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Wavelet Processes and Adaptive Estimation of the Evolutionary Wavelet Spectrum
, 1998
"... This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated wavelets which generalizes the Cramer (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power va ..."
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Cited by 73 (28 self)
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This article defines and studies a new class of nonstationary random processes constructed from discrete nondecimated 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 timelocalized autocovariance. We illustrate our theory with a pedagogical example based on discrete nondecimated Haar wavelets and also a real medical time series example.
Wavelet Analysis and Its Statistical Applications
, 1999
"... 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 ..."
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Cited by 61 (13 self)
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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 article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date 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 e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise 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...
A survey on wavelet applications in data mining
 SIGKDD Explor. Newsl
"... Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework tha ..."
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Cited by 37 (4 self)
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Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a highlevel datamining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications. 1.
Forecasting nonstationary time series by wavelet process modelling
 ANNALS OF THE INSTITUTE OF STATISTICAL MATHEMATICS
, 2002
"... ..."
Dimension reduction in function regression with applications
 Computational Statistics & Data Analysis
, 2005
"... Two dimensional reduction regression methods to predict a scalar response from a discretized sample path of a continuous time covariate process are presented. The methods take into account the functional nature of the predictor and are both based on appropriate wavelet decompositions. Using such dec ..."
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Cited by 16 (1 self)
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Two dimensional reduction regression methods to predict a scalar response from a discretized sample path of a continuous time covariate process are presented. The methods take into account the functional nature of the predictor and are both based on appropriate wavelet decompositions. Using such decompositions, we derive prediction methods that are similar to minimum average variance estimation (MAVE) or functional sliced inverse regression (FSIR). We describe their practical implementation and we apply the method both in simulation and on real data analyzing three calibration examples of near infrared spectra.
Wavelets in Statistics: Beyond the Standard Assumptions
 Phil. Trans. Roy. Soc. Lond. A
, 1999
"... this paper, attention has been focused on methods that treat coe#cients at least as if they were independent. However, it is intuitively clear that if one coe#cient in the wavelet array is nonzero, then it is more likely #in some appropriate sense# that neighbouring coe#cients will be also. One way ..."
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Cited by 8 (2 self)
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this paper, attention has been focused on methods that treat coe#cients at least as if they were independent. However, it is intuitively clear that if one coe#cient in the wavelet array is nonzero, then it is more likely #in some appropriate sense# that neighbouring coe#cients will be also. One way of incorporating this notion is by some form of block thresholding, where coe#cients are considered in neighbouring blocks; see for example Hall et al. #1998# and Cai & Silverman #1998#. An obvious question for future consideration is integrate the ideas of block thresholding and related methods within the range of models and methods considered in this paper.
Wavelet Packet Modelling of Infant Sleep State Using Heart Rate Data
, 2001
"... We show how a recently developed wavelet packet modelling methodology could be useful for infant sleep state classification using heart rate data. The suggested approach produces adequate classification rates when applied to recordings from an infant who was placed to bed at night at different age ..."
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Cited by 6 (2 self)
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We show how a recently developed wavelet packet modelling methodology could be useful for infant sleep state classification using heart rate data. The suggested approach produces adequate classification rates when applied to recordings from an infant who was placed to bed at night at different ages. As well as classification, this approach gives us valuable information about the relationship between sleep state and heart rate. The statistical model tells us which sorts of wavelet packets of heart rate are most important for classifying sleep state. Keywords: ANTEDEPENDENCE MODELS; INFANT SLEEP STATE CLASSIFICATION; LINEAR DISCRIMINANT ANALYSIS; VARIABLE SELECTION; WAVELETS; WAVELET PACKETS 1
A test for second order stationarity of a time series based on the Discrete Fourier Transform,” arXiv:0911.4744
, 2009
"... We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical Fourier frequencies. It is well known that the discrete Fourier transform is asymptotically uncorrelated at the canonical frequencies if and if only the time series is second order stationary. Ex ..."
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Cited by 5 (0 self)
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We consider a zero mean discrete time series, and define its discrete Fourier transform at the canonical Fourier frequencies. It is well known that the discrete Fourier transform is asymptotically uncorrelated at the canonical frequencies if and if only the time series is second order stationary. Exploiting this important property, we construct a Portmanteau type test statistic for testing stationarity of the time series. It is shown that under the null of stationarity, the test statistic is approximately a chi square distribution. To examine the power of the test statistic, the asymptotic distribution under the locally stationary alternative is established. It is shown to be a type of noncentral chisquare, where the noncentrality parameter measures the deviation from stationarity. The test is illustrated with simulations, where is it shown to have good power. Some real examples are also included to illustrate the test. Kew words and phrases Discrete Fourier Transform, local stationarity, Portmanteau test, test for second order stationarity. 1
Locally stationary processes
 In Handbook of Statistics 30, Time Series Analysis: Methods and Applications 30 351–408
, 2012
"... The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time vary ..."
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Cited by 5 (0 self)
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The article contains an overview over locally stationary processes. At the beginning time varying autoregressive processes are discussed in detail both as as a deep example and an important class of locally stationary processes. In the next section a general framework for time series with time varying finite dimensional parameters is discussed with special emphasis on nonlinear locally stationary processes. Then the paper focusses on linear processes where a more general theory is possible. First a general definition for linear processes is given and time varying spectral densities are discussed in detail. Then the Gaussian likelihood theory is presented for locally stationary processes. In the next section the relevance of empirical spectral processes for locally stationary time series is discussed. Empirical spectral processes play a major role in proving theoretical results and provide a deeper understanding of many techniques. The article concludes with an overview of other results for locally stationary processes.