corrected confidence bands for functional data using principal components functional principal components fpc analysis is widely used to decompose and express functional observations curve estimates implicitly condition on basis functions and other quantities derived from fpc decompositions however these objects are unknown in practice in this article we propose a method for obtaining correct curve estimates by accounting for uncertainty in fpc decompositions additionally pointwise and simultaneous confidence intervals that account for both model and decomposition based variability are constructed standard mixed model representations of functional expansions are used to construct curve estimates and variances conditional on a specific decomposition iterated expectation and variance formulas combine model based conditional estimates across the distribution of decompositions a bootstrap procedure is implemented to understand the uncertainty in principal component decomposition quantities our method compares favorably to competing approaches in simulation studies that include both densely and sparsely observed functions we apply our method to sparse observations of cd4 cell counts and to dense white matter tract profiles code for the analyses and simulations is publicly available and our method is implemented in the r package refund on cran

on a decoding algorithm for codes on maximal curves a decoding algorithm for algebraic geometric codes that was given by a n skorobogatov and s g vladut preprint inst problems of information transmission 1988 is considered the author gives a modified algorithm with improved performance which he obtains by applying the above algorithm a number of times in parallel he proves the existence of the decoding algorithm on maximal curves by showing the existence of certain divisors however he has so far been unable to give an efficient procedure of finding these divisors

entropy formulation of optimal and adaptive control the use of entropy as the common measure to evaluate the different levels of intelligent machines is reported at the execution level the design of the desirable control can be expressed by the uncertainty of selecting the optimal control that minimizes a given performance index by choosing a density function over the set of admissible controls to minimize the differential control entropy it can be shown that the optimal control problem is equivalent to the problem of minimization of the assigned entropy function with respect to the association control the adaptive control problem can be analyzed by considering the same entropy over extended space that includes the uncertain parameters it is shown that the optimal entropy is decomposed into three terms the optimal control term with given parameters the parameter identification term and the equivocation term which accounts for the active transition of dual control the equivocation when calculated can serve as a measure of optimality of the adaptive control algorithms that involve only distinct identification and optimal control algorithms an upper bound can be used instead when the equivocation is hard to calculate an example illustrates the method

deciding the dimension of effective dimension reduction space for functional and high dimensional data in this paper we consider regression models with a hilbert space valued predictor and a scalar response where the response depends on the predictor only through a finite number of projections the linear subspace spanned by these projections is called the effective dimension reduction edr space to determine the dimensionality of the edr space we focus on the leading principal component scores of the predictor and propose two sequential 2 testing procedures under the assumption that the predictor has an elliptically contoured distribution we further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores the proposed procedures are supported by theory validated by simulation studies and illustrated by a real data example our methods and theory are applicable to functional data and high dimensional multivariate data

brief paper fault isolation for multivariate nonlinear non gaussian systems using generalized entropy optimization principle this paper is concerned with the fault isolation fi problem for multivariate nonlinear non gaussian systems by using a novel filtering method the generalized entropy optimization principle geop is established for non gaussian systems with multiple faults and disturbances where the statistic information including entropy and mean of the residual variable is maximized in the presence of the target fault as well as all the nuisance faults and disturbances and is minimized in the absence of the target fault but in the presence of the nuisance faults and disturbances different from the existing results where the output is measurable for feedback the fault isolation filter is designed and driven by the joint output stochastic distributions rather than its deterministic value the error dynamics is represented by a multivariate nonlinear non gaussian system for which new recursive relationships are proposed to formulate the joint probability density functions jpdfs of the residual variable in terms of the jpdfs of the noises and the faults finally a simulation example is given to demonstrate the effectiveness of the proposed multivariate fi algorithms

construction and decoding of a class of algebraic geometry codes a class of codes derived from algebraic plane curves is constructed the concepts and results from algebraic geometry that were used are explained in detail no further knowledge of algebraic geometry is needed parameters generator and parity check matrices are given the main result is a decoding algorithm which turns out to be a generalization of the peterson algorithm for decoding bch decoder codes

decoding of codes defined by a single point on a curve a decoding algorithm for certain codes from algebraic curves is presented the dual code which is used for decoding is formed by evaluating rational functions having poles at a single point a theoretical foundation is developed from which an improved bound for the minimum distance and results on decoding up to that bound are derived the decoding is done by a computationally efficient berlekamp massey type algorithm which is intrinsic to the curve

unique decoding of general ag codes a unique decoding algorithm for general ag codes namely multipoint evaluation codes on algebraic curves is presented it is a natural generalization of the previous decoding algorithm which was only for one point ag codes as such it retains the same advantages of fast speed regular structure and direct message recovery upon this generalization we add a technique from the guruswami sudan list decoding that boosts the decoding speed significantly compared with other known decoding algorithms for general ag codes it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure

unique decoding of plane ag codes via interpolation we present a unique decoding algorithm of algebraic geometry ag codes on plane curves hermitian codes in particular from an interpolation point of view the algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the ag code it is the first decoding algorithm to combine some features of the interpolation based list decoding with the performance of the syndrome decoding with the majority voting scheme the regular structure of the algorithm allows a straightforward parallel implementation

a geometric approach to confidence regions and bands for functional parameters summary functional data analysis is now a well established discipline of statistics with its core concepts and perspectives in place despite this there are still fundamental statistical questions which have received relatively little attention one of these is the systematic construction of confidence regions for functional parameters this work is concerned with developing understanding and visualizing such regions we provide a general strategy for constructing confidence regions in a real separable hilbert space by using hyperellipsoids and hyper rectangles we then propose specific implementations which work especially well in practice they provide powerful hypothesis tests and useful visualization tools without relying on simulation we also demonstrate the negative result that nearly all regions including our own have zero coverage when working with empirical covariances to overcome this challenge we propose a new paradigm for evaluating confidence regions by showing that the distance between an estimated region and the desired region with proper coverage tends to 0 faster than the regions shrink to a point we call this phenomena ghosting and refer to the empirical regions as ghost regions we illustrate the proposed methods in a simulation study and an application to fractional anisotropy tract profile data

an ilc based adaptive control for general stochastic systems with strictly decreasing entropy in this paper a new method for adaptive control of general nonlinear and non gaussian unknown stochastic systems has been proposed the method applies the minimum entropy control scheme to decrease the closed loop randomness of the output under an iterative learning control ilc basis both modeling and control of the plant are performed using dynamic neural networks for this purpose the whole control horizon is divided into a certain number of time domain subintervals called batches and a pseudo d type ilc law is employed to train the plant model and controller parameters so that the entropy of the closed loop tracking error is made to decrease batch by batch the method has the advantage of decreasing the output uncertainty versus the advances of batches along the time horizon the analysis on the proposed ilc convergence is made and a set of demonstrable experiment results is also provided to show the effectiveness of the obtained control algorithm where encouraging results have been obtained

majority coset decoding a majority coset decoding mcd procedure that can be applied to an arbitrary geometric code is discussed in general the basic algorithm for decoding of algebraic geometric codes does not correct up to the designed minimum distance in mcd a reduction step is added to the basic algorithm in case the basic algorithm fails a majority scheme is used to obtain an additional syndrome for the error vector thus a strictly smaller cost containing the error vector is obtained in this way the basic algorithm is applied to a decreasing chain of cosets and after finitely many steps the coset will be small enough for successful application of the basic algorithm

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

improved two point codes on hermitian curves one point codes on the hermitian curve produce long codes with excellent parameters feng and rao introduced a modified construction that improves the parameters while still using one point divisors a separate improvement of the parameters was introduced by matthews considering the classical construction but with two point divisors those two approaches are combined to describe an elementary construction of two point improved codes upon analysis of their minimum distance and redundancy it is observed that they improve on the previous constructions for a large range of designed distances

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

minimum entropy control of closed loop tracking errors for dynamic stochastic systems the entropy has been used to characterize the uncertainty of the tracking error for general nonlinear and non gaussian stochastic systems a recursive optimization solution has been developed and the local stability condition of the closed loop system has been established the generality of this algorithm has been proved by the special case study of the minimum variance control for linear gaussian systems

from sparse to dense functional data and beyond nonparametric estimation of mean and covariance functions is important in functional data analysis we investigate the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme which includes two commonly used schemes equal weight per observation obs and equal weight per subject subj as two special cases we provide a comprehensive analysis of their asymptotic properties on a unified platform for all types of sampling plan be it dense sparse or neither three types of asymptotic properties are investigated in this paper asymptotic normality l 2 convergence and uniform convergence the asymptotic theories are unified on two aspects 1 the weighing scheme is very general 2 the magnitude of the number n i of measurements for the i th subject relative to the sample size n can vary freely based on the relative order of n i to n functional data are partitioned into three types non dense dense and ultra dense functional data for the obs and subj schemes these two weighing schemes are compared both theoretically and numerically we also propose a new class of weighing schemes in terms of a mixture of the obs and subj weights of which theoretical and numerical performances are examined and compared

algebraic decoding using special divisors the basic algorithm for decoding of algebraic geometric codes corrects up to d sub c 1 2 g 2 errors where d sub c denotes the designed minimum distance of a code and g denotes the genus of a curve the modified algorithm improves on this but applies to a restricted class of codes an extended modified algorithm that applies to all codes is formulated it will correct up to d sub c 1 2 s errors s is called the clifford defect of a curve for curves with g or 1 this defect satisfies 0

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

optimal policies for passive learning controllers abstract this paper deals with the optimal control of unknown parameter partially observed linear discrete time systems the objective of the paper is to find a suitable cost functional for which passive learning controllers are optimal it is shown that if there are parameter uncertainties in the observation equation then a quadratic functional can be found such that a passively adaptive closed loop control law is optimal with respect to this functional if there are also parameter uncertainties in the system equation then a quadratic functional can be found such that a passively adaptive open loop feedback control law is optimal with respect to this functional

functional models for time varying random objects functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued in recent years samples of time varying object data such as time varying networks that are not in a vector space have been increasingly collected these data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data such as functional principal component analysis cannot be applied we propose metric covariance a novel association measure for paired object data lying in a metric space d that we use to define a metric autocovariance function for a sample of random valued curves where generally will not have a vector space or manifold structure the proposed metric autocovariance function is non negative definite when the squared semimetric d2 is of negative type then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for valued functional data including time varying probability distributions covariance matrices and time dynamic networks analogues of functional principal components for time varying objects are obtained by applying frechet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions leading to real valued frechet scores using the notion of generalized frechet integrals we construct object functional principal components that lie in the metric space we establish asymptotic consistency of the sample based estimators for the corresponding population targets under mild metric entropy conditions on and continuity of the valued random curves these concepts are illustrated with samples of time varying probability distributions for human mortality time varying covariance matrices derived from trading patterns and time varying networks that arise from new york taxi trips

extracting state information form a quantized output record abstract we consider the problem of manipulating the input to a discrete time state space linear system with the goal of obtaining information at each time about the system s current state from a record of past quantized measurements of the system s output we find that if the system is not excessively unstable there exist feedback control strategies that allow one to make an asymtotically perfect determination of the current stage based on the output records that result even if the system is too unstable to apply such strategies there are feedback control laws that make the system s output record more informative about the system s state evolution than one might expect in deriving these control laws we regard quantized measurements of real numbers more as partial observations than as strict approximations and employ techniques from information theory and the theory of markov chains with countable state spaces

oversampled sigma delta modulation oversampled sigma delta modulation has been proposed as a practical implementation for high rate analog to digital conversion because of its simplicity and its robustness against circuit imperfections to date mathematical developments of the basic properties of such systems have been based either on simplified continuous time approximate models or on linearized discrete time models where the quantizer is replaced by an additive white uniform noise source in this paper we rigorously derive several basic properties of a simple discrete time single integrator loop sigma delta modulator with an accumulate and dump demodulator the derivation does not require any assumptions on the correlation or distribution of the quantizer error and hence involves no linearization of the nonlinear system but it does show that when the input is constant the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space two basic properties are developed 1 the behavior of the sigma delta quantizer when driven by a constant input and its relation to uniform quantization and 2 the rate distortion tradeoffs between the oversampling ratio and the average mean squared quantization error

finite wordlength design of digital kalman filters for state estimation the optimal design of a kalman filter is considered with respect to its finite wordlength fwl characteristics taking into account the roundoff noise due to state quantization the issues are particularly relevant in the design of fwl kalman filters for continuous time systems operating under a fast sampling rate in this respect the results demonstrate one compromise between the selection of the sampling rate and the selection of the state wordlength the optimum filter structure includes state residue feedback compensation which can result in the saving of many bits of additional state wordlength

optimal estimation of the mean function based on discretely sampled functional data phase transition the problem of estimating the mean of random functions based on discretely sampled data arises naturally in functional data analysis in this paper we study optimal estimation of the mean function under both common and independent designs minimax rates of convergence are established and easily implementable rate optimal estimators are introduced the analysis reveals interesting and different phase transition phenomena in the two cases under the common design the sampling frequency solely determines the optimal rate of convergence when it is relatively small and the sampling frequency has no effect on the optimal rate when it is large on the other hand under the independent design the optimal rate of convergence is determined jointly by the sampling frequency and the number of curves when the sampling frequency is relatively small when it is large the sampling frequency has no effect on the optimal rate another interesting contrast between the two settings is that smoothing is necessary under the independent design while somewhat surprisingly it is not essential under the common design

clt in functional linear regression models we propose in this work to derive a clt in the functional linear regression model the main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill posed inverse problem we consider estimators that belong to a large class of regularizing methods and we first show that contrary to the multivariate case it is not possible to state a clt in the topology of the considered functional space however we show that we can get a clt for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made

kernel regression estimation using repeated measurements data abstract the estimation of growth curves has been studied extensively in parametric situations here we consider the nonparametric estimation of an average growth curve suppose that there are observations from several experimental units each following the regression model y xi f xj e j 1 n where e1 e n are correlated zero mean errors and 0 x1 xn 1 are fixed constants we study some of the properties of a kernel estimator of f x asymptotic and finite sample results concerning the mean squared error of the estimator are obtained in particular the influence of correlation on the bandwidth minimizing mean squared error is discussed a data based method for selecting the bandwidth is illustrated in a data analysis most previous research on kernel regression estimators has involved uncorrelated errors we investigate how dependence of the errors changes the behavior of a kernel estimator our theorems concerning the asymptotic mean squared error show that the estimator cannot be consistent un

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

optimal state estimation for stochastic systems an information theoretic approach in this paper we examine the problem of optimal state estimation or filtering in stochastic systems using an approach based on information theoretic measures in this setting the traditional minimum mean square measure is compared with information theoretic measures kalman filtering theory is reexamined and some new interpretations are offered we show that for a linear gaussian system the kalman filter is the optimal filter not only for the mean square error measure but for several information theoretic measures which are introduced in this work for nonlinear systems these same measures generally are in conflict with each other and the feedback control policy has a dual role with regard to regulation and estimation for linear stochastic systems with general noise processes a lower bound on the achievable mutual information between the estimation error and the observation are derived the properties of an optimal probing control law and the associated optimal filter which achieve this lower bound and their relationships are investigated it is shown that for a linear stochastic system with an affine linear filter for the homogeneous system under some reachability and observability conditions zero mutual information between estimation error and observations can be achieved only when the system is gaussian

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

optimal control of jump linear gaussian systems this paper investigates the problem of controlling a discrete time linear system with jump parameters a review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems dynamic programming has been applied by many researchers and it was observed that no closed form analytical solution could be constructed because of the dual aspects of the controller the main contribution of the present work is an algorithm suitable for computer implementation for the optimal dual control the construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite dimensional optimization procedure to obtain the optimal control as a function of the statistics this is achieved by first developing a suitable recursive realization of a filter which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations

on the periodicity of symbolic observations of piecewise smooth discrete time systems a study is made of the behavior of discrete time systems composed of a set of smooth transition maps coupled by a quantized feedback function the feedback function partitions the state space into disjoint regions and assigns a smooth transition function to each region the main result is that under a constraint on the norm of the derivative of the transition maps a bounded state trajectory with limit points in the interior of the switching regions leads to a region index sequence that is eventually periodic under these assumptions it is shown that eventually the feedback function is determined by a finite state automaton a similar result is proved in the case of finite state dynamic feedback

active probing for information in control systems with quantized state measurements a minimum entropy approach in this paper the effect of state quantization in scaler discrete time linear control systems is studied by analyzing the system as a partially observed stochastic system the problem of optimal state information gathering and filtering is investigated using information theoretic measures and formulating the state estimation problem as an entropy optimization problem the active probing effect of the feedback control is thoroughly studied optimal feedback controls which minimize various types of entropy costs are determined and it is shown that this problem is equivalent to an optimal control problem for a controlled markov chain

brief paper minimum entropy of b spline pdf systems with mean constraint for the b spline approximation of the continuous probability density function pdf the relationships between the b spline weights and entropy both in general and under the mean constraint have been analyzed it provides the conditions under which the minimum entropy can be achieved subject to the mean constraint the difference between the entropy of continuous and discrete distributions has also been clarified a minimum entropy controller with the mean constraint is then developed and several simulations are performed to verify the main results

on semigroups generated by two consecutive integers and improved hermitian codes analysis of the berlekamp massey sakata algorithm for decoding one point codes leads to two methods for improving code rate one method due to feng and rao removes parity checks that may be recovered by their majority voting algorithm the second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic in this work formulae are given for the redundancies of hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance the results proceed from an analysis of numerical semigroups generated by two consecutive integers

entropy optimization filtering for fault isolation of nonlinear non gaussian stochastic systems in this paper the fault isolation fi problem is investigated for nonlinear non gaussian systems with multiple faults or abrupt changes of system parameters in the presence of noises by constructing a filter to estimate the states the fi problem can be reduced to an entropy optimization problem subjected to the non gaussian estimation error systems the design objective for the fi purpose is that the entropy of the estimation error is maximized in the presence of diagnosed fault and is minimized in the presence of the nuisance faults or noises it is shown that the error dynamics is represented by a nonlinear non gaussian stochastic system for which new relationships are applied to formulate the probability density functions pdfs of the stochastic error in terms of the pdfs of the noises and the faults the renyi s entropy has been used to simplify the computations in the filtering for the recursive design algorithms it is noted that the output can be supposed to be immeasurable but with known stochastic distributions which is different from the existing results where the output is always measurable for feedback finally simulations are given to demonstrate the effectiveness of the proposed data driven fi filtering algorithms

on the decoding of algebraic geometric codes over f sub q for q or 16 it is proved that for algebraic geometric codes on a curve over f sub q for q or 37 or on a curve of sufficiently large genus over f sub q for q or 16 there exists a polynomial decoding algorithm up to d 1 2 errors d being the designed minimum distance

a smooth simultaneous confidence corridor for the mean of sparse functional data functional data analysis fda has become an important area of statistics research in the recent decade yet a smooth simultaneous confidence corridor scc does not exist in the literature for the mean function of sparse functional data scc is a powerful tool for making statistical inference on an entire unknown function nonetheless classic hungarian embedding techniques for establishing asymptotic correctness of scc completely fail for sparse functional data we propose a local linear scc and a shoal of confidence intervals sci for the mean function of sparse functional data and establish that it is asymptotically equivalent to the scc of independent regression data using new results from gaussian process extreme value theory the scc procedure is examined in simulations for its superior theoretical accuracy and performance and used to analyze growth curve data confirming findings with quantified high significance levels supplementary materials for this article are available online