parametric and nonparametric systems is also established for an alphabet-based cost function.

A new statistical method is proposed for deblurring two-tone images, i.e., images with two unknown grey levels, that are blurred by an unknown linear filter. The key idea of the proposed method is to adjust a deblurring filter until its output becomes two tone. Two optimization criteria are proposed for the adjustment of the deblurring filter. A three-step iterative algorithm (TSIA) is also proposed to minimize the criteria. It is proven mathematically that by minimizing either of the criteria, the original (nonblurred) image, along with the blur filter, will be recovered uniquely (only with possible scale/shift ambiguities) at high SNR. The recovery is guaranteed not only for i.i.d. images but also for correlated and nonstationary images. It does not require a priori knowledge of the statistical parameters or the tone values of the original image; neither does it require a priori knowledge of the phase or other special information (e.g., FIR, symmetry, nonnegativity, etc.) about the blur filter. Numerical experiments are carried out to test the method on synthetic and real images.

The wavelet spectrum of a random process comprises the variances of the wavelet coefficients of the process computed within each scale. This paper investigates the possibility of using the wavelet spectrum, obtained from a continuous wavelet transform (CWT), to uniquely represent the second-order statistical properties of random processes-particularly, stationary processes and long-memory nonstationary processes. As is well known, the Fourier spectrum of a stationary process is mathematically equivalent to the autocovariance function (ACF) and thus uniquely determines the second-order statistics of the process. This characterization property is shown to be possessed also by the wavelet spectrum under very mild regularity conditions that are easily satisfied by many widely used wavelets. It is also shown that under suitable regularity conditions, the characterization property remains valid for processes with stationary increments including 1/f noise.

Based on an asymptotic analysis of the contraction mapping (CM) method of Li and Kedem (1993), a bandwidth shrinkage rule is proposed for fast and accurate estimation of the frequencies of multiple sinusoids from noisy measurements. The CM frequency estimates are defined as the fixed points of a contractive mapping formed by the lag-one autocorrelation coefficient calculated from the output. of a parametric filter applied to the observed time series. With bandwidth parameters judiciously chosen according to the asymptotic analysis, the algorithm is shown to be able to accommodate possibly poor initial values of precision O(n/sup -1/3/) and converge to a final estimate whose accuracy is arbitrarily close to O(n/sup -3/2/), the optimal error rate for frequency estimation under the Gaussian assumption. The total computational complexity of the algorithm is shown to be O(n log n), which is comparable to that of n-point fast Fourier transform (FFT). A novelty in the asymptotic analysis is that it accommodates closely spaced frequencies by allowing not only the filter bandwidth but also the frequency separation to be functions of the sample size n. This enables an assessment of the accuracy of the frequency estimates for given bandwidths and initial values in situations where some or all of the frequencies are close to each other.

It is hypothesized that the perception of an alternative image in ambiguous figures would be manifested as an increase in synchronization of EEG signals over multiple scalp sites as "cognitive binding" occurs. A statistical time-frequency analysis (STFA) approach is taken to detect cognitive binding. The STFA approach consists of an analysis filter bank that extracts useful components from the input EEG signals and certain statistical parameters at the output that characterize the synchronization patterns over multiple scalp sites. With the help of wavelet-packet filters and wavelet coherence measures, cognitive binding is detected over many frequency bands by statistical testing.

A new method of nonstationary signal analysis, called time-correlation analysis (TCA), is applied to a class of nonstationary random signals containing time distortion. The method reveals a relationship between the TCA summary statistics and the distortion and leads to two nonparametric estimators for the distortion function. An example is given that demonstrates the application of the TCA method to motion compensation problems in radar imaging.

A new set of digital signal processing techniques for detecting changes in a speech signal is considered. The overall approach is called parametric filtering, and it yields several promising new diagnostics for speech analysis and segmentation including in particular, the demodulated lag-one autocorrelation /spl gamma//sub /spl theta//(/spl eta/), the time-correlation analysis plot, and the /spl gamma//sub /spl theta//(/spl eta/)-based distortion measures. Initial experiments described in this paper establish the potential significance of the parametric filtering method and these new diagnostics for speech analysis and segmentation.

The problem of simultaneous wavelet estimation and deconvolution is investigated with a Bayesian approach under the assumption that the reflectivity obeys a Bernoulli-Gaussian distribution. Unknown quantities, including the seismic wavelet, the reflection sequence, and the statistical parameters of reflection sequence and noise are all treated as realizations of random variables endowed with suitable prior distributions. Instead of deterministic procedures that can be quite computationally burdensome, a simple Monte Carlo method, called Gibbs sampler, is employed to produce random samples iteratively from the joint posterior distribution of the unknowns. Modifications are made in the Gibbs sampler to overcome the ambiguity problems inherent in seismic deconvolution. Simple averages of the random samples are used to approximate the minimum mean-squared error (MMSE) estimates of the unknowns. Numerical examples are given to demonstrate the performance of the method.

Some statistical properties in regard to the contraction mapping (CM) method are discussed. One of the requirements in this method is that the filter be parameterized to satisfy a certain fundamental property. The parameterization clearly depends on the normalized noise spectrum which theoretically has to be known or estimated a priori. If this information is available, one can first whiten the noise with a linear filter and then apply the CM method to the filtered data. In this way, the parameterization only needs to be done under the white noise assumption and filters like the AR(2) can be used by the CM method. In applications, however, prewhitening may not always be necessary.>

The problem of blind identification and deconvolution of linear systems with independent binary inputs is addressed. To solve the problem, a linear system is applied to the observed data and adjusted so as to produce binary outputs. It is proved that the system coincides with the inverse of the unknown system (with scale and shift ambiguities), whether it is minimum or nonminimum phase. These results are derived for nonstationary independent binary inputs of infinite or finite length. Based on these results, an identification method is proposed for parametric linear systems. It is shown that under some mild conditions, a consistent estimator of the parameter can be obtained by minimizing a binariness criterion for the output data. Unlike many other blind identification and deconvolution methods, this criterion handles nonstationary signals and does not utilize any moment information of the inputs. Three numerical examples are presented to demonstrate the effectiveness of the proposed method.>