forecasting multifractal volatility this paper develops analytical methods to forecast the distribution of future returns for a new continuous time process the poisson multifractal the process captures the thick tails volatility persistence and moment scaling exhibited by many financial time series it can be interpreted as a stochastic volatility model with multiple frequencies and a markov latent state we assume for simplicity that the forecaster knows the true generating process with certainty but only observes past returns the challenge in this environment is long memory and the corresponding infinite dimension of the state space we introduce a discretized version of the model that has a finite state space and an analytical solution to the conditioning problem as the grid step size goes to zero the discretized model weakly converges to the continuous time process implying the consistency of the density forecasts

a test for second order stationarity and approximate confidence intervals for localized autocovariances for locally stationary time series summary many time series are not second order stationary and it is not appropriate to analyse them by using methods designed for stationary series the paper introduces a new test for second order stationarity that detects kinds of departures from stationarity that are different from those based on fourier methods the new test is also computationally fast designed to work with gaussian and a wide range of non gaussian time series and can locate non stationarities in time and scale the test is demonstrated on earthquake explosion infant electrocardiogram and simulated time series showing varying degrees of stationarity the second main contribution develops approximate confidence intervals for time varying autocovariances for locally stationary series as the usual bands computed for stationary series are not appropriate our new bands enable practitioners to assess time varying autocovariances statistically and are exhibited on localized autocovariances of explosion and simulated time series

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

consistent classification of nonstationary time series using stochastic wavelet representations consider the situation when we have training data containing many time series having known group membership and testing data with unknown group membership the goals are to find timescale features using training data that can best separate the groups and to use these highly discriminant features to classify test data we propose a method for classification using a bias corrected nondecimated wavelet transform wavelets are ideal for identifying highly discriminant local time and scale features the observed signals will be treated as realizations of locally stationary wavelet processes under which we define and rigorously estimate the evolutionary wavelet spectrum timescale decomposition of variance the evolutionary wavelet spectrum which contains the second moment information on the signals is used as the classification signature for each test time series we compute the empirical wavelet spectrum and its divergence from the wavelet spectrum of each group the test time series is then assigned

analytical blind channel identification a novel analytical blind single input single output siso identification algorithm is presented based on the noncircular second order statistics of the output it is shown that statistics of order higher than two are not mandatory to restore identifiability our approach is valid for instance when the channel is excited by phase shift keying psk inputs it is shown that the channel taps need to satisfy a polynomial system of degree 2 and that identification amounts to solving the system we describe the algorithm that is able to solve this particular system entirely analytically thus avoiding local minima computer results eventually show the robustness with respect to noise and to channel length overdetermination identifiability issues are also addressed

blind equalization of digital communication channels using high order moments the authors describe algorithms for blind equalization of digital communication channels of the quadrature amplitude modulation qam type these algorithms are based on the fourth order statistical moments of the received data sequence the first of the two is a linear least squares type algorithm the second algorithm is of the nonlinear least squares type the algorithms use the fourth order statistical moments of the symbol sequence to explicitly estimate the channel impulse response the estimated impulse response is used in turn to construct a linear mean square error equalizer the performance of this equalizer is not optimal in any sense but it is adequate for channels with mild intersymbol interference or when the number of data points available for estimating the channel response is very large

spectrum estimation by wavelet thresholding of multitaper estimators current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram unfortunately the periodogram is a very poor estimate when the true spectrum has a high dynamic range and or is rapidly varying in addition because the distribution of the log periodogram is markedly non gaussian special wavelet dependent thresholding schemes are needed these difficulties can be bypassed by starting with a multitaper spectrum estimator the logarithm of this estimator is close to gaussian distributed if a moderate number spl ges 5 of tapers are used in contrast to the log periodogram log multitaper estimates are not approximately pairwise uncorrelated at the fourier frequencies but the form of the correlation can be accurately and simply approximated for scale independent thresholding the correlation acts in accordance with the wavelet shrinkage paradigm to suppress small scale noise spikes while leaving informative coarse scale coefficients relatively unattenuated this simple approach to spectrum estimation is demonstrated to work very well in practice additionally the progression of the variance of wavelet coefficients with scale can be accurately calculated allowing the use of scale dependent thresholds this more involved approach also works well in practice but is not uniformly preferable to the scale independent approach

nonparametric quasi maximum likelihood estimation for gaussian locally stationary processes this paper deals with nonparametric maximum likelihood estimation for gaussian locally stationary processes our nonparametric mle is constructed by minimizing a frequency domain likelihood over a class of functions the asymptotic behavior of the resulting estimator is studied the results depend on the richness of the class of functions both sieve estimation and global estimation are considered our results apply in particular to estimation under shape constraints as an example autoregressive model fitting with a monotonic variance function is discussed in detail including algorithmic considerations a key technical tool is the time varying empirical spectral process indexed by functions for this process a bernstein type exponential inequality and a central limit theorem are derived these results for empirical spectral processes are of independent interest

existence and energy decay rates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback abstract this paper is concerned with the existence and energy decay estimates of solutions to the variable coefficient euler bernoulli plate with a delay in localized nonlinear internal feedback the existence of solutions to the nonlinear system is obtained by using galerkin method combined with some energy estimates furthermore applying riemannian geometry method we establish uniform decay rates of the plate equation with variable coefficients and a nonlinear delay term

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 thresholding techniques for power spectrum estimation estimation of the power spectrum s f of a stationary random process can be viewed as a nonparametric statistical estimation problem we introduce a nonparametric approach based on a wavelet representation for the logarithm of the unknown s f this approach offers the ability to capture statistically significant components of ln s f at different resolution levels and guarantees nonnegativity of the spectrum estimator the spectrum estimation problem is set up as a problem of inference on the wavelet coefficients of a signal corrupted by additive non gaussian noise we propose a wavelet thresholding technique to solve this problem under specified noise resolution tradeoffs and show that the wavelet coefficients of the additive noise may be treated as independent random variables the thresholds are computed using a saddle point approximation to the distribution of the noise coefficients

a test for the absence of aliasing or local white noise in locally stationary wavelet time series aliasing is often overlooked in time series analysis but can seriously distort the spectrum autocovariance and their estimates we show that dyadic subsampling of a locally stationary wavelet process which can cause aliasing results in a process that is the sum of asymptotic white noise and another locally stationary wavelet process with a modified spectrum we develop a test for the absence of aliasing in a locally stationary wavelet series at a fixed location and illustrate it on simulated data and a wind energy time series a useful by product is a new test for local white noise the tests are robust to model misspecification in that it is unnecessary for the analysis and synthesis wavelets to be identical hence in principle the tests work irrespective of which wavelet is used to analyze the time series though in practice there is a tradeoff between increasing statistical power and time localization of the test

estimating linear dependence between nonstationary time series using the locally stationary wavelet model large volumes of neuroscience data comprise multiple nonstationary electrophysiological or neuroimaging time series recorded from different brain regions accurately estimating the dependence between such neural time series is critical since changes in the dependence structure are presumed to reflect functional interactions between neuronal populations we propose a new dependence measure derived from a bivariate locally stationary wavelet time series model since wavelets are localized in both time and scale this approach leads to a natural local and multi scale estimate of nonstationary dependence our methodology is illustrated by application to a simulated example and to electrophysiological data relating to interactions between the rat hippocampus and prefrontal cortex during working memory and decision making

multiple change point detection for high dimensional time series via sparsified binary segmentation time series segmentation a k a multiple change point detection is a well established problem however few solutions are designed specifically for high dimensional situations in this paper our interest is in segmenting the second order structure of a high dimensional time series in a generic step of a binary segmentation algorithm for multivariate time series one natural solution is to combine cusum statistics obtained from local periodograms and cross periodograms of the components of the input time series however the standard maximum and average methods for doing so often fail in high dimensions when for example the change points are sparse across the panel or the cusum statistics are spuriously large in this paper we propose the sparsified binary segmentation sbs algorithm which aggregates the cusum statistics by adding only those that pass a certain threshold this sparsifying step reduces the impact of irrelevant noisy contributions which is particularly beneficial in high dimensions in order to show the consistency of sbs we introduce the multivariate locally stationary wavelet model for time series which is a separate contribution of this work

structural break estimation for nonstationary time series models this article considers the problem of modeling a class of nonstationary time series using piecewise autoregressive ar processes the number and locations of the piecewise ar segments as well as the orders of the respective ar processes are assumed unknown the minimum description length principle is applied to compare various segmented ar fits to the data the goal is to find the best combination of the number of segments the lengths of the segments and the orders of the piecewise ar processes such a best combination is implicitly defined as the optimizer of an objective function and a genetic algorithm is implemented to solve this difficult optimization problem numerical results from simulation experiments and real data analyses show that the procedure has excellent empirical properties the segmentation of multivariate time series is also considered assuming that the true underlying model is a segmented autoregression this procedure is shown to be consistent for estimating the location of

prediction error method for second order blind identification blind channel identification methods based on the oversampled channel output are a problem of current theoretical and practical interest in this paper we introduce a second order blind identification technique based on a linear prediction approach in contrast to eigenstructure based methods it will be shown that the linear prediction error method is robust to order overdetermination an asymptotic performance analysis of the proposed estimation method is carried out consistency and asymptotic normality of the estimates is established a closed form expression for the asymptotic covariance of the estimates is given numerical simulations and investigations are finally presented to demonstrate the potential and the robustness of the proposed method

estimation of ambiguity functions with limited spread in this correspondence we propose a class of estimators of the ambiguity function of a discrete time process using thresholding to this purpose the distributional and moment properties of the empirical ambiguity function for a realization of an analytic random process are derived and the moments for the special cases of stationary uniformly modulated and deterministic signals immersed in noise processes are determined suitable thresholds are chosen based on the distributional properties of the empirical ambiguity function and the procedure is tested on simulated data

multiscale variance stabilization via maximum likelihood this article proposes maximum likelihood approaches for multiscale variance stabilization mvs tranforms for independently and identically distributed data for two mvs transforms we present new unified theoretical results on their jacobians which are a key component of the likelihood the new results provide i a deeper theoretical understanding of the transforms and ii the ability to compute the likelihood in linear time compared to previous numerical methods which required quadratic effort the new mvs transforms are shown empirically to compare favourably to the well known box cox transform and almost dominate it we show how the new transforms like box cox can facilitate simpler models by an example involving the famous wool data

approaches of fir system identification with noisy data using higher order statistics the problem of estimating the parameters of a moving average model from the cumulant statistics of the noisy observations of the system output is discussed the system is driven by an independently identically distributed nongaussian sequence that is not observed the noise is additive and may be colored and nongaussian following some existing linear parametric approaches to this problem a linear method and a modification to an existing linear method are proposed for consistent parameter estimation in measurement noise under the assumption that the system order is known both recursive closed form and batch least squares versions of the parameter estimators are presented the existing and the proposed linear methods utilize only a partial set of the relevant output statistics this restriction being necessary to obtain a linear estimator whereas there exist nonlinear method that exploit a much larger set of output statistics a simulation example where two existing linear methods and the two new methods are compared to two existing nonlinear methods is presented

the uncertainty of storm season changes quantifying the uncertainty of autocovariance changepoints in oceanography there is interest in determining storm season changes for logistical reasons such as equipment maintenance scheduling in particular there is interest in capturing the uncertainty associated with these changes in terms of the number and location of them such changes are associated with autocovariance changes this article proposes a framework to quantify the uncertainty of autocovariance changepoints in time series motivated by this oceanographic application more specifically the framework considers time series under the locally stationary wavelet lsw framework deriving a joint density for scale processes in the raw wavelet periodogram by embedding this density within a hidden markov model hmm framework we consider changepoint characteristics under this multiscale setting such a methodology allows us to model changepoints and their uncertainty for a wide range of models including piecewise second order stationary processes for example piecewise moving average processes

exploiting input cyclostationarity for blind channel identification in ofdm systems transmitter induced cyclostationarity has been explored previously as an alternative to fractional sampling and antenna array methods for blind identification of fir communication channels an interesting application of these ideas is in ofdm systems which induce cyclostationarity due to the cyclic prefix we develop a novel subspace approach for blind channel identification using cyclic correlations at the ofdm receiver even channels with equispaced unit circle zeros are identifiable in the presence of any nonzero length cyclic prefix with adequate block length simulations of the proposed channel estimator along with its performance in ofdm systems combined with impulse response shortening and reed solomon coding are presented

a haar fisz technique for locally stationary volatility estimation summary we consider a locally stationary model for financial log returns whereby the returns are independent and the volatility is a piecewise constant function with jumps of an unknown number and locations defined on a compact interval to enable a meaningful estimation theory we demonstrate that the model explains well the common characteristics of log returns we propose a new wavelet thresholding algorithm for volatility estimation in this model in which haar wavelets are combined with the variance stabilising fisz transform the resulting volatility estimator is mean square consistent with a near parametric rate does not require any pre estimates is rapidly computable and is easily implemented we also discuss important variations on the choice of estimation parameters we show that our approach both gives a very good fit to selected currency exchange datasets and achieves accurate long and short term volatility forecasts in comparison to the garch 1 1 and moving window techniques

a cholesky factorization based approach for blind fir channel identification for blind channel identification various techniques including transmitter induced cyclostationarity based approaches have been proposed in literature to induce cyclostationarity at the transmitter zero padding to every symbol packet can be considered some subspace methods are proposed for the channel estimation by exploiting the induced cyclostationarity due to zero padding the covariance matrix of received signal vectors can have a special structure utilizing this structure we can estimate channel impulse responses through factorization in this correspondence we propose a factorization based approach to estimate channel impulse responses in general the proposed factorization based approach can work with a small number of samples and becomes much more computationally efficient than the subspace method when the length of packets is long

adaptive covariance estimation of locally stationary processes it is shown that the covariance operator of a locally stationary process has approximate eigenvectors that are local cosine functions we model locally stationary processes with pseudo differential operators that are time varying convolutions an adaptive covariance estimation is calculated by searching first for a best local cosine basis which approximates the covariance by a band or a diagonal matrix the estimation is obtained from regularized versions of the diagonal coefficients in the best basis

how much information does dependence between wavelet coefficients contain abstractthis article is motivated by several articles that propose statistical inference where the independence of wavelet coefficients for both short as well as long range dependent time series is assumed we focus on the sample variance and investigate the influence of the dependence between wavelet coefficients and this statistic to this end we derive asymptotic distributional properties of the sample variance for a time series that is synthesized ignoring some or all dependence between wavelet coefficients we show that the second order properties differ from the those of the true time series whose wavelet coefficients have the same marginal distribution except in the independent gaussian case this holds true even if the dependency is correct within each level and only the dependence between levels is ignored in the case of sample autocovariances and sample autocorrelations at lag one we indicate that first order properties are erroneous in a second step we investigate several nonparametric b

simultaneous multiple change point and factor analysis for high dimensional time series abstract we propose the first comprehensive treatment of high dimensional time series factor models with multiple change points in their second order structure we operate under the most flexible definition of piecewise stationarity and estimate the number and locations of change points consistently as well as identifying whether they originate in the common or idiosyncratic components through the use of wavelets we transform the problem of change point detection in the second order structure of a high dimensional time series into the relatively easier problem of change point detection in the means of high dimensional panel data also our methodology circumvents the difficult issue of the accurate estimation of the true number of factors in the presence of multiple change points by adopting a screening procedure we further show that consistent factor analysis is achieved over each segment defined by the change points estimated by the proposed methodology in extensive simulation studies we observe that factor analysis prior to change point detection improves the detectability of change points and identify and describe an interesting spillover effect in which substantial breaks in the idiosyncratic components get naturally enough identified as change points in the common components which prompts us to regard the corresponding change points as also acting as a form of factors our methodology is implemented in the r package factorcpt available from cran

locally adaptive estimation of evolutionary wavelet spectra we introduce a wavelet based model of local stationarity this model enlarges the class of locally stationary wavelet processes and contains processes whose spectral density function may change very suddenly in time a notion of time varying wavelet spectrum is uniquely de ned as a wavelet type transform of the autocovariance function with respect to so called autocorrelation wavelets this leads to a natural representation of the autocovariance which is localized on scales we propose a pointwise adaptive estimator of the time varying spectrum the behavior of the estimator studied in homogeneous and inhomogeneous regions of the wavelet spectrum

haar fisz estimation of evolutionary wavelet spectra summary we propose a new haar fisz technique for estimating the time varying piecewise constant local variance of a locally stationary gaussian time series we apply our technique to the estimation of the spectral structure in the locally stationary wavelet model our method combines haar wavelets and the variance stabilizing fisz transform the resulting estimator is mean square consistent rapidly computable and easy to implement and performs well in practice we also introduce the haar fisz transform a device for stabilizing the variance of scaled x2 data and bringing their distribution close to gaussianity

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed