Guy P. Nason

The Stationary Wavelet Transform and some Statistical Applications

Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A ‘stationary wavelet transform’, where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy.

Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This article defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramer (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.

Wavelet Methods in Statistics with R

Wavelet methods have recently undergone a rapid period of development with important implications for a number of disciplines including statistics. This book has three main objectives: (i) providing an introduction to wavelets and their uses in statistics; (ii) acting as a quick and broad reference to many developments in the area; (iii) interspersing R code that enables the reader to learn the methods, to carry out their own analyses, and further develop their own ideas. The book code is designed to work with the freeware R package WaveThresh4, but the book can be read independently of R. The book introduces the wavelet transform by starting with the simple Haar wavelet transform, and then builds to consider more general wavelets, complex-valued wavelets, non-decimated transforms, multidimensional wavelets, multiple wavelets, wavelet packets, boundary handling, and initialization. Later chapters consider a variety of wavelet-based nonparametric regression methods for different noise models and designs including density estimation, hazard rate estimation, and inverse problems; the use of wavelets for stationary and non-stationary time series analysis; and how wavelets might be used for variance estimation and intensity estimation for non-Gaussian sequences. The book is aimed both at Masters/Ph.D. students in a numerate discipline (such as statistics, mathematics, economics, engineering, computer science, and physics) and postdoctoral researchers/users interested in statistical wavelet methods.

The Discrete Wavelet Transform in S

Abstract The theory of wavelets has recently undergone a period of rapid development. We introduce a software package called wavethresh that works within the statistical language S to perform one- and two-dimensional discrete wavelet transforms. The transforms and their inverses can be computed using any particular wavelet selected from a range of different families of wavelets. Pictures can be drawn of any of the one- or two-dimensional wavelets available in the package. The wavelet coefficients can be presented in a variety of ways to aid in the interpretation of data. The packages wavelet transform “engine” is written in C for speed and the object-oriented functionality of S makes wavethresh easy to use. We provide a tutorial introduction to wavelets and the wavethresh software. We also discuss how the software may be used to carry out nonlinear regression and image compression. In particular, thresholding of wavelet coefficients is a method for attempting to extract signal from noise and wavethresh i...

Coronary revascularization with or without cardiopulmonary bypass in patients with preoperative nondialysis-dependent renal insufficiency

BACKGROUND Preoperative renal insufficiency is a predictor of acute renal failure in patients undergoing conventional coronary artery bypass grafting. Off-pump coronary artery bypass operations have been shown to reduce renal dysfunction in patients with normal renal function, but the effect of this technique in patients with preoperative nondialysis-dependent renal insufficiency is unknown. METHODS From June 1996 to December 1999, data of 3,250 consecutive patients undergoing coronary artery bypass grafting were prospectively entered into the Patient Analysis & Tracking Systems (PATS, Dendrite Clinical Systems, London, UK). Two hundred and fifty-three patients with preoperative serum creatinine more than 150 micromol/L were identified (202 patients on-pump, 51 patients off-pump), and clinical outcomes were analyzed. Serum creatinine and urea, in-hospital mortality, and morbidity were compared between groups. The association of perioperative factors with acute renal failure was investigated by multiple logistic regression analysis. RESULTS Preoperative characteristics were similar between the groups. Mean number of grafts was 2.9 +/- 0.8 and 2.3 +/- 0.8 in the on-pump and off-pump groups, respectively (p < 0.0001). Comparison between groups showed a significantly higher incidence of stroke, inotropic requirement, blood loss, and transfusion of red packed cell and platelets in the on-pump group (all p < 0.05). Postoperative serum creatinine and urea were higher in the on-pump group with a significant difference at 12 hours postoperatively (p < 0.05). Logistic regression analysis identified cardiopulmonary bypass, serum creatinine level 60 hours postoperatively, inotropic requirement, need for intraaortic balloon pump, transfusion of red packed cell, and hours of ventilation as predictors of postoperative acute renal failure. CONCLUSIONS This study suggests that off-pump coronary artery bypass operations reduce in-hospital morbidity and the likelihood of acute renal failure in patients with preoperative nondialysis-dependent renal insufficiency undergoing myocardial revascularization.

A Haar-Fisz Algorithm for Poisson Intensity Estimation

This article introduces a new method for the estimation of the intensity of an inhomogeneous one-dimensional Poisson process. The Haar-Fisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the Haar-Fisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the Haar-Fisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed state-of-the-art competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current state-of-the-art.

Wavelets in time-series analysis

This article reviews the role of wavelets in statistical time–series analysis. We survey work that emphasizes scale, such as estimation of variance, and the scale exponent of processes with a specific scale behaviour, such as 1/f processes. We present some of our own work on locally stationary wavelet (LSW) processes, which model both stationary and some kinds of non–stationary processes. Analysis of time–series assuming the LSW model permits identification of an evolutionary wavelet spectrum (EWS) that quantifies the variation in a time–series over a particular scale and at a particular time. We address estimation of the EWS and show how our methodology reveals phenomena of interest in an infant electrocardiogram series.

Density and hazard rate estimation for right-censored data by using wavelet methods

This paper describes a wavelet method for the estimation of density and hazard rate functions from randomly right-censored data. We adopt a nonparametric approach in assuming that the density and hazard rate have no specific parametric form. The method is based on dividing the time axis into a dyadic number of intervals and then counting the number of events within each interval. The number of events and the survival function of the observations are then separately smoothed over time via linear wavelet smoothers, and then the hazard rate function estimators are obtained by taking the ratio. We prove that the estimators have pointwise and global mean-square consistency, obtain the best possible asymptotic mean integrated squared error convergence rate and are also asymptotically normally distributed. We also describe simulation experiments that show that these estimators are reasonably reliable in practice. The method is illustrated with two real examples. The first uses survival time data for patients with liver metastases from a colorectal primary tumour without other distant metastases. The second is concerned with times of unemployment for women and the wavelet estimate, through its flexibility, provides a new and interesting interpretation.

Choice of the threshold parameter in wavelet function estimation

The procedures of Donoho, Johnstone, Kerkyacharian and Picard [DJKP] estimate functions by inverting thresholded wavelet transform coefficients of the data. The choice of threshold is crucial to the success of the method and is currently subject to an intense research effort. We describe how we have applied the statistical technique of cross-validation to choose a threshold and we present results that indicate that its performance for correlated data. Finally, to illustrate the techniques, we apply various wavelet-based estimation methods to some noisy one- and two-dimensional signals and display the results.

Choice of wavelet smoothness, primary resolution and threshold in wavelet shrinkage

This article introduces a fast cross-validation algorithm that performs wavelet shrinkage on data sets of arbitrary size and irregular design and also simultaneously selects good values of the primary resolution and number of vanishing moments.We demonstrate the utility of our method by suggesting alternative estimates of the conditional mean of the well-known Ethanol data set. Our alternative estimates outperform the Kovac-Silverman method with a global variance estimate by 25% because of the careful selection of number of vanishing moments and primary resolution. Our alternative estimates are simpler than, and competitive with, results based on the Kovac-Silverman algorithm equipped with a local variance estimate.We include a detailed simulation study that illustrates how our cross-validation method successfully picks good values of the primary resolution and number of vanishing moments for unknown functions based on Walsh functions (to test the response to changing primary resolution) and piecewise polynomials with zero or one derivative (to test the response to function smoothness).