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 nondecimated Haar wavelets and also a real medical time series example.

The aggregation procedure is a natural way to analyse signals which exhibit long-range dependent features and has been used as a basis for estimation of the Hurst parameter, H. In this paper it is shown how aggregation can be naturally rephrased within the wavelet transform framework, being directly related to approximations of the signal in the sense of a Haar-multiresolution analysis. A natural wavelet based generalisation to traditional aggregation is then proposed: "a-aggregation". It is shown that a-aggregation cannot lead to good estimators of H, and so a new kind of aggregation, "d-aggregation", is defined, which is related to the details rather than the approximations of a multiresolution analysis. An estimator of H based on d-aggregation has excellent statistical and computational properties, whilst preserving the spirit of aggregation. The estimator is applied to telecommunications network data.

Self-similar and long-range dependent processes are the most important kinds of random processes possessing scale invariance. We describe how to analyze them using the discrete wavelet transform. We have chosen a didactic approach, useful to practitioners. Focusing on the Discrete Wavelet Transform, we describe the nature of the wavelet coefficients and their statistical properties. Pitfalls in understanding and key features are highlighted and we sketch some proofs to provide additional insight. The Logscale Diagram is introduced as a natural means to study scaling data and we show how it can be used to obtain unbiased semi-parametric estimates of the scaling exponent. We then focus on the case of long-range dependence and address the problem of defining a lower cutoff scale corresponding to where scaling starts. We also discuss some related problems arising from the application of wavelet analysis to discrete time series. Numerical examples using many discrete time models are th...

Wavelet packets are a useful extension of wavelets providing an adaptive timescale analysis. In using noisy observations of a signal of interest, the criteria for best bases representation are random variables. The search may thus be very sensitive to noise. In this paper, we characterize the asymptotic statistics of the criteria to gain insight which can in turn, be used to improve on the performance of the analysis. By way of a well-known information-theoretic principle, namely the Minimum Description Length, we provide an alternative approach to Minimax methods for deriving various attributes of nonlinear wavelet packet estimates. 1 Introduction Research interest in wavelets and their applications have tremendously grown over the last five years. Only, more recently, however, have their applications been considered in a stochastic setting [Fl1, Wo1, BB + , CH1]. A number of papers which have addressed the optimal representation of a signal in a wavelet/wavelet packet basis, have...

In this work we define and study a new class of non--stationary random processes which are characterized by a representation with respect to a family of localized basis functions. Using non--decimated or "stationary" wavelets this generalizes the Cram'er (Fourier) spectral representation of stationary time series. We provide a time--scale instead of a time--frequency decomposition and, hence, instead of thinking as scale in terms of "inverse frequency" we start from genuine time--scale building blocks or "atoms". Using our new model of "locally stationary wavelet" processes, we develop a theory how to define and estimate an "evolutionary wavelet spectrum". Our asymptotics are based on rescaling in time--location which allows us to perform rigorous estimation starting from a single stretch of observations of the process. This wavelet spectrum measures the local power in the variance--covariance decomposition of the process at a certain scale and a (rescaled) time location. To estimate t...

Abstract—Statistically self-similar (SSS) processes can be used to describe a variety of physical phenomena, yet modeling these phenomena has proved challenging. Most of the proposed models for SSS and approximately SSS processes have power spectra that behave as 1=f, such as fractional Brownian motion (fBm), fractionally differenced noise, and wavelet-based syntheses. The most flexible framework is perhaps that based on wavelets, which provides a powerful tool for the synthesis and estimation of 1=f processes, but assumes a particular distribution of the measurements. An alternative framework is the class of multiresolution processes proposed by Chou et al. [1994], which has already been shown to be useful for the identification of the parameters of fBm. These multiresolution processes are defined by an autoregression in scale that makes them naturally suited to the representation of SSS (and approximately SSS) phenomena, both stationary and nonstationary. Also, this multiresolution framework is accompanied by an efficient estimator, likelihood calculator, and conditional simulator that make no assumptions about the distribution of the measurements. In this paper, we show how to use the multiscale framework to represent SSS (or approximately SSS) processes such as fBm and fractionally differenced Gaussian noise. The multiscale models are realized by using canonical correlations (CC) and by exploiting the selfsimilarity and possible stationarity or stationary increments of the desired process. A number of examples are provided to demonstrate the utility of the multiscale framework in simulating and estimating SSS processes. Index Terms—Canonical correlations, fractional Brownian motion, multiscale, self-similarity. I.

The infinite source Poisson model is a fluid queue approximation of network data transmission that assumes that sources begin constant rate transmissions of data at Poisson time points for random lengths of time. This model has been a popular one as analysts attempt to provide explanations for observed features in telecommunications data such as self-similarity, long range dependence and heavy tails. We survey some features of this model in cases where transmission length distributions have (a) tails so heavy that means are infinite, (b) heavy tails with finite mean and infinite variance and (c) finite variance. We survey the self-similarity properties of various descriptor processes in this model and then present analyses of four data sets which show that certain features of the model are consistent with the data while others are contradicted. The data sets are 1) the Boston University 1995 study of web sessions, 2) the UC Berkeley home IP HTTP data collected in November 1996, 3) tr...

Processing nonstationary signals is an important and challenging problem. We focus on the class of nonstationary processes with stationary increments of an arbitrary order, and place them in a multiscale framework. Unlike other related studies, we concentrate on the discrete time analysis and derive a number of new results in addition to placing the related existing ones in the same framework. We extend the study to various parametric models for which we derive the resulting multiresolution description. We show that wide-sense stationarity may be achieved by adequately selecting the analysis wavelet. After generalizing the study to wavelet packet analysis, we show that the latter possesses additional properties which are useful in the presence of other types of nonstationarities. Keywords: wavelets, wavelet packets, nonstationarity, stationary increments, ARIMA. The work of the first author is partially supported by grants from ARO (DAAL03-92G -0115), AFOSR (F49620-92-J-0002), and ...

. In this paper we are concerned with estimating the fractional order of integration associated with a long-memory stochastic volatilitymodel. Wedevelop a new Bayesian estimator based on the Markov chain Monte Carlo sampler and the wavelet representation of the log-squared returns to drawvalues of the fractional order of integration and latentvolatilities from their joint posterior distribution. Unlike shortmemory stochastic volatility models, long-memory stochastic volatilitymodelsdonothave a state-space representation, and thus their sampler cannot employ the Kalman filters simulation smoother to update the chain of latentvolatilities. Instead, we design a simulator where the latent long-memory volatilities are drawn quickly and efficiently from the near independentmultivariate distribution of the long-memory volatility's wavelet coefficients. We find that sampling volatility in the wavelet domain, rather than in the time domain, leads to a fast and simulation-efficient sampler of the posterior distribution for the volatility's long-memory parameter and serves as a promising alternative estimator to the existing frequentist based estimators of long-memory volatility. Keywords: Long-memory# Markovchain Monte Carlo# Metropolis-Hastings# Semiparametric# Stochastic volatility# Wavelets JEL Classification: C11# C14# C22# 1

We describe several aspects of wavelet analysis and more general methods of time-frequency analysis, emphasizing applications to signal analysis and processing problems.