number of degrees of freedom of two dimensional turbulence we derive upper bounds for the number of degrees of freedom of two dimensional navier stokes turbulence freely decaying from a smooth initial vorticity field omega x y 0 omega0 this number denoted by n is defined as the minimum dimension such that for n or n arbitrary n dimensional balls in phase space centered on the solution trajectory omega x y t for t 0 contract under the dynamics of the system linearized about omega x y t in other words n is the minimum number of greatest lyapunov exponents whose sum becomes negative it is found that n or c1re when the phase space is endowed with the energy norm and n or c2re 1 ln re 1 3 when the phase space is endowed with the enstrophy norm here c1 and c2 are constant and re is the reynolds number defined in terms of omega0 the system length scale and the viscosity nu the linear or nearly linear dependence of n on re is consistent with the estimate for the number of active modes deduced from a recent mathematical bound for the viscous dissipation wave number this result is in a sharp contrast to the forced case for which well known estimates for the hausdorff dimension dh of the global attractor scale highly superlinearly with nu 1 we argue that the extra dependence of dh on nu 1 is not an intrinsic property of the turbulent dynamics rather it is a removable artifact brought about by the use of a time independent forcing as a model for energy and enstrophy injection that drives the turbulence

extensivity of two dimensional turbulence this study is concerned with how the attractor dimension of the two dimensional navier stokes equations depends on characteristic length scales including the system integral length scale the forcing length scale and the dissipation length scale upper bounds on the attractor dimension derived by constantin foias and temam are analysed it is shown that the optimal attractor dimension estimate grows linearly with the domain area suggestive of extensive chaos for a sufficiently large domain if the kinematic viscosity and the amplitude and length scale of the forcing are held fixed for sufficiently small domain area a slightly super extensive estimate becomes optimal in the extensive regime the attractor dimension estimate is given by the ratio of the domain area to the square of the dissipation length scale defined on physical grounds in terms of the average rate of shear this dissipation length scale which is not necessarily the scale at which the energy or enstrophy dissipation takes place can be identified with the dimension correlation length scale the square of which is interpreted according to the concept of extensive chaos as the area of a subsystem with one degree of freedom furthermore these length scales can be identified with a minimum length scale of the flow which is rigorously deduced from the concept of determining nodes

on multiple regression models with nonstationary correlated errors summary we consider the estimation of parameters of a multiple regression model with non stationary errors we assume the nonstationary errors satisfy a time dependent auto regressive process and describe a method for estimating the parameters of the regressors and the time dependent autoregressive parameters the parameters are rescaled as in nonparametric regression to obtain the asymptotic sampling properties of the estimators the method is illustrated with an example taken from global temperature anomalies

optimal data based kernel estimation of evolutionary spectra complex demodulation of evolutionary spectra is formulated as a two dimensional kernel smoother in the time frequency domain first a tapered fourier transform y sub v f t is calculated then the log spectral estimate is smoothed as the characteristic widths of the kernel smoother increase the bias from the temporal and frequency averaging increases while the variance decreases the demodulation parameters such as the order length and bandwidth of spectral taper and the kernel smoother are determined by minimizing the expected error for well resolved evolutionary spectra the optimal taper length is a small fraction of the optimal kernel halfwidth the optimal frequency bandwidth w for the spectral window scales as w sup 2 approximately lambda tau where tau is the characteristic time and lambda sub f is the characteristic frequency scalelength in contrast the optimal halfwidths for the second stage kernel smoother scales as h approximately 1 tau lambda sub f sup 1 p 2 where p is the order of the kernel smoother the ratio of the optimal frequency halfwidth to the optimal time halfwidth is determined

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

multiscale clustering of nonparametric regression curves abstract in a wide range of modern applications one observes a large number of time series rather than only a single one it is often natural to suppose that there is some group structure in the observed time series when each time series is modeled by a nonparametric regression equation one may in particular assume that the observed time series can be partitioned into a small number of groups whose members share the same nonparametric regression function we develop a bandwidth free clustering method to estimate the unknown group structure from the data more precisely speaking we construct multiscale estimators of the unknown groups and their unknown number which are free of classical bandwidth or smoothing parameters in the theoretical part of the paper we analyze the statistical properties of our estimators our theoretical results are derived under general conditions which allow the data to be dependent both in time series direction and across different time series the technical analysis of the paper is complemented by simulated and real data examples

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

narrowband acoustic doppler volume backscattering signal ii spectral centroid estimation for pt i see ibid vol 50 p 2644 2655 2002 this paper proposes a novel algorithm to estimate the time varying spectral centroid of the narrowband doppler volume backscattering signal it is constructed in a semi parametric way that is modeling parametrically the local narrowband evolutionary spectrum using an ar 2 and nonparametrically adapting its time varying coefficients using the wavelet shrinkage the improved performance is gained by the underlying linear prediction function of the ar 2 and the minimax optimality of the unknown smoothness adaptation of the wavelet shrinkage procedure

low dimensional attractors arise from forcing at small scales abstract standard estimates of the dimension of the attractor of the 2d navier stokes equations are given in terms of a dimensionless grashof number that measures the total amount of energy injected into the flow however this result takes no account of whether the forcing is concentrated at large or small scales by a simple modification of the usual argument this paper provides a bound that is linear in a modified grashof number that reflects the spatial structure of the forcing in particular as a fixed amount of energy is injected at progressively smaller scales then the dimension of the attractor decreases

thick pen transformation for time series traditional visualisation of time series data often consists of plotting the time series values against time and connecting the dots we propose an alternative multiscale visualisation technique motivated by the scale space approach in computer vision in brief our method also connects the dots but uses a range of pens of varying thicknesses for this purpose the resulting multiscale map termed the thick pen transform tpt corresponds to viewing the time series from a range of distances we formally prove that the tpt is a discriminatory statistic for two gaussian time series with distinct correlation structures further we show interesting possible applications of the tpt to measuring cross dependence in multivariate time series classifying time series and testing for stationarity in particular we derive the asymptotic distribution of our test statistic and argue that the test is applicable to both linear and nonlinear processes under low moment assumptions various other aspects of the methodology including other possible applications are also discussed

adaptive drift estimation for nonparametric diffusion model we consider a nonparametric diffusion process whose drift and diffusion coefficients are nonparametric functions of the state variable the goal is to estimate the unknown drift coefficient we apply a locally linear smoother with a data driven bandwidth choice the procedure is fully adaptive and nearly optimal up to a log log factor the results about the quality of estimation are nonasymptotic and do not require any ergodic or mixing properties of the observed process

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

a smoothness priors time varying ar coefficient modeling of nonstationary covariance time series a smoothness priors time varying ar coefficient model approach for the modeling of nonstationary in the covariance time series is shown smoothness priors in the form of a difference equation constraint excited by an independent white noise are imposed on each ar coefficient the unknown white noise variances are hyperparameters of the ar coefficient distribution the critical computation is of the likelihood of the hyperparameters of the bayesian model this computation is facilitated by a state space representation kalman filter implementation the best difference equation order best ar model order best hyperparameter model locally in time is selected using the minimum aic method also an instantaneous spectral density is defined in terms of the instantaneous ar model coefficients and a smoothed estimate of the instantaneous time series variance an earthquake record is analyzed the changing spectral analysis of the original data and of simulations from a time varying ar coefficient model of that data are shown

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

detecting gradual changes in locally stationary processes in a wide range of applications the stochastic properties of the observed time series change over time the changes often occur gradually rather than abruptly the properties are approximately constant for some time and then slowly start to change in such situations it is frequently of interest to locate the time point where the properties start to vary in contrast to the analysis of abrupt changes methods for detecting smooth or gradual change points are less developed and often require strong parametric assumptions in this paper we develop a fully nonparametric method to estimate a smooth change point in a locally stationary framework we set up a general procedure which allows to deal with a wide variety of stochastic properties including the mean auto covariances and higher order moments the theoretical part of the paper establishes the convergence rate of the new estimator in addition we examine its finite sample performance by means of a simulation study and illustrate the methodology by applications to temperature and financial return data

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

statistical inference for time varying arch processes in this paper the class of arch oo models is generalized to the nonstationary class of arch oo models with time varying coefficients for fixed time points a stationary approximation is given leading to the notation locally stationary arch oo process the asymptotic properties of weighted quasi likelihood estimators of time varying arch p processes p oo are studied including asymptotic normality in particular the extra bias due to nonstationarity of the process is investigated moreover a taylor expansion of the nonstationary arch process in terms of stationary processes is given and it is proved that the time varying arch process can be written as a time varying volterra series

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

determining modes for continuous data assimilation in 2d turbulence we study the number of determining modes necessary for continuous data assimilation in the two dimensional incompressible navier stokes equations our focus is on how the spatial structure of the body forcing affects the rate of continuous data assimilation and the number of determining modes we treat this problem analytically by proving a convergence result depending on the h 1 norm of f and computationally by considering a family of forcing functions with identical grashof numbers that are supported on different annuli in fourier space the rate of continuous data assimilation and the number of determining modes is shown to depend strongly on the length scales present in the forcing

an efficient estimator for locally stationary gaussian long memory processes this paper addresses the estimation of locally stationary long range dependent processes a methodology that allows the statistical analysis of time series data exhibiting both nonstationarity and strong dependency a time varying parametric formulation of these models is introduced and a whittle likelihood technique is proposed for estimating the parameters involved large sample properties of these whittle estimates such as consistency normality and efficiency are established in this work furthermore the finite sample behavior of the estimators is investigated through monte carlo experiments as a result from these simulations we show that the estimates behave well even for relatively small sample sizes

testing for smooth structural changes in time series models via nonparametric regression checking parameter stability of econometric models is a long standing problem almost all existing structural change tests in econometrics are designed to detect abrupt breaks little attention has been paid to smooth structural changes which may be more realistic in economics we propose a consistent test for smooth structural changes as well as abrupt structural breaks with known or unknown change points the idea is to estimate smooth time varying parameters by local smoothing and compare the fitted values of the restricted constant parameter model and the unrestricted time varying parameter model the test is asymptotically pivotal and does not require prior information about the alternative a simulation study highlights the merits of the proposed test relative to a variety of popular tests for structural changes in an application we strongly reject the stability of univariate and multivariate stock return prediction models in the postwar and post oil shocks periods

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

an example of absence of turbulence for any reynolds number ii we study a viscous incompressible fluid moving in a two dimensional flat torus 0 l 0 2 l 2 we show a set of external forces for which the stationary state is attractive for any reynolds numberr moreover the size of this set and the basin of attraction are independent ofr

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

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

should annual maximum temperatures follow a generalized extreme value distribution summary motivated by the study of annual temperature extremes two new results on the limiting distribution of block maxima of random variables with varying upper bounds are obtained one gives a generalized extreme value distribution as the limit but with a different shape parameter from that obtained when the bound on the random variables does not vary the other gives a limiting distribution that is only a generalized extreme value in certain cases both results consider triangular arrays of random variables in order to mimic the property of an upper bound that changes slowly with the day of the year as seems to occur for temperature data at many locations an analysis of 140 years of daily temperatures in new york city shows mixed results in terms of the ability of the theory presented here to provide new insights into the behaviour of extreme temperatures at this location

small sample effects in time series analysis a new asymptotic theory and a new estimate on introduit un modele mathematique qui couvre les effets des petits echantillons dans l approximation de whittle 1953 on montre que l estimation du maximum de vraisemblance exacte est toujours optimale dans ce modele

online bayesian inference in some time frequency representations of non stationary processes the use of bayesian inference in the inference of time frequency representations has thus far been limited to offline analysis of signals using a smoothing spline based model of the time frequency plane in this paper we introduce a new framework that allows the routine use of bayesian inference for online estimation of the time varying spectral density of a locally stationary gaussian process the core of our approach is the use of a likelihood inspired by a local whittle approximation this choice along with the use of a recursive algorithm for non parametric estimation of the local spectral density permits the use of a particle filter for estimating the time varying spectral density online we provide demonstrations of the algorithm through tracking chirps and the analysis of musical data

asymptotic analysis of the navier stokes equations abstract new bounds are established on the number of modes which determine the solutions of the navier stokes equations in two dimensions the best bound available at present is nearly proportional to the generalized grashof number defined in the paper and less than logarithmically dependent on the spatial structure or the shape of the force driving the flow to the extent than for the case of 2 dimensional rayleigh benard convection the generalized grashof number may be identified with the usual grashof number the resulting bound on the number of modes is found to differ only slightly from a bound obtained earlier on heuristic grounds

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

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

simultaneous inference of linear models with time varying coefficients summary the paper considers construction of simultaneous confidence tubes for time varying regression coefficients in functional linear models using a gaussian approximation result for non stationary multiple time series we show that the constructed simultaneous confidence tubes have asymptotically correct nominal coverage probabilities our results are applied to the problem of testing whether the regression coefficients are of certain parametric forms which is a fundamental problem in the inference of functional linear models as an application we analyse an environmental data set and study the association between levels of pollutants and hospital admissions