on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

redundancy of the context tree weighting method on renewal and markov renewal processes the context tree weighting method ctw is shown to be almost adaptive on the classes of renewal and markov renewal processes up to logarithmic factor ctw achieves the minimax pointwise redundancy described by i csiszaacuter and p shields in ieee trans inf theory vol 42 no 6 pp 2065 2072 nov 1996 this result not only complements previous results on the adaptivity of the context tree weighting method on the relatively small class of all finite context tree sources which encompasses the class of all finite order markov sources it shows that almost minimax redundancy can be achieved on massive classes of sources classes that cannot be smoothly parameterized by subsets of finite dimensional spaces moreover it shows that almost adaptive compression can be achieved in a computationally efficient way on those massive classes while previous adaptivity results for ctw could rely on the fact that any markov source is a finite context tree source this is no longer the case for renewal sources in order to prove almost adaptivity of ctw over renewal sources it is necessary to establish that ctw carefully balances estimation error and approximation error

implementation of a variable stepsize variable formula method in the time integration part of a code for treatment of long range transport of air pollutants abstract a mathematical model consisting of two partial differential equations is used to study the long range transport of sulphur di oxide and sulphate over europe the discretization of the first order space derivatives the advection terms is carried out by a pseudospectral fourier algorithm a special technique is applied in the discretization of the second order space derivatives the diffusion terms two large systems of ordinary differential equations are solved at each time step it is shown that these systems can efficiently be treated by a variable stepsize variable formula method vsvfm based on the use of predictor corrector schemes the stepsize selection strategy and the formula selection strategy are discussed in detail an attempt to carry out both an accuracy control and a stability control is made at each time step the great efficiency of the vsvfm implemented in our software as well as the reliability of the results are illustrated by numerical experiments in which real meteorological data for 1979 at the grid points of a space domain covering the whole of europe were used the main ideas implemented in the time integration part might be applied in many other situations where the systems of ordinary differential equations arising after the space discretization are only moderately stiff so that the stability requirements are dominant over the accuracy requirements on a large part of the time interval but the use of implicit time integration algorithms that require solving systems of algebraic equations at each time step is not justified as an illustration only it should be mentioned that such an application has been carried out in connection with models describing long range transport of nitrogen pollutants over europe

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

three level fourier spectral approximations for fluid flow with low mach number fourier spectral approximation combined with a second order time differencing technique for fluid flow with low mach number is considered in this paper its generalized stability and convergence are strictly proven

consistent estimation of the basic neighborhood of markov random fields for markov random fields on z d with finite state space we address the statistical estimation of the basic neighborhood the smallest region that determines the conditional distribution at a site on the condition that the values at all other sites are given a modification of the bayesian information criterion replacing likelihood by pseudo likelihood is proved to provide strongly consistent estimation from observing a realization of the field on increasing finite regions the estimated basic neighborhood equals the true one eventually almost surely not assuming any prior bound on the size of the latter stationarity of the markov field is not required and phase transition does not affect the results

variable length markov chains we study estimation in the class of stationary variable length markov chains vlmc on a finite space the processes in this class are still markovian of high order but with memory of variable length yielding a much bigger and structurally richer class of models than ordinary high order markov chains from an algorithmic view the vlmc model class has attracted interest in information theory and machine learning but statistical properties have not yet been explored provided that good estimation is available the additional structural richness of the model class enhances predictive power by finding a better tradeoff between model bias and variance and allowing better structural description which can be of specific interest the latter is exemplified with some dna data a version of the tree structured context algorithm proposed by rissanen in an information theoretical setup is shown to have new good asymptotic properties for estimation in the class of vlmcs this remains true even when the underlying model increases in dimensionality fur thermore consistent estimation of minimal state spaces and mixing properties of fitted models are given we also propose a new bootstrap scheme based on fitted vlmcs we show its validity for quite general stationary categorical time series and for a broad range of statistical procedures

the fourier pseudospectral method with a restrain operator for the korteweg de vries equation abstract in this paper we develop a fourier pseudospectral method with a restrain operator for the korteweg de vries equation we prove the generalized stability of the schemes and give convergence estimations depending on the smoothness of the solution of the p d e

the sampling window corresp the concept of the sampling window is introduced for the central interpolation of finite energy band limited functions the sampling window does not increase the rate of convergence of the truncation error series as do various convergence factors but does significantly reduce the truncation error bound

spectral methods in time for parabolic problems a pseudospectral explicit scheme for solving linear periodic parabolic problems is described it has infinite accuracy both in time and in space the high accuracy is achieved while the time resolution parameter m m o 1 delta t for time marching algorithm and the space resolution parameter n n o 1 delta x must satisfy m o n 1 varepsilon varepsilon 0 compared to the common stability condition m o n 2 which must be satisfied in any explicit finite order time algorithm

dynamic adaptive partitioning for nonlinear time series summary we propose a dynamic adaptive partitioning scheme for nonparametric analysis of stationary nonlinear time series it yields estimates of the whole probability distribution of the underlying process we use information from past values to construct adaptive partitioning in a dynamic fashion which is then different from the more common static schemes in the regression set up the idea of dynamic partitioning is new we make it constructive by an approach based on quantisation of the data and adaptively modelling partition cells with a parsimonious markov chain the methodology is formulated in terms of a new model class the so called quantised variable length markov chains it is a new extension of finite valued variable length markov chains to processes with values in rd we discuss estimation explore asymptotic properties of the new method and give some numerical results which reflect the finite sample behaviour

stochastic processes with random contexts a characterization and adaptive estimators for the transition probabilities this paper introduces the concept of random context representations for the transition probabilities of a finite alphabet stochastic process processes with these representations generalize context tree processes also known as variable length markov chains and are proved to coincide with processes whose transition probabilities are almost surely continuous functions of the infinite past this is similar to a classical result by kalikow about continuous transition probabilities existence and uniqueness of a minimal random context representation are shown in the sense that there exists a unique representation that looks into the past as little as possible in order to determine the next symbol both this representation and the transition probabilities can be consistently estimated from data and some finite sample adaptivity properties are also obtained including an oracle inequality in particular the estimator achieves minimax performance up to logarithmic factors for the class of binary renewal processes whose arrival distributions have bounded moments of order 2 gamma

the context tree weighting method basic properties describes a sequential universal data compression procedure for binary tree sources that performs the double mixture using a context tree this method weights in an efficient recursive way the coding distributions corresponding to all bounded memory tree sources and achieves a desirable coding distribution for tree sources with an unknown model and unknown parameters computational and storage complexity of the proposed procedure are both linear in the source sequence length the authors derive a natural upper bound on the cumulative redundancy of the method for individual sequences the three terms in this bound can be identified as coding parameter and model redundancy the bound holds for all source sequence lengths not only for asymptotically large lengths the analysis that leads to this bound is based on standard techniques and turns out to be extremely simple the upper bound on the redundancy shows that the proposed context tree weighting procedure is optimal in the sense that it achieves the rissanen 1984 lower bound

sieve bootstrap with variable length markov chains for stationary categorical time series a bootstrap for stationary categorical time series based on the method of sieves is studied here the data generating process is approximated by the so called variable length markov chain vlmc a flexible class of markov models that allows for parsimonious structure then the resampling is given by simulating from the fitted model it is shown that for a whole class of stationary categorical time series that is more general than vlmc the vlmc sieve has faster rate of convergence for variance estimation than the more general block bootstrap results are illustrated from a theoretical and empirical perspective for the latter a real data application about in homogeneity classification of a dna strand is also presented finally the vlmc sieve scheme enjoys an implementational advantage of using the plug in rule for bootstrapping a statistical procedure which generally is not the case for the block method

a sequential algorithm for the universal coding of finite memory sources the estimation and universal compression of discrete sources are considered and a sequential algorithm for the universal coding of finite memory sources attaining asymptotically minimum redundancy is presented the algorithm performs an online estimation of the source states and uses an arithmetic code

type classes of context trees it is well known that a tree model does not always admit a finite state machine fsm representation with the same minimal number of parameters therefore known characterizations of type classes for fsms do not apply in general to tree models in this paper the type class of a sequence with respect to a given context tree is studied an exact formula is derived for the size of the class extending whittle s formula for type classes with respect to fsms the derivation is more intricate than in the fsm case since some basic properties of fsm types do not hold in general for tree types the derivation also yields an efficient enumeration of the tree type class a formula for the number of type classes with respect to is also derived the formula is asymptotically tight up to a multiplicative constant and also extends the corresponding result for fsms the asymptotic behavior of the number of type classes and of the size of a class is expressed in terms of the so called minimal canonical extension of t a tree that is generally larger than but smaller than its fsm closure

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

estimating the lengths of memory words for a stationary stochastic process xn with values in some set a a finite word w isin ak is called a memory word if the conditional probability of x 0 given the past is constant on the cylinder set defined by x k 1 w it is a called a minimal memory word if no proper suffix of w is also a memory word for example in a k step markov processes all words of length k are memory words but not necessarily minimal we consider the problem of determining the lengths of the longest minimal memory words and the shortest memory words of an unknown process xn based on sequentially observing the outputs of a single sample xi1 xi2 xin we will give a universal estimator which converges almost surely to the length of the longest minimal memory word and show that no such universal estimator exists for the length of the shortest memory word the alphabet a may be finite or countable

modified b splines for the sampling of bandlimited functions it is shown that there is an optimal finite linear combination of b splines denominated modified b splines such that a pertinent low frequency condition called m flatness is satisfied a profound relationship of the modified b splines with the beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling

polynomial spline signal approximations filter design and asymptotic equivalence with shannon s sampling theorem the least squares polynomial spline approximation of a signal g t in l sub 2 r is obtained by projecting g t on s sup n r the space of polynomial splines of order n it is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g t with a b spline of order n more specifically the coefficients of the b spline interpolation of order 2n 1 of the sampled filtered sequence are identical to the coefficients of the least squares approximation of g t of order n it is shown that this approximation can be obtained from a succession of three basic operations prefiltering sampling and postfiltering which confirms the parallel with the classical sampling reconstruction procedure for bandlimited signals the frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift invariant basis functions of s sup n r the standard expansion in terms of b spline coefficients a representation in terms of sampled signal values and a representation using orthogonal basis functions

a universal finite memory source an irreducible parameterization for a finite memory source is constructed in the form of a tree machine a universal information source for the set of finite memory sources is constructed by a predictive modification of an earlier studied algorithm context it is shown that this universal source incorporates any minimal data generating tree machine in an asymptotically optimal manner in the following sense the negative logarithm of the probability it assigns to any long typical sequence generated by any tree machine approaches that assigned by the tree machine at the best possible rate

large scale typicality of markov sample paths and consistency of mdl order estimators for markov chains of arbitrary order with finite alphabet a almost sure sense limit theorems are proved on relative frequencies of k blocks and of symbols preceded by a given k block when k is permitted to grow as the sample size n grows as an application the consistency of two kinds of minimum description length mdl markov order estimators is proved with upper bound o log n respectively spl alpha log n with spl alpha 1 log a on the permissible value of the estimated order it was shown by csiszar and shields see ann statist vol 28 p 1601 1619 2000 that in the absence of any bound or with bound spl alpha log n with large spl alpha consistency fails

finding words with unexpected frequencies in deoxyribonucleic acid sequences considering a markov chain model for deoxyribonucleic acid sequences this paper proposes two asymptotically normal statistics to test whether the frequency of a given word is concordant with the first order markov chain model or not the problem is to choose estimates w of the expectation of the frequency m w of a word w in the observed sequence such that the asymptotic variance of m w w is easily computable the first estimator is derived from the frequency of w 1 which is w with its last letter deleted the second following an idea of cowan is the conditional expectation m w given the observed frequencies of all two letter words two examples on phage lambda and phage t7 are shown

a universal data compression system a universal data compression algorithm is described which is capable of compressing long strings generated by a finitely generated source with a near optimum per symbol length without prior knowledge of the source this class of sources may be viewed as a generalization of markov sources to random fields moreover the algorithm does not require a working storage much larger than that needed to describe the source generating parameters

complexity of strings in the class of markov sources shannon s self information of a string is generalized to its complexity relative to the class of finite state machine fsm defined sources unlike an earlier generalization the new one is valid for both short and long strings the definition is justified in part by a theorem stating that asymptotically the mean complexity provides a tight lower bound for the mean length of all so called regular codes this also generalizes shannon s noiseless coding theorem for a large subclass of fsm sources a simple algorithm is described for computing the complexity

context tree estimation for not necessarily finite memory processes via bic and mdl the concept of context tree usually defined for finite memory processes is extended to arbitrary stationary ergodic processes with finite alphabet these context trees are not necessarily complete and may be of infinite depth the familiar bayesian information criterion bic and minimum description length mdl principles are shown to provide strongly consistent estimators of the context tree via optimization of a criterion for hypothetical context trees of finite depth allowed to grow with the sample size n as o logn algorithms are provided to compute these estimators in o n time and to compute them on line for all i les n in o nlogn time

universal prediction of individual sequences the problem of predicting the next outcome of an individual binary sequence using finite memory is considered the finite state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite state fs predictor it is proven that this fs predictability can be achieved by universal sequential prediction schemes an efficient prediction procedure based on the incremental parsing procedure of the lempel ziv data compression algorithm is shown to achieve asymptotically the fs predictability some relations between compressibility and predictability are discussed and the predictability is proposed as an additional measure of the complexity of a sequence