estimating direction fields in autonomous equation models with an application to system identification from cross sectional data summary we consider a situation where cross sectional observations follow an underlying longitudinal model with a population mean function measurements are available made at unknown observation times of the mean function values without noise and of derivatives with noise the population mean function lies in a nonparametric smoothness class and while not fully identifiable is a trajectory in the direction field of an autonomous ordinary differential equation an efficient reconstruction of that field is proposed which proceeds from nonparametric estimation of the function driving the differential equation rates of convergence are investigated a simulation is included showing the recovery of the mean function signal and noise are chosen to be typical for t4 cell counts in a small cohort of undated hiv seroconverters and the reconstruction of the mean function is seen to be satisfactory

finding numerical derivatives for unstructured and noisy data by multiscale kernels the recently developed multiscale kernel of r opfer adv comput math 25 2006 pp 357 380 is applied to approximate numerical derivatives the proposed method is truly mesh free and can handle unstructured data with noise in any dimension the method of tikhonov and the method of l curve are employed for regularization no information about the noise level is required an error analysis is provided in a general setting for all dimensions numerical comparisons are given in two dimensions which show competitive results with recently published thin plate spline methods

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

derivative estimation for longitudinal data analysis examining features of blood pressure measured repeatedly during pregnancy estimating velocity and acceleration trajectories allows novel inferences in the field of longitudinal data analysis such as estimating change regions rather than change points and testing group effects on nonlinear change in an outcome ie a nonlinear interaction in this article we develop derivative estimation for 2 standard approaches polynomial mixed models and spline mixed models we compare their performance with an established method principal component analysis through conditional expectation through a simulation study we then apply the methods to repeated blood pressure bp measurements in a uk cohort of pregnant women where the goals of analysis are to i identify and estimate regions of bp change for each individual and ii investigate the association between parity and bp change at the population level the penalized spline mixed model had the lowest bias in our simulation study and we identified evidence for bp change regions in over 75 of pregnant women using mean velocity difference revealed differences in bp change between women in their first pregnancy compared with those who had at least 1 previous pregnancy we recommend the use of penalized spline mixed models for derivative estimation in longitudinal data analysis

functional quadratic regression summary we extend the common linear functional regression model to the case where the dependency of a scalar response on a functional predictor is of polynomial rather than linear nature focusing on the quadratic case we demonstrate the usefulness of the polynomial functional regression model which encompasses linear functional regression as a special case our approach works under mild conditions for the case of densely spaced observations and also can be extended to the important practical situation where the functional predictors are derived from sparse and irregular measurements as is the case in many longitudinal studies a key observation is the equivalence of the functional polynomial model with a regression model that is a polynomial of the same order in the functional principal component scores of the predictor processes theoretical analysis as well as practical implementations are based on this equivalence and on basis representations of predictor processes we also obtain an explicit representation of the regression surface that defines quadratic functional regression and provide functional asymptotic results for an increasing number of model components as the number of subjects in the study increases the improvements that can be gained by adopting quadratic as compared to linear functional regression are illustrated with a case study that includes absorption spectra as functional predictors

modeling price dynamics in ebay auctions using differential equations empirical research of online auctions has grown dramatically in recent years studies using publicly available bid data from such websites as ebay com have found many divergences of bidding behavior and auction outcomes compared with ordinary offline auctions and auction theory among the main differences between online and offline auctions are the former s longer duration anonymity of bidders and sellers and low barriers of entry all of these factors lead to dynamics in the bid arrival and price process that change throughout the auction in this work we examine the price process in a large and diverse set of ebay auctions for both low and high valued items in terms of item auction bidder and seller characteristics we propose a family of differential equation models that captures online auction dynamics in particular we show that a second order linear differential equation well approximates the dynamics that occur in our diverse set of auctions we also introduce a multiple comparisons test fo

parameter estimation for differential equations a generalized smoothing approach summary we propose a new method for estimating parameters in models that are defined by a system of non linear differential equations such equations represent changes in system outputs by linking the behaviour of derivatives of a process to the behaviour of the process itself current methods for estimating parameters in differential equations from noisy data are computationally intensive and often poorly suited to the realization of statistical objectives such as inference and interval estimation the paper describes a new method that uses noisy measurements on a subset of variables to estimate the parameters defining a system of non linear differential equations the approach is based on a modification of data smoothing methods along with a generalization of profiled estimation we derive estimates and confidence intervals and show that these have low bias and good coverage properties respectively for data that are simulated from models in chemical engineering and neurobiology the performance of the method is demonstrated by using real world data from chemistry and from the progress of the autoimmune disease lupus

construction of low density parity check codes based on balanced incomplete block designs this correspondence presents a method for constructing structured regular low density parity check ldpc codes based on a special type of combinatoric designs known as balance incomplete block designs codes constructed by this method have girths at least 6 and they perform well with iterative decoding furthermore several classes of these codes are quasi cyclic and hence their encoding can be implemented with simple feedback shift registers

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

estimating derivatives for samples of sparsely observed functions with application to online auction dynamics it is often of interest to recover derivatives of a sample of random functions from sparse and noise contaminated measurements especially when the dynamics of underlying processes is of interest we propose a novel approach based on estimating derivatives of eigenfunctions and expansions of random functions into their eigenfunctions to obtain a representation for derivatives in combination with estimates for functional principal component scores for sparse data this leads to a viable solution of the challenging problem to recover derivatives for sparsely observed functions we establish consistency results and demonstrate in simulations that the method is superior to alternative approaches derivative estimation with random effects models based on b spline bases kernel smoothing smoothing splines or p splines our study is motivated by an analysis of bidding histories for ebay auctions for which bids are typically very sparse in the middle and somewhat more frequent near the beginning and end of a

on the construction of local parities for r t availability in distributed storage motivated by applications involving hot data the notion of r t availability was introduced where a symbol is said to have r t availability if it can be reconstructed respectively from t disjoint repair alternatives of other symbols each of size at most r the key to constructing a family of locally repairable codes with r t availability is the design of a membership matrix for dividing global parities into local ones such that each repair group contains only one parity symbol although explicit designs of membership matrices are available it remains unclear whether membership matrices with significantly better parameters can be constructed which was a question left open to tackle the open problem this paper first provides a connection between the design of membership matrices and the design of a combinatorial object in the well known combinatorial design theory called resolvable configurations this paper then proposes several direct designs of membership matrices based on euclidean geometry finite fields circulant permutation matrices affine permutation matrices and reed solomon codes by leveraging the notion of resolvable configurations and the kronecker product this paper further proposes a two step design of membership matrices the proposed direct and two step designs of membership matrices are shown to have significantly better parameters than those recently available

bandwidth choice and confidence intervals for derivatives of noisy data summary we propose a method for automatic bandwidth selection for kernel estimators of derivatives of a regression function the finite sample behaviour of this new method is compared with that of other methods in a monte carlo study the automatic estimation of derivatives can be employed for the construction of asymptotic local confidence intervals for the nonparametric estimate of the regression function and its first derivatives

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

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

the design and performance analysis for several new classes of codes for optical synchronous cdma and for arbitrary medium time hopping synchronous cdma communication systems new families of spread spectrum codes are constructed that are applicable to optical synchronous code division multiple access cdma communications as well as to arbitrary medium time hopping synchronous cdma communications proposed constructions are based on the mappings from integer sequences into binary sequences the authors use the concept of number theoretic quadratic congruences and a subset of reed solomon codes similar to the one utilized in the welch costas frequency hop fh patterns the properties of the codes are as good as or better than the properties of existing codes for synchronous cdma communications both the number of code sequences within a single code family and the number of code families with good properties are significantly increased when compared to the known code designs possible applications are presented to evaluate the performance of the proposed codes a new class of hit arrays called cyclical hit arrays is recalled which give insight into the previously unknown properties of the few classes of number theoretic fh patterns cyclical hit arrays and the proposed mappings are used to determine the exact probability distribution functions of random variables that represent interference between users of a time hopping or optical cdma system expressions for the bit error probability in multi user cdma systems are derived as a function of the number of simultaneous cdma system users the length of signature sequences and the threshold of a matched filter detector the performance results are compared with the results for some previously known codes

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

mean and covariance estimation for functional snippets we consider estimation of mean and covariance functions of functional snippets which are short segments of functions possibly observed irregularly on an individual specific subinterval that is muc

properties of principal component methods for functional and longitudinal data analysis the use of principal component methods to analyze functional data is appropriate in a wide range of different settings in studies of functional data analysis it has often been assumed that a sample of random functions is observed precisely in the continuum and without noise while this has been the traditional setting for functional data analysis in the context of longitudinal data analysis a random function typically represents a patient or subject who is observed at only a small number of randomly distributed points with nonnegligible measurement error nevertheless essentially the same methods can be used in both these cases as well as in the vast number of settings that lie between them how is performance affected by the sampling plan in this paper we answer that question we show that if there is a sample of n functions or subjects then estimation of eigenvalues is a semiparametric problem with root n consistent estimators even if only a few observations are made of each function and if each observation is encumbered by noise however estimation of eigenfunctions becomes a nonparametric problem when observations are sparse the optimal convergence rates in this case are those which pertain to more familiar function estimation settings we also describe the effects of sampling at regularly spaced points as opposed to random points in particular it is shown that there are often advantages in sampling randomly however even in the case of noisy data there is a threshold sampling rate depending on the number of functions treated above which the rate of sampling either randomly or regularly has negligible impact on estimator performance no matter whether eigenfunctions or eigenvectors are being estimated

recovering gradients from sparsely observed functional data summary the recovery of gradients of sparsely observed functional data is a challenging ill posed inverse problem given observations of smooth curves e g growth curves at isolated time points the aim is to provide estimates of the underlying gradients or growth velocities to address this problem we develop a bayesian inversion approach that models the gradient in the gaps between the observation times by a tied down brownian motion conditionally on its values at the observation times the posterior mean and covariance kernel of the growth velocities are then found to have explicit and computationally tractable representations in terms of quadratic splines the hyperparameters in the prior are specified via nonparametric empirical bayes with the prior precision matrix at the observation times estimated by constrained l1 minimization the infinitessimal variance of the brownian motion prior is selected by cross validation the approach is illustrated using both simulated and real data examples

construction of one coincidence sequence quasi cyclic ldpc codes of large girth one approach for designing the one coincidence sequence ocs low density parity check ldpc codes of large girth is investigated these ocs ldpc codes are quasi cyclic and their parity check matrices are composed of circulant permutation matrices generally the cycle structures in these codes are determined by the shift values of circulant permutation matrices and the existence of cycles in the corresponding tanner graph is governed by certain cycle governing equations cges therefore finding the proper shift values is the key point to increase the girth of these codes in this paper we provide an effective method to systematically find out the cges for these codes of girth 6 8 and 10 respectively then one less computation intensive algorithm is used to generate the proper shift values for constructing the ocs ldpc codes of large girth simulation results show that significant gains in signal to noise ratio over an additive white gaussian noise channel can be achieved by increasing the girth of the ocs ldpc codes

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

explicit construction of families of ldpc codes with no 4 cycles low density parity check ldpc codes are serious contenders to turbo codes in terms of decoding performance one of the main problems is to give an explicit construction of such codes whose tanner graphs have known girth for a prime power q and m spl ges 2 lazebnik and ustimenko construct a q regular bipartite graph d m q on 2q sup m vertices which has girth at least 2 spl lceil m 2 spl rceil 4 we regard these graphs as tanner graphs of binary codes lu m q we can determine the dimension and minimum weight of lu 2 q and show that the weight of its minimum stopping set is at least q 2 for q odd and exactly q 2 for q even we know that d 2 q has girth 6 and diameter 4 whereas d 3 q has girth 8 and diameter 6 we prove that for an odd prime p lu 3 p is a p sup 3 k code with k spl ges p sup 3 2p sup 2 3p 2 2 we show that the minimum weight and the weight of the minimum stopping set of lu 3 q are at least 2q and they are exactly 2q for many lu 3 q codes we find some interesting ldpc codes by our partial row construction we also give simulation results for some of our codes

on the girth of tanner 3 5 quasi cyclic ldpc codes in this correspondence the cycles of tanner 3 5 quasi cyclic qc low density parity check ldpc codes are analyzed and their girth values are derived the conditions for the existence of cycles of lengths 4 6 8 and 10 in tanner 3 5 qc ldpc codes of length 5p are expressed in terms of polynomial equations in a 15th root of unity of the prime field f sub p by checking the existence of solutions for these equations over f sub p the girths of tanner 3 5 qc ldpc codes are derived

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

an optimal selection of regression variables summary an asymptotically optimal selection of regression variables is proposed the key assumption is that the number of control variables is infinite or increases with the sample size it is also shown that mallows s qp akaike s fpe and aic methods are all asymptotically equivalent to this method

shortened array codes of large girth one approach to designing structured low density parity check ldpc codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity check matrix contain all the variables involved in short cycles this approach is especially effective if the parity check matrix of a code is a matrix composed of blocks of circulant permutation matrices as is the case for the class of codes known as array codes we show how to shorten array codes by deleting certain columns of their parity check matrices so as to increase their girth the shortening approach is based on the observation that for array codes and in fact for a slightly more general class of ldpc codes the cycles in the corresponding tanner graph are governed by certain homogeneous linear equations with integer coefficients consequently we can selectively eliminate cycles from an array code by only retaining those columns from the parity check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles we provide ramsey theoretic estimates for the maximum number of columns that can be retained from the original parity check matrix with the property that the sequence of their indices avoid solutions to various types of cycle governing equations this translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles simulation results show that for the codes considered shortening them to increase the girth can lead to significant gains in signal to noise ratio snr in the case of communication over an additive white gaussian noise awgn channel

effective dimension reduction for sparse functional data we propose a method of effective dimension reduction for functional data emphasizing the sparse design where one observes only a few noisy and irregular measurements for some or all of the subjects the proposed method borrows strength across the entire sample and provides a way to characterize the effective dimension reduction space via functional cumulative slicing our theoretical study reveals a bias variance trade off associated with the regularizing truncation and decaying structures of the predictor process and the effective dimension reduction space a simulation study and an application illustrate the superior finite sample performance of the method

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

empirical dynamics for longitudinal data we demonstrate that the processes underlying on line auction price bids and many other longitudinal data can be represented by an empirical first order stochastic ordinary differential equation with time varying coefficients and a smooth drift process this equation may be empirically obtained from longitudinal observations for a sample of subjects and does not presuppose specific knowledge of the underlying processes for the nonparametric estimation of the components of the differential equation it suffices to have available sparsely observed longitudinal measurements which may be noisy and are generated by underlying smooth random trajectories for each subject or experimental unit in the sample the drift process that drives the equation determines how closely individual process trajectories follow a deterministic approximation of the differential equation we provide estimates for trajectories and especially the variance function of the drift process at each fixed time point the proposed empirical dynamic model implies a decomposition of the derivative of the process underlying the longitudinal data into a component explained by a linear component determined by a varying coefficient function dynamic equation and an orthogonal complement that corresponds to the drift process an enhanced perturbation result enables us to obtain improved asymptotic convergence rates for eigenfunction derivative estimation and consistency for the varying coefficient function and the components of the drift process we illustrate the differential equation with an application to the dynamics of on line auction data

quantitative estimates of strong unique continuation for wave equations the main results of the present paper consist in some quantitative estimates for solutions to the wave equation partial 2 t u text div a x nabla x u 0 such estimates imply the following strong unique continuation properties a if u is a solution to the the wave equation and u is flat on a segment x 0 times j on the t axis then u vanishes in a neighborhood of x 0 times j b let u be a solution of the above wave equation in omega times j that vanishes on a a portion z times j where z is a portion of partial omega and u is flat on a segment x 0 times j x 0 in z then u vanishes in a neighborhood of x 0 times j the property a has been proved by lebeau commun partial differ equ 24 777 783 1999