least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

an accurate fractional brownian motion generator abstract a new algorithm for the simulation of fractional brownian motion is suggested this algorithm has been tested and compared with previous ones clear improvements in the statistical and scaling properties of the process built with our algorithm are shown large improvements have been achieved for short times and for hurst s exponents less then 0 5 i e for path with high fractal dimension a new strategy to generate fractional brownian motions based on the use of moments of increasing size blocks is then described this numerical method proved to be fast and accurate

fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix geostatistical simulations often require the generation of numerous realizations of a stationary gaussian process over a regularly meshed sample grid omega this paper shows that for many important correlation functions in geostatistics realizations of the associated process over m 1 equispaced points on a line can be produced at the cost of an initial fft of length 2m with each new realization requiring an additional fft of the same length in particular the paper first notes that if an m 1 times m 1 toeplitz correlation matrix r can be embedded in a nonnegative definite 2m times2m circulant matrix s exact realizations of the normal multivariate y sim cal n 0 r can be generated via ffts of length 2m theoretical results are then presented to demonstrate that for many commonly used correlation structures the minimal embedding in which m m is nonnegative definite extensions to simulations of stationary fields in higher dimensions are also provided and illustrated

multi fractional generalized cauchy process and its application to teletraffic abstract the contributions given in this paper are in two aspects the first is to introduce a novel random function which we call the multi fractional generalized cauchy mgc process the second is to dissertate its application to network traffic for studying the multi fractal behavior of traffic on a point by point basis the introduced mgc process is with the time varying fractal dimension d t and the time varying hurst parameter h t the representations of the autocorrelation function acf and the power spectrum density psd of the mgc process are proposed besides the asymptotic expressions of the acf and psd of the mgc process are presented the computation formula of d t is given the mgc model may be a new tool to describe the multi fractal behavior of traffic precisely it may be used to reveal the local irregularity or local self similarity lss which is a small time scale behavior of traffic and global long term persistence or long range dependence lrd which is a large time scale behavior of traffic on a point by point basis the cast study with real traffic traces exhibits that the variance of d t is much greater than that of h t thus the present mgc model may provide a novel way to explain the fact that traffic has highly local irregularity while its lrd is robust

simulating sample paths of linear fractional stable motion an algorithm for generating sample paths of linear fractional stable motion lfsm is introduced it is based on the approximation of lfsm by a linear process and exhibits low computational complexity a detailed analysis of the error term involved in the approximation is provided which in turn guides the user on selecting the size of the generated sequence

optimal control problem for viscous cahn hilliard equation abstract this paper is concerned with the viscous cahn hilliard equation which arises in the dynamics of viscous first order phase transitions in cooling binary solutions the optimal control under boundary condition is given and the existence of optimal solution to the equation is proved

generalized cauchy model of sea level fluctuations with long range dependence abstract this article suggests the contributions with two highlights one is to propose a novel model of sea level fluctuations sea level for short which is called the generalized cauchy gc process it provides a new outlook for the description of local and global behaviors of sea level from a view of fractal in that the fractal dimension d that measures the local behavior of sea level and the hurst parameter h which characterizes the global behavior of sea level are independent of each other the other is to show that sea level appears multi fractal in both spatial and time such a meaning of multi fractal is new in the sense that a pair of fractal parameters d h of sea level is varying with measurement sites and time this research exhibits that the ranges of d and h of sea level in general are 1 d 2 and 0 5 h 1 respectively but d is independent of h with respect to the global behavior of sea level we shall show that h 0 96 for all data records at all measurement sites implying that strong lrd may be a general phenomenon of sea level on the other side regarding with the local behavior we will reveal that there appears d 1 or d 1 for data records at a few stations and at some time but d 0 96 at most stations and at most time meaning that sea level may appear highly local irregularity more frequently than weak local one

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

the screening effect in kriging when predicting the value of a stationary random field at a location x in some region in which one has a large number of observations it may be difficult to compute the optimal predictor one simple way to reduce the computational burden is to base the predictor only on those observations nearest to x as long as the number of observations used in the predictor is sufficiently large one might generally expect the best predictor based on these observations to be nearly optimal relative to the best predictor using all observations indeed this phenomenon has been empirically observed in numerous circumstances and is known as the screening effect in the geostatistical literature for linear predictors when observations are on a regular grid this work proves that there generally is a screening effect as the grid becomes increasingly dense this result requires that at high frequencies the spectral density of the random field not decay faster than algebraically and not vary too quickly examples demonstrate that there may be no screening effect if these conditions on the spectral density are violated

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

estimating the dimension of a fractal summary we suggest refinements of the box counting method which address the obvious problems caused by the incomplete information and inaccessibility of the limit a method for the statistical analysis of these corrected data is developed and tested on simulated and real data

optimal controls of 3 dimensional navier stokes equations with state constraints this work is concerned with the maximum principles for optimal control problems governed by 3 dimensional navier stokes equations some types of state constraints time variables are considered

local error estimates for least squares finite element methods for first order system we present local energy type error estimates for first order system div least squares ls finite element methods the estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution the estimate is similar to the one for the standard galerkin methods however our estimate needs to consider the effect of error of dual flux variables since ls methods approximate the primary and dual variables simultaneously the effect of error of the dual variables is shown to be of higher order moreover our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest as an elementary consequence of the estimate asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions

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

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

local stationarity and simulation of self affine intrinsic random functions gaussian intrinsic random functions with power law generalized covariance functions which in one dimension essentially correspond to fractional and integrated fractional brownian motions form a class of self affine models for random fields with a wide range of smoothness properties these random fields are nonstationary but appropriately filtered versions of them are stationary this work proves that most such random functions are locally stationary in a certain well defined sense this result yields an efficient and exact method of simulating all fractional and integrated fractional brownian motions

estimating the fractal dimension of a locally self similar gaussian process by using increments summary consider the problem of estimating the parameter ca of a stationary gaussian process with

spectral representation of fractional brownian motion in n dimensions and its properties fractional brownian motion fbm provides a useful model for processes with strong long term dependence such as 1 f sup spl beta spectral behavior however fbm s are nonstationary processes so that the interpretation of such a spectrum is still a matter of speculation to facilitate the study of this problem another model is provided for the construction of fbm from a white noise like process by means of a stochastic or ito integral in frequency of a stationary uncorrelated random process also a generalized power spectrum of the nonstationary fbm process is defined this new approach to fbm can be used to compute all of the correlations power spectra and other properties of fbm in this paper a number of these fbm properties are developed from this model such as the t sup h law of scaling the power law of fractional order the correlation of two arbitrary fbm s and the evaluation of the fractal dimension under various transformations this new treatment of fbm using a spectral representation is extended also for the first time to two or more topological dimensions in order to analyze the features of isotropic n dimensional fbm

optimal control for an adsorbate induced phase transition model this paper is concerned with the optimal control problem for an adsorbate induced phase transition model that is we show the existence of the optimal control and derive the optimality conditions by showing the differentiability of the cost functional with respect to the control

numerical solutions for nonlinear elliptic problems based on first order system we present a theoretical analysis of numerical solutions for nonlinear elliptic problems our analysis is based on the abstract approximation theory for branches of nonsingular solutions developed by brezzi rappaz and raviart brr in most cases of finite element analysis the same spaces are used for the domain and range of the nonlinear operator concerning the application of brr theory this results in a loss of accuracy in the error estimates while numerical experiments show optimal convergence rates the main contribution of this paper is a theoretical analysis for the optimal convergence rates this is achieved by choosing different spaces for the domain and range of the nonlinear operator used in the application of brr theory previously this idea was used for an analysis of petrov galerkin formulation for the brr theory our analysis is developed for the first order least squares finite element methods and the mixed galerkin methods based on first order systems of equations

nonparametric identification of anisotropic elliptic correlations in spatially distributed data sets random fields are useful models of spatially variable quantities such as those occurring in environmental processes and medical imaging the fluctuations obtained in most natural data sets are typically anisotropic the parameters of anisotropy are often determined from the data by means of empirical methods or the computationally expensive method of maximum likelihood in this paper we propose a systematic method for the identification of geometric elliptic anisotropy parameters of scalar fields the proposed method is computationally efficient nonparametric noniterative and it applies to differentiable random fields with normal or lognormal probability density functions our approach uses sample based estimates of the random field spatial derivatives that we relate through closed form expressions to the anisotropy parameters this paper focuses on two spatial dimensions we investigate the performance of the method on synthetic samples with gaussian and matern correlations both on regular and irregular lattices the systematic anisotropy detection provides an important preprocessing stage of the data knowledge of the anisotropy parameters followed by suitable rotation and rescaling transformations restores isotropy thus allowing classical interpolation and signal processing methods to be applied

global attractor of a modified swift hohenberg equation in hk spaces abstract in this paper by using an iteration procedure regularity estimates for the linear semigroups and a classical existence theorem of global attractor we prove that a modified swift hohenberg equation u t 2 u g u possesses a global attractor in sobolev space h k for all k 0 which attracts any bounded subset of h k in the h k norm

record length requirement of long range dependent teletraffic abstract this article contributes the highlights mainly in two folds on the one hand it presents a formula to compute the upper bound of the variance of the correlation periodogram measurement of teletraffic traffic for short with long range dependence lrd for a given record length t and a given value of the hurst parameter h theorems 1 and 2 on the other hand it proposes two formulas for the computation of the variance upper bound of the correlation periodogram measurement of traffic of fractional gaussian noise fgn type and the generalized cauchy gc type respectively corollaries 1 and 2 they may constitute a reference guideline of record length requirement of traffic with lrd in addition record length requirement for the correlation periodogram measurement of traffic with either the schuster type or the bartlett one is studied and the present results about it show that both types of periodograms may be used for the correlation measurement of traffic with a pre desired variance bound of correlation estimation moreover real traffic in the internet archive by the special interest group on data communication under the association for computing machinery of us acm sigcomm is analyzed in the case study in this topic

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

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

testing non existence of input output relationships by estimating fractal dimensions our aim is to propose tests for non existence of nonlinear relationships between signals which after passing a test can be interpreted as input and output signals of a certain system if its characteristic is sufficiently smooth the proposed tests are based on the theoretical results on equality of fractal dimensions of these signals as well as on estimation of fractal dimensions from observations they are applicable when at least one of these signals has the fractal dimensions strictly larger than one i e it is rough enough the tests are then verified on simulated data their applicability is illustrated by two sets of real data namely observations of two financial time series and samples of displacement force signals in a magneto hydrological damper

generalized ornstein uhlenbeck processes and associated self similar processes we consider three types of generalized ornstein uhlenbeck processes the stationary process obtained from the lamperti transformation of fractional brownian motion the process with stretched exponential covariance and the process obtained from the solution of the fractional langevin equation these stationary gaussian processes have many common properties such as the fact that their local covariances share a similar structure and they exhibit identical spectral densities at large frequency limit in addition the generalized ornstein uhlenbeck processes can be shown to be local stationary representations of fractional brownian motion two new self similar gaussian processes in addition to fractional brownian motion are obtained by applying the inverse lamperti transformation to the generalized ornstein uhlenbeck processes we study some of the properties of these self similar processes such as the long range dependence we give a simulation of their sample paths based on numerical karhunan loeve expansion

continuity properties for random fields l auteur donne pour un champ aleatoire n dimensionnel donne une condition simple sur la fonction de covariance qui garantit l existence d une version du champ aleatoire dans laquelle les realisations sont partout continues la demonstration implique une approximation assez delicate du champ aleatoire par des polynomes d interpellation d ordre eleve

optimal controls of multidimensional modified swift hohenberg equation in this paper we deal with the optimal control problem governed by multidimensional modified swift hohenberg equation after showing the relationship between the control problem and its approximation we derive the optimality conditions for an optimal control of our original problem by using one of the approximate problems

on the performance of box counting estimators of fractal dimension summary box counting estimators are popular for estimating fractal dimension however very little is known of their stochastic properties despite increasing statistical interest in their application we show that if the irregular curve to which the estimators are applied is modelled by a gaussian process concise formulae may be developed for asymptotic bias and variance of box counting estimators these formulae point to critical differences between a naive form of the box counting estimator based directly on the capacity definition of fractal dimension and a regression inspired version of that estimator

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

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

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

superconvergence of least squares methods for a coupled system of elliptic equations abstract we present a numerical method for solving a coupled system of elliptic partial differential equations pdes our method is based on the least squares ls approach we develop ellipticity estimates and error bounds for the method the main idea of the error estimates is the establishment of supercloseness of the ls solutions and solutions of the mixed finite element methods and ritz projections using the supercloseness property we obtain l 2 norm error estimates and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces moreover we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions application to optimal control problems is briefly considered

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

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

large time behavior of solution to a fully parabolic chemotaxis haptotaxis model in higher dimensions abstract this paper deals with the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r n with zero flux boundary conditions where and are positive parameters it is shown that if is suitably large then for all sufficiently smooth initial data the associated initial boundary value problem possesses a unique global in time classical solution that is bounded in 0 and if the initial data w 0 is small w becomes asymptotically negligible moreover we prove that when domain is convex 1 1 0 is globally asymptotically stable provided that u 0 0 and thereby extends the result of hillen et al 2013 18 to the higher space dimensions

estimation of time dependent hurst exponents with variational smoothing and application to forecasting foreign exchange rates abstract hurst exponents depict the long memory of a time series for human dependent phenomena as in finance this feature may vary in the time it justifies modelling dynamics by multifractional brownian motions which are consistent with time dependent hurst exponents we improve the existing literature on estimating time dependent hurst exponents by proposing a smooth estimate obtained by variational calculus this method is very general and not restricted to the sole hurst framework it is globally more accurate and easier than other existing non parametric estimation techniques besides in the field of hurst exponents it makes it possible to make forecasts based on the estimated multifractional brownian motion the application to high frequency foreign exchange markets gbp chf sek usd cad aud jpy cny and sgd all against eur shows significantly good forecasts when the hurst exponent is higher than 0 5 what depicts a long memory feature the accuracy is higher

inference on fractal processes using multiresolution approximation we consider bayesian inference via markov chain monte carlo for a variety of fractal gaussian processes on the real line these models have unknown parameters in the covariance matrix requiring inversion of a new covariance matrix at each markov chain monte carlo iteration the processes have no suitable independence properties so this becomes computationally prohibitive we surmount these difficulties by developing a computational algorithm for likelihood evaluation based on a multiresolution approximation to the original process the method is computationally very efficient and widely applicable making likelihood based inference feasible for large datasets a simulation study indicates that this approach leads to accurate estimates for underlying parameters in fractal models including fractional brownian motion and fractional gaussian noise and functional parameters in the recently introduced multifractional brownian motion we apply the method to a variety of real datasets and illustrate its application to prediction and to model selection

kriging and splines with derivative information summary spline fitting is a popular method of interpolating a real valued function given its values at a set of points in w other linear constraints such as derivative information can also be incorporated as we show here spline fitting is well known to be a special case of kriging using the kriging framework we give a full description of the theory including algorithms for computation and various special cases are discussed an application is given to the construction of deformations with landmark tangent and curvature constraints

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

characterizing surface smoothness via estimation of effective fractal dimension the fractal dimension d of stationary gaussian surfaces may be expressed very simply in terms of behaviour of the covariance function near the origin indeed only the covariance of line transect samples is required and that fact makes practical estimation of d relatively straightforward the case of non gaussian surfaces is more poorly understood but we might define effective fractal dimension in terms of the covariance function as though the surface were gaussian in the present paper we suggest simple practical methods for estimating effective fractal dimension based on the variogram like techniques proposed recently by taylor and taylor ours are founded on log linear regression

generalized whittle matern random field as a model of correlated fluctuations this paper considers a generalization of the gaussian random field with covariance function of the whittle matern family such a random field can be obtained as the solution to the fractional stochastic differential equation with two fractional orders asymptotic properties of the covariance functions belonging to this generalized whittle matern family are studied which are used to deduce the sample path properties of the random field the whittle matern field has been widely used in modeling geostatistical data such as sea beam data wind speed field temperature and soil data in this paper we show that the generalized whittle matern field provides a more flexible model for wind speed data

fractal analysis of surface roughness by using spatial data summary we develop fractal methodology for data taking the form of surfaces an advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale free component fractal dimension and properties that depend purely on scale particular emphasis is given to anisotropy where we show that for many surfaces the fractal dimension of line transects across a surface must either be constant in every direction or be constant in each direction except one this virtual direction invariance of fractal dimension provides another canonical feature of fractal analysis complementing its scale invariance properties and enhancing its attractiveness as a method for summarizing properties of roughness the dependence of roughness on direction may be explained in terms of scale rather than dimension and can vary with orientation scale may be described by a smooth periodic function and may be estimated nonparametrically our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrapping for the soil data interest centres on the effect of surface roughness on retention of rain water and data are recorded as a series of digital images over time our analysis captures the way in which both the fractal dimension and the scale change with rainfall or equivalently with time the food wrapping data are on a much finer scale than the soil data and are particularly anisotropic the analysis allows us to determine the manufacturing process which produces the smoothest wrapping with least tendency for micro organisms to adhere

synthesis of accurate fractional gaussian noise by filtering this paper describes a method for generating long sample paths of accurate fractional gaussian noise fgn the increment process of fractional brownian motion fbm the method is based on a wold decomposition in which fgn is expressed as the output of a finite impulse response filter with discrete white gaussian noise as input the form of the ideal filter is derived analytically in the continuous time case for the finite length discrete time case an iterative projection algorithm incorporating a newton raphson step is described for computing the coefficients of a length n filter in a time approximately proportional to n fast convolution of discrete white gaussian noise with the computed filter impulse response yields arbitrarily long sequences which exactly match the correlation structure of fgn over a finite range of lags for values of the hurst parameter h smaller than a critical value hcritap0 85 and large n the finite length autocorrelation sequence of fgn is positive definite and this range of lags can be as large as the filter autocorrelation length when h h crit the finite length autocorrelation sequence of fgn is no longer positive definite and a modification is made to allow a wold decomposition the generated sequences then exactly match the autocorrelation structure of fgn over a more restricted range of lags which becomes smaller as h approaches unity

feedback stabilization and optimal control for the cahn hilliard equation this paper is concerned with the control problem for the cahn hilliard equation 4 the physical background is as follows suppose there is a binary solution in a container sz c ir n 5 3 and suppose the mole fraction of the a component is u then the mole fraction for the other component b is 1 u it is well known that the function u of x e sz and t 1 0 due to mass conservation satisfies the cahn hilliard equation 4 u ya2u ap u in sz x r 1 1 where y 0 is a small positive constant and typically p u u3 24 1 2 suppose u has spatial distribution uo x initially i e u x 0 u x 1 3 then the evolution of u x t is determined by 1 l 1 3 and the boundary conditions au i i aau o an x 1 4

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

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

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

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

sharp l_2 norm error estimates for first order div least squares methods a theoretical analysis of a first order div least squares finite element method is presented our main interest is providing l 2 norm error estimates for the primary function u and the flux mbox boldmath mathcal a nabla u while there has been significant progress in the error estimates for the flux in l 2 norm these estimates have drawbacks such as requiring smooth solutions or that the error estimate for mbox boldmath mbox boldmath h 0 is coupled with the error estimate for u u h 1 where u h mbox boldmath h is the least squares approximate solution for u mbox boldmath mathcal a nabla u in this paper with the minimum regularity assumption u in h 1 alpha alpha 1 2 we separate the error estimate for mbox boldmath mbox boldmath h 0 from u u h 1 and establish that mbox boldmath mbox boldmath h 0 is bounded by the best approximation for mbox boldmath in the finite element space and u u h 0 then we proceed to obtain two new estimates for u u h 0 the first estimate is useful to understand how the accuracy of u u h 0 affects the accuracy of mbox boldmath mbox boldmath h 0 and the other sheds new light on the convergence behavior of u u h 0 our analysis does not require smooth solutions and the domain is allowed to be nonconvex the resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the raviart thomas spaces for the flux mbox boldmath

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