Non-parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., is time variation in the data. Thus, the recovery of a 'smooth' regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear 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, non-stationarity 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 time-varying 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 near-optimal rate for the -risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non-stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear non-stationarity in its variance.     

Smoothed detrended fluctuation analysis

ABSTRACT

The method of detrended fluctuation analysis (DFA) is useful in revealing the extent of long-range dependence, it has successfully been applied to different fields of interest. In this paper we proposed a smoothed detrended fluctuation analysis method based on the principle of wavelet shrinkage. The procedure is illustrated and compared with the DFA method by Monte Carlo simulations on fractional Gaussian noise models.     

Deconvolution of supersmooth densities with smooth noise

The author considers the estimation of the common probability density of independent and identically distributed random variables observed with added white noise. She assumes that the unknown density belongs to some class of supersmooth functions, and that the error distribution is ordinarily smooth, meaning that its characteristic function decays polynomially asymptotically. In this context, the author evaluates the minimax rate of convergence of the pointwise risk and describes a kernel estimator having this rate. She computes upper bounds for the L₂ risk of this estimator.     

ON A STRONGLY CONSISTENT WAVELET DENSITY ESTIMATOR FOR THE DECONVOLUTION PROBLEM

ABSTRACT

The problem of wavelet density estimation is studied when the sample observations are contaminated with random noise. In this paper a linear wavelet estimator based on Meyer-type wavelets is shown to be strongly consistent when Fourier transform of random noise has polynomial descent or exponential descent.     

上海股票市场的基于小波去噪的混沌性检验

由于经济混沌需要大样本、低噪声的时间序列,所以文章首先利用小波变换对上证指数日收盘价序列进行去噪处理,然后由去噪后的日收盘价序列计算出日收益率序列,姑且称其为去噪后的日收益率序列,并把它同未经过去噪处理得到的日收益率序列进行比较,发现该方法较好地保留了序列自身固有的特性,只是剔除了由于日常细微波动产生的噪声,为有效地探测我国上海证券市场的混沌性打下了基础。最后分别计算去噪前后收益率的关联维数和Lyapunov指数,发现小波去噪并未改变上海证券市场的混沌性,但是去噪后的市场的复杂度要小于去噪前的市场的复杂度。所以进行混沌性探测的时候必须对数据进行去噪处理。     

On Estimating the Integrated Co-Volatility Using Noisy High-Frequency Data with Jumps

In this article, we consider the estimation of covariation of two asset prices which contain jumps and microstructure noise, based on high-frequency data. We propose a realized covariance estimator, which combines pre-averaging method to remove the microstructure noise and the threshold method to reduce the jumps effect. The asymptotic properties, such as consistency and asymptotic normality, are investigated. The estimator allows very general structure of jumps, for example, infinity activity or even infinity variation. Simulation is also included to illustrate the performance of the proposed procedure.     

Minimax A‐ and D‐optimal integer‐valued wavelet designs for estimation

The author discusses integer‐valued designs for wavelet estimation of nonparametric response curves in the possible presence of heteroscedastic noise using a modified wavelet version of the Gasser‐Müller kernel estimator or weighted least squares estimation. The designs are constructed using a minimax treatment and the simulated annealing algorithm. The author presents designs for three case studies in experiments for investigating Gompertz's theory on mortality rates, nitrite utilization in bush beans and the impact of crash helmets in motorcycle accidents.     

Assessing direct and indirect seasonal decomposition in state space

The problem of whether seasonal decomposition should be used prior to or after aggregation of time series is quite old. We tackle the problem by using a state-space representation and the variance/covariance structure of a simplified one-component model. The variances of the estimated components in a two-series system are compared for direct and indirect approaches and also to a multivariate method. The covariance structure between the two time series is important for the relative efficiency. Indirect estimation is always best when the series are independent, but when the series or the measurement errors are negatively correlated, direct estimation may be much better in the above sense. Some covariance structures indicate that direct estimation should be used while others indicate that an indirect approach is more efficient. Signal-to-noise ratios and relative variances are used for inference.     

Detection of Change Point in Nonparametric Function with Unit-Root Noise by Wavelet

A wavelet method is proposed to detect jumps in a function which is observed with unit-root noise. We obtain critical values at any scale and prove the consistency of wavelet detection when the nonparametric function is smooth. It shows that the estimation of the number and locations of change points are consistent when there are change points in the nonparametric function. Simulation study supports our method.     

Full Bayesian wavelet inference with a nonparametric prior

In this paper, we introduce a new Bayesian nonparametric model for estimating an unknown function in the presence of Gaussian noise. The proposed model involves a mixture of a point mass and an arbitrary (nonparametric) symmetric and unimodal distribution for modeling wavelet coefficients. Posterior simulation uses slice sampling ideas and the consistency under the proposed model is discussed. In particular, the method is shown to be computationally competitive with some of best Empirical wavelet estimation methods.     

Multiscale detection and location of multiple variance changes in the presence of long memory

Procedures for detecting change points in sequences of correlated observations (e.g., time series) can help elucidate their complicated structure. Current literature on the detection of multiple change points emphasizes the analysis of sequences of independent random variables. We address the problem of an unknown number of variance changes in the presence of long-range dependence (e.g., long memory processes). Our results are also applicable to time series whose spectrum slowly varies across octave bands. An iterated cumulative sum of squares procedure is introduced in order to look at the multiscale stationarity of a time series; that is, the variance structure of the wavelet coefficients on a scale by scale basis. The discrete wavelet transform enables us to analyze a given time series on a series of physical scales. The result is a partitioning of the wavelet coefficients into locally stationary regions. Simulations are performed to validate the ability of this procedure to detect and locate multiple variance changes. A ‘time’ series of vertical ocean shear measurements is also analyzed, where a variety of nonstationary features are identified.     

Simultaneous Wavelet Deconvolution in Periodic Setting

Abstract.  The paper proposes a method of deconvolution in a periodic setting which combines two important ideas, the fast wavelet and Fourier transform-based estimation procedure of Johnstone et al . [ J. Roy. Statist. Soc. Ser. B 66 (2004) 547] and the multichannel system technique proposed by Casey and Walnut [ SIAM Rev . 36 (1994) 537]. An unknown function is estimated by a wavelet series where the empirical wavelet coefficients are filtered in an adapting non-linear fashion. It is shown theoretically that the estimator achieves optimal convergence rate in a wide range of Besov spaces. The procedure allows to reduce the ill-posedness of the problem especially in the case of non-smooth blurring functions such as boxcar functions: it is proved that additions of extra channels improve convergence rate of the estimator. Theoretical study is supplemented by an extensive set of small-sample simulation experiments demonstrating high-quality performance of the proposed method.     

Wavelet Threshold Estimators for Data with Correlated Noise

Wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level-dependent soft threshold to the coefficients in the wavelet transform. A variety of threshold choices is proposed, including one based on an unbiased estimate of mean-squared error. The practical performance of the method is demonstrated on examples, including data from a neurophysiological context. The theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable `bench-mark', that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity, and more generally is obtained by the use of an `oracle' that provides information that is not actually available in the data. It is shown that the level-dependent threshold estimator performs well relative to the bench-mark risk, and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator. The wavelet domain structure of both short- and long-range dependent noise is considered, and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes, adapting automatically to the regularity properties of the underlying model. The proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem.     

Empirical Bayes approach to wavelet regression using ϵ-contaminated priors

We consider an empirical Bayes approach to standard nonparametric regression estimation using a nonlinear wavelet methodology. Instead of specifying a single prior distribution on the parameter space of wavelet coefficients, which is usually the case in the existing literature, we elicit the ?-contamination class of prior distributions that is particularly attractive to work with when one seeks robust priors in Bayesian analysis. The type II maximum likelihood approach to prior selection is used by maximizing the predictive distribution for the data in the wavelet domain over a suitable subclass of the ?-contamination class of prior distributions. For the prior selected, the posterior mean yields a thresholding procedure which depends on one free prior parameter and it is level- and amplitude-dependent, thus allowing better adaptation in function estimation. We consider an automatic choice of the free prior parameter, guided by considerations on an exact risk analysis and on the shape of the thresholding rule, enabling the resulting estimator to be fully automated in practice. We also compute pointwise Bayesian credible intervals for the resulting function estimate using a simulation-based approach. We use several simulated examples to illustrate the performance of the proposed empirical Bayes term-by-term wavelet scheme, and we make comparisons with other classical and empirical Bayes term-by-term wavelet schemes. As a practical illustration, we present an application to a real-life data set that was collected in an atomic force microscopy study.     

Adaptive wavelet series estimation in separable nonparametric regression models

It is well-known that multivariate curve estimation suffers from the curse of dimensionality. However, reasonable estimators are possible, even in several dimensions, under appropriate restrictions on the complexity of the curve. In the present paper we explore how much appropriate wavelet estimators can exploit a typical restriction on the curve such as additivity. We first propose an adaptive and simultaneous estimation procedure for all additive components in additive regression models and discuss rate of convergence results and data-dependent truncation rules for wavelet series estimators. To speed up computation we then introduce a wavelet version of functional ANOVA algorithm for additive regression models and propose a regularization algorithm which guarantees an adaptive solution to the multivariate estimation problem. Some simulations indicate that wavelets methods complement nicely the existing methodology for nonparametric multivariate curve estimation.     

Robust estimation for boundary correction in wavelet regression

For boundary problems present in wavelet regression, two common methods are usually considered: polynomial wavelet regression (PWR) and hybrid local polynomial wavelet regression (LPWR). Normality assumption played a key role for making such choices for the order of the low-order polynomial, the wavelet thresholding value and other calculations involved in LPWR. However, in practice, the normality assumption may not be valid. In this paper, for PWR, we propose three automatic robust methods based on: MM-estimator, bootstrap and robust threshold procedure. For LPWR, the use of a robust local polynomial (RLP) estimator with a robust threshold procedure has been investigated. The proposed methods do not require any knowledge of noise distribution, are easy to implement and achieve high performances when only a small amount of data is in hand. A simulation study is conducted to assess the numerical performance of the proposed methods.     

Denoising radiocommunications signals by using iterative wavelet shrinkage

Summary. Radiocommunications signals pose particular problems in the context of statistical signal processing. This is because short-term fluctuations (noise) are a consequence of atmospheric effects whose characteristics vary in both the short and the longer term. We contrast traditional time domain and frequency domain filters with wavelet methods. We also propose an iterative wavelet procedure which appears to provide benefits over existing wavelet methods.     

A new process for modeling heartbeat signals during exhaustive run with an adaptive estimator of its fractal parameters

This paper is devoted to a new study of the fractal behavior of heartbeats during a marathon. Such a case is interesting since it allows the examination of heart behavior during a very long exercise in order to reach reliable conclusions on the long-term properties of heartbeats. Three points of this study can be highlighted. First, the whole race heartbeats of each runner are automatically divided into several stages where the signal is nearly stationary and these stages are detected with an adaptive change points detection method. Secondly, a new process called the locally fractional Gaussian noise (LFGN) is proposed to fit such data. Finally, a wavelet-based method using a specific mother wavelet provides an adaptive procedure for estimating low frequency and high frequency fractal parameters as well as the corresponding frequency bandwidths. Such an estimator is theoretically proved to converge in the case of LFGNs, and simulations confirm this consistency. Moreover, an adaptive chi-squared goodness-of-fit test is also built, using this wavelet-based estimator. The application of this method to marathon heartbeat series indicates that the LFGN fits well data at each stage and that the low frequency fractal parameter increases during the race. A detection of a too large low frequency fractal parameter during the race could help prevent the too frequent heart failures occurring during marathons.     

Minimax estimation via wavelets for indirect long-memory data

In this paper, we model linear inverse problems with long-range dependence by a fractional Gaussian noise model and study function estimation based on observations from the model. By using two wavelet-vaguelette decompositions, one for the inverse problem which simultaneously quasi-diagonalizes both the operator and the prior information, and one for long-range dependence which decorrelates fractional Gaussian noise, we establish asymptotics for minimax risks, and show that the wavelet shrinkage estimate can be tuned to achieve the minimax convergence rate and significantly outperform linear estimates.     

Block thresholding wavelet regression using SCAD penalty

This paper concerns wavelet regression using a block thresholding procedure. Block thresholding methods utilize neighboring wavelet coefficients information to increase estimation accuracy. We propose to construct a data-driven block thresholding procedure using the smoothly clipped absolute deviation (SCAD) penalty. A simulation study demonstrates competitive finite sample performance of the proposed estimator compared to existing methods. We also show that the proposed estimator achieves optimal convergence rates in Besov spaces.