a decomposition theorem for polymeasures we prove that every countably additive polymeasure can be decomposed in a unique way as the sum of a discrete polymeasure plus a continuous polymeasure this generalizes a previous result of saeki

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

characterization of dual mixed volumes via polymeasures we prove a characterization of the dual mixed volume in terms of functional properties of the polynomial associated with it to do this we use tools from the theory of multilinear operators on spaces of continuous functions along the way we reprove with these same techniques a recently found characterization of the dual mixed volume

oscillation analysis of numerical solutions for nonlinear delay differential equations of hematopoiesis with unimodal production rate this paper is concerned with oscillation and non oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate by investigating the roots of characteristic equation some conditions under which the numerical solution is oscillatory and non oscillatory are obtained the properties of non oscillatory numerical solutions are investigated to verify our results we give numerical experiments

stochastic joint time frequency response analysis of nonlinear structural systems abstract a novel approximate analytical approach for determining the response evolutionary power spectrum eps of nonlinear hysteretic structural systems subject to stochastic excitation is developed specifically relying on the theory of locally stationary processes and utilizing a recently proposed representation of non stationary stochastic processes via wavelets a versatile formula for determining the nonlinear system response eps is derived this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets further the nonlinear system non stationary response amplitude probability density function pdf which is required as input for the developed approach is determined either by utilizing a numerical path integral scheme or by employing a time dependent rayleigh pdf approximation technique a significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non separable in time and frequency eps as well the hardening duffing and the versatile preisach hysteretic oscillators are considered in the numerical examples section comparisons with pertinent monte carlo simulations demonstrate the reliability of the approach

oscillation criteria of first order linear difference equations with delay argument abstract this paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument u k p k u k 0 k n where u k u k 1 u k p n r n n and lim k k examples illustrating the results are given it is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument k

covariance matrix estimation and linear process bootstrap for multivariate time series of possibly increasing dimension multivariate time series present many challenges especially when they are high dimensional the paper s focus is twofold first we address the subject of consistently estimating the autocovariance sequence this is a sequence of matrices that we conveniently stack into one huge matrix we are then able to show consistency of an estimator based on the so called flat top tapers most importantly the consistency holds true even when the time series dimension is allowed to increase with the sample size secondly we revisit the linear process bootstrap lpb procedure proposed by mcmurry and politis journal of time series analysis 2010 for univariate time series based on the aforementioned stacked autocovariance matrix estimator we are able to define a version of the lpb valid for multivariate time series under rather general assumptions we show that our multivariate linear process bootstrap mlpb has asymptotic validity for the sample mean in two important cases a when the time series dimension is fixed and b when it is allowed to increase with sample size as an aside in case a we show that the mlpb works also for spectral density estimators which is a novel result even in the univariate case we conclude with a simulation study that demonstrates the superiority of the mlpb in some important cases

bimeasure algebras on locally compact groups for locally compact groups g and h let bm g h denote the banach space of bounded bilinear forms on c0 g c0 h using a consequence of the fundamental inequality of a grothendieck a multiplication and an adjoint operation are introduced on bm g h which generalize the convolution structure of m g h and which make bm g h into a kg2 banach algebra where kg is grothendieck s universal constant various topics relating to the ideal structure of bm g h and the lifting of unitary representations of g h to representations of bm g h are investigated

estimating the null and the proportion of nonnull effects in large scale multiple comparisons an important issue raised by efron in the context of large scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect and seemingly negligible differences in the null may result in large differences in subsequent studies this suggests that a careful study of estimation of the null is indispensable in this article we consider the problem of estimating a null normal distribution and a closely related problem estimation of the proportion of nonnull effects we develop an approach based on the empirical characteristic function and fourier analysis the estimators are shown to be uniformly consistent over a wide class of parameters we investigate the numerical performance of the estimators using both simulated and real data in particular we apply our procedure to the analysis of breast cancer and human immunodeficiency virus microarray datasets the estimators perform favorably compared with existing methods

evolutionary coherence of nonstationary signals coherence is a widely used measure for characterizing linear dependence between a pair of signals for nonstationary signals the autospectrum cross spectrum and coherence between signals may evolve over time a standard approach is to divide the signals into overlapping blocks of fixed width and then smooth over frequency the periodogram matrix at each time block in this paper a consistent estimation procedure is developed using time localized linear filtering the proposed method automatically selects via repeated tests of homogeneity the optimal window width for estimating local coherence it is pointwise adaptive in the sense that the width of the optimal interval is allowed to change across time under the locally stationary process framework we develop a central limit theorem on the fisher z transform of our time localized band coherence we apply our method to a pair of highly dynamic brain waves signals whose coherence is shown to evolve during an epileptic seizure

discrimination and classification of nonstationary time series using the slex model statistical discrimination for nonstationary random processes is important in many applications our goal was to develop a discriminant scheme that can extract local features of the time series is consistent and is computationally efficient here we propose a discriminant scheme based on the slex smooth localized complex exponential library the slex library forms a collection of fourier type bases that are simultaneously orthogonal and localized in both time and frequency domains thus the slex library has the ability to extract local spectral features of the time series the first step in our procedure which is the feature extraction step based on work by saito is to find a basis from the slex library that can best illuminate the difference between two or more classes of time series in the next step we construct a discriminant criterion that is related to the kullback leibler divergence between the slex spectra of the different classes the discrimination criterion is based on estimates of the

wavelet threshold estimators for data with correlated noise wavelet threshold estimators for data with stationary correlated noise are constructed by applying a level dependent soft threshold to the coefficients in the wavelet transform a variety of threshold choices is proposed including one based on an unbiased estimate of mean squared error the practical performance of the method is demonstrated on examples including data from a neurophysiological context the theoretical properties of the estimators are investigated by comparing them with an ideal but unattainable bench mark that can be considered in the wavelet context as the risk obtained by ideal spatial adaptivity and more generally is obtained by the use of an oracle that provides information that is not actually available in the data it is shown that the level dependent threshold estimator performs well relative to the bench mark risk and that its minimax behaviour cannot be improved on in order of magnitude by any other estimator the wavelet domain structure of both short and long range dependent noise is considered and in both cases it is shown that the estimators have near optimal behaviour simultaneously in a wide range of function classes adapting automatically to the regularity properties of the underlying model the proofs of the main results are obtained by considering a more general multivariate normal decision theoretic problem

semiparametric estimation by model selection for locally stationary processes over recent decades increasingly more attention has been paid to the problem of how to fit a parametric model of time series with time varying parameters a typical example is given by autoregressive models with time varying parameters we propose a procedure to fit such time varying models to general non stationary processes the estimator is a maximum whittle likelihood estimator on sieves the results do not assume that the observed process belongs to a specific class of time varying parametric models we discuss in more detail the fitting of time varying ar p processes for which we treat the problem of the selection of the order p and we propose an iterative algorithm for the computation of the estimator a comparison with model selection by akaike s information criterion is provided through simulations

extensions of multilinear operators and banach space properties a new characterization of the dunford pettis property in terms of the extensions of multilinear operators to the biduals is obtained for the first time multilinear characterizations of the reciprocal dunford pettis property and pelczynski s property v are also found polynomial and holomorphic versions of these properties are given as well

validating stationarity assumptions in time series analysis by rolling local periodograms we propose a simple and powerful procedure to validate the assumption of weak stationarity in time series analysis our focus is on processes with a slowly varying autocovariance structure the procedure evaluates the supremum over time of the l2 distance between the local sample spectral density local periodogram calculated using a segment of observations falling within a rolling window and an estimator of the spectral density obtained using the entire time series at hand large sample properties of a basic deviation process are investigated and critical values of a supremum type test are obtained using an appropriate bootstrap procedure the finite sample size and power properties of the procedure are investigated by means of simulations real data examples demonstrate the ability of the procedure to detect possible changes in the autocovariance structure of a time series and to understand their nature

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

oscillation criteria for delay and difference equations with non monotone arguments consider the first order linear delay differential equation 1 x t p t x t t 0 t t 0 where p email protected c t 0 r t t t 0 and lim t email protected t and the discrete analogue difference equation 1 dx n p n x t n 0 n 0 1 2 where d denotes the forward difference operator dx n x n 1 x n p n is a sequence of nonnegative real numbers and t n is a sequence of integers such that t n 0 and lim n email protected n the state of the art on the oscillation of all solutions to these equations are established especially in the case of non monotone arguments examples illustrating the results are given

wavelet based estimations of fractional brownian sheet least squares versus maximum likelihood abstract the scaling behavior or self similarity and long range dependence has been studied in numerous fields such as computer telecommunication hydrology and economics and can be characterized by the hurst parameter h fractional brownian sheet plays an important role in modeling the scaling behavior in multi dimensional signals a wavelet based maximum likelihood estimator for the hurst parameter h h 1 h d of fractional brownian sheet is proposed it is asymptotically normal and compared with the wavelet based least squares estimators introduced in the previous article from theoretical analysis and numerical simulation the theoretical results show that the proposed estimator has the same asymptotic covariance as the least squares estimator of which the weight is equal to the number of available wavelet coefficients and the convergence rates of two estimators are both o 1 i 1 d t i t i is sample length in the i th dimension the simulation results show that the proposed estimator performs better than the least squares estimators in bias and costs less runtime than the generalized least squares estimator realized by the two step method

estimation of fractal signals from noisy measurements using wavelets the role of the wavelet transformation as a whitening filter for 1 f processes is exploited to address problems of parameter and signal estimations for 1 f processes embedded in white background noise robust computationally efficient and consistent iterative parameter estimation algorithms are derived based on the method of maximum likelihood and cramer rao bounds are obtained included among these algorithms are optimal fractal dimension estimators for noisy data algorithms for obtaining bayesian minimum mean square signal estimates are also derived together with an explicit formula for the resulting error these smoothing algorithms find application in signal enhancement and restoration the parameter estimation algorithms find application in signal enhancement and restoration the parameter estimation algorithms in addition to solving the spectrum estimation problem and to providing parameters for the smoothing process are useful in problems of signal detection and classification results from simulations are presented to demonstrated the viability of the algorithms

wavelet based parameter estimation for polynomial contaminated fractionally differenced processes we consider the problem of estimating the parameters for a stochastic process using a time series containing a trend component trend i e large scale variations in the series that are best modeled outside of a stochastic framework is often confounded with low frequency stochastic fluctuations this problem is particularly evident in models such as fractionally differenced fd processes which exhibit slowly decaying autocorrelations and can be extended to encompass nonstationary processes with substantial low frequency components we use the discrete wavelet transform dwt to estimate parameters for stationary and nonstationary fd processes in a model of polynomial trend plus fd noise using daubechies wavelet filters allows for automatic elimination of polynomial trends due to embedded differencing operations parameter estimation is based on an approximate maximum likelihood approach made possible by the fact that the dwt decorrelates fd processes approximately we consider this decorrelation in detail examining the between and within scale wavelet correlations separately better between scale decorrelation can be achieved by increasing the length of the wavelet filter whereas the within scale correlations can be handled via explicit modeling by a low order autoregressive process we demonstrate our methodology by applying it to a popular climate dataset

inference for non stationary time series regression with or without inequality constraints summary we consider statistical inference for time series linear regression where the response and predictor processes may experience general forms of abrupt and smooth non stationary behaviours over time meanwhile the regression parameters may be subject to linear inequality constraints a simple and unified procedure for structural stability checks and parameter inference is proposed in the case where the regression parameters are constrained the methodology proposed is shown to be consistent whether or not the true regression parameters are on the boundary of the restricted parameter space via utilizing an asymptotically invariant geometric property of polyhedral cones

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

asymptotic decorrelation of between scale wavelet coefficients in recent years there has been much interest in the analysis of time series using a discrete wavelet transform dwt based upon a daubechies wavelet filter part of this interest has been sparked by the fact that the dwt approximately decorrelates certain stochastic processes including stationary fractionally differenced fd processes with long memory characteristics and certain nonstationary processes such as fractional brownian motion it is shown that as the width of the wavelet filter used to form the dwt increases the covariance between wavelet coefficients associated with different scales decreases to zero for a wide class of stochastic processes these processes are gaussian with a spectral density function sdf that is the product of the sdf for a not necessarily stationary fd process multiplied by any bounded function that can serve as an sdf on its own we demonstrate that this asymptotic theory provides a reasonable approximation to the between scale covariance properties of wavelet coefficients based upon filter widths in common use our main result is one important piece of an overall strategy for establishing asymptotic results for certain wavelet based statistics

applications of complex valued wavelet transforms to subband decomposition a method for constructing complex valued linear phase fir conjugate quadrature filters and associated wavelet bases is described each filter is derived by replating certain zeros of a real valued fir conjugate quadrature filter by their reciprocal conjugates the derived filters have the same frequency response magnitudes as the original filters and their linear phase property permits the use of symmetrization in subband decomposition to avoid border discontinuities that result from signal periodization subband decomposition and reconstruction using both a length 6 filter associated with a daubechies 1988 wavelet bases and a related length 6 complex valued linear phase filter are compared to illustrate the reduced border effects

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

functional classification in hilbert spaces let x be a random variable taking values in a separable hilbert space x with label y spl isin 0 1 we establish universal weak consistency of a nearest neighbor type classifier based on n independent copies x sub i y sub i of the pair x y extending the classical result of stone to infinite dimensional hilbert spaces under a mild condition on the distribution of x we also prove strong consistency we reduce the infinite dimension of x by considering only the first d coefficients of a fourier series expansion of each x sub i and then we perform k nearest neighbor classification in spl ropf sup d both the dimension and the number of neighbors are automatically selected from the data using a simple data splitting device an application of this technique to a signal discrimination problem involving speech recordings is presented

a radon nikodým theorem for frechet measures abstract we apply results in operator space theory to the setting of multidimensional measure theory using the extended haagerup tensor product of effros and ruan we derive a radon nikod m theorem for bimeasures and then extend the result to general frechet measures scalar valued polymeasures we also prove a measure theoretic grothendieck inequality provide a characterization of the injective tensor product of two spaces of lebesgue integrable functions and discuss the possibility of a bounded convergence theorem for frechet measures

a necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments abstract in this article we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions in particular we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non negative examples and tables are given to illustrate our results

decorrelation of wavelet coefficients for long range dependent processes we consider a discrete time stationary long range dependent process xk kisinz such that its spectral density equals phi lambda 2d where phi is a smooth function such that phi 0 phi 0 0 and phi lambda gesclambda for lambdaisin 0 pi then for any wavelet psi with n vanishing moments the lag k within level covariance of wavelet coefficients decays as o k2d 2n 1 when krarrinfin the result applies to fractionally integrated autoregressive moving average arma processes as well as to fractional gaussian noise

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

discrimination and clustering for multivariate time series abstract minimum discrimination information provides a useful generalization of likelihood methodology for classification and clustering of multivariate time series discrimination between different classes of multivariate time series that can be characterized by differing covariance or spectral structures is of importance in applications occurring in the analysis of geophysical and medical time series data for discrimination between such multivariate series kullback leibler discrimination information and the chernoff information measure are developed for the multivariate non gaussian case asymptotic error rates and limiting distributions are given for a generalized spectral disparity measure that includes the foregoing criteria as special cases applications to problems of clustering and classifying earthquakes and mining explosions are given

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

discrimination of locally stationary time series based on the excess mass functional discrimination of time series is an important practical problem with applications in various scientific fields we propose and study a novel approach to this problem our approach is applicable to cases where time series in different categories have a different shape although based on the idea of feature extraction our method is not distance based and as such does not require aligning the time series instead features are measured for each time series and discrimination is based on these individual measures an ar process with a time varying variance is used as an underlying model our method then uses shape measures or better measures of concentration of the variance function as a criterion for discrimination it is this concentration aspect or shape aspect that makes the approach intuitively appealing we provide some mathematical justification for our proposed methodology as well as a simulation study and an application to the problem of discriminating earthquakes and explosions

on the range of validity of the autoregressive sieve bootstrap we explore the limits of the autoregressive ar sieve bootstrap and show that its applicability extends well beyond the realm of linear time series as has been previously thought in particular for appropriate statistics the ar sieve bootstrap is valid for stationary processes possessing a general wold type autoregressive representation with respect to a white noise in essence this includes all stationary purely nondeterministic processes whose spectral density is everywhere positive our main theorem provides a simple and effective tool in assessing whether the ar sieve bootstrap is asymptotically valid in any given situation in effect the large sample distribution of the statistic in question must only depend on the first and second order moments of the process prominent examples include the sample mean and the spectral density as a counterexample we show how the ar sieve bootstrap is not always valid for the sample autocovariance even when the underlying process is linear

automatic statistical analysis of bivariate nonstationary time series we propose a new method for analyzing bivariate nonstationary time series the proposed method is a statistical procedure that automatically segments the time series into approximately stationary blocks and selects the span to be used to obtain the smoothed estimates of the time varying spectra and coherence it is based on the smooth localized complex exponential slex transform which forms a library of orthogonal complex valued transforms that are simultaneously localized in time and frequency we show that the smoothed slex periodograms are consistent estimators report simulation results and apply the method to a two channel electroencephalogram dataset recorded during an epileptic seizure

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

fractal estimation from noisy data via discrete fractional gaussian noise dfgn and the haar basis the authors show that the application of the discrete wavelet transform dwt using the haar basis to the increments of fractional brownian motion fbm also known as discrete fractional gaussian noise dfgn yields coefficients which are weakly correlated and have a variance that is exponentially related to scale similar results were derived by flandrin 1989 tewfik and kim for a continuous time fbm going through a continuous wavelet transform cwt the new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by wornell and oppenheim the performance of the new algorithm is compared with that of wornell and oppenheim s see ieee trans signal processing vol 40 p 611 623 mar 1992 algorithm in numerical simulation

wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra we derive minimax rates for estimation in anisotropic smoothness classes this rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension it is shown that this basis is superior to its one scale multiresolution analog if different degrees of smoothness in different directions are present as an important application we introduce a new adaptive wavelet estimator of the time dependent spectrum of a locally stationary time series using this model which was resently developed by dahlhaus we show that the resulting estimator attains nearly the rate which is optimal in gaussian white noise simultaneously over a wide range of smoothness classes moreover by our new approach we overcome the difficulty of how to choose the right amount of smoothing i e how to adapt to the appropriate resolution for reconstructing the local structure of the evolutionary spectrum in the time frequency plane

eigenvalues and pseudo eigenvalues of toeplitz matrices the eigenvalues of a nonhermitian toeplitz matrix a are usually highly sensitive to perturbations having condition numbers that increase exponentially with the dimension n an equivalent statement is that the resolvent zz a of a toeplitz matrix may be much larger in norm than the eigenvalues alone would suggest exponentially large as a function of n even when z is far from the spectrum because of these facts the meaningfulness of the eigenvalues of nonhermitian toeplitz matrices for any but the most theoretical purposes should be considered suspect in many applications it is more meaningful to investigate the e pseudo eigenvalues the complex numbers z with ll zz a 11 l this paper analyzes the pseudospectra of toeplitz matrices and in particular relates them to the symbols of the matrices and thereby to the spectra of the associated toeplitz and laurent operators our results are reasonably complete in the triangular case and preliminary in the cases of nontriangular toeplitz matrices block toeplitz matrices and toeplitz like matrices with smoothly varying coefficients computed examples of pseudospectra are presented throughout and applications in numerical analysis are mentioned

wavelet transforms versus fourier transforms this note is a very basic introduction to wavelets it starts with an orthogonal basis of piecewise constant functions constructed by dilation and translation the wavelet transform maps each f x to its coefficients with respect to this basis the mathematics is simple and the transform is fast faster than the fast fourier transform which we briefly explain but approximation by piecewise constants is poor to improve this first wavelet we are led to dilation equations and their unusual solutions higher order wavelets are constructed and it is surprisingly quick to compute with them always indirectly and recursively

on oscillation of differential and difference equations with non monotone delays abstract for the delay differential equation x t p t x h t 0 p t 0 h t t t 0 lim t h t we demonstrate that the inequality limsup t h t t p u d u 1 is not sufficient for oscillation moreover for any a 0 the relation limsup t h t t p u d u a generally does not imply oscillation a similar result is obtained for difference equations in terms of the maximal argument new sufficient oscillation conditions are presented for differential and difference equations

oracle and adaptive compound decision rules for false discovery rate control we develop a compound decision theory framework for multiple testing problems and derive an oracle rule based on the z values that minimizes the false nondiscovery rate fnr subject to a constraint on the false discovery rate fdr we show that many commonly used multiple testing procedures which are p value based are inefficient and propose an adaptive procedure based on the z values the z value based adaptive procedure asymptotically attains the performance of the z value oracle procedure and is more efficient than the conventional p value based methods we investigate the numerical performance of the adaptive procedure using both simulated and real data in particular we demonstrate our method in an analysis of the microarray data from a human immunodeficiency virus study that involves testing a large number of hypotheses simultaneously

estimating the population local wavelet spectrum with application to non stationary functional magnetic resonance imaging time series functional magnetic resonance imaging fmri is a dynamic four dimensional imaging modality however in almost all fmri analyses the time series elements of this data are assumed to be second order stationary in this paper we examine using time series spectral methods whether such stationary assumptions can be made and whether estimates of non stationarity can be used to gain understanding into fmri experiments a non stationary version of replicated stationary time series analysis is proposed that takes into account the replicated time series that are available from nearby voxels in a region of interest roi these are used to investigate non stationarities in both the roi itself and the variations within the roi the proposed techniques are applied to simulated data and to an anxiety inducing fmri experiment

clustering high dimensional time series based on parallelism this article considers the problem of clustering high dimensional time series based on trend parallelism the underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors a special class of nonstationary processes for each group where the parallelism holds i semiparametrically estimate its representative trend function and vertical shifts of group members and establish their central limit theorems an information criterion consisting of in group similarities and number of groups is then proposed for the purpose of clustering i prove its theoretical consistency and propose a splitting coalescence algorithm to reduce the computational burden in practice the method is illustrated by both simulation and a real data example

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

on the approximate decorrelation property of the discrete wavelet transform for fractionally differenced processes in this correspondence we develop an asymptotic theory for the correlations of wavelet coefficients of the discrete wavelet transform dwt for fractionally differenced processes it provides a theoretical justification for the approximate decorrelation property of the dwt for fractionally differenced processes in addition it provides insights on how the length of the wavelet filter affects the within scale correlations and the between scale correlations differently for within scale correlations increasing the length of the wavelet filter increases the rate of decay as the two wavelet coefficients get further apart while for between scale correlations using a wavelet filter that is long enough can reduce the between scale correlations even for wavelet coefficients that are close together

autoregressive aided periodogram bootstrap for time series a bootstrap methodology for the periodogram of a stationary process is proposed which is based on a combination of a time domain parametric and a frequency domain nonparametric bootstrap the parametric fit is used to generate periodogram ordinates that imitate the essential features of the data and the weak dependence structure of the periodogram while a nonparametric kernel based correction is applied in order to catch features not represented by the parametric fit the asymptotic theory developed shows validity of the proposed bootstrap procedure for a large class of periodogram statistics for important classes of stochastic processes validity of the new procedure is also established for periodogram statistics not captured by existing frequency domain bootstrap methods based on independent periodogram replicates

wavelet variance components in image space for spatiotemporal neuroimaging data neuroimaging studies place great emphasis on not only the estimation but also the standard error estimates of underlying parameters derived from a temporal model this allows inferences to be made about the signal estimates and resulting conclusions to be drawn about the underlying data it can often be advantageous to interrogate temporal models after spatial transformation of the data into the wavelet domain wavelet bases provide a multiresolution decomposition of the spatial data dimension and an ensuing reduction in spatial correlation however widespread acceptance of these wavelet techniques has been hampered by the limited ability to reconstruct both parametric and error estimates into the image domain after analysis of temporal models in the wavelet domain this paper introduces a derivation and a fast implementation of a method for the calculation of the variance of the parametric images obtained from wavelet filters the technique is proposed for a class of estimators that have been shown to be useful in neuroimaging studies the techniques are demonstrated for both functional magnetic resonance imaging fmri and positron emission tomography pet data sets

unbalanced haar technique for nonparametric function estimation the discrete unbalanced haar uh transform is a decomposition of one dimensional data with respect to an orthonormal haar like basis where jumps in the basis vectors do not necessarily occur in the middle of their support we introduce a multiscale procedure for estimation in gaussian noise that consists of three steps a uh transform thresholding of the decomposition coefficients and the inverse uh transform we show that our estimator is mean squared consistent with near optimal rates for a wide range of functions uniformly over uh bases that are not too unbalanced a vital ingredient of our approach is basis selection we choose each basis vector so that it best matches the data at a specific scale and location where the latter parameters are determined by the parent basis vector our estimator is computable in o n log n operations a simulation study demonstrates the good performance of our estimator compared with state of the art competitors we apply our method to the estimation of the mea

a frequency domain bootstrap for ratio statistics in time series analysis the asymptotic properties of the bootstrap in the frequency domain based on studentized periodogram ordinates are studied it is proved that this bootstrap approximation is valid for ratio statistics such as autocorrelations by using edgeworth expansions it is shown that the bootstrap approximation even outperforms the normal approximation the results carry over to whittle estimates in a simulation study the behavior of the bootstrap is studied for empirical correlations and whittle estimates

wavelet shrinkage asymptopia much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects curves densities spectral densities images from noisy data a now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems unfortunately the results have rarely moved into practice for a variety of reasons among them being similarity to known methods computational intractability and lack of spatial adaptivity we discuss a method for curve estimation based on n noisy data translate the empirical wavelet coefficients towards the origin by an amount 2 log n n the proposal differs from those in current use is computationally practical and is spatially adaptive it thus avoids several of the previous objections further the method is nearly minimax both for a wide variety of loss functions pointwise error global error measured in l p norms pointwise and global error in estimation of derivatives and for a wide range of smoothness classes including standard holder and sobolev classes and bounded variation this is a much broader near optimality than anything previously proposed we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves finally the theory underlying the method is interesting as it exploits a correspondence between statistical questions and questions of optimal recovery and information based complexity

resampling hierarchical processes in the wavelet domain a case study using atmospheric turbulence abstract there is a growing need for statistical methods that generate an ensemble of plausible realizations of a hierarchical process from a single run or experiment the main challenge is how to construct such an ensemble in a manner that preserves the internal dynamics e g intermittency and temporal persistency a popular hierarchical process often used as a case study in such problems is atmospheric turbulent flow analogies to turbulence are often called upon when information flow from large to small scales non gaussian statistics and intermittency are inherent attributes of the hierarchical process under consideration these attributes are key defining syndromes of the turbulent cascade thereby making turbulence time series ideal for testing such ensemble generation schemes in this study we propose a wavelet based resampling scheme wb and compare it to the traditional fourier based phase randomization bootstrap fb approach within the context of the turbulence energy cascade the comparison between the two resampling methods and observed ensemble statistics constructed by clustering similar meteorological conditions demonstrate that the wb reproduces several features related to intermittency of the ensemble series when compared to fb in particular the wb exhibited an increase in wavelet energy activity and an increase in the wavelet flatness factor with increasing frequency consistent with the cluster of ensemble statistics on the other hand the fb yielded no increase in such energy activity with scale and resulted in near gaussian wavelet coefficients at all frequencies within the inertial subrange the scaling behavior of the longitudinal u and vertical w velocity structure functions of various order p 0 confirms that wb preserves the small scale intermittency whereas fb completely destroys it the extension of wb to the multivariate case is also demonstrated via the conservation of co spectra between u and w time series because the resampling strategy proposed here is conducted in the wavelet domain gap infected and uneven sampled time series can be readily accommodated within the wb finally recommendations about the filter and block sizes are discussed

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

semiparametric bayesian inference for time series with mixed spectra summary a bayesian analysis is presented of a time series which is the sum of a stationary component with a smooth spectral density and a deterministic component consisting of a linear combination of a trend and periodic terms the periodic terms may have known or unknown frequencies the advantage of our approach is that different features of the data such as the regression parameters the spectral density unknown frequencies and missing observations are combined in a hierarchical bayesian framework and estimated simultaneously a bayesian test to detect deterministic components in the data is also constructed by using an asymptotic approximation to the likelihood the computation is carried out efficiently using the markov chain monte carlo method in o mn operations where n is the sample size and m is the number of iterations we show empirically that our approach works well on real and simulated samples

on the continuous wavelet transform of second order random processes some second order properties of random processes such as periodic correlation stationarity harmonizability self similarity are characterized via corresponding properties of their wavelet transform any one of these properties of the wavelet transform characterizes the corresponding property of the increments of the random process of order equal to the order of regularity of the analyzing wavelet these results are then specialized to fractional brownian motion and other self similar processes

false discovery control with p value weighting we present a method for multiple hypothesis testing that maintains control of the false discovery rate while incorporating prior information about the hypotheses the prior information takes the form of p value weights if the assignment of weights is positively associated with the null hypotheses being false the procedure improves power except in cases where power is already near one even if the assignment of weights is poor power is only reduced slightly as long as the weights are not too large we also provide a similar method to control false discovery exceedance

orthogonally additive polynomials on spaces of continuous functions we show that for every orthogonally additive homogeneous polynomial p on a space of continuous functions c k with values in a banach space y there exists a linear operator s c k y such that p f s f n this is the c k version of a related result of sundaresam for polynomials on l p spaces

asymptotic distributions of the sample mean autocovariances and autocorrelations of long memory time series we derive the asymptotic distributions of the sample mean autocovariances and autocorrelations for a time series whose autocovariance function has the powerlaw decay 0 o 1 as the results differ in important respects from the corresponding results for short memory processes whose autocovariance functions are absolutely summable for long memory processes the variances of the sample mean and of the sample autocovariances and autocorrelations for 0 12 are not of asymptotic order n 1 when 0 12 the asymptotic distributions of the sample autocovariances and autocorrelations are not normal

bootstrapping locally stationary processes summary we propose a non parametric method to bootstrap locally stationary processes which combines a time domain wild bootstrap approach with a non parametric frequency domain approach the method generates pseudotime series which mimic asymptotically correct the local second and to the necessary extent the fourth order moment structure of the underlying process thus it can be applied to approximate the distribution of several statistics that are based on observations of the locally stationary process we prove a bootstrap central limit theorem for a general class of statistics that can be expressed as functionals of the preperiodogram the latter being a useful tool for inferring properties of locally stationary processes some simulations and a real data example shed light on the finite sample properties and illustrate the ability of the bootstrap method proposed

on the correlation structure of the wavelet coefficients of fractional brownian motion shows that the interdependence of the discrete wavelet coefficients of fractional brownian motion defined by normalized correlation decays exponentially fast across scales and hyperbolically fast along time

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

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

construction of multivariate surrogate sets from nonlinear data using the wavelet transform abstract the use of surrogate data has become a crucial first step in the study of nonlinearity in time series data a widely used technique to construct surrogate data is to randomize the phases of the data in the fourier domain in this paper an alternative technique based on the resampling of wavelet coefficients is discussed this approach exploits between scale correlations that exist within nonlinear data but which are either absent or weak in stochastic data it proceeds by transforming the data into the wavelet domain and permuting the wavelet coefficients experimental and numerical time series data are used to demonstrate that the performance of the wavelet resampling technique is comparable to phase randomization in terms of the preservation of linear properties removal of nonlinear structure and computational demands however the wavelet technique may have specific and distinct advantages in the application to complex data sets such as numerical analysis of turbulence and experimental brain imaging data where wavelets give a more parsimonious representation of spatio temporal patterns than fourier modes it is shown that different techniques of resampling the data in the wavelet domain may optimize the construction of surrogate data according to the properties of the experimental time series and computational constraints

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

wavelet based estimation of 1 f type signal parameters confidence intervals using the bootstrap we propose to construct confidence intervals of parameters of 1 f type signals using a nonparametric wavelet based bootstrap method bootstrap based confidence intervals of maximum likelihood parameter estimates are compared to the confidence intervals derived from the cramer rao lower bound crlb for moderately large data sample sizes the bootstrap approach achieves the nominal coverage and may perform better than the crlb based parametric approach

multifractality tests using bootstrapped wavelet leaders multifractal analysis which mostly consists of measuring scaling exponents is becoming a standard technique available in most empirical data analysis toolboxes making use of the most recent theoretical results it is based here on the estimation of the cumulants of the log of the wavelet leaders an elaboration on the wavelet coefficients these log cumulants theoretically enable discrimination between mono and multifractal processes as well as between simple log normal multifractal models and more advanced ones the goal of the present contribution is to design nonparametric bootstrap hypothesis tests aiming at testing the nature of the multifractal properties of stochastic processes and empirical data bootstrap issues together with six declinations of test designs are analyzed their statistical performance significances powers and p values are assessed and compared by means of monte carlo simulations performed on synthetic stochastic processes whose multifractal properties and log cumulants are known theoretically a priori we demonstrate that the joint use of wavelet leaders log cumulants and bootstrap procedures enable us to obtain a powerful tool for testing the multifractal properties of data this tool is practically effective and can be applied to a single observation of data with finite length

high dimensional variable selection this paper explores the following question what kind of statistical guarantees can be given when doing variable selection in high dimensional models in particular we look at the error rates and power of some multi stage regression methods in the first stage we fit a set of candidate models in the second stage we select one model by cross validation in the third stage we use hypothesis testing to eliminate some variables we refer to the first two stages as screening and the last stage as cleaning we consider three screening methods the lasso marginal regression and forward stepwise regression our method gives consistent variable selection under certain conditions

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

remark on positive solutions of discrete equation δu k n p k u k abstract a delayed discrete equation u k n p k u k with positive function p k is considered sufficient conditions are formulated to guarantee the existence of a positive solution if k

modeling and estimation of covariance of replicated modulated cyclical time series this paper introduces the novel class of modulated cyclostationary processes a class of nonstationary processes exhibiting frequency coupling and proposes a method of their estimation from repeated trials cyclostationary processes also exhibit frequency correlation but have loeve spectra whose support lies only on parallel lines in the dual frequency plane such extremely sparse structure does not adequately represent many biological processes thus we propose a model that in the time domain modulates the covariance of cyclostationary processes and consequently broadens their frequency support in the dual frequency plane the spectra and the cross coherence of the proposed modulated cyclostationary process are first estimated using multitaper methods a shrinkage procedure is then applied to each trial specific estimate to reduce the estimation risk multiple trials of each series are observed when combining information across trials we carefully take into account the bias that may be introduced by phase misalignment and the fact that the loeve spectra and cross coherence across replicates may only be similar but not necessarily identical in a simulation study we illustrate the performance of our estimation method on a model that realistically captures the features observed in the electroencephalogram eeg data the application of the inference methods developed for the modulated cyclostationary model to eeg data also demonstrates that the proposed model captures statistically significant cross frequency interactions that ought to be further examined by neuroscientists

oscillation of first order delay equations we study the oscillation of first order delay equations using a method that parallels the use of riccati equations in the study of the oscillation of second order ordinary differential equations without delay we give an alternative proof of a comparison theorem first established by kwong and patula obtain some asymptotic behaviors of nonoscillatory solutions prove a new criterion that falls within a gap left open by established results and confirm a conjecture raised by hunt and yorke

integral operators on the product of c k spaces we study and characterize the integral multilinear operators on a product of c k spaces in terms of the representing polymeasure of the operator some applications are given in particular we characterize the borel polymeasures that can be extended to a measure in the product algebra generalizing previous results for bimeasures we also give necessary conditions for the weak compactness of the extension of an integral multilinear operator on a product of c k spaces

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