Theofanis Sapatinas

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the functions regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a functions regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.

Wavelet Analysis and its Statistical Applications

Summary. In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this paper gives a relatively accessible introduction to standard wavelet analysis and provides a review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be effective, rather than to researchers who are already familiar with the field. Given that objective, we do not emphasize mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in greater detail and generality if required. The paper first establishes some necessary basic mathematical background and terminology relating to wavelets. It then reviews the more well-established applications of wavelets in statistics including their use in nonparametric regression, density estimation, inverse problems, changepoint problems and in some specialized aspects of time series analysis. Possible extensions to the uses of wavelets in statistics are then considered. The paper concludes with a brief reference to readily available software packages for wavelet analysis.

Wavelet methods for continuous-time prediction using Hilbert-valued autoregressive processes

We consider the prediction problem of a continuous-time stochastic process on an entire time-interval in terms of its recent past. The approach we adopt is based on the notion of autoregressive Hilbert processes that represent a generalization of the classical autoregressive processes to random variables with values in a Hilbert space. A careful analysis reveals, in particular, that this approach is related to the theory of function estimation in linear ill-posed inverse problems. In the deterministic literature, such problems are usually solved by suitable regularization techniques. We describe some recent approaches from the deterministic literature that can be adapted to obtain fast and feasible predictions. For large sample sizes, however, these approaches are not computationally efficient.With this in mind, we propose three linear wavelet methods to efficiently address the aforementioned prediction problem. We present regularization techniques for the sample paths of the stochastic process and obtain consistency results of the resulting prediction estimators. We illustrate the performance of the proposed methods in finite sample situations by means of a real-life data example which concerns with the prediction of the entire annual cycle of climatological El Nino-Southern Oscillation time series 1 year ahead. We also compare the resulting predictions with those obtained by other methods available in the literature, in particular with a smoothing spline interpolation method and with a SARIMA model.

Estimation and inference in functional mixed-effects models

Functional mixed-effects models are very useful in analyzing functional data. A general functional mixed-effects model that inherits the flexibility of linear mixed-effects models in handling complex designs and correlation structures is considered. A wavelet decomposition approach is used to model both fixed-effects and random-effects in the same functional space, meaning that the population-average curve and the subject-specific curves have the same smoothness property. A linear mixed-effects representation is then obtained that is used for estimation and inference in the general functional mixed-effects model. Adapting recent methodologies in linear mixed-effects and nonparametric regression models, hypothesis testing procedures for both fixed-effects and random-effects are provided. Using classical linear mixed-effects estimation techniques, the linear mixed-effects representation is also used to obtain wavelet-based estimates for both fixed-effects and random-effects in the general functional mixed-effects model. The usefulness of the proposed estimation and hypothesis testing procedures is illustrated by means of a small simulation study and a real-life dataset arising from physiology.

Empirical Bayes approach to block wavelet function estimation

Wavelet methods have demonstrated considerable success in function estimation through term-by-term thresholding of the empirical wavelet coefficients. However, it has been shown that grouping the empirical wavelet coefficients into blocks and making simultaneous threshold decisions about all the coefficients in each block has a number of advantages over term-by-term wavelet thresholding, including asymptotic optimality and better mean squared error performance in finite sample situations. An empirical Bayes approach to incorporating information on neighbouring empirical wavelet coefficients into function estimation that results in block wavelet shrinkage and block wavelet thresholding estimators is considered. Simulated examples are used to illustrate the performance of the resulting estimators, and to compare these estimators with several existing non-Bayesian block wavelet thresholding estimators. It is observed that the proposed empirical Bayes block wavelet shrinkage and block wavelet thresholding estimators outperform the non-Bayesian block wavelet thresholding estimators in finite sample situations. An application to a data set that was collected in an anaesthesiological study is also presented.

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, Yt, in terms of an explanatory time series, Xt. 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.

Short-Term Load Forecasting: The Similar Shape Functional Time-Series Predictor

A novel functional time-series methodology for short-term load forecasting is introduced. The prediction is performed by means of a weighted average of past daily load segments, the shape of which is similar to the expected shape of the load segment to be predicted. The past load segments are identified from the available history of the observed load segments by means of their closeness to a so-called reference load segment. The latter is selected in a manner that captures the expected qualitative and quantitative characteristics of the load segment to be predicted. As an illustration, the suggested functional time-series forecasting methodology is applied to historical daily load data in Cyprus. Its performance is compared with some recently proposed alternative methodologies for short-term load forecasting.

Signal detection in underwater sound using wavelets

Abstract This article considers the use of wavelet methods in relation to a common signal processing problem, that of detecting transient features in sound recordings that contain interference or distortion. In this particular case, the data are various types of underwater sounds, and the objective is to detect intermittent departures (potential “signals”) from the background sound environment in the data (“noise”), where the latter may itself be evolving and changing over time. We develop an adaptive model of the background interference, using recursive density estimation of the joint distribution of certain summary features of its wavelet decomposition. Observations considered to be outliers from this density estimate at any time are then flagged as potential “signals.” The performance of our method is illustrated on artificial data, where a known “signal” is contaminated with simulated underwater “noise” using a range of different signal-to-noise ratios, and a “baseline” comparison is made with results...

FUNCTIONAL DECONVOLUTION IN A PERIODIC SETTING : UNIFORM CASE

We extend deconvolution in a periodic setting to deal with functional data. The resulting functional deconvolution model can be viewed as a generalization of a multitude of inverse problems in mathematical physics where one needs to recover initial or boundary conditions on the basis of observations from a noisy solution of a partial differential equation. In the case when it is observed at a finite number of distinct points, the proposed functional deconvolution model can also be viewed as a multichannel deconvolution model. We derive minimax lower bounds for the L 2 -risk in the proposed functional deconvolution model when f(·) is assumed to belong to a Besov ball and the blurring function is assumed to possess some smoothness properties, including both regular-smooth and super-smooth convolutions. Furthermore, we propose an adaptive wavelet estimator of f(·) that is asymptotically optimal (in the minimax sense), or near-optimal within a logarithmic factor, in a wide range of Besov balls. In addition, we consider a discretization of the proposed functional deconvolution model and investigate when the availability of continuous data gives advantages over observations at the asymptotically large number of points. As an illustration, we discuss particular examples for both continuous and discrete settings.

Bayesian Approach to Wavelet Decomposition and Shrinkage

We consider Bayesian approach to wavelet decomposition. We show how prior knowledge about a function’s regularity can be incorporated into a prior model for its wavelet coefficients by establishing a relationship between the hyperparameters of the proposed model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation may be seen as giving insight into the meaning of the Besov space parameters themselves. Furthermore, we consider Bayesian wavelet-based function estimation that gives rise to different types of wavelet shrinkage in non-parametric regression. Finally, we discuss an extension of the proposed Bayesian model by considering random functions generated by an overcomplete wavelet dictionary.