small subset sums abstract let be a norm in r d whose unit ball is b assume that v b is a finite set of cardinality n with v v v 0 we show that for every integer k with 0 k n there exists a subset u of v consisting of k elements such that v u v d 2 we also prove that this bound is sharp in general we improve the estimate to o d for the euclidean and the max norms an application on vector sums in the plane is also given

modeling conditional distributions for functional responses with application to traffic monitoring via gps enabled mobile phones motivated by problems involving a traffic monitoring system in which trajectory data are obtained from global positioning system enabled mobile phones we propose a novel approach to functional regression modeling where instead of the usual mean regression the entire distribution of functional responses is modeled conditionally on predictors an approach that sensibly balances flexibility and stability is obtained by assuming that the response functions are drawn from a gaussian process the mean and covariance function of which depend on predictors the dependence of the mean function and covariance function of the response on the predictors is modeled additively we demonstrate the proposed methods by constructing predicted curves and corresponding prediction regions for traffic velocity trajectories for a future time period using current traffic velocity fields as predictor functions the proposed functional regression and conditional distribution approach is of general interest for functional respon

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

k means inverse regression li suggested the method of sliced inverse regression for dimension reduction in regressions with a univariate response in this article we extend that method to multivariate regressions by introducing a new way of performing the slicing this method applies for any number of response variables and may be particularly useful at the outset of an analysis before positing a multivariate model the emphasis is on application no new asymptotic theory is presented

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

cook s distance measures for varying coefficient models with functional responses the aim of this article is to develop cook s distance measures for assessing the influence of both atypical curves and observations under varying coefficient model with functional responses our cook s distance measures include cook s distances for deleting multiple curves and for deleting multiple grid points and their scaled cook s distances we systematically investigate some theoretical properties of these diagnostic measures simulation studies are conducted to evaluate the finite sample properties of these cook s distances under different scenarios a real diffusion tensor tract dataset is analyzed to illustrate the use of our diagnostic measures

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

pfhc a clustering algorithm based on data partitioning for unevenly distributed datasets recently many researchers exert their effort on clustering as a primary data mining method for knowledge discovery but only few of them have focused on uneven dataset in the last research we proposed an efficient hierarchical algorithm based on fuzzy graph connectedness fhc to discover clusters with arbitrary shapes in this paper we present a novel clustering algorithm for uneven dataset pfhc which is an extended version based on fhc in pfhc dataset is divided into several local spaces firstly according to the data density of distribution where the data density in any local space is nearly uniform in order to achieve the goal local and l are used in each local domain to acquire local clustering result by fhc then boundary between local areas needs being taken into consideration for combination finally local clusters need to be merged to obtain global clusters as an extension of fhc pfhc can deal with uneven datasets more effectively and efficiently and generate better quality clusters than other methods as experiment shows furthermore pfhc is found to be able to process incremental data as well in this work

modeling hazard rates as functional data for the analysis of cohort lifetables and mortality forecasting as world populations age the analysis of demographic mortality data and demographic predictions of future mortality have met with increasing interest the study of mortality patterns and the forecasting of future mortality with its associated impacts on social welfare health care and societal planning has become a more pressing issue an ideal set of data to study patterns of change in long term mortality is the well known historical swedish cohort mortality data because of its high quality and long span of more than two centuries we explore the use of functional data analysis to model these data and to derive mortality forecasts specifically we address the challenge of flexibly modeling these data while including the effect of the birth year by regarding log hazard functions derived from observed cohort lifetables as random functions a functional model for the analysis of these cohort log hazard functions extending functional principal component approaches by introducing time varying eigenfunc

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

adaptive fuzzy c shells clustering and detection of ellipses several generalizations of the fuzzy c shells fcs algorithm are presented for characterizing and detecting clusters that are hyperellipsoidal shells an earlier generalization the adaptive fuzzy c shells afcs algorithm is examined in detail and is found to have global convergence problems when the shapes to be detected are partial new formulations are considered wherein the norm inducing matrix in the distance metric is unconstrained in contrast to the afcs algorithm the resulting algorithm called the afcs u algorithm performs better for partial shapes another formulation based on the second order quadrics equation is considered these algorithms can detect ellipses and circles in 2d data they are compared with the hough transform ht based methods for ellipse detection existing ht based methods for ellipse detection are evaluated and a multistage method incorporating the good features of all the methods is used for comparison numerical examples of real image data show that the afcs algorithm requires less memory than the ht based methods and it is at least an order of magnitude faster than the ht approach

continuously additive models for nonlinear functional regression summary we introduce continuously additive models which can be viewed as extensions of additive regression models with vector predictors to the case of infinite dimensional predictors this approach produces a class of flexible functional nonlinear regression models where random predictor curves are coupled with scalar responses in continuously additive modelling integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion we use tensor product basis expansions to fit the smooth regression surface that characterizes the model in a theoretical investigation we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal we also consider extensions to generalized responses the proposed class of models outperforms existing functional regression models in simulations and real data examples

assessing extrema of empirical principal component functions the difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite dimensional settings local maxima and minima in principal component functions are of direct importance they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign we explore statistical properties of the relationship between extrema of empirical principal component functions and their counterparts for the true principal component functions it is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema but can experience difficulty distinguishing a shoulder in a curve from a small bump for example when the true principal component function has a shoulder the probability that the empirical principal component function has instead a bump is approximately equal to 1 2 we suggest and describe the performance of bootstrap methods for assessing the strength of extrema it is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard a bootstrap likelihood is proposed for measuring extremum strength exploratory numerical methods are suggested

a hierarchical clustering algorithm based on fuzzy graph connectedness many clustering methods have been proposed in the area of data mining but only few of them focused on the incremental databases in this paper an algorithm for hierarchical clustering based on fuzzy graph connectedness algorithm fhc is investigated the presented algorithm applies fuzzy set theory to hierarchical clustering method so as to discover clusters with arbitrary shape it first partitions the data sets into several sub clusters using a partitioning method and then constructs a fuzzy graph of sub clusters by analyzing the fuzzy connectedness degree among sub clusters by computing the cut graph the connected components of the fuzzy graph can be obtained hence resulting the desired clustering the algorithm can be performed in high dimensional data sets finding clusters of arbitrary shapes such as the spherical linear elongated or concave ones also rendered in this research is the incremental algorithm ifhc applicable to periodically incremental environments not only can fhc and ifhc handle data with numerical attributes but categorical attributes can be dealt with as well the results of our experimental study for data sets with arbitrary shape and size are very encouraging the experimental study in web log files is also conducted that can help discover the user access patterns efficiently the investigation demonstrates that the proposed method generates better quality clusters than traditional algorithms and scales up well for large databases

horvitz thompson estimators for functional data asymptotic confidence bands and optimal allocation for stratified sampling when dealing with very large datasets of functional data survey sampling approaches are useful in order to obtain estimators of simple functional quantities without being obliged to store all the data we propose a horvitz thompson estimator of the mean trajectory in the context of a superpopulation framework we prove under mild regularity conditions that we obtain uniformly consistent estimators of the mean function and of its variance function with additional assumptions on the sampling design we state a functional central limit theorem and obtain asymptotic confidence bands stratified sampling is studied in detail and we also obtain a functional version of the usual optimal allocation rule considering a mean variance criterion these techniques are illustrated by a test population of n 18 902 electricity meters for which we have individual electricity consumption measures every 30 minutes over one week we show that stratification can substantially improve both the accuracy of the estimators and reduce the width of the global confidence bands compared with simple random sampling without replacement

helly type theorems for the sum of vectors in a normed plane abstract the main results here are two helly type theorems for the sum of at most unit vectors in a normed plane also we give a new characterization of centrally symmetric convex sets in the plane

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

additive colourful caratheodory type results with an application to radii abstract in this paper we study the behaviour of the circumradius with respect to the minkowski addition in generalized minkowski spaces to do so we solve additive colourful caratheodory type results under certain equilibria conditions

a quasi f test for functional linear models with functional covariates and its application to longitudinal data functional linear models are useful in analyzing data from designed experiments and observational studies with functional responses as well as longitudinal data with a large number of repeated measures on each subject we propose a quasi f test for functional linear models with functional covariates and outcomes we develop a numerical procedure and an efficient approximation for computing p values and present a simple way to test individual predictors for illustration we apply the proposed procedure to a longitudinal depression data set with repeatedly measured methamphetamine use as a predictor we conduct a simulation study to assess the size and the power of the test

functional regression analysis using an f test for longitudinal data with large numbers of repeated measures analysis abstract longitudinal data sets from certain fields of biomedical research often consist of several variables repeatedly measured on each subject yielding large number of observations this characteristic complicates the use of standard longitudinal modeling strategies such as random effects models and marginal models where rigorous assumptions on intra subject correlation structure are required an innovative way to model the data is to apply functional regression analysis an emerging statistical approach in which observations of the same subject are viewed as a sample from a functional space no assumptions are needed for the intra subject correlation structure shen and faraway satistica sinica 2004 1239 1257 introduced an f test for linear models with functional responses this paper illustrates how to apply this f test and functional regression analysis to the setting of longitudinal data where intra subject repeated measures are viewed as discrete samples from an underlying curve with continuous function forms a smoking cessation study for methadone maintained tobacco smokers is analyzed for demonstration in estimating the treatment effects the functional regression analysis provides meaningful clinical interpretations and the functional f test provides consistent results supported by a random effects linear regression model

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

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

robustness of density based clustering methods with various neighborhood relations cluster analysis is one of the most crucial techniques in statistical data analysis among the clustering methods density based methods have great importance due to their ability to recognize clusters with arbitrary shape in this paper robustness of the clustering methods is handled these methods use distance based neighborhood relations between points in particular dbscan density based spatial clustering of applications with noise algorithm and fn dbscan fuzzy neighborhood dbscan algorithm are analyzed fn dbscan algorithm uses fuzzy neighborhood relation whereas dbscan uses crisp neighborhood relation the main characteristic of the fn dbscan algorithm is that it combines the speed of the dbscan and robustness of the nrfjp noise robust fuzzy joint points algorithms it is observed that the fn dbscan algorithm is more robust than the dbscan algorithm to datasets with various shapes and densities

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

operator valued kernels for learning from functional response data in this paper we consider the problems of supervised classification and regression in the case where attributes and labels are functions a data is represented by a set of functions and the label is also a function we focus on the use of reproducing kernel hilbert space theory to learn from such functional data basic concepts and properties of kernel based learning are extended to include the estimation of function valued functions in this setting the representer theorem is restated a set of rigorously defined infinite dimensional operator valued kernels that can be valuably applied when the data are functions is described and a learning algorithm for nonlinear functional data analysis is introduced the methodology is illustrated through speech and audio signal processing experiments

canonical correlation analysis when the data are curves summary it is not immediately straightforward to extend canonical correlation analysis to the context of functional data analysis where the data are themselves curves or functions the obvious approach breaks down and it is necessary to use a method involving smoothing in some way such a method is introduced and discussed with reference to a data set on human gait the breakdown of the unsmoothed method is illustrated in a practical context and is demonstrated theoretically a consistency theorem for the smoothed method is proved in an increasing number of problems in a wide range of fields the data observed are not the univariate or multivariate observations of classical statistics but are func tions observed continuously ramsay and dalzell 1991 give the name functional data analysis to the analysis of data of this kind in most cases the observations will be functions of time or a closely related variable but there are clearly applications where the functions are surfaces observed over two or three dimensional space the motivating example for the present paper is a set of data on children s gait collected by the motion analysis laboratory at children s hospital san diego california see olshen et al 1989 for full details for each of a number of children several angles made by the child s joints knee hip etc are observed during the child s gait cycle one aim of the study is to gain understanding of the gait cycle of a normal child to make comparisons with children suffering from walking diffi culties these data motivated rice and silverman 1991 to discuss the extension of principal component analysis to the functional setting and to explain how smoothing can be incorporated into the analysis in a natural way in a substantial complemen tary paper ramsay and dalzell 1991 discuss different approaches to regression and principal component analysis in functional data analysis illustrating their work by a meteorological example the insights of functional data analysis may be helpful in chemometrics for instance a standard chemometric problem is the analysis of spectra observed in chromatography and these are to all intents and purposes functional observations another obvious area of potential relevance is the analysis of growth curve data the focus in the present paper will be on canonical correlation analysis cca in the gait example one might ask how variability in the knee angle cycle is related to

multivariate varying coefficient model for functional responses motivated by recent work studying massive imaging data in the neuroimaging literature we propose multivariate varying coefficient models mvcm for modeling the relation between multiple functional responses and a set of covariates we develop several statistical inference procedures for mvcm and systematically study their theoretical properties we first establish the weak convergence of the local linear estimate of coefficient functions as well as its asymptotic bias and variance and then we derive asymptotic bias and mean integrated squared error of smoothed individual functions and their uniform convergence rate we establish the uniform convergence rate of the estimated covariance function of the individual functions and its associated eigenvalue and eigenfunctions we propose a global test for linear hypotheses of varying coefficient functions and derive its asymptotic distribution under the null hypothesis we also propose a simultaneous confidence band for each individual effect curve we conduct monte carlo simulation to examine the finite sample performance of the proposed procedures we apply mvcm to investigate the development of white matter diffusivities along the genu tract of the corpus callosum in a clinical study of neurodevelopment

kernel smoothing of data with correlated errors abstract kernel smoothing is a common method of estimating the mean function in the nonparametric regression model y f x e where f x is a smooth deterministic mean function and e is an error process with mean zero in this article the mean squared error of kernel estimators is computed for processes with correlated errors and the estimators are shown to be consistent when the sequence of error processes converges to a mixing sequence the standard techniques for bandwidth selection such as cross validation and generalized cross validation are shown to perform very badly when the errors are correlated standard selection techniques are shown to favor undersmoothing when the correlations are predominantly positive and oversmoothing when negative the selection criteria can however be adjusted to correct for the effect of correlation in simulations the standard selection criteria are shown to behave as predicted the corrected criteria are shown to be very effective when the correlation functi

statistical inferences for functional data with modem technology development functional data are being observed frequently in many scientific fields a popular method for analyzing such functional data is smoothing first then estimation that is statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another however little is known about this substitution effect on functional data analysis in this paper this problem is investigated when the local polynomial kernel lpk smoothing technique is used for individual function reconstructions we find that under some mild conditions the substitution effect can be ignored asymptotically based on this we construct lpk reconstruction based estimators for the mean covariance and noise variance functions of a functional data set and derive their asymptotics we also propose a gcv rule for selecting good bandwidths for the lpk reconstructions when the mean function also depends on some time independent covariates we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time the lpk reconstruction based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived moreover we propose a l 2 norm based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression the effect of the bandwidths selected by the proposed gcv rule on the accuracy of the lpk reconstructions and the mean function estimator is investigated via a simulation study the proposed methodologies are illustrated via an application to a real functional data set collected in climatology

interpretable support vector machines for functional data support vector machines svm has been shown to be a powerful nonparametric classication technique even for high dimensional data although predictive ability is important obtaining an easy to interpret classier

generalized semiparametrically structured ordinal models motivated by recent work studying massive imaging data in the neuroimaging literature we propose multivariate varying coefficient models mvcm for modeling the relation between multiple functional responses and a set of covariates we develop several statistical inference procedures for mvcm and systematically study their theoretical properties we first establish the weak convergence of the local linear estimate of coefficient functions as well as its asymptotic bias and variance and then we derive asymptotic bias and mean integrated squared error of smoothed individual functions and their uniform convergence rate we establish the uniform convergence rate of the estimated covariance function of the individual functions and its associated eigenvalue and eigenfunctions we propose a global test for linear hypotheses of varying coefficient functions and derive its asymptotic distribution under the null hypothesis we also propose a simultaneous confidence band for each individual effect curve we conduct monte carlo simulation to examine the finite sample performance of the proposed procedures we apply mvcm to investigate the development of white matter diffusivities along the genu tract of the corpus callosum in a clinical study of neurodevelopment

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