On simple wavelet estimators of random signals and their small-sample properties

In the paper we suggest certain nonparametric estimators of random signals based on the wavelet transform. We consider stochastic signals embedded in white noise and extractions with wavelet denoizing algorithms utilizing the non-decimated discrete wavelet transform and the idea of wavelet scaling. We evaluate properties of these estimators via extensive computer simulations and partially also analytically. Our wavelet estimators of random signals have clear advantages over parametric maximum likelihood methods as far as computational issues are concerned, while at the same time they can compete with these methods in terms of precision of estimation in small samples. An illustrative example concerning smoothing of survey data is also provided.     

On wavelet analysis of the nth order fractional Brownian motion

In this paper, we investigate the use of wavelet techniques in the study of the nth order fractional Brownian motion (n-fBm). First, we exploit the continuous wavelet transform??s capabilities in derivative calculation to construct a two-step estimator of the scaling exponent of the n-fBm process. We show, via simulation, that the proposed method improves the estimation performance of the n-fBm signals contaminated by large-scale noise. Second, we analyze the statistical properties of the n-fBm process in the time-scale plan. We demonstrate that, for a convenient choice of the wavelet basis, the discrete wavelet detail coefficients of the n-fBm process are stationary at each resolution level whereas their variance exhibits a power-law behavior. Using the latter property, we discuss a weighted least squares regression based-estimator for this class of stochastic process. Experiments carried out on simulated and real-world datasets prove the relevance of the proposed method.     

Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective

We present theoretical results on the random wavelet coefficients covariance structure. We use simple properties of the coefficients to derive a recursive way to compute the within- and across-scale covariances. We point out a useful link between the algorithm proposed and the two-dimensional discrete wavelet transform. We then focus on Bayesian wavelet shrinkage for estimating a function from noisy data. A prior distribution is imposed on the coefficients of the unknown function. We show how our findings on the covariance structure make it possible to specify priors that take into account the full correlation between coefficients through a parsimonious number of hyperparameters. We use Markov chain Monte Carlo methods to estimate the parameters and illustrate our method on bench-mark simulated signals.     

Denoising ozone concentration measurements with BAMS filtering

We propose a method for filtering self-similar geophysical signals infected by an autoregressive noise using a combination of non-decimated wavelet transform and a Bayesian model. In the application part, we consider separating the instrumentation noise from high frequency ozone concentration measurements sampled in the atmospheric boundary layer. The elicitation of priors needed to specify the statistical model in this application is guided by the well-known Kolmogorov K41-theory, which describes the statistical structure of turbulent high frequency scalar concentration fluctuations.     

Wavelet Regression Technique for Streamflow Prediction

In order to explain many secret events of natural phenomena, analyzing non-stationary series is generally an attractive issue for various research areas. The wavelet transform technique, which has been widely used last two decades, gives better results than former techniques for the analysis of earth science phenomena and for feature detection of real measurements. In this study, a new technique is offered for streamflow modeling by using the discrete wavelet transform. This new technique depends on the feature detection characteristic of the wavelet transform. The model was applied to two geographical locations with different climates. The results were compared with energy variation and error values of models. The new technique offers a good advantage through a physical interpretation. This technique is applied to streamflow regression models, because they are simple and widely used in practical applications. However, one can apply this technique to other models.     

Detection of glitches and signal reconstruction using Hölder and wavelet analysis

We present a method based on a local regularity analysis for detecting and removing artefact signatures in noisy interferometric signals. Using Hölder and wavelet transform modulus maxima lines analysis (WTMML) [S. Mallat, W. Hwang, Singularities detection and processing with wavelets, IEEE Transaction on Information Theory 38 (1992) 617–643] in suitably selected regions of the time-scale half-plane, we can estimate the regularity degree of the signal. Glitches that are considered as a discontinuity on the signal show Hölder component lower than a fixed threshold defined for a continuous signal. After detection and signature removal, the signal is then locally reconstructed using Mallat reconstruction formulae. The method has been tested with Herschel SPIRE FTS proto-flight model calibration observations and shows remarkable results. Optimization of the detection parameters has been performed on the correlation coefficient, the scale domain for Hölder exponent estimation and reconstruction for SPIRE FTS signals.     

A ‘nondecimated’ lifting transform

Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new ‘nondecimated’ lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a ‘best’ representation, which we shall explore.     

Wavelet goodness-of-fit test for dependent data

We introduce a new goodness-of-fit test which can be applied to hypothesis testing about the marginal distribution of dependent data. We derive a new test for the equivalent hypothesis in the space of wavelet coefficients. Such properties of the wavelet transform as orthogonality, localisation and sparsity make the hypothesis testing in wavelet domain easier than in the domain of distribution functions. We propose to test the null hypothesis separately at each wavelet decomposition level to overcome the problem of bi-dimensionality of wavelet indices and to be able to find the frequency where the empirical distribution function differs from the null in case the null hypothesis is rejected. We suggest a test statistic and state its asymptotic distribution under the null and under some of the alternative hypotheses.     

Wavelet shrinkage for unequally spaced data

Wavelet shrinkage (WaveShrink) is a relatively new technique for nonparametric function estimation that has been shown to have asymptotic near-optimality properties over a wide class of functions. As originally formulated by Donoho and Johnstone, WaveShrink assumes equally spaced data. Because so many statistical applications (e.g., scatterplot smoothing) naturally involve unequally spaced data, we investigate in this paper how WaveShrink can be adapted to handle such data. Focusing on the Haar wavelet, we propose four approaches that extend the Haar wavelet transform to the unequally spaced case. Each approach is formulated in terms of continuous wavelet basis functions applied to a piecewise constant interpolation of the observed data, and each approach leads to wavelet coefficients that can be computed via a matrix transform of the original data. For each approach, we propose a practical way of adapting WaveShrink. We compare the four approaches in a Monte Carlo study and find them to be quite comparable in performance. The computationally simplest approach (isometric wavelets) has an appealing justification in terms of a weighted mean square error criterion and readily generalizes to wavelets of higher order than the Haar.     

Multiscale Bayesian state-space model for Granger causality analysis of brain signal

Modelling time-varying and frequency-specific relationships between two brain signals is becoming an essential methodological tool to answer theoretical questions in experimental neuroscience. In this article, we propose to estimate a frequency Granger causality statistic that may vary in time in order to evaluate the functional connections between two brain regions during a task. We use for that purpose an adaptive Kalman filter type of estimator of a linear Gaussian vector autoregressive model with coefficients evolving over time. The estimation procedure is achieved through variational Bayesian approximation and is extended for multiple trials. This Bayesian State Space (BSS) model provides a dynamical Granger-causality statistic that is quite natural. We propose to extend the BSS model to include the à trous Haar decomposition. This wavelet-based forecasting method is based on a multiscale resolution decomposition of the signal using the redundant à trous wavelet transform and allows us to capture short- and long-range dependencies between signals. Equally importantly it allows us to derive the desired dynamical and frequency-specific Granger-causality statistic. The application of these models to intracranial local field potential data recorded during a psychological experimental task shows the complex frequency-based cross-talk between amygdala and medial orbito-frontal cortex.     

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.     

Posterior probability intervals for wavelet thresholding

Summary. We use cumulants to derive Bayesian credible intervals for wavelet regression estimates. The first four cumulants of the posterior distribution of the estimates are expressed in terms of the observed data and integer powers of the mother wavelet functions. These powers are closely approximated by linear combinations of wavelet scaling functions at an appropriate finer scale. Hence, a suitable modification of the discrete wavelet transform allows the posterior cumulants to be found efficiently for any given data set. Johnson transformations then yield the credible intervals themselves. Simulations show that these intervals have good coverage rates, even when the underlying function is inhomogeneous, where standard methods fail. In the case where the curve is smooth, the performance of our intervals remains competitive with established nonparametric regression methods.     

Real nonparametric regression using complex wavelets

Summary.  Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients.     

Forecasting Study of Shanghai’s and Shenzhen’s Stock Markets Using a Hybrid Forecast Method

In this article, a novel hybrid method to forecast stock price is proposed. This hybrid method is based on wavelet transform, wavelet denoising, linear models (autoregressive integrated moving average (ARIMA) model and exponential smoothing (ES) model), and nonlinear models (BP Neural Network and RBF Neural Network). The wavelet transform provides a set of better-behaved constitutive series than stock series for prediction. Wavelet denoising is used to eliminate some slight random fluctuations of stock series. ARIMA model and ES model are used to forecast the linear component of denoised stock series, and then BP Neural Network and RBF Neural Network are developed as tools for nonlinear pattern recognition to correct the estimation error of the prediction of linear models. The proposed method is examined in the stock market of Shanghai and Shenzhen and the results are compared with some of the most recent stock price forecast methods. The results show that the proposed hybrid method can provide a considerable improvement for the forecasting accuracy. Meanwhile, this proposed method can also be applied to analysis and forecast reliability of products or systems and improve the accuracy of reliability engineering.     

A non linear wavelet based estimator for long memory processes

Two wavelet based estimators are considered in this paper for the two parameters that characterize long range dependence processes. The first one is linear and is based on the statistical properties of the coefficients of a discrete wavelet transform of long range dependence processes. The estimator consists in measuring the slope (related to the long memory parameter) and the intercept (related to the variance of the process) of a linear regression after a discrete wavelet transform is performed (Veitch and Abry, 1999). In this paper its properties are reviewed, and analytic evidence is produced that the linear estimator is applicable only when the second parameter is unknown. To overcome this limitation a non linear wavelet based estimator - that takes into account that the intercept depends on the long memory parameter - is proposed here for the cases in which the second parameter is known or the only parameter of interest is the long memory parameter. Under the same hypothesis assumed for the linear estimator, the non linear estimator is shown to be asymptotically more efficient for the long memory parameter. Numerical simulations show that, even for small data sets, the bias is very small and the variance close to optimal. An application to ATM based Internet traffic is presented.Financial support from the Italian Ministry of University and Scientific Research (MIUR), also in the context of the COFIN 2002 ALINWEB (Algorithms for the Internet and the Web) Project, is gratefully acknowledged.     

Time–Threshold Maps: Using information from wavelet reconstructions with all threshold values simultaneously

Wavelets are a commonly used tool in science and technology. Often, their use involves applying a wavelet transform to the data, thresholding the coefficients and applying the inverse transform to obtain an estimate of the desired quantities. In this paper, we argue that it is often possible to gain more insight into the data by producing not just one, but many wavelet reconstructions using a range of threshold values and analysing the resulting object, which we term the Time–Threshold Map (TTM) of the input data. We discuss elementary properties of the TTM, in its “basic” and “derivative” versions, using both Haar and Unbalanced Haar wavelet families. We then show how the TTM can help in solving two statistical problems in the signal + noise model: breakpoint detection, and estimating the longest interval of approximate stationarity. We illustrate both applications with examples involving volatility of financial returns. We also briefly discuss other possible uses of the TTM.     

Wavelet fuzzy hybrid model for physico-financial signals

In the present paper, a fuzzy logic-based method is combined with wavelet decomposition to develop a step-by-step dynamic hybrid model to analyze and approximate one-dimensional physico-financial signals characterized by fuzzy values. Computational tests based on a well-known signal and conducted with the pure fuzzy model, the wavelet one and the new hybrid model, are developed and result in an efficient hybrid one.     

Matching pursuit by undecimated discrete wavelet transform for non-stationary time series of arbitrary length

We describe how to formulate a matching pursuit algorithm which successively approximates a periodic non-stationary time series with orthogonal projections onto elements of a suitable dictionary. We discuss how to construct such dictionaries derived from the maximal overlap (undecimated) discrete wavelet transform (MODWT). Unlike the standard discrete wavelet transform (DWT), the MODWT is equivariant under circular shifts and may be computed for an arbitrary length time series, not necessarily a multiple of a power of 2. We point out that when using the MODWT and continuing past the level where the filters are wrapped, the norms of the dictionary elements may, depending on N, deviate from the required value of unity and require renormalization.We analyse a time series of subtidal sea levels from Crescent City, California. The matching pursuit shows in an iterative fashion how localized dictionary elements (scale and position) account for residual variation, and in particular emphasizes differences in construction for varying parts of the series.     

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.     

工业取用水监测奇异数据挖掘与重构方法

提高工业取用水监测数据质量是目前国家水资源监控能力建设的重要内容,而奇异值问题已成为影响监测数据质量的关键短板。本文在解析现阶段工业取用水监测数据奇异值主要类型基础上,以国家水资源管理系统数据库中工业取用水监测数据为样本,利用小波变换模极大值模型提取工业取用水监测数据时频变化特征,并利用傅里叶函数对其残差序列进行修正,进而运用相对误差控制方法挖掘监测数据奇异值。在此基础上,采用混沌粒子群优化的最小二乘支持向量机模型重构填补奇异值数据。研究结果表明：小波变换模极大值模型能够较好地提取工业取用水监测数据序列的时频变化特征,但是同时容易导致监测数据的信息损失,利用傅里叶函数对小波变换进行残差修正则可进一步提升取用水监测数据序列的特征提取效果;以小波变换模极大值特征序列为基础,通过相对误差控制可实现对监测数据奇异值的高效挖掘;对于挖掘出的奇异值重构填补问题,可选取混沌粒子群优化的最小二乘支持向量机模型,其重构精度要优于多项式曲线拟合等传统统计学方法和普通最小二乘支持向量机模型。上述工业取用水监测数据奇异值挖掘重构策略为现阶段国家水资源监控能力建设的推进提供了重要技术方法支持。