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Wavelets in Time Series Analysis
 Phil. Trans. R. Soc. Lond. A
, 1999
"... This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process with a specific scale behaviour such as 1/f processes. We present some of our own work on locally stationary wavelet ..."
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Cited by 28 (7 self)
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This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process 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 nonstationary 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.
Wavelet Based Estimation of Local Kolmogorov Turbulence
"... We present a new approach for analyzing locally stationary processes with power law spectral densities. We divide the data into segments over which the process is essentially stationary and then use the wavelet scale spectrum to estimate the parameters of the power law, which are the scale factor ..."
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Cited by 11 (2 self)
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We present a new approach for analyzing locally stationary processes with power law spectral densities. We divide the data into segments over which the process is essentially stationary and then use the wavelet scale spectrum to estimate the parameters of the power law, which are the scale factor and the exponent. These parameters vary from segment to segment, with part of the variation due to the nonstationarity of the data and part due to estimation errors that depend on the length of the segments. In the approach we introduce here, segmentation effects due to estimation errors are removed by filtering. We also estimate an effective local inertial range, that is, the set of scales over which the process can be modeled by a power law. We apply our estimation method to atmospheric temperature data that are expected to have Kolmogorov power law spectra. We find that there are significant fluctuations about the Kolmogorov law and analyze them in detail.
Assessing Nonstationary Time Series Using Wavelets
, 1998
"... The discrete wavelet transform has be used extensively in the field of Statistics, mostly in the area of "denoising signals" or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series  especially l ..."
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Cited by 10 (4 self)
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The discrete wavelet transform has be used extensively in the field of Statistics, mostly in the area of "denoising signals" or nonparametric regression. This thesis provides a new application for the discrete wavelet transform, assessing nonstationary events in time series  especially long memory processes. Long memory processes are those which exhibit substantial correlations between events separated by a long period of time. Departures from stationarity in these heavily autocorrelated time series, such as an abrupt change in the variance at an unknown location or "bursts" of increased variability, can be detected and accurately located using discrete wavelet transforms  both orthogonal and overcomplete. A cumulative sum of squares method, utilizing a KolomogorovSmirnovtype
Locally SelfSimilar Processes and Their Wavelet Analysis
"... Introduction A stochastic process Y (t) is defined as selfsimilar with selfsimilarity parameter H if for any positive stretching factor c, the distribution of the rescaled and reindexed process c Y (c t) is equivalent to that of the original process Y (t). This means for any sequence of time p ..."
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Cited by 2 (0 self)
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Introduction A stochastic process Y (t) is defined as selfsimilar with selfsimilarity parameter H if for any positive stretching factor c, the distribution of the rescaled and reindexed process c Y (c t) is equivalent to that of the original process Y (t). This means for any sequence of time points t 1 ; : : : ; t n and any positive constant c, the collections fc Y (c t 1 ); : : : ; c Y (ct n )g and fY (t 1 ); : : : ; Y (t n )g are governed by the same probability law. As a consequence, the qualitative features of a sample path of a selfsimilar process are invariant to magnification or shrinkage, so that the path will retain the same general appearance regardless of the distance from which it is observed. Although selfsimilar processes were first introduced in a theoretical context by Kolmogorov (1941), statisticians were made aware of the practical applicability of such processes through the work of B.B. Mandelbrot (Mandelbrot and van Ness, 1968; Mandelbrot and Wallis,
Wavelets in Time Series Analysis
, 1999
"... This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process with a specific scale behaviour such as 1/f processes. We present some of our own work on locally stationary wavelet ..."
Abstract
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This article reviews the role of wavelets in statistical time series analysis. We survey work that emphasises scale such as estimation of variance and the scale exponent of a process 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 nonstationary 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.
MEMORY PARAMETER dt
, 2008
"... Avril 2008A simple fractionally integrated model with a timevarying long memory parameter dt ∗† ..."
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Avril 2008A simple fractionally integrated model with a timevarying long memory parameter dt ∗†
Bayesian WaveletBased Methods for the Detection of Multiple Changes of the Long Memory Parameter
"... Abstract—Long memory processes are widely used in many scientific fields, such as economics, physics, and engineering. Change point detection problems have received considerable attention in the literature because of their wide range of possible applications. Here we describe a waveletbased Bayesia ..."
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Abstract—Long memory processes are widely used in many scientific fields, such as economics, physics, and engineering. Change point detection problems have received considerable attention in the literature because of their wide range of possible applications. Here we describe a waveletbased Bayesian procedure for the estimation and location of multiple change points in the long memory parameter of Gaussian autoregressive fractionally integrated moving average models (ARFIMA ()), with unknown autoregressive and moving average parameters. Our methodology allows the number of change points to be unknown. The reversible jump Markov chain Monte Carlo algorithm is used for posterior inference. The method also produces estimates of all model parameters. Performances are evaluated on simulated data and on the benchmark Nile river dataset. Index Terms—ARFIMA models, Bayesian inference, change point, reversible jump, wavelets.
Preface Acknowledgments
, 2002
"... at the division of Design, Manufacture, and Industrial Innovation. This supplement and the S+Wavelets module are products of the efforts made by the Insightful Corporation. Many of the mathematical and statistical ideas behind this software are described more thoroughly in the book coauthored by on ..."
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at the division of Design, Manufacture, and Industrial Innovation. This supplement and the S+Wavelets module are products of the efforts made by the Insightful Corporation. Many of the mathematical and statistical ideas behind this software are described more thoroughly in the book coauthored by one of us (Percival) and Andrew Walden entitled, Wavelet Methods for Time Series Analysis,
Final Report: SpecLab Phase I, STTR Supported by AFOSR
, 2000
"... This Phase I Strategic Technology Transfer Report (STTR) nal report summarizes the development of a prototype interactive software environment, SpecLab, for analyzing nonstationary processes that admit local powerlaw representations. Standard spectral analysis procedures assume stationarity, wherea ..."
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This Phase I Strategic Technology Transfer Report (STTR) nal report summarizes the development of a prototype interactive software environment, SpecLab, for analyzing nonstationary processes that admit local powerlaw representations. Standard spectral analysis procedures assume stationarity, whereas most naturally occuring processes admit random departures from strict stationarity. SpecLab estimates and synthesizes the nonstationary process by allowing both the powerlaw parameters and the powerlaw scale range to vary over a data segmentation chosen interactively by the user. The SpecLab procedures are accessible via a graphical user interface that guides the user through the steps involved in selecting data segmentations and executing the estimation procedures. In its nal form SpecLab will provides ecient user access to leadingedge analysis procedures for nonstationary processes. SpecLab is also congured to provide reproducible research that would ordinarily be available only ...