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

new estimation and model selection procedures for semiparametric modeling in longitudinal data analysis semiparametric regression models are very useful for longitudinal data analysis the complexity of semiparametric models and the structure of longitudinal data pose new challenges to parametric inferences and model selection that frequently arise from longitudinal data analysis in this article two new approaches are proposed for estimating the regression coefficients in a semiparametric model the asymptotic normality of the resulting estimators is established an innovative class of variable selection procedures is proposed to select significant variables in the semiparametric models the proposed procedures are distinguished from others in that they simultaneously select significant variables and estimate unknown parameters rates of convergence of the resulting estimators are established with a proper choice of regularization parameters and penalty functions the proposed variable selection procedures are shown to perform as well as an oracle estimator a robust standard error formula is derived usi

on the cox model with time varying regression coefficients in the analysis of censored failure time observations the standard cox proportional hazards model assumes that the regression coefficients are time invariant often these parameters vary over time and the temporal covariate effects on the failure time are of great interest in this article following previous work of cai and sun we propose a simple estimation procedure for the cox model with time varying coefficients based on a kernel weighted partial likelihood approach we construct pointwise and simultaneous confidence intervals for the regression parameters over a properly chosen time interval via a simple resampling technique we derive a prediction method for future patients survival with any specific set of covariates building on the estimates for the time varying coefficients we also consider the mixed case and present an estimation procedure for time independent parameters in the model furthermore we show how to use an integrated function of the estimate for a specific regression coeffic

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

the effect of spatial separation on informational and energetic masking of speech the effect of spatial separation of sources on the masking of a speech signal was investigated for three types of maskers ranging from energetic to informational normal hearing listeners performed a closed set speech identification task in the presence of a masker at various signal to noise ratios stimuli were presented in a quiet sound field the signal was played from 0 azimuth and a masker was played either from the same location or from 90 to the right signals and maskers were derived from sentences that were preprocessed by a modified cochlear implant simulation program that filtered each sentence into 15 frequency bands extracted the envelopes from each band and used these envelopes to modulate pure tones at the center frequencies of the bands in each trial the signal was generated by summing together eight randomly selected frequency bands from the preprocessed signal sentence three maskers were derived from the preprocessed masker sentences 1 different band sentence which was generated by summing together six randomly selected frequency bands out of the seven bands not present in the signal resulting in primarily informational masking 2 different band noise which was generated by convolving the different band sentence with gaussian noise and 3 same band noise which was generated by summing the same eight bands from the preprocessed masker sentence that were used in the signal sentence and convolving the result with gaussian noise resulting in primarily energetic masking results revealed that in the different band sentence masker the effect of spatial separation averaged 18 db at 51 correct while in the different band and same band noise maskers the effect was less than 10 db these results suggest that in these conditions the advantage due to spatial separation of sources is greater for informational masking than for energetic masking

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

covariate adjusted functional principal components analysis for longitudinal data classical multivariate principal component analysis has been extended to functional data and termed functional principal component analysis fpca most existing fpca approaches do not accommodate covariate information and it is the goal of this paper to develop two methods that do in the first approach both the mean and covariance functions depend on the covariate z and time scale t while in the second approach only the mean function depends on the covariate z both new approaches accommodate additional measurement errors and functional data sampled at regular time grids as well as sparse longitudinal data sampled at irregular time grids the first approach to fully adjust both the mean and covariance functions adapts more to the data but is computationally more intensive than the approach to adjust the covariate effects on the mean function only we develop general asymptotic theory for both approaches and compare their performance numerically through simulation studies and a data set

quadratic inference functions for varying coefficient models with longitudinal data nonparametric smoothing methods are used to model longitudinal data but the challenge remains to incorporate correlation into nonparametric estimation procedures in this article we propose an efficient estimation procedure for varying coefficient models for longitudinal data the proposed procedure can easily take into account correlation within subjects and deal directly with both continuous and discrete response longitudinal data under the framework of generalized linear models the proposed approach yields a more efficient estimator than the generalized estimation equation approach when the working correlation is misspecified for varying coefficient models it is often of interest to test whether coefficient functions are time varying or time invariant we propose a unified and efficient nonparametric hypothesis testing procedure and further demonstrate that the resulting test statistics have an asymptotic chi squared distribution in addition the goodness of fit test is applied to test whether the model assumption is satisfied the corresponding test is also useful for choosing basis functions and the number of knots for regression spline models in conjunction with the model selection criterion we evaluate the finite sample performance of the proposed procedures with monte carlo simulation studies the proposed methodology is illustrated by the analysis of an acquired immune deficiency syndrome aids data set

failure time regression with continuous covariates measured with error summary we consider failure time regression analysis with an auxiliary variable in the presence of a validation sample we extend the nonparametric inference procedure of zhou and pepe to handle a continuous auxiliary or proxy covariate we estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates we present some asymptotic properties for the kernel estimator and provide some simulation results the method proposed is illustrated with a data set from an on going epidemiologic study

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

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

predicting binaural speech intelligibility using the signal to noise ratio in the envelope power spectrum domain this study proposes a binaural extension to the multi resolution speech based envelope power spectrum model mr sepsm jorgensen ewert and dau 2013 j acoust soc am 134 436 446 it consists of a combination of better ear be and binaural unmasking processes implemented as two monaural realizations of the mr sepsm combined with a short term equalization cancellation process and uses the signal to noise ratio in the envelope domain snrenv as the decision metric the model requires only two parameters to be fitted per speech material and does not require an explicit frequency weighting the model was validated against three data sets from the literature which covered the following effects the number of maskers the masker types speech shaped noise ssn speech modulated ssn babble and reversed speech the masker s azimuths reverberation on the target and masker and the interaural time difference of the target and masker the pearson correlation coefficient between the simulated spee

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

the effect of spatial separation in distance on the intelligibility of speech in rooms the influence of spatial separation in source distance on speech reception thresholds srts is investigated in one scenario the target was presented at 0 5 m distance and the masker varied from 0 5 m distance up to 10 m in a second scenario the masker was presented at 0 5 m distance and the target distance varied the stimuli were synthesized using convolution with binaural room impulse responses brirs measured on a dummy head in a reverberant auditorium and were equalized to compensate for distance dependent spectral and intensity changes all sources were simulated directly in front of the listener srts decreased monotonically when the target was at 0 5 m and the speech masker was moved further away resulting in a srt improvement of up to 10 db when the speech masker was at 0 5 m and the target was moved away a large variation across subjects was observed neither short term signal to noise ratio snr improvements nor cross ear glimpsing could account for the observed improvement in intelligibility however the effect might be explained by an improvement in the snr in the modulation domain and a decrease in informational masking this study demonstrates that distance related cues can play a significant role when listening in complex environments

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

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

the effect of better ear glimpsing on spatial release from masking the current experiment investigated whether better ear glimpsing can explain the spatial release achieved by normal hearing adults when situations are high in informational masking both modeling and behavioral methods were used the speech reception thresholds of 38 young adults were measured for co located spatially separated and two better ear glimpsed conditions in the better ear glimpsed conditions the binaural signals were processed so that in each time frequency segment the signal with the better snr left or right ear was presented diotically to investigate the effect of widening auditory filters on better ear glimpsing adjacent frequency bands were combined in one of the better ear glimpsing conditions twenty two participants were tested with maskers high in informational masking while 16 participants were tested with maskers lower in informational masking the mean speech reception thresholds achieved in the glimpsed conditions were significantly worse than in the spatially separated condition this suggests that better ear glimpsing can explain some but not all of the observed spatial release from masking the difference between performance in the spatially separated and glimpsed conditions was largest when informational masking was high suggesting better ear glimpsing may release energetic rather than informational masking reducing the number of frequency bands sampled had a small effect on performance

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

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

checking the cox model with cumulative sums of martingale based residuals summary this paper presents a new class of graphical and numerical methods for checking the adequacy of the cox regression model the procedures are derived from cumulative sums of martingale based residuals over follow up time and or covariate values the distributions of these stochastic processes under the assumed model can be approximated by zero mean gaussian processes each observed process can then be compared both visually and analytically with a number of simulated realizations from the approximate null distribution these comparisons enable the data analyst to assess objectively how unusual the observed residual patterns are special attention is given to checking the functional form of a covariate the form of the link function and the validity of the proportional hazards assumption an omnibus test consistent against any model misspecification is also studied the proposed techniques are illustrated with two real data sets

a simple root n bandwidth selector the asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related fourier transform methods the point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process the essential idea is derived through careful study of factorizations of the pilot bandwidth in terms of the original bandwidth

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

classical and modular approaches to exponential diophantine equations i fibonacci and lucas perfect powers this is the first in a series of papers whereby we combine the classical approach to exponential diophantine equations linear forms in logarithms thue equations etc with a modular approach based on some of the ideas of the proof of fermat s last theorem in this paper we give new improved bounds for linear forms in three logarithms we also apply a combination of classical techniques with the modular approach to show that the only perfect powers in the fibonacci sequence are 0 1 8 and 144 and the only perfect powers in the lucas sequence are 1 and 4

analysis of longitudinal data with semiparametric estimation of covariance function improving efficiency for regression coefficients and predicting trajectories of individuals are two important aspects in the analysis of longitudinal data both involve estimation of the covariance function yet challenges arise in estimating the covariance function of longitudinal data collected at irregular time points a class of semiparametric models for the covariance function by that imposes a parametric correlation structure while allowing a nonparametric variance function is proposed a kernel estimator for estimating the nonparametric variance function is developed two methods for estimating parameters in the correlation structure a quasi likelihood approach and a minimum generalized variance method are proposed a semiparametric varying coefficient partially linear model for longitudinal data is introduced and an estimation procedure for model coefficients using a profile weighted least squares approach is proposed sampling properties of the proposed estimation procedures are studied and asy

time varying functional regression for predicting remaining lifetime distributions from longitudinal trajectories a recurring objective in longitudinal studies on aging and longevity has been the investigation of the relationship between age at death and current values of a longitudinal covariate trajectory that quantifies reproductive or other behavioral activity we propose a novel technique for predicting age at death distributions for situations where an entire covariate history is included in the predictor the predictor trajectories up to current time are represented by time varying functional principal component scores which are continuously updated as time progresses and are considered to be time varying predictor variables that are entered into a class of time varying functional regression models that we propose we demonstrate for biodemographic data how these methods can be applied to obtain predictions for age at death and estimates of remaining lifetime distributions including estimates of quantiles and of prediction intervals for remaining lifetime estimates and predictions are obtained for individual subjects based on their observed behavioral trajectories and include a dimension reduction step that is implemented by projecting on a single index the proposed techniques are illustrated with data on longitudinal daily egg laying for female medflies predicting remaining lifetime and age at death distributions from individual event histories observed up to current time

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

generalized semiparametric varying coefficient model for longitudinal data with applications to adaptive treatment randomizations summary this article investigates a generalized semiparametric varying coefficient model for longitudinal data that can flexibly model three types of covariate effects time constant effects time varying effects and covariate varying effects different link functions can be selected to provide a rich family of models for longitudinal data the model assumes that the time varying effects are unspecified functions of time and the covariate varying effects are parametric functions of an exposure variable specified up to a finite number of unknown parameters the estimation procedure is developed using local linear smoothing and profile weighted least squares estimation techniques hypothesis testing procedures are developed to test the parametric functions of the covariate varying effects the asymptotic distributions of the proposed estimators are established a working formula for bandwidth selection is discussed and examined through simulations our simulation study shows that the proposed methods have satisfactory finite sample performance the proposed methods are applied to the actg 244 clinical trial of hiv infected patients being treated with zidovudine to examine the effects of antiretroviral treatment switching before and after hiv develops the t215y f drug resistance mutation our analysis shows benefits of treatment switching to the combination therapies as compared to continuing with zdv monotherapy before and after developing the 215 mutation

covers of the integers with odd moduli and their applications to the forms x m 2 n and x 2 f_ 3n 2 in this paper we construct a cover a s mod n s k s 1 of with odd moduli such that there are distinct primes p 1 p k dividing 2 n1 1 nk 1 respectively using this cover we show that for any positive integer m divisible by none of 3 5 7 11 13 there exists an infinite arithmetic progression of positive odd integers the mth powers of whose terms are never of the form 2 n p a with a n 0 1 2 and p a prime we also construct another cover of z with odd moduli and use it to prove that x 2 f 3n 2 has at least two distinct prime factors whenever n 0 1 2 and x a mod m where f i i 0 is the fibonacci sequence and a and m are suitable positive integers having 80 decimal digits

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

alignment of curves by dynamic time warping when studying some process or development in different subjects or units be it biological chemical or physical we usually see a typical pattern common to all curves yet there is variation both in amplitude and dynamics between curves following some ideas of structural analysis introduced by kneip and gasser we study a method dynamic time warping with a proper cost function for estimating the shift or warping function from one curve to another to align the two functions for some models this method can identify the true shift functions if the data are noise free noisy data are smoothed by a nonparametric function estimate such as a kernel estimate it is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity some simulation results are presented to illustrate the performance of this method

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

penalized spline models for functional principal component analysis summary we propose an iterative estimation procedure for performing functional principal component analysis the procedure aims at functional or longitudinal data where the repeated measurements from the same subject are correlated an increasingly popular smoothing approach penalized spline regression is used to represent the mean function this allows straightforward incorporation of covariates and simple implementation of approximate inference procedures for coefficients for the handling of the within subject correlation we develop an iterative procedure which reduces the dependence between the repeated measurements that are made for the same subject the resulting data after iteration are theoretically shown to be asymptotically equivalent in probability to a set of independent data this suggests that the general theory of penalized spline regression that has been developed for independent data can also be applied to functional data the effectiveness of the proposed procedure is demonstrated via a simulation study and an application to yeast cell cycle gene expression data

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

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

sentence reception in noise from one versus two sources effects of aging and hearing loss sentence reception thresholds srts and babble detection thresholds bdts were measured for signals presented from loudspeakers located at 0 deg and 90 deg azimuth in addition the s b ratios necessary to achieve srt in the presence of a 12 talker babble were determined under three conditions in which the speech and babble were presented from the same or spatially separated loudspeakers in the first experiment normal hearing subjects were tested on two occasions the results were highly reliable with standard errors of the test retest differences of less than 1db for all conditions in the second experiment srts bdts and s b ratios were measured for three normal hearing groups 39 40 54 and 55 years and for a fourth group with presbycusis 55 years presbycusics had higher srts and bdts than all other groups regardless of age and the oldest normals had higher thresholds than younger subjects presbycusics required higher s b ratios than any of the normal groups and the oldest normal group needed a higher s b ratio compared to the youngest group all groups demonstrated a significant advantage in s b ratio when the sentences and babble sources were spatially separated compared to when they were presented from the same loudspeaker there was no significant difference in the magnitude of advantage due to spatial separation of speech and noise between the normal groups regardless of age but the presbycusics had significantly smaller advantages than all normals these findings suggest that aging alone does not impair the ability to take advantage of the spatial separation of speech and noise and that the speech reception problems of aging individuals involve various components that depend upon the nature of the listening task

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

the effects of age and cochlear hearing loss on temporal fine structure sensitivity frequency selectivity and speech reception in noise temporal fine structure tfs sensitivity frequency selectivity and speech reception in noise were measured for young normal hearing nhy old normal hearing nho and hearing impaired hi subjects two measures of tfs sensitivity were used the tfs lf test interaural phase difference discrimination and the tfs2 test discrimination of harmonic and frequency shifted tones these measures were not significantly correlated with frequency selectivity after partialing out the effect of audiometric threshold suggesting that insensitivity to tfs cannot be wholly explained by a broadening of auditory filters the results of the two tests of tfs sensitivity were significantly but modestly correlated suggesting that performance of the tests may be partly influenced by different factors the nho group performed significantly more poorly than the nhy group for both measures of tfs sensitivity but not frequency selectivity suggesting that tfs sensitivity declines with age in the absence of elevated audiometric thresholds or broadened auditory filters when the effect of mean audiometric threshold was partialed out speech reception thresholds in modulated noise were correlated with tfs2 scores but not measures of frequency selectivity or tfs lf test scores suggesting that a reduction in sensitivity to tfs can partly account for the speech perception difficulties experienced by hearing impaired subjects

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

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

sieving by large integers and covering systems of congruences a finite set of congruences where each integer satisfies at least one them is called a covering system a famous problem of erd s from 1950 4 is to determine whether for every n there is a covering system with distinct moduli greater than n in other words can the minimum modulus in a covering system with distinct moduli be arbitrarily large in regards to this problem erd s writes in 6 this is perhaps my favourite problem it is easy to see that in a covering system the reciprocal sum of the moduli is at least 1 examples with distinct moduli are known with least modulus 2 3 and 4 where this reciprocal sum can be arbitrarily close to 1 see 10 f13 erd s and selfridge 5 conjectured that this fails for all large enough choices of the least modulus in fact they made the following much stronger conjecture

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

smoothing spline models for the analysis of nested and crossed samples of curves abstract we introduce a class of models for an additive decomposition of groups of curves stratified by crossed and nested factors generalizing smoothing splines to such samples by associating them with a corresponding mixed effects model the models are also useful for imputation of missing data and exploratory analysis of variance we prove that the best linear unbiased predictors blups from the extended mixed effects model correspond to solutions of a generalized penalized regression where smoothing parameters are directly related to variance components and we show that these solutions are natural cubic splines the model parameters are estimated using a highly efficient implementation of the em algorithm for restricted maximum likelihood reml estimation based on a preliminary eigenvector decomposition variability of computed estimates can be assessed with asymptotic techniques or with a novel hierarchical bootstrap resampling scheme for nested mixed effects models our methods are applied to me

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

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