Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (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 time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.     

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.     

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.     

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.     

A Bayesian wavelet approach to estimation of a change-point in a nonlinear multivariate time series

ABSTRACT

We propose a semiparametric approach to estimate the existence and location of a statistical change-point to a nonlinear multivariate time series contaminated with an additive noise component. In particular, we consider a p-dimensional stochastic process of independent multivariate normal observations where the mean function varies smoothly except at a single change-point. Our approach involves conducting a Bayesian analysis on the empirical detail coefficients of the original time series after a wavelet transform. If the mean function of our time series can be expressed as a multivariate step function, we find our Bayesian-wavelet method performs comparably with classical parametric methods such as maximum likelihood estimation. The advantage of our multivariate change-point method is seen in how it applies to a much larger class of mean functions that require only general smoothness conditions.     

Wavelet packet transfer function modelling of nonstationary time series

This article shows how a non-decimated wavelet packet transform (NWPT) can be used to model a response time series, Y _t, in terms of an explanatory time series, X _t. The proposed computational technique transforms the explanatory time series into a NWPT representation and then uses standard statistical modelling methods to identify which wavelet packets are useful for modelling the response time series. We exhibit S-Plus functions from the freeware WaveThresh package that implement our methodology.The proposed modelling methodology is applied to an important problem from the wind energy industry: how to model wind speed at a target location using wind speed and direction from a reference location. Our method improves on existing target site wind speed predictions produced by widely used industry standard techniques. However, of more importance, our NWPT representation produces models to which we can attach physical and scientific interpretations and in the wind example enable us to understand more about the transfer of wind energy from site to site.     

A wavelet-based approach for modelling exchange rates

This paper proposes a new approach, based on the recent developments of the wavelet theory, to model the dynamic of the exchange rate. First, we consider the maximum overlap discrete wavelet transform (MODWT) to decompose the level exchange rates into several scales. Second, we focus on modelling the conditional mean of the detrended series as well as their volatilities. In particular, we consider the generalized fractional, one-factor, Gegenbauer process (GARMA) to model the conditional mean and the fractionally integrated generalized autoregressive conditional heteroskedasticity process (FIGARCH) to model the conditional variance. Moreover, we estimate the GARMA-FIGARCH model using the wavelet-based maximum likelihood estimator (Whitcher in Technometrics 46:225–238, 2004). To illustrate the usefulness of our methodology, we carry out an empirical application using the daily Tunisian exchange rates relative to the American Dollar, the Euro and the Japanese Yen. The empirical results show the relevance of the selected modelling approach which contributes to a better forecasting performance of the exchange rate series.     

Estimation of Time-Varying Long Memory Parameter Using Wavelet Method

Stationary long memory processes have been extensively studied over the past decades. When we deal with financial, economic, or environmental data, seasonality and time-varying long-range dependence can often be observed and thus some kind of non-stationarity exists. To take into account this phenomenon, we propose a new class of stochastic processes: locally stationary k-factor Gegenbauer process. We present a procedure to estimate consistently the time-varying parameters by applying discrete wavelet packet transform. The robustness of the algorithm is investigated through a simulation study. And we apply our methods on Nikkei Stock Average 225 (NSA 225) index series.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Undercoverage of Wavelet-Based Resampling Confidence Intervals

The decorrelating property of the discrete wavelet transformation (DWT) appears valuable because one can avoid estimating the correlation structure in the original data space by bootstrap resampling of the DWT. Several authors have shown that the wavestrap approximately retains the correlation structure of observations. However, simply retaining the same correlation structure of original observations does not guarantee enough variation for regression parameter estimators. Our simulation studies show that these wavestraps yield undercoverage of parameters for a simple linear regression for time series data of the type that arise in functional MRI experiments. It is disappointing that the wavestrap does not even provide valid resamples for both white noise sequences and fractional Brownian noise sequences. Thus, the wavestrap method is not completely valid in obtaining resamples related to linear regression analysis and should be used with caution for hypothesis testing as well. The reasons for these undercoverages are also discussed. A parametric bootstrap resampling in the wavelet domain is introduced to offer insight into these previously undiscovered defects in wavestrapping.     

A wavelet-based approach to the analysis and modelling of financial time series exhibiting strong long-range dependence: the case of Southeast Europe

This paper demonstrates the utilization of wavelet-based tools for the analysis and prediction of financial time series exhibiting strong long-range dependence (LRD). Commonly emerging markets' stock returns are characterized by LRD. Therefore, we track the LRD evolvement for the return series of six Southeast European stock indices through the application of a wavelet-based semi-parametric method. We further engage the á trous wavelet transform in order to extract deeper knowledge on the returns term structure and utilize it for prediction purposes. In particular, a multiscale autoregressive (MAR) model is fitted and its out-of-sample forecast performance is benchmarked to that of ARMA. Additionally, a data-driven MAR feature selection procedure is outlined. We find that the wavelet-based method captures adequately LRD dynamics both in calm as well as in turmoil periods detecting the presence of transitional changes. At the same time, the MAR model handles with the complicated autocorrelation structure implied by the LRD in a parsimonious way achieving better performance.     

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.     

Wavelet and short-time Fourier transform comparison-based analysis of myoelectric signals

In this investigation, extracted features ofsignals have been analyzed for the recognition of arm movements. Short-time Fourier transform and wavelet transform based on Euclidian distance were applied to reordered signals. Results show that wavelet is a more useful and powerful tool for analyzing signals, since it shows multiresolution property with a significant reduction in the computation time for eliminating resolution problems. Finally, a statistical technique of repeated factorial analysis of variance for experimental recorded data was implemented in a way to investigate the effect of class separability for multiple motions for establishing surface electromyogram–muscular force relationship.     

Tests for Trend: A Simulation Study

In this study, we use the wavelet analysis to construct a test statistic to test for the existence of a trend in the series. We also propose a new approach for testing the presence of trend based on the periodogram of the data. Since we are also interested in the presence of a long-memory process among the data, we study the properties of our test statistics under different degrees of dependency. We compare the results when using the band periodogram test and the wavelet test with results obtained by applying the ordinary least squares (OLS) method under the same conditions.     

A multi-resolution census algorithm for calculating vortex statistics in turbulent flows

Summary.  The fundamental equations that model turbulent flow do not provide much insight into the size and shape of observed turbulent structures. We investigate the efficient and accurate representation of structures in two-dimensional turbulence by applying statistical models directly to the simulated vorticity field. Rather than extract the coherent portion of the image from the background variation, as in the classical signal-plus-noise model, we present a model for individual vortices using the non-decimated discrete wavelet transform. A template image, which is supplied by the user, provides the features to be extracted from the vorticity field. By transforming the vortex template into the wavelet domain, specific characteristics that are present in the template, such as size and symmetry, are broken down into components that are associated with spatial frequencies. Multivariate multiple linear regression is used to fit the vortex template to the vorticity field in the wavelet domain. Since all levels of the template decomposition may be used to model each level in the field decomposition, the resulting model need not be identical to the template. Application to a vortex census algorithm that records quantities of interest (such as size, peak amplitude and circulation) as the vorticity field evolves is given. The multiresolution census algorithm extracts coherent structures of all shapes and sizes in simulated vorticity fields and can reproduce known physical scaling laws when processing a set of vorticity fields that evolve over time.     

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.     

Discrimination Measures for Fractional Integrated Models Based on Wavelets

Discrimination measures have been well developed for stationary time series. However in a large number of phenomena, long-term dependencies are involved. In this article, we are dealing with discrimination of fractional integrated models. Kullback–Leibler and Chernoff's discrimination measures are approximated, using the discrete wavelet transform (DWT) for discrimination of these time series classes. The simulation study indicates low misclassification rate, related to the approximations of Kullback–Leibler and Chernoff discrimination measures. Application to problem of classifying seismic data showed that our procedure performs as well as other procedures.     

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.     

Testing for climate warming in Sweden during 1850-1999, using wavelets analysis

This paper describes an alternative approach for testing for the existence of trend among time series. The test method has been constructed using wavelet analysis which has the ability of decomposing a time series into low frequencies (trend) and high-frequency (noise) components. Under the normality assumption, the test is distributed as F. However, using generated empirical critical values, the properties of the test statistic have been investigated under different conditions and different types of wavelet. The Harr wavelet has shown to exhibit the highest power among the other wavelet types.

The methodology here has been applied to real temperature data in Sweden for the period 1850-1999. The results indicate a significant increasing trend which agrees with the 'global warming' hypothesis during the last 100 years.     

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.