optimal eigen expansions and uniform bounds let x k k in mathbb z in mathbb l 2 mathcal t be a stationary process with associated lag operators varvec mathcal c h uniform asymptotic expansions of the corresponding empirical eigenvalues and eigenfunctions are established under almost optimal conditions on the lag operators in terms of the eigenvalues spectral gap in addition the underlying dependence assumptions are optimal in a certain sense including both short and long memory processes this allows us to study the relative maximum deviation of the empirical eigenvalues under very general conditions among other things convergence to an extreme value distribution is shown we also discuss how the asymptotic expansions transfer to the long run covariance operator varvec mathcal g in a general framework

vector quantizers with direct sum codebooks the use of direct sum codebooks to minimize the memory requirements of vector quantizers is investigated assuming arbitrary fixed partitions necessary conditions for minimum distortion codebooks are derived first for scalar codebooks assuming mean squared error distortion and then for vector codebooks and a broader class of distortion measures an iterative procedure is described for designing locally optimal direct sum codebooks both optimal and computationally efficient suboptimal encoding schemes are considered it is shown that although an optimal encoding can be implemented by a sequential encoder the complexity of implementing optimal stagewise partitions generally exceeds the complexity of an exhaustive search of the direct sum codebook it is also shown that sequential nearest neighbor encoders can be extremely inefficient the m search method is explored as one method of improving the effectiveness of suboptimal sequential encoders representative results for simulated direct sum quantizers are presented

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier the data functions that are studied in the course of functional data analysis are assembled from discrete data and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually smooth functions that were not actually observed existing literature shows that this approach is effective and even optimal when using functional data methods for prediction or hypothesis testing however in the present paper we show that this approach is not effective in classification problems there a useful rule of thumb is that undersmoothing is often desirable but there are several surprising qualifications to that approach first the effect of smoothing the training data can be more significant than that of smoothing the new data set to be classified second undersmoothing is not always the right approach and in fact in some cases using a relatively large bandwidth can be more effective and third these perverse results are the consequence of very unusual properties of error rates expressed as functions of smoothing parameters for example the orders of magnitude of optimal smoothing parameter choices depend on the signs and sizes of terms in an expansion of error rate and those signs and sizes can vary dramatically from one setting to another even for the same classifier

3 d shape estimation of dna molecules from stereo cryo electron micro graphs using a projection steerable snake we introduce a three dimensional 3 d parametric active contour algorithm for the shape estimation of dna molecules from stereo cryo electron micrographs we estimate the shape by matching the projections of a 3 d global shape model with the micrographs we choose the global model as a 3 d filament with a b spline skeleton and a specified radial profile the active contour algorithm iteratively updates the b spline coefficients which requires us to evaluate the projections and match them with the micrographs at every iteration since the evaluation of the projections of the global model is computationally expensive we propose a fast algorithm based on locally approximating it by elongated blob like templates we introduce the concept of projection steerability and derive a projection steerable elongated template since the two dimensional projections of such a blob at any 3 d orientation can be expressed as a linear combination of a few basis functions matching the projections of such a 3 d template involves evaluating a weighted sum of inner products between the basis functions and the micrographs the weights are simple functions of the 3 d orientation and the inner products are evaluated efficiently by separable filtering we choose an internal energy term that penalizes the average curvature magnitude since the exact length of the dna molecule is known a priori we introduce a constraint energy term that forces the curve to have this specified length the sum of these energies along with the image energy derived from the matching process is minimized using the conjugate gradients algorithm we validate the algorithm using real as well as simulated data and show that it performs well

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

dispersion operators and resistant second order functional data analysis inferences related to the second order properties of functional data as expressed by covariance structure can become unreliable when the data are non gaussian or contain unusual observations in the functional setting it is often difficult to identify atypical observations as their distinguishing characteristics can be manifold but subtle in this paper we introduce the notion of a dispersion operator investigate its use in probing the second order structure of functional data and develop a test for comparing the second order characteristics of two functional samples that is resistant to atypical observations and departures from normality the proposed test is a regularized m test based on a spectrally truncated version of the hilbert schmidt norm of a score operator defined via the dispersion operator we derive the asymptotic distribution of the test statistic investigate the behaviour of the test in a simulation study and illustrate the method on a structural biology dataset

variable rate vector quantization for medical image compression three techniques for variable rate vector quantizer design are applied to medical images the first two are extensions of an algorithm for optimal pruning in tree structured classification and regression due to breiman et al the code design algorithms find subtrees of a given tree structured vector quantizer tsvq each one optimal in that it has the lowest average distortion of all subtrees of the tsvq with the same or lesser average rate since the resulting subtrees have variable depth natural variable rate coders result the third technique is a joint optimization of a vector quantizer and a noiseless variable rate code this technique is relatively complex but it has the potential to yield the highest performance of all three techniques

entropy constrained vector quantization an iterative descent algorithm based on a lagrangian formulation for designing vector quantizers having minimum distortion subject to an entropy constraint is discussed these entropy constrained vector quantizers ecvqs can be used in tandem with variable rate noiseless coding systems to provide locally optimal variable rate block source coding with respect to a fidelity criterion experiments on sampled speech and on synthetic sources with memory indicate that for waveform coding at low rates about 1 bit sample under the squared error distortion measure about 1 6 db improvement in the signal to noise ratio can be expected over the best scalar and lattice quantizers when block entropy coded with block length 4 even greater gains are made over other forms of entropy coded vector quantizers for pattern recognition it is shown that the ecvq algorithm is a generalization of the k means and related algorithms for estimating cluster means in that the ecvq algorithm estimates the prior cluster probabilities as well experiments on multivariate gaussian distributions show that for clustering problems involving classes with widely different priors the ecvq outperforms the k means algorithm in both likelihood and probability of error

detecting and localizing differences in functional time series dynamics a case study in molecular biophysics abstractmotivated by the problem of inferring the molecular dynamics of dna in solution and linking them with its base pair composition we consider the problem of comparing the dynamics of functional time series fts and of localizing any inferred differences in frequency and along curvelength the approach we take is one of fourier analysis where the complete second order structure of the fts is encoded by its spectral density operator indexed by frequency and curvelength the comparison is broken down to a hierarchy of stages at a global level we compare the spectral density operators of the two fts across frequencies and curvelength based on a hilbert schmidt criterion then we localize any differences to specific frequencies and finally we further localize any differences along the length of the random curves that is in physical space a hierarchical multiple testing approach guarantees control of the averaged false discovery rate over the selected frequencies in this sense we are ab

data driven version of neyman s smooth test of fit abstract neyman s smooth test for testing uniformity is a recognized goodness of fit procedure as stated by lariccia the test can be viewed as a compromise between omnibus test procedures with generally low power in all directions and procedures whose power is focused in the direction of a specific alternative the basic idea behind this test is to embed the null density into say a k dimensional exponential family and then to construct an asymptotically optimal test for the parametric testing problem the resulting procedure is neyman s test with k components the most difficult problem related with using this test is the choice of k recommendations in statistical literature are sometimes confusing some authors advocate a small number of components whereas others show that in some situations a larger number of components is profitable all existing suggestions concerning how to select k exploit in fact some preliminary knowledge about a possible alternative in this article a new data driven met

fourier analysis of stationary time series in function space we develop the basic building blocks of a frequency domain framework for drawing statistical inferences on the second order structure of a stationary sequence of functional data the key element in such a context is the spectral density operator which generalises the notion of a spectral density matrix to the functional setting and characterises the second order dynamics of the process our main tool is the functional discrete fourier transform fdft we derive an asymptotic gaussian representation of the fdft thus allowing the transformation of the original collection of dependent random functions into a collection of approximately independent complex valued gaussian random functions our results are then employed in order to construct estimators of the spectral density operator based on smoothed versions of the periodogram kernel the functional generalisation of the periodogram matrix the consistency and asymptotic law of these estimators are studied in detail as immediate consequences we obtain central limit theorems for the mean and the long run covariance operator of a stationary functional time series our results do not depend on structural modelling assumptions but only functional versions of classical cumulant mixing conditions and are shown to be stable under discrete observation of the individual curves

new tests for equality of several covariance functions for functional data in this article we propose two new tests for the equality of the covariance functions of several functional populations namely a quasi gpf test and a quasi fmax test whose test statistics are o

a wilcoxon mann whitney type test for infinite dimensional data the wilcoxon mann whitney test is a robust competitor of the t test in the univariate setting for finite dimensional multivariate non gaussian data several extensions of the wilcoxon mann whitney test have been shown to outperform hotelling s t 2 test in this paper we study a wilcoxon mann whitney type test based on spatial ranks in infinite dimensional spaces we investigate its asymptotic properties and compare it with several existing tests the proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails we study its performance using real and simulated data

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

distances and inference for covariance operators a framework is developed for inference concerning the covariance operator of a functional random process where the covariance operator itself is an object of interest for statistical analysis distances for comparing positive definite covariance matrices are either extended or shown to be inapplicable to functional data in particular an infinite dimensional analogue of the procrustes size and shape distance is developed convergence of finite dimensional approximations to the infinite dimensional distance metrics is also shown for inference a frechet estimator of both the covariance operator itself and the average covariance operator is introduced a permutation procedure to test the equality of the covariance operators between two groups is also considered additionally the use of such distances for extrapolation to make predictions is explored as an example of the proposed methodology the use of covariance operators has been suggested in a philological study of cross linguistic dependence as a way to incorporate quantitative phonetic information it is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the romance languages

bootstrap based testing of equality of mean functions or equality of covariance operators for functional data we investigate the properties of a simple bootstrap method for testing the equality of mean functions or of covariance operators in functional data theoretical size and power results are derived for certain test statistics whose limiting distributions depend on unknown infinite dimensional parameters simulations demonstrate good size and power of the bootstrap based functional tests

estimation of the mean of functional time series and a two sample problem summary the paper is concerned with inference based on the mean function of a functional time series we develop a normal approximation for the functional sample mean and then focus on the estimation of the asymptotic variance kernel using these results we develop and asymptotically justify testing procedures for the equality of means in two functional samples exhibiting temporal dependence evaluated by means of a simulation study and application to a real data set these two sample procedures enjoy good size and power in finite samples

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

searching for structure in curve samples abstract the shape of a regression curve can to a large extent be characterized by the succession of structural features like extrema inflection points and so on when analyzing a sample of regression curves it is often important to know at an early stage of data analysis which structural features are occurring consistently in each curve of the sample such a definition is usually not easy due to substantial interindividual variation both in the x and the y axis and due to the influence of noise a method is proposed for identifying typical features without relying on an a priori specified functional model for the curves the approach is based on the frequencies of occurrence of structural features as for example maxima in the curve sample along the x axis important tools are nonparametric regression and differentiation and kernel density estimation apart from a theoretical foundation the usefulness of the method is documented by application to two interesting biomedical areas growth and develop

variable rate vector quantization for speech image and video compression the performance of a vector quantizer can be improved by using a variable rate code three variable rate vector quantization systems are applied to speech image and video sources and compared to standard vector quantization and noiseless variable rate coding approaches the systems range from a simple and flexible tree based vector quantizer to a high performance but complex jointly optimized vector quantizer and noiseless code the systems provide significant performance improvements for subband speech coding predictive image coding and motion compensated video but provide only marginal improvements for vector quantization of linear predictive coefficients in speech and direct vector quantization of images criteria are suggested for determining when variable rate vector quantization may provide significant performance improvement over standard approaches

tests for error correlation in the functional linear model in this chapter we consider two tests for error correlation in the fully functional linear model which we call methods i and ii they complement the tools described in section 8 6 and the graphical goodness of fit checks used in chapter 9 to construct the test statistics finite dimensional residuals are computed in two different ways and then their autocorrelations are suitably defined from these autocorrelation matrices two quadratic forms are constructed whose limiting distribution are chi squared with known numbers of degrees of freedom different for the two forms the test statistics can be relatively easily computed using the r package fda

detecting changes in the mean of functional observations principal component analysis has become a fundamental tool of functional data analysis it represents tne functional data as x i t t 1 i l v l t wnere is the common mean v l are the eigenfunctions of the covariance operator and the i l are the scores inferential procedures assume that the mean function t is the same for all values of i if in fact the observations do not come from one population but rather their mean changes at some point s the results of principal component analysis are confounded by the change s it is therefore important to develop a methodology to test the assumption of a common functional mean we develop such a test using quantities which can be readily computed in the r package fda the null distribution of the test statistic is asymptotically pivotal with a well known asymptotic distribution the asymptotic test has excellent finite sample performance its application is illustrated on temperature data from england

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

constrained storage multistage vector quantization based on genetic algorithms multistage vector quantization msvq and their variants have been recently proposed before msvq is designed the user must artificially determine the number of codewords in each vq stage however the users usually have no idea regarding the number of codewords in each vq stage and thus doubt whether the resulting msvq is optimal this paper proposes the genetic design gd algorithm to design the msvq the gd algorithm can automatically find the number of codewords to optimize each vq stage according to the rate distortion performance thus the msvq based on the gd algorithm namely msvq gd is proposed here furthermore using a sharing codebook sc can further reduce the storage size of msvq combining numerous similar codewords in the vq stages of msvq produces the codewords of the sharing codebook this paper proposes the genetic merge gm algorithm to design the sc of msvq therefore the constrained storage msvq using a sc namely csmsvq is proposed and outperforms other msvqs in the experiments presented here

robust functional estimation using the median and spherical principal components summary we present robust estimators for the mean and the principal components of a stochastic process in l2 r robustness and asymptotic properties of the estimators are studied theoretically by simulation and by example it is shown that the proposed estimators are generally more robust to outliers than the commonly used sample mean and principal components although their properties depend on the spacings of the eigenvalues of the covariance function

the structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets multiwavelets and multiframelets are of interest in several applications such as numerical algorithms and signal processing due to their desirable properties such as high smoothness and vanishing moments with relatively small supports of their generating functions and masks in order to process and represent vector valued discrete data efficiently and sparsely by a multiwavelet transform a multiwavelet has to be prefiltered or balanced balanced orthonormal univariate multiwavelets and multivariate biorthogonal multiwavelets have been studied and constructed in the literature dual multiframelets include bi orthogonal multiwavelets as special cases but their fundamental prefiltering and balancing property has not yet been investigated in the literature in this paper we shall study the balancing property of multivariate multiframelets from the point of view of the discrete multiframelet transform this approach to our best knowledge has not been considered so far in the literature even for multiwavelets but it reveals the essential structure of prefiltering and the balancing property of multiwavelets and multiframelets we prove that every biorthogonal multiwavelet can be prefiltered with the balancing order matching the order of its vanishing moments that is from every given compactly supported multivariate biorthogonal multiwavelet one can always build another essentially equivalent compactly supported biorthogonal multiwavelets with its balancing order matching the order of the vanishing moments of the original one more generally we show that if a dual multiframelet can be prefiltered then it can be equivalently transformed into a balanced dual multiframelet with the same balancing order however we notice that most available dual multiframelets in the literature cannot be simply prefiltered with its balancing order matching its order of vanishing moments and they must be designed to possess high balancing orders the key ingredient of our approach is based on investigating some properties of the subdivision and transition operators acting on discrete vector polynomial sequences which play a central role in a discrete multiframelet transform and are of interest in their own right we also establish a new canonical form of a matrix mask which greatly facilitates the investigation and construction of multiwavelets and multiframelets in this paper we obtain a complete criterion and the essential structure for balanced or prefiltered dual multiframelets in the most general setting our investigation of the balancing property of a multiframelet deepens our understanding of the multiframelet transform in signal processing and scientific computation

second order comparison of gaussian random functions and the geometry of dna minicircles given two samples of continuous zero mean iid gaussian processes on 0 1 we consider the problem of testing whether they share the same covariance structure our study is motivated by the problem of determining whether the mechanical properties of short strands of dna are significantly affected by their base pair sequence though expected to be true had so far not been observed in three dimensional electron microscopy data the testing problem is seen to involve aspects of ill posed inverse problems and a test based on a karhunen loeve approximation of the hilbert schmidt distance of the empirical covariance operators is proposed and investigated when applied to a dataset of dna minicircles obtained through the electron microscope our test seems to suggest potential sequence effects on dna shape supplemental material available online

necessary conditions for the optimality of variable rate residual vector quantizers necessary conditions for the optimality of variable rate residual vector quantizers are derived and an iterative descent algorithm based on a lagrangian formulation is introduced for designing residual vector quantizers having minimum average distortion subject to an entropy constraint simulation results for entropy constrained residual vector quantizers are presented for memoryless gaussian laplacian and uniform sources a gauss markov source is also considered the rate distortion performance is shown to be competitive with that of entropy constrained vector quantization and entropy constrained trellis coded quantization

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

enhanced multistage vector quantization by joint codebook design multistage vector quantization msvq can achieve very low encoding and storage complexity in comparison to unstructured vector quantization however the conventional stage by stage design of the codebooks in msvq is suboptimal with respect to the overall performance measure the authors introduce an algorithm for the joint design of the stage codebooks to optimize the overall performance the performance improvement although modest is achieved with no effect on encoding or storage complexity and only a slight increase in design effort

portmanteau test of independence for functional observations in various fields observations are curves over some natural time interval these curves often arise from finely spaced measurements e g in physical sciences and finance or from smoothing unequally spaced sparse measurements recent years have seen the development of tools for analyzing such data in the growing field of functional data analysis to validate the assumptions underlying these tools it is important to verify that the functional observations form a simple random sample if the curves form a functional time series then model validation typically relies on checking whether model residuals are independent and identically distributed we propose a test for independence and identical distribution of functional observations to reduce dimension curves are projected on the most important functional principal components then a test statistic based on lagged cross covariances of the resulting vectors is constructed we show that this dimension reduction step introduces asymptotically negligib

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

self modelling warping functions the paper introduces a semiparametric model for functional data the warping functions are assumed to be linear combinations ofqcommon components which are estimated from the data hence the name self modelling even small values ofqprovide remarkable model flexibility comparable with nonparametric methods at the same time this approach avoids overfitting because the common components are estimated combining data across individuals as a convenient by product component scores are often interpretable and can be used for statistical inference an example of classification based on scores is given abstract from author

balanced multi wavelets in ℝ the notion of k balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi wavelets to ensure the property of preservation annihilation of scalarvalued discrete polynomial data of order k or degree k 1 when decomposed by the corresponding matrix valued low pass high pass filters while this condition is indeed precise to the best of our knowledge only the proof for k 1 is known in addition the formulation of the k balancing condition for k 2 is so prohibitively difficult to satisfy that only a very few examples for k 2 and vector dimension 2 have been constructed in the open literature the objective of this paper is to derive various characterizations of the k balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose these results are established in the general multivariate and bi orthogonal settings for any k 1

image coding using entropy constrained residual vector quantization an entropy constrained residual vector quantization design algorithm is used to design codebooks for image coding entropy constrained residual vector quantization has several important advantages it can outperform entropy constrained vector quantization in terms of rate distortion performance memory and computation requirements it can also be used to design vector quantizers with relatively large vector sizes and high output rates experimental results indicate that good image reproduction quality can be achieved at relatively low bit rates for example a peak signal to noise ratio of 30 09 db is obtained for the 512 spl times 512 lena image at a bit rate of 0 145 b p

common functional principal components functional principal component analysis fpca based on the karhunen loeve decomposition has been successfully applied in many applications mainly for one sample problems in this paper we con sider common functional principal components for two sample prob lems our research is motivated not only by the theoretical challenge of this data situation but also by the actual question of dynamics of implied volatility iv functions for different maturities the log returns of ivs are samples of smooth random functions and the methods proposed here study the similarities of their stochastic be havior first we present a new method for estimation of functional principal components from discrete noisy data next we present the two sample inference for fpca and develop the two sample theory we propose bootstrap tests for testing the equality of eigenvalues eigenfunctions and mean functions of two functional samples illus trate the test properties by simulation study and apply the method to the iv analysis

components and completion of partially observed functional data functional data are traditionally assumed to be observed on the same domain motivated by a data set of heart rate temporal profiles we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere we formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework estimation of parameters mean and covariance operator and principal component analysis principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory next we propose a functional completion procedure that recovers the missing part by using the observed part of the curve we construct prediction intervals for principal scores and bands for missing parts of trajectories the prediction problems are seen to be ill posed inverse problems regularization techniques are used to obtain a stable solution a simulation study shows the good performance of our methods we illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding