Wavelet-based Estimation Research Articles

A reexamination of fractional integrating dynamics in foreign currency markets

This paper reexamines foreign currency markets for evidence of fractional integration, and extends the extant literature in several important dimensions. First, we utilize a new semiparametric wavelet-based estimator, which is far superior to the more prevalent GPH estimator on the basis of mean squared error. Second, we utilize a broader and longer sample, which better facilitates the detection of long memory dynamics. Our analysis yields interesting empirical results that contrast with other recent studies. In particular, we find new evidence that a large proportion (14 out of 19) of exchange rate series display evidence of long memory, with little variation over alternative sample periods and alternative frequencies.

Wavelet-based estimation with multiple sampling rates

We suggest an adaptive sampling rule for obtaining information from noisy signals using wavelet methods. The technique involves increasing the sampling rate when relatively high-frequency terms are incorporated into the wavelet estimator, and decreasing it when, again using thresholded terms as an empirical guide, signal complexity is judged to have decreased. Through sampling in this way the algorithm is able to accurately recover relatively complex signals without increasing the long-run average expense of sampling. It achieves this level of performance by exploiting the opportunities for near-real time sampling that are available if one uses a relatively high primary resolution level when constructing the basic wavelet estimator. In the practical problems that motivate the work, where signal to noise ratio is particularly high and the long-run average sampling rate may be several hundred thousand operations per second, high primary resolution levels are quite feasible.

CHARACTERIZING DYNAMIC SPECKLE TIME SERIES WITH THE HURST COEFFICIENT CONCEPT

We propose in this work a dynamic speckle descriptor based on a Hurst wavelet estimator. The dynamic speckle or biospeckle is a phenomenon produced by laser illumination of active materials, such as biological tissue or the drying process of paint. Dynamic speckle interferometry is a useful technique for assessing the time evolution of surfaces as also to segment the loci of different activity in living tissues. Considering previous biospeckle characterization based on the autocorrelation function and its relation with the Hurst coefficient, a wavelet-based estimator is proposed as a feature extraction of the dynamic speckle characteristic. Encouraging results of the descriptor performance are obtained via three different experiments: a time history of speckle pattern applied to the drying of painting, segmenting regions in whole field image applied to the viability test of a corn seed and also to the bruising in fruits.

Fast Real-Time Hurst Parameter Estimation via Adaptive Wavelet Lifting

A new real-time estimator of the Hurst parameter of a long-range dependent process is developed based on the lifting scheme for wavelet transform. Compared with the existing wavelet-based estimator, the new method performs inplace computation and reduces the computational complexity by about half. We also propose a median-based nonlinear lifting scheme to mitigate the border effects and the noise in the data, by adaptively adjusting the number of vanishing moments of the wavelet. The proposed algorithms are applied to estimate the Hurst parameters in two types of long-range dependent processes, namely, the multiple-access interference in a code-division multiple-access packet data network, and the measurement data of Internet traffic.

Non-stationarity and high-order scaling in TCP flow arrivals

The last decade has been a very fruitful period in important discoveries in network traffic modeling, uncovering various scaling behaviors. Self-similarity, long-range dependence, multifractal behavior and finally cascades have been studied and convincingly matched to real traffic. The first purpose of this paper is to provide a methodology to go beyond the naive analysis of the second-order wavelet-based estimators of scaling, by performing non-stationarity checks and relying on the information contained in the high-order properties of the wavelet coefficients. Then, we apply this methodology to study the scaling properties of the TCP flow arrivals based on several traffic traces spanning the years from 1993 to early 2002. Our study reveals that the second-order scaling properties of this process describe its dynamics quite well. However, our analysis also provides evidence that high-order scaling in this process appears due to pathological behaviors like rate limitation and non-stationarity.

Multiscale likelihood analysis and complexity penalized estimation

We describe here a framework for a certain class of multiscale likelihood factorizations wherein, in analogy to a wavelet decomposition of an L2 function, a given likelihood function has an alternative representation as a product of conditional densities reflecting information in both the data and the parameter vector localized in position and scale. The framework is developed as a set of sufficient conditions for the existence of such factorizations, formulated in analogy to those underlying a standard multiresolution analysis for wavelets, and hence can be viewed as a multiresolution analysis for likelihoods. We then consider the use of these factorizations in the task of nonparametric, complexity penalized likelihood estimation. We study the risk properties of certain thresholding and partitioning estimators, and demonstrate their adaptivity and near-optimality, in a minimax sense over a broad range of function spaces, based on squared Hellinger distance as a loss function. In particular, our results provide an illustration of how properties of classical wavelet-based estimators can be obtained in a single, unified framework that includes models for continuous, count and categorical data types.

Extracting symptoms of bearing faults in the wavelet domain

Improving the signal-to-noise (S/N) ratio is one of the classic machinery failure diagnostic problems. A new failure detection procedure using a wavelet-based estimator is proposed for extracting the symptoms of bearing faults under environmental noise conditions. The paper discusses the relationship between de-noising efficiency and S/N ratio under different threshold values and wavelet decompositions. Efficient ranges of wavelet coefficients and levels of wavelet decomposition for extracting the symptoms of bearing faults are obtained. A comparison of the failure detection capabilities of an envelope detector and a wavelet-based estimator under significant environmental noise is performed and the results show that the wavelet-based estimator is an effective method for extracting the symptoms of a bearing fault under such conditions.

Extracting symptoms of bearing faults in the wavelet domain

Improving the signal-to-noise (S/N) ratio is one of the classic machinery failure diagnostic problems. A new failure detection procedure using a wavelet-based estimator is proposed for extracting the symptoms of bearing faults under environmental noise conditions. The paper discusses the relationship between de-noising efficiency and S/N ratio under different threshold values and wavelet decompositions. Efficient ranges of wavelet coefficients and levels of wavelet decomposition for extracting the symptoms of bearing faults are obtained. A comparison of the failure detection capabilities of an envelope detector and a wavelet-based estimator under significant environmental noise is performed and the results show that the wavelet-based estimator is an effective method for extracting the symptoms of a bearing fault under such conditions.

Counting process intensity estimation by orthogonal wavelet methods

In recent years wavelet-based methods have been proposed to estimate density and regression functions, most often in settings with independent and identically distributed observations. An attractive feature of these methods is that the rate of convergence is unaffected by the presence of discontinuities in the function being estimated. We provide structure and mean-square analyses of wavelet-based estimators of counting process intensities in the context of the multiplicative intensity model.

Wavelets and functional magnetic resonance imaging of the human brain

The discrete wavelet transform (DWT) is widely used for multiresolution analysis and decorrelation or "whitening" of nonstationary time series and spatial processes. Wavelets are naturally appropriate for analysis of biological data, such as functional magnetic resonance images of the human brain, which often demonstrate scale invariant or fractal properties. We provide a brief formal introduction to key properties of the DWT and review the growing literature on its application to fMRI. We focus on three applications in particular: (i) wavelet coefficient resampling or "wavestrapping" of 1-D time series, 2- to 3-D spatial maps and 4-D spatiotemporal processes; (ii) wavelet-based estimators for signal and noise parameters of time series regression models assuming the errors are fractional Gaussian noise (fGn); and (iii) wavelet shrinkage in frequentist and Bayesian frameworks to support multiresolution hypothesis testing on spatially extended statistic maps. We conclude that the wavelet domain is a rich source of new concepts and techniques to enhance the power of statistical analysis of human fMRI data.

Wavelet-based estimation of a semiparametric generalized linear model of fMRI time-series

This paper addresses the problem of detecting significant changes in fMRI time series that are correlated to a stimulus time course. This paper provides a new approach to estimate the parameters of a semiparametric generalized linear model of fMRI time series. The fMRI signal is described as the sum of two effects: a smooth trend and the response to the stimulus. The trend belongs to a subspace spanned by large scale wavelets. The wavelet transform provides an approximation to the Karhunen-Loève transform for the long memory noise and we have developed a scale space regression that permits to carry out the regression in the wavelet domain while omitting the scales that are contaminated by the trend. In order to demonstrate that our approach outperforms the state-of-the art detrending technique, we evaluated our method against a smoothing spline approach. Experiments with simulated data and experimental fMRI data, demonstrate that our approach can infer and remove drifts that cannot be adequately represented with splines.

Asymptotic Theory for the Multiscale Wavelet Density Derivative Estimator

In this paper we consider the wavelet-based estimation of density derivatives. The multiscale density derivative estimator is proposed which is constructed by using a number of scaling functions. Asymptotic theory is developed in which asymptotic expressions for the bias, the variance and the mean integrated squared error are included. In addition, asymptotic normality of the proposed estimator is proved. Theoretical and numerical comparisons with the usual kernel-based estimators are also reported.

Bearing Fault Diagnostic Using Wavelet Transforms (2nd Report, Application of the Wavelet De-Noising for Fault Diagnostic)

Extracting symptom of bearing faults from noisy signal is a key problem to detect early faults of machinery. An effective method is presented for improving the signal to noise ratio by the wavelet transforms. The paper, which is Part II of a pair, illustrates the advantage of the wavelet de-noising procedure on a range of practical examples. We have made a comparison of failure detection-ability between envelope estimator and wavelet-based estimator under the noise condition of machinery. Experiment results have shown that using wavelet de-noising procedure is an effective means to extract early symptoms of machine fault form the heavy noise condition of machinery.

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.

Wavelet-based estimators of scaling behavior

Various wavelet-based estimators of self-similarity or long-range dependence scaling exponent are studied extensively. These estimators mainly include the (bi)orthogonal wavelet estimators and the wavelet transform modulus maxima (WTMM) estimator. This study focuses both on short and long time-series. In the framework of fractional autoregressive integrated moving average (FARIMA) processes, we advocate the use of approximately adapted wavelet estimators. For these ideal processes, the scaling behavior actually extends down to the smallest scale, i.e., the sampling period of the time series, if an adapted decomposition is used. But in practical situations, there generally exists a cutoff scale below which the scaling behavior no longer holds. We test the robustness of the set of wavelet-based estimators with respect to that cutoff scale as well as to the specific density of the underlying law of the process. In all situations, the WTMM estimator is shown to be the best or among the best estimators in terms of the mean-squared error (MSE). We also compare the wavelet estimators with the detrended fluctuation analysis (DFA) estimator which was previously proved to be among the best estimators which are not wavelet-based estimators. The WTMM estimator turns out to be a very competitive estimator which can be further generalized to characterize multiscaling behavior.

From Blackman–Tukey pilot estimators to wavelet packet estimators: a modern perspective on an old spectrum estimation idea

The ubiquitous problem of good spectrum estimation has been researched for many years, and in their seminal 1958 book Blackman and Tukey describe a ‘pilot spectrum,’ a ‘simple and cheap preliminary estimate’ of the spectral density function. They also define a modification to the simple pilot estimator, the ‘complete version.’ Remarkably, both these estimators are in fact wavelet-based estimators, using the Haar wavelet filter. In this paper we update and extend the ‘complete’ pilot estimator of Blackman and Tukey by firstly utilising the discrete wavelet packet transform rather than the discrete wavelet transform previously implicitly used in the basic pilot estimator, thus enabling uniform frequency resolution, and secondly by using the maximal overlap, or undecimated, version (MODWPT), the equivalent of a ‘complete’ estimator. Both equiripple and maximally flat wavelet filter designs are considered. For rapidly varying or high dynamic range power spectra the technique of rejection filtration, also discussed in the same book, is implemented, but again using the MODWPT. The result is a spectrum estimator easily derivable from wavelet software, and which is extremely competitive with other existing spectrum estimation approaches.

Bayesian tree-structured image modeling using wavelet-domain hidden Markov models

Wavelet-domain hidden Markov models have proven to be useful tools for statistical signal and image processing. The hidden Markov tree (HMT) model captures the key features of the joint probability density of the wavelet coefficients of real-world data. One potential drawback to the HMT framework is the need for computationally expensive iterative training to fit an HMT model to a given data set (e.g., using the expectation-maximization algorithm). We greatly simplify the HMT model by exploiting the inherent self-similarity of real-world images. The simplified model specifies the HMT parameters with just nine meta-parameters (independent of the size of the image and the number of wavelet scales). We also introduce a Bayesian universal HMT (uHMT) that fixes these nine parameters. The uHMT requires no training of any kind, while extremely simple, we show using a series of image estimation/denoising experiments that these new models retain nearly all of the key image structure modeled by the full HMT. Finally, we propose a fast shift-invariant HMT estimation algorithm that outperforms other wavelet-based estimators in the current literature, both visually and in mean square error.

Robust wavelet denoising

For extracting a signal from noisy data, waveshrink and basis pursuit are powerful tools both from an empirical and asymptotic point of view. They are especially efficient at estimating spatially inhomogeneous signals when the noise is Gaussian. Their performance is altered when the noise has a long tail distribution, for instance, when outliers are present. We propose a robust wavelet-based estimator using a robust loss function. This entails solving a nontrivial optimization problem and appropriately choosing the smoothing and robustness parameters. We illustrate the advantage of the robust wavelet denoising procedure on simulated and real data.

A statistical test for the time constancy of scaling exponents

A statistical test is described for determining if scaling exponents vary over time. It is applicable to diverse scaling phenomena including long-range dependence and exactly self-similar processes in a uniform framework without the need for prior knowledge of the type in question. It is based on the special properties of wavelet-based estimates of the scaling exponent, strongly motivating an idealized inference problem: the equality or otherwise of means of independent Gaussian variables with known variances. A uniformly most powerful invariant test exists for this problem and is described. A separate uniformly most powerful invariant test is also given for when the scaling exponent undergoes a level change. The power functions of both tests are given explicitly and compared. Using simulation, the effect, in practice, of deviations from the idealizations made of the statistical properties of the wavelet detail coefficients are analyzed and found to be small. The tests inherit the significant robustness and computational advantages of the underlying wavelet-based estimator. A detailed methodology is given, describing its use in practical situations. The use and benefits of the test are illustrated on the Bellcore Ethernet data sets.

Asymptotics for Wavelet Based Estimates of Piecewise Smooth Regression for Stationary Time Series

Wavelet methods are used to estimate density and (auto-) regression functions that are possibly discontinuous. For stationary time series that satisfy appropriate mixing conditions, we derive mean integrated squared errors (MISEs) of wavelet-based estimators. In contrast to the case for kernel methods, the MISEs of wavelet-based estimators are not affected by the presence of discontinuities in the curves. Applications of this approach to problems of identification of nonlinear time series models are discussed.