generalized functional extended redundancy analysis functional extended redundancy analysis fera was recently developed to integrate data reduction into functional linear models this technique extracts a component from each of multiple sets of predictor data in such a way that the component accounts for the maximum variance of response data moreover it permits predictor and or response data to be functional fera can be of use in describing overall characteristics of each set of predictor data and in summarizing the relationships between predictor and response data in this paper we extend fera into the framework of generalized linear models glm so that it can deal with response data generated from a variety of distributions specifically the proposed method reduces each set of predictor functions to a component and uses the component for explaining exponential family responses as in glm we specify the random systematic and link function parts of the proposed method we develop an iterative algorithm to maximize a penalized log likelihood criterion that is derived in combination with a basis function expansion approach we conduct two simulation studies to investigate the performance of the proposed method based on synthetic data in addition we apply the proposed method to two examples to demonstrate its empirical usefulness

on fuzzy spaces and fuzzy group theory abstract the concept of fuzzy space is introduced this concept corresponds to the concept of the universal set in the ordinary case the algebra of fuzzy spaces and fuzzy subspaces is studied using the concept of fuzzy space and fuzzy binary operation a new approach to study fuzzy group theory is given this approach can be considered as a generalization and reformulation of the rosenfeld theory of fuzzy subgroups therefore it is an active tool to develop the theory of fuzzy groups

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

the racah algebra for groups with time reversal symmetry iii the elements of the racah algebra for a general compact group with time reversal symmetry are developed as time reversal is antilinear it is not possible to treat these groups by representation theory but we may instead use wigner s theory of corepresentations the results we obtain often parallel those for linear groups but there are some important divergences we illustrate these with the grey double point groups

modeling repeated functional observations we introduce a new methodological framework for repeatedly observed and thus dependent functional data aiming at situations where curves are recorded repeatedly for each subject in a sample our methodology covers the case where the recordings of the curves are scheduled on a regular and dense grid and also situations more typical for longitudinal studies where the timing of recordings is often sparse and random the proposed models lead to an interpretable and straightforward decomposition of the inherent variation in repeatedly observed functional data and are implemented through a straightforward two step functional principal component analysis we provide consistency results and asymptotic convergence rates for the estimated model components we compare the proposed model with an alternative approach via a two dimensional karhunen loeve expansion and illustrate it through the analysis of longitudinal mortality data from period lifetables that are repeatedly observed for a sample of countries over ma

inference of fuzzy relational tableaux from fuzzy exemplifications the paper describes a method for deriving fuzzy relational tableaux from fuzzy exemplifications in particular the work by thole zimmerman and zysno 4 is cited as they derive an operator for and using data derived from human subjects the method results in an operator or operators which is a maximal restriction of the operators implied by the data results showing the process are described and a value of and is derived using data presented in 4

some properties of fuzzy groups on definit des sous groupes flous normaux des sous groupes flous normaux de niveau et leurs homomorphismes et on donne certaines de leurs proprietes

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

doubly functional graphical models in high dimensions we consider estimating a functional graphical model from multivariate functional observations in functional data analysis the classical assumption is that each function has been measured over a densely sampled grid however in practice the functions have often been observed with measurement error at a relatively small number of points we propose a class of doubly functional graphical models to capture the evolving conditional dependence relationship among a large number of sparsely or densely sampled functions our approach first implements a nonparametric smoother to perform functional principal components analysis for each curve then estimates a functional covariance matrix and finally computes sparse precision matrices which in turn provide the doubly functional graphical model we derive some novel concentration bounds uniform convergence rates and model selection properties of our estimator for both sparsely and densely sampled functional data in the high dimensional large p small n regime we demonstrate via simulations that the proposed method significantly outperforms possible competitors our proposed method is applied to a brain imaging dataset

incorporating parametric effects into functional principal components analysis summary the ideas of functional principal component analysis are extended to deal with data that are hybrids of functional and parametric effects the parametric effects may be more general than just the addition of a multiple of a given function a detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval and some remarks are made for the extension to a far more general case given data a procrustes fitting method can be used to estimate the parametric effects several possible ways of treating the estimated parameter values are discussed the methods are illustrated by reference to temperature data at 35 canadian weather stations

gait analysis and the bootstrap this paper is about random coefficient trigonometric regression models and their use in gait analysis here gait analysis means free speed walking on a level surface our study focuses on bootstrap based prediction regions for the angular rotation curves of test children when the relevant training data are gathered from normal children of comparable ages considerations that led to our choice of model and use of the bootstrap are given prediction regions and empirical bootstrap distributions are displayed as is their application to several test cases also included is a study of the almost sure asymptotic behavior of the theoretical bootstrap probability of the prediction regions

plurality and resemblance in fmri data analysis abstract we apply nine analytic methods employed currently in imaging neuroscience to simulated and actual bold fmri signals and compare their performances under each signal type starting with baseline time series generated by a resting subject during a null hypothesis study we compare method performance with embedded focal activity in these series of three different types whose magnitudes and time courses are simple convolved with spatially varying hemodynamic responses and highly spatially interactive we then apply these same nine methods to bold fmri time series from contralateral primary motor cortex and ipsilateral cerebellum collected during a sequential finger opposition study paired comparisons of results across methods include a voxel specific concordance correlation coefficient for reproducibility and a resemblance measure that accommodates spatial autocorrelation of differences in activity surfaces receiver operating characteristic curves show considerable model differences in ranges less than 10 significance level false positives and greater than 80 power true positives concordance and resemblance measures reveal significant differences between activity surfaces in both data sets these measures can assist researchers by identifying groups of models producing similar and dissimilar results and thereby help to validate consolidate and simplify reports of statistical findings a pluralistic strategy for fmri data analysis can uncover invariant and highly interactive relationships between local activity foci and serve as a basis for further discovery of organizational principles of the brain results also suggest that a pluralistic empirical strategy coupled formally with substantive prior knowledge can help to uncover new brain behavior relationships that may remain hidden if only a single method is employed

how far are automatically chosen regression smoothing parameters from their optimum abstract we address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation call the optimal bandwidth the minimizer of the average squared error we consider several automatically selected bandwidths that approximate the optimum how far are the automatically selected bandwidths from the optimum the answer is studied theoretically and through simulations the theoretical results include a central limit theorem that quantifies the convergence rate and gives the differences asymptotic distribution the convergence rate turns out to be excruciatingly slow this is not too disappointing because this rate is of the same order as the convergence rate of the difference between the minimizers of the average squared error and the mean average squared error in some simulations by john rice the selectors considered here performed quite differently from each other we anticipated that these differences would be reflected in differ

interpretable functional principal component analysis functional principal component analysis fpca is a popular approach to explore major sources of variation in a sample of random curves these major sources of variation are represented by functional principal components fpcs the intervals where the values of fpcs are significant are interpreted as where sample curves have major variations however these intervals are often hard for naive users to identify because of the vague definition of significant values in this article we develop a novel penalty based method to derive fpcs that are only nonzero precisely in the intervals where the values of fpcs are significant whence the derived fpcs possess better interpretability than the fpcs derived from existing methods to compute the proposed fpcs we devise an efficient algorithm based on projection deflation techniques we show that the proposed interpretable fpcs are strongly consistent and asymptotically normal under mild conditions simulation studies confirm that with a competitive performance in explaining variations of sample curves the proposed fpcs are more interpretable than the traditional counterparts this advantage is demonstrated by analyzing two real datasets namely electroencephalography data and canadian weather data

on the symmetries of the 6j symbol the 6j tensor for compact groups is shown to transform as a basis vector for the identity representation of the permutation group s4 this allows character theory to be used to determine the minimum number of independent components and a projection operator to determine the relations between components the symmetry properties

empirical likelihood semiparametric regression analysis for longitudinal data a semiparametric regression model for longitudinal data is considered the empirical likelihood method is used to estimate the regression coefficients and the baseline function and to construct confidence regions and intervals it is proved that the maximum empirical likelihood estimator of the regression coefficients achieves asymptotic efficiency and the estimator of the baseline function attains asymptotic normality when a bias correction is made two calibrated empirical likelihood approaches to inference for the baseline function are developed we propose a groupwise empirical likelihood procedure to handle the inter series dependence for the longitudinal semiparametric regression model and employ bias correction to construct the empirical likelihood ratio functions for the parameters of interest this leads us to prove a nonparametric version of wilks theorem compared with methods based on normal approximations the empirical likelihood does not require consistent estimators for the asymptotic variance and bias a simulation compares the empirical likelihood and normal based methods in terms of coverage accuracies and average areas lengths of confidence regions intervals

uniform convergence rates for nonparametric regression and principal component analysis in functional longitudinal data we consider nonparametric estimation of the mean and covariance functions for functional longitudinal data strong uniform convergence rates are developed for estimators that are local linear smoothers our results are obtained in a unified framework in which the number of observations within each curve cluster can be of any rate relative to the sample size we show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve for sparse functional data these rates are equivalent to the optimal rates in nonparametric regression for dense functional data root n rates of convergence can be achieved with proper choices of bandwidths we further derive almost sure rates of convergence for principal component analysis using the estimated covariance function the results are illustrated with simulation studies

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

functional mixed effects models in this article a new class of functional models in which smoothing splines are used to model fixed effects as well as random effects is introduced the linear mixed effects models are extended to nonparametric mixed effects models by introducing functional random effects which are modeled as realizations of zero mean stochastic processes the fixed functional effects and the random functional effects are modeled in the same functional space which guarantee the population average and subject specific curves have the same smoothness property these models inherit the flexibility of the linear mixed effects models in handling complex designs and correlation structures can include continuous covariates as well as dummy factors in both the fixed or random design matrices and include the nested curves models as special cases two estimation procedures are proposed the first estimation procedure exploits the connection between linear mixed effects models and smoothing splines and can be fitted using existing software the second procedure is a sequential estimation procedure using kalman filtering this algorithm avoids inversion of large dimensional matrices and therefore can be applied to large data sets a generalized maximum likelihood gml ratio test is proposed for inference and model selection an application to comparison of cortisol profiles is used as an illustration

generalized quasi likelihood ratio tests for semiparametric analysis of covariance models in longitudinal data abstractwe model generalized longitudinal data from multiple treatment groups by a class of semiparametric analysis of covariance models which take into account the parametric effects of time dependent covariates and the nonparametric time effects in these models the treatment effects are represented by nonparametric functions of time and we propose a generalized quasi likelihood ratio test procedure to test if these functions are identical our estimation procedure is based on profile estimating equations combined with local linear smoothers we find that the much celebrated wilks phenomenon which is well established for independent data still holds for longitudinal data if a working independence correlation structure is assumed in the test statistic however this property does not hold in general especially when the working variance function is misspecified our empirical study also shows that incorporating correlation into the test statistic does not necessarily improve the power of the test the

semiparametric regression analysis for recurrent event interval counts this paper deals with analysis of data from longitudinal studies where the rate of a recurrent event characterizing morbidity is the primary criterion for treatment evaluation we consider clinical trials which require patients to visit their clinical center at successive scheduled times as part of follow up at each visit the patient reports the number of events that occurred since the previous visit or an examination reveals the number of accumulated events such as skin cancers the exact occurrence times of the events are unavailable and the actual patient visit times typically vary randomly about the scheduled follow up times each patient s record thus consists of a sequence of clinic visit dates event counts corresponding to the successive time intervals between clinic visits and baseline covariates we propose a semiparametric regression model extending the fully parametric model of thall 1988 biometrics 44 197 209 to estimate and test for covariate effects on the rate of events over time while also accounting for the possibly time varying nature of the underlying event rate covariate effects enter the model parametrically while the underlying time varying event rate is modelled nonparametrically the method of severini and wong 1992 annals of statistics 20 1768 1802 is used to construct asymptotically efficient estimators of the parametric component and to specify their asymptotic distribution a simulation study and application to a data set are provided

functional convex averaging and synchronization for time warped random curves data that can be best described as a sample of curves are now fairly common in science and engineering when the dynamics of development growth or response over time are at issue subjects or experimental units may experience events at different temporal paces for functional data where trajectories may be individually time transformed it is usually inadequate to use commonly used sample statistics such as the cross sectional mean or median or the cross sectional sample variance if one observes time warped curve data i e random curves or random trajectories that exhibit random transformations of the time scale then the usual l2 norm and metric typically are inadequate one may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis dynamic time warping alignment curve registration and landmark based methods have been put forward with the goal of finding adequate empirical time transforma

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

structured functional additive regression in reproducing kernel hilbert spaces summary functional additive models provide a flexible yet simple framework for regressions involving functional predictors the utilization of a data driven basis in an additive rather than linear structure naturally extends the classical functional linear model however the critical issue of selecting non linear additive components has been less studied in this work we propose a new regularization framework for structure estimation in the context of reproducing kernel hilbert spaces the approach proposed takes advantage of functional principal components which greatly facilitates implementation and theoretical analysis the selection and estimation are achieved by penalized least squares using a penalty which encourages the sparse structure of the additive components theoretical properties such as the rate of convergence are investigated the empirical performance is demonstrated through simulation studies and a real data application

high dimensional analysis of semidefinite relaxations for sparse principal components principal component analysis pca is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix however pca is well known to behave poorly in the large p small n setting in which the problem dimension p is comparable to or larger than the sample size n this paper studies pca in this high dimensional regime but under the additional assumption that the maximal eigenvector is sparse say with at most k nonzero components we consider a spiked covariance model in which a base matrix is perturbed by adding a k sparse maximal eigenvector and we analyze two computationally tractable methods for recovering the support set of this maximal eigenvector as follows a a simple diagonal thresholding method which transitions from success to failure as a function of the rescaled sample size dia n p k n k 2 log p k and b a more sophisticated semidefinite programming sdp relaxation which succeeds once the rescaled sample size sdp n p k n klog p k is larger than a critical threshold in addition we prove that no method including the best method which has exponential time complexity can succeed in recovering the support if the order parameter sdp n p k is below a threshold our results thus highlight an interesting trade off between computational and statistical efficiency in high dimensional inference

on the 3j symmetries the symmetry properties of the 3jm tensor for any finite or compact linear group are discussed using a wreath product construction this is shown to provide a complete group theoretic explanation for all symmetry properties whether essential or arbitrary the link with the similar but distinct method of inner plethysms is considered

sampled forms of functional pca in reproducing kernel hilbert spaces space of smooth functions this model includes time and frequency sampling as special cases in contrast to classical approach in fpca in which access to entire functions is assumed having a limited number m of functional samples places limitations on the performance of statistical procedures we study these effects by analyzing the rate of convergence of an m estimator for the subspace spanned by the leading components in a multi spiked covariance model the estimator takes the form of regularized pca and hence is computationally attractive we analyze the behavior of this estimator within a nonasymptotic framework and provide bounds that hold with high probability as a function of the number of statistical samples n and the number of functional samples m we also derive lower bounds showing that the rates obtained are minimax optimal

fuzzy ideals and fuzzy bi ideals in fuzzy semigroups abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra such as fuzzy left right ideal and fuzzy bi ideal in fuzzy semigroup through the new approach of fuzzy space and fuzzy group introduced by dib 1994 our notion of fuzzy ideal and fuzzy bi ideal includes the classical concepts of fuzzy ideal and fuzzy bi ideal of ordinary semigroup many counterexamples are also given

some character theory for groups of linear and antilinear operators elementary group concepts are recast into a form applicable to finite magnetic groups of linear and antilinear operators analogs of useful definitions for linear groups such as the frobenius schur invariant commutator subgroups and ambivalent classes are considered these are applied to the 180 magnetic single and double point groups and it is shown that only seven require independent treatment of characters

the fuzzy normal subgroup using the concept of fuzzy spaces we define the fuzzy normal subgroup in a fuzzy group its relation with the classical fuzzy normal subgroup is studied

generating relations for reducing matrices ii corepresentations the auxiliary group approach developed in paper 1 r dirl p kasperkovitz m i aroyo j n kotzev and m angelova tjurkedjieva j math phys 27 37 1986 is generalized for the case of corepresentations of antiunitary groups it allows us to reduce the multiplicity problem and to derive consistent generating relations for the elements of the reducing matrices for coreps two examples are worked out to illustrate the general scheme

asymptotic confidence regions for kernel smoothing of a varying coefficient model with longitudinal data abstract we consider the estimation of the k 1 dimensional nonparametric component t of the varying coefficient model y t x t t t e t based on longitudinal observations yij x i tij tij i 1 n j 1 ni where tij is the jth observed design time point t of the ith subject and yij and x i tij are the real valued outcome and r k 1 valued covariate vectors of the ith subject at tij the subjects are independently selected but the repeated measurements within subject are possibly correlated asymptotic distributions are established for a kernel estimate of t that minimizes a local least squares criterion these asymptotic distributions are used to construct a class of approximate pointwise and simultaneous confidence regions for t applying these methods to an epidemiological study we show that our procedures are useful for predicting cd4 t helper lymphocytes cell changes among hiv human immunodeficiency virus infected persons the finite sample properties of o

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

functional data analysis with application to periodically stimulated foetal heart rate data ii functional logistic regression functional regression is used to model longitudinal data where the number of measurements on the functional covariate is much greater than the number of subjects in the study thus functional regression can be thought of as singular longitudinal analysis we have modified existing functional regression techniques to the case of a functional covariate with a repeated stimulus we applied this modified functional regression to periodically stimulated foetal heart rates the heart rate tracings were used as a predictor of the child s psychomotor development at approximately 18 months of age in the past this type of data has been analysed using the partially subjective concept of habituation by using the entire heart rate tracings through functional regression we have the advantage that habituation does not need to be defined and all available information is used to predict later child development

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

an rkhs formulation of the inverse regression dimension reduction problem suppose that y is a scalar and x is a second order stochastic process where y and x are conditionally independent given the random variables 1 p which belong to the closed span l2 of x this paper investigates a unified framework for the inverse regression dimension reduction problem it is found that the identification of l2 with the reproducing kernel hilbert space of x provides a platform for a seamless extension from the finite to infinite dimensional settings it also facilitates convenient computational algorithms that can be applied to a variety of models

tolerance bands for functional data often the object of inference in biomedical applications is a range that brackets a given fraction of individual observations in a population a classical estimate of this range for univariate measurements is a tolerance interval this article develops its natural extension for functional measurements a tolerance band and proposes a methodology for constructing its pointwise and simultaneous versions that incorporates both sparse and dense functional data assuming that the measurements are observed with noise the methodology uses functional principal component analysis in a mixed model framework to represent the measurements and employs bootstrapping to approximate the tolerance factors needed for the bands the proposed bands also account for uncertainty in the principal components decomposition simulations show that the methodology has generally acceptable performance unless the data are quite sparse and unbalanced in which case the bands may be somewhat liberal the methodology is illustrated using two real datasets a sparse dataset involving cd4 cell counts and a dense dataset involving core body temperatures

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

multilevel matrix variate analysis and its application to accelerometry measured physical activity in clinical populations the number of studies where the primary measurement is a matrix is exploding in response to this we propose a statistical framework for modeling populations of repeatedly observed matrix variate

nonparametric regression analysis of longitudinal data abstract nonparametric methods are developed for estimating the dose effect when a response consists of correlated observations over time measured in a dose response experiment the methods can also be applied to data collected from a completely randomized design experiment methods are developed for the detection and description of the effects of dose time and their interaction the methods allow for individual variation in the timing and number of observations a generalization allowing baseline covariates to be incorporated is addressed these results may be used in an exploratory fashion in the process of building a random effects model for longitudinal data

modelling sparse generalized longitudinal observations with latent gaussian processes summary in longitudinal data analysis one frequently encounters non gaussian data that are repeatedly collected for a sample of individuals over time the repeated observations could be binomial poisson or of another discrete type or could be continuous the timings of the repeated measurements are often sparse and irregular we introduce a latent gaussian process model for such data establishing a connection to functional data analysis the functional methods proposed are non parametric and computationally straightforward as they do not involve a likelihood we develop functional principal components analysis for this situation and demonstrate the prediction of individual trajectories from sparse observations this method can handle missing data and leads to predictions of the functional principal component scores which serve as random effects in this model these scores can then be used for further statistical analysis such as inference regression discriminant analysis or clustering we illustrate these non parametric methods with longitudinal data on primary biliary cirrhosis and show in simulations that they are competitive in comparisons with generalized estimating equations and generalized linear mixed models

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

on fuzzy subgroups and fuzzy ideals under triangular norms abstract in this communication we explicitly point out that the principal results of liu 1982 basically deduced from the definition of binary operation on the set f x of all fuzzy subsets of x also hold if one uses the weaker definition of product under triangular norm t fuzzy ideals with respect to the triangular norms are also defined

unified inference for sparse and dense longitudinal models in longitudinal data analysis statistical inference for sparse data and dense data could be substantially different for kernel smoothing estimate of the mean function the convergence rates and limiting variance functions are different under the two scenarios the latter phenomenon poses challenges for statistical inference as a subjective choice between the sparse and dense cases may lead to wrong conclusions we develop self normalization based methods that can adapt to the sparse and dense cases in a unified framework simulations show that the proposed methods outperform some existing methods

stringing high dimensional data for functional analysis we propose stringing a class of methods where one views high dimensional observations as functional data stringing takes advantage of the high dimension by representing such data as discretized and noisy observations that originate from a hidden smooth stochastic process assuming that the observations result from scrambling the original ordering of the observations of the process stringing reorders the components of the high dimensional vectors followed by transforming the high dimensional vector observations into functional data established techniques from functional data analysis can be applied for further statistical analysis once an underlying stochastic process and the corresponding random trajectory for each subject have been identified stringing of high dimensional data is implemented with distance based metric multidimensional scaling mapping high dimensional data to locations on a real interval such that predictors that are close in a suitable sample metric also are located close to each

a functional data analytic approach to signal discrimination motivated by specific problems involving radar range profiles we suggest techniques for real time discrimination in the context of signal analysis the key to our approach is to regard the signals as curves in the continuum and employ a functional data analytic fda method for dimension reduction based on the fda technique for principal coordinates analysis this has the advantage relative to competing methods such as canonical variates analysis of providing a signal approximation that is best possible in an l2 sense for a given dimension as a result it produces particularly good discrimination we explore the use of both nonparametric and gaussian based discriminators applied to the dimensionreduced data

the character table for the corepresentations of magnetic groups a square character table is shown to exist for all finite magnetic groups the table possesses row and column orthogonality properties similar to the character table for linear groups

when the data are functions a datum is often a continuous functionx t of a variable such as time observed over some interval one or more such functions are observed for each subject or unit of observation the extension of classical data analytic techniques designed forp variate observations to such data is discussed the essential step is the expression of the classical problem in the language of functional analysis after which the extension to functions is a straightforward matter a schematic device called the duality diagram is a very useful tool for describing an analysis and for suggesting new possibilities least squares approximation descriptive statistics principal components analysis and canonical correlation analysis are discussed within this broader framework

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

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

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

functional graphical models graphical models have attracted increasing attention in recent years especially in settings involving high dimensional data in particular gaussian graphical models are used to model the conditio

functional quasi likelihood regression models with smooth random effects summary we propose a class of semiparametric functional regression models to describe the influence of vector valued covariates on a sample of response curves each observed curve is viewed as the realization of a random process composed of an overall mean function and random components the finite dimensional covariates influence the random components of the eigenfunction expansion through single index models that include unknown smooth link and variance functions the parametric components of the single index models are estimated via quasi score estimating equations with link and variance functions being estimated nonparametrically we obtain several basic asymptotic results the functional regression models proposed are illustrated with the analysis of a data set consisting of egg laying curves for 1000 female mediterranean fruit flies medflies

functional multiple set canonical correlation analysis we propose functional multiple set canonical correlation analysis for exploring associations among multiple sets of functions the proposed method includes functional canonical correlation analysis as a special case when only two sets of functions are considered as in classical multiple set canonical correlation analysis computationally the method solves a matrix eigen analysis problem through the adoption of a basis expansion approach to approximating data and weight functions we apply the proposed method to functional magnetic resonance imaging fmri data to identify networks of neural activity that are commonly activated across subjects while carrying out a working memory task

dynamic conditionally linear mixed models for longitudinal data we develop a new class of models dynamic conditionally linear mixed models for longitudinal data by decomposing the within subject covariance matrix using a special cholesky decomposition here dynamic means using past responses as covariates and conditional linearity means that parameters entering the model linearly may be random but nonlinear parameters are nonrandom this setup offers several advantages and is surprisingly similar to models obtained from the first order linearization method applied to nonlinear mixed models first it allows for flexible and computationally tractable models that include a wide array of covariance structures these structures may depend on covariates and hence may differ across subjects this class of models includes e g all standard linear mixed models antedependence models and vonesh carter models second it guarantees the fitted marginal covariance matrix of the data is positive definite we develop methods for bayesian inference and motivate the usefulness of these models using a series of longitudinal depression studies for which the features of these new models are well suited

a new approach to fuzzy groupoids abstract we establish a new approach to fuzzy groupoids a concept of fuzzy binary operation is introduced in terms of the concepts of fuzzy cartesian product and fuzzy function introduced by the authors in a foregoing paper having defined a fuzzy binary operation the concepts of fuzzy groupoid subgroupoid homomorphism etc are introduced and investigated a theory of fuzzy groupoids is built up this approach is both intrinsic and more general than the well known current approach

mixture inner product spaces and their application to functional data analysis we introduce the concept of mixture inner product spaces associated with a given separable hilbert space which feature an infinite dimensional mixture of finite dimensional vector spaces and are dense in the underlying hilbert space any hilbert valued random element can be arbitrarily closely approximated by mixture inner product space valued random elements while this concept can be applied to data in any infinite dimensional hilbert space the case of functional data that are random elements in the l 2 space of square integrable functions is of special interest for functional data mixture inner product spaces provide a new perspective where each realization of the underlying stochastic process falls into one of the component spaces and is represented by a finite number of basis functions the number of which corresponds to the dimension of the component space in the mixture representation of functional data the number of included mixture components used to represent a given random element in l 2 is specifically adapted to each random trajectory and may be arbitrarily large key benefits of this novel approach are first that it provides a new perspective on the construction of a probability density in function space under mild regularity conditions and second that individual trajectories possess a trajectory specific dimension that corresponds to a latent random variable making it possible to use a larger number of components for less smooth and a smaller number for smoother trajectories this enables flexible and parsimonious modeling of heterogeneous trajectory shapes we establish estimation consistency of the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation maximization algorithm simulations confirm that in comparison to traditional functional principal component analysis the proposed method achieves similar or better data recovery while using fewer components on average its practical merits are also demonstrated in an analysis of egg laying trajectories for medflies

nonparametric estimation of covariance structure in longitudinal data summary in longitudinal studies the effect of various treatments over time is usually of prime interest however observations on the same subject are usually correlated and any analysis should account for the underlying covariance structure a nonparametric estimate of the covariance structure is useful either as a guide to the formulation of a parametric model or as the basis for formal inference without imposing parametric assumptions the sample covariance matrix provides such an estimate when the data consist of a short sequence of measurements at a common set of time points on each of many subjects but is impractical when the data are severely unbalanced or when the sequences of measurements on individual subjects are long relative to the number of subjects the variogram of residuals from a saturated model for the mean response has previously been suggested as a nonparametric estimator for covariance structure assuming stationarity in this paper we consider kernel weighted local linear regression smoothing of sample variogram ordinates and of squared residuals to provide a nonparametric estimator for the covariance structure without assuming stationarity the value of the estimator as a diagnostic tool is demonstrated in two applications one to a set of data concerning the blood pressure of newborn babies in an intensive care unit and the other to data on the time evolution of cd4 cell numbers in hiv seroconverters

a bayesian model for sparse functional data we propose a method for analyzing data which consist of curves on multiple individuals i e longitudinal or functional data we use a bayesian model where curves are expressed as linear combinations of b splines with random coefficients the curves are estimated as posterior means obtained via markov chain monte carlo mcmc methods which automatically select the local level of smoothing the method is applicable to situations where curves are sampled sparsely and or at irregular time points we construct posterior credible intervals for the mean curve and for the individual curves this methodology provides unified efficient and flexible means for smoothing functional data

free knot spline smoothing for functional data summary the paper introduces free knot regression spline estimators for the mean and the variance components of a sample of curves the asymptotic distribution of the mean estimator is derived and asymptotic confidence bands are constructed a comparative simulation study shows that free knot splines estimate salient features of the functions such as sharp peaks more accurately than smoothing splines this adaptive behaviour is also illustrated by an analysis of weather data

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

products of fuzzy subgroups fuzzy subgroups are different from ordinary subgroups in that one cannot tell with certainty which group elements belong and which do not of course this requires an appropriate modification of the closure property which takes the form of an inequality in this paper a study of products of fuzzy subgroups is begun some results are expected the product of fuzzy subgroups is a fuzzy subgroup products with the same but permuted factors are isomorphic and there is a natural isomorphism from any factor into the product functions called t norms are used to construct products of fuzzy subgroups different t norms yielding different products since a fuzzy subgroup must satisfy an inequality generalizing closure and the strength of that inequality is important some results relate the t norm used in forming the product to the strength of the ensuing inequality when can complicated fuzzy subgroups be expressed as products of simple ones this question is addressed for fuzzy subgroups of finite abelian groups

fuzzy cartesian product fuzzy relations and fuzzy functions abstract a new approach to cartesian product relations and functions in fuzzy set theory is established a concept of fuzzy cartesian product is introduced using a suitable lattice a fuzzy relation is then defined as a subset of the fuzzy cartesian product analogously to the crisp case fuzzy equivalence relations are introduced and investigated results corresponding to those on ordinary equivalence relations are proved for fuzzy equivalence relations finally the notion of a fuzzy function is introduced as a special type of fuzzy relations and is then investigated this fuzzy function generalizes zadeh s definition of functions in the fuzzy context

gaussian process modeling of a functional output with information from boundary and initial conditions and analytical approximations a partial differential equation pde models a physical quantity as a function of space and time these models are often solved numerically with the finite element fe method and the computer outp

partially functional linear regression in high dimensions summary in modern experiments functional and nonfunctional data are often encountered simultaneously when observations are sampled from random processes and high dimensional scalar covariates it is difficult to apply existing methods for model selection and estimation we propose a new class of partially functional linear models to characterize the regression between a scalar response and covariates of both functional and scalar types the new approach provides a unified and flexible framework that simultaneously takes into account multiple functional and ultrahigh dimensional scalar predictors enables us to identify important features and offers improved interpretability of the estimators the underlying processes of the functional predictors are considered to be infinite dimensional and one of our contributions is to characterize the effects of regularization on the resulting estimators we establish the consistency and oracle properties of the proposed method under mild conditions demonstrate its performance with simulation studies and illustrate its application using air pollution data

fuzzy hyperrings hv rings based on fuzzy universal sets the aim of this paper is to define and study the concept of a fuzzy hyperring which depends on the concept of a fuzzy universal set a correspondence relation between a fuzzy hyperring based on a fuzzy universal set and both ordinary and fuzzy hyperideal in the sense of davvaz is obtained