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Adaptive Covariance Estimation Of Locally Stationary Processes
, 1995
"... this paper so we will not mention this explicitly. The ideas and methods of Calderon and Zygmund [7] in harmonic analysis have shown that although we are not able to find the ..."
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Cited by 67 (7 self)
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this paper so we will not mention this explicitly. The ideas and methods of Calderon and Zygmund [7] in harmonic analysis have shown that although we are not able to find the
Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum
, 1998
"... This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power va ..."
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Cited by 46 (27 self)
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This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable timelocalized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.
Thresholding Estimators for Linear Inverse Problems and Deconvolutions
, 2003
"... Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Theta, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconv ..."
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Cited by 25 (1 self)
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Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Theta, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new "mirror wavelet" basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.
Nonparametric Curve Estimation By Wavelet Thresholding With Locally Stationary Errors
, 1998
"... In the modeling of biological phenomena, in living organisms whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., one of the important aspects is time variation in the data. Thus, the recovery of a "smooth" regression or trend function from nois ..."
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Cited by 18 (7 self)
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In the modeling of biological phenomena, in living organisms whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., one of the important aspects is time variation in the data. Thus, the recovery of a "smooth" regression or trend function from noisy timevarying sampled data becomes a problem of particular interest. Here we use nonlinear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, nonstationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g., that of a timevarying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a nearoptimal rate for the L 2 risk between the unknown fu...
Adapted Bases Of TimeFrequency Local Cosines
 MODM (Section IIC) O(L 2 · N log N) O(N3 log N) O(N log N) SImodM (Section IIE) O((M + L 2 ) · N log N) O(N3 log N) O(N2 log N) MINM (Section IIIA) O(N 2 · N log N) O(N3 log N) O(N3 log N) BLOCKS with blockwise MODM (Section IIIB) O(NL 2 log M2 +
, 1999
"... . We develop and analyze a best basis algorithm for orthonormal bases of local cosines which satisfy a uniform bound on their timefrequency concentration. All waveforms are obtained from three elementary window functions by shifts, rescaling, and modulation. For a discrete signal of length N , the ..."
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Cited by 6 (0 self)
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. We develop and analyze a best basis algorithm for orthonormal bases of local cosines which satisfy a uniform bound on their timefrequency concentration. All waveforms are obtained from three elementary window functions by shifts, rescaling, and modulation. For a discrete signal of length N , the complexity is of order N(logN ) 2 . 1. Introduction A natural version of local Fourier analysis is to start with a segmentation of the line in disjoint intervals R = [ I2S I and attempt to decompose a given function in timefrequency atoms of the form v I (t) e i!t ; (1.1) where v I (t) is a smooth window function localized around the interval I. A major breakthrough in this field was the discovery that orthonormal bases of L 2 (R) can be obtained if the complex exponentials in (1.1) are replaced by cosines or sines. The first step is to use window functions whose supports are finite length intervals that overlap only in pairs. Orthogonality is then achieved by exploiting even/odd pr...
SLEX Analysis of Multivariate Nonstationary Time Series
"... We develop a procedure for analyzing multivariate nonstationary time series using the SLEX library (smooth localized complex exponentials), which is a collection of bases, each basis consisting of waveforms that are orthogonal and timelocalized versions of the Fourier complex exponentials. Under th ..."
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Cited by 6 (1 self)
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We develop a procedure for analyzing multivariate nonstationary time series using the SLEX library (smooth localized complex exponentials), which is a collection of bases, each basis consisting of waveforms that are orthogonal and timelocalized versions of the Fourier complex exponentials. Under the SLEX framework, we build a family of multivariate models that can explicitly characterize the timevarying spectral and coherence properties. Every model has a spectral representation in terms of a unique SLEX basis. Before selecting a model, we first decompose the multivariate time series into nonstationary components with uncorrelated (nonredundant) spectral information. The best SLEX model is selected using the penalized log energy criterion, which we derive in this article to be the Kullback–Leibler distance between a model and the SLEX principal components of the multivariate time series. The model selection criterion takes into account all of the pairwise crosscorrelation simultaneously in the multivariate time series. The proposed SLEX analysis gives results that are easy to interpret, because it is an automatic timedependent generalization of the classical Fourier analysis of stationary time series. Moreover, the SLEX method uses computationally efficient algorithms and hence is able to provide a systematic framework for extracting spectral features from a massive dataset. We illustrate the SLEX analysis with an application to a multichannel brain wave dataset recorded during an epileptic seizure. KEY WORDS: Multivariate nonstationary time series; SLEX model; SLEX principal components; SLEX transform; Timefrequency analysis; Timevarying spectral matrix. 1.
Locally Stationary Covariance and Signal Estimation with Macrotiles
 IEEE Trans. Signal Process
, 2001
"... A macrotile estimation algorithm is introduced to estimate the covariance of locally stationary processes. A macrotile algorithm uses a penalized method to optimize the partition of the space in orthogonal subspaces, and the estimation is computed with a projection operator. ..."
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Cited by 3 (0 self)
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A macrotile estimation algorithm is introduced to estimate the covariance of locally stationary processes. A macrotile algorithm uses a penalized method to optimize the partition of the space in orthogonal subspaces, and the estimation is computed with a projection operator.
A NONPARAMETRIC TEST FOR STATIONARITY BASED ON LOCAL FOURIER ANALYSIS
"... In this paper we propose a nonparametric hypothesis test for stationarity based on local Fourier analysis. We employ a test statistic that measures the variation of timelocalized estimates of the power spectral density of an observed random process. For the case of a white Gaussian noise process, w ..."
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Cited by 3 (1 self)
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In this paper we propose a nonparametric hypothesis test for stationarity based on local Fourier analysis. We employ a test statistic that measures the variation of timelocalized estimates of the power spectral density of an observed random process. For the case of a white Gaussian noise process, we characterize the asymptotic distribution of this statistic under the null hypothesis of stationarity, and use it to directly set test thresholds corresponding to constant false alarm rates. For other cases, we introduce a simple procedure to simulate from the null distribution of interest. After validating the procedure on synthetic examples, we demonstrate one potential use for the test as a method of obtaining a signaladaptive means of local Fourier analysis and corresponding signal enhancement scheme. Index Terms — Hypothesis testing, stationarity, adaptive STFT, nonparametric spectral estimation, Wold decomposition
Timefrequency spectral estimation of multichannel EEG using the autoSLEX method
 IEEE Transactions on Biomedical Engineering
, 2002
"... Abstract—In this paper, we apply a new timefrequency spectral estimation method for multichannel data to epileptiform electroencephalography (EEG). The method is based on the smooth localized complex exponentials (SLEX) functions which are timefrequency localized versions of the Fourier functions ..."
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Abstract—In this paper, we apply a new timefrequency spectral estimation method for multichannel data to epileptiform electroencephalography (EEG). The method is based on the smooth localized complex exponentials (SLEX) functions which are timefrequency localized versions of the Fourier functions and, hence, are ideal for analyzing nonstationary signals whose spectral properties evolve over time. The SLEX functions are simultaneously orthogonal and localized in time and frequency because they are obtained by applying a projection operator rather than a window or taper. In this paper, we present the AutoSLEX method which is a statistical method that 1) computes the periodogram using the SLEX transform, 2) automatically segments the signal into approximately stationary segments using an objective criterion that is based on log energy, and 3) automatically selects the optimal bandwidth of the spectral smoothing window. The method is applied to the intracranial EEG from a patient with temporal lobe epilepsy. This analysis reveals a reduction in average duration of stationarity in preseizure epochs of data compared to baseline. These changes begin up to hours prior to electrical seizure onset in this patient. Index Terms—Electroencephalography, spectral analysis, stochastic processes, timefrequency analysis. I.
Local Spectral Envelope: An Approach Using Dyadic TreeBased Adaptive Segmentation
, 2002
"... The concept of the spectral envelope was introduced as a statistical basis for the frequency domain analysis and scaling of qualitativevalued time series. ..."
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Cited by 1 (0 self)
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The concept of the spectral envelope was introduced as a statistical basis for the frequency domain analysis and scaling of qualitativevalued time series.