estimation of functional regression model via functional dimension reduction abstract the functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common since the predictor is infinite dimensional some form of dimension reduction is essential there are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases such as b spline wavelet in the present paper we estimate the functional linear model by using the functional sufficient dimension reduction fsdr basis compared to the existing methods the proposed method is appealing because the fsdr basis is related to both the functional predictor and the response variable whereas the functional principal component basis is only related to the functional predictor and the fixed basis b spline wavelet is independent from both the functional predictor and the response variable our techniques involve methods for giving a new expansion for the predictor for giving a specific expression for the regression parameter function for estimating the fsdr space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples numerical studies including both simulation studies and applications on real life data are presented to demonstrate the accuracy of the proposed method

unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier the data functions that are studied in the course of functional data analysis are assembled from discrete data and the level of smoothing that is used is generally that which is appropriate for accurate approximation of the conceptually smooth functions that were not actually observed existing literature shows that this approach is effective and even optimal when using functional data methods for prediction or hypothesis testing however in the present paper we show that this approach is not effective in classification problems there a useful rule of thumb is that undersmoothing is often desirable but there are several surprising qualifications to that approach first the effect of smoothing the training data can be more significant than that of smoothing the new data set to be classified second undersmoothing is not always the right approach and in fact in some cases using a relatively large bandwidth can be more effective and third these perverse results are the consequence of very unusual properties of error rates expressed as functions of smoothing parameters for example the orders of magnitude of optimal smoothing parameter choices depend on the signs and sizes of terms in an expansion of error rate and those signs and sizes can vary dramatically from one setting to another even for the same classifier

functional clustering by bayesian wavelet methods summary we propose a nonparametric bayes wavelet model for clustering of functional data the wavelet based methodology is aimed at the resolution of generic global and local features during clustering and is suitable for clustering high dimensional data based on the dirichlet process the nonparametric bayes model extends the scope of traditional bayes wavelet methods to functional clustering and allows the elicitation of prior belief about the regularity of the functions and the number of clusters by suitably mixing the dirichlet processes posterior inference is carried out by gibbs sampling with conjugate priors which makes the computation straightforward we use simulated as well as real data sets to illustrate the suitability of the approach over other alternatives

classification of non parametric regression functions in longitudinal data models summary we investigate a longitudinal data model with non parametric regression functions that may vary across the observed individuals in a variety of applications it is natural to impose a group structure on the regression curves specifically we may suppose that the observed individuals can be grouped into a number of classes whose members all share the same regression function we develop a statistical procedure to estimate the unknown group structure from the data moreover we derive the asymptotic properties of the procedure and investigate its finite sample performance by means of a simulation study and a real data example

correlation based functional clustering via subspace projection a correlation based functional clustering method is proposed for grouping curves with similar shapes a correlation between two random functions defined through the functional inner product is used as a similarity measure curves with similar shapes are embedded in the cluster subspace spanned by a mean shape function and eigenfunctions of the covariance kernel the cluster membership prediction for each curve attempts to maximize the functional correlation between the observed and predicted curves via shape standardization and subspace projection among all possible clusters the proposed method accounts for shape differentials through the functional multiplicative random effects shape function model for each cluster which regards random scales and intercept shifts as a nuisance a consistent estimate is proposed for the random scale effect whose sample variance estimate is also consistent the derived identifiability conditions for the clustering procedure unravel the predictability of cluster membersh

frechet integration and adaptive metric selection for interpretable covariances of multivariate functional data summary for multivariate functional data recorded for a sample of subjects on a common domain one 10 is often interested in the covariance between pairs of the component functions extending the notion of a covariance matrix for multivariate data to the functional case a straightforward approach is to integrate the pointwise covariance matrices over the functional time domain we generalize this approach by defining the frintegral which depends on the metric chosen for the space of covariance matrices and demonstrate that the ordinary integration is a special 15 case when the frobenius metric is used as the space of covariance matrices is nonlinear we propose a class of power metrics as alternatives to the frobenius metric for any such power metric the calculation of frintegrals is equivalent to transforming the covariance matrices with the chosen power applying the classical riemann integral to the transformed matrices and finally applying the inverse transformation to return to the original scale we also propose data 20 adaptive metric selection with respect to a user specified target criterion for example fastest decline of the eigenvalues establish consistency of the proposed procedures and demonstrate their effectiveness in a simulation the proposed functional covariance approach through fr echet integration is illustrated in a comparison of connectivity between brain voxels for normal subjects and alzheimer s patients based on fmri data 25

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

classification of functional fragments by regularized linear classifiers with domain selection we consider classification of functional data into two groups by linear classifiers based on one dimensional projections of functions we reformulate the task of finding the best classifier as an optimization problem and solve it by the conjugate gradient method with early stopping the principal component method and the ridge method we study the empirical version with finite training samples consisting of incomplete functions observed on different subsets of the domain and show that the optimal possibly zero misclassification probability can be achieved in the limit along a possibly nonconvergent empirical regularization path we propose a domain extension and selection procedure that finds the best domain beyond the common observation domain of all curves in a simulation study we compare the different regularization methods and investigate the performance of domain selection our method is illustrated on a medical dataset where we observe a substantial improvement of classification accuracy due to domain extension

power spectrum of mass and activity fluctuations in a sandpile we consider a directed abelian sandpile on a strip of size 2 ifmmode times else texttimes fi n driven by adding a grain randomly at the left boundary after every t timesteps we establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary base representation of the position of a random walker on a ring of size 3 n we find that while the fluctuations of mass have a power spectrum that varies as 1 f for frequencies in the range 3 ensuremath 2n ensuremath ll f ensuremath ll 1 t the activity fluctuations in the same frequency range have a power spectrum that is linear in f

on a conjecture of ilmonen haukkanen and merikoski concerning the smallest eigenvalues of certain gcd related matrices abstract let k n be the set of all n n lower triangular 0 1 matrices with each diagonal element equal to 1 l n y y t y k n and let c n be the minimum of the smallest eigenvalue of y y t as y goes through k n the ilmonen haukkanen merikoski conjecture the ihm conjecture states that c n is equal to the smallest eigenvalue of y 0 y 0 t where y 0 k n with y 0 i j 1 1 i j 2 for i j in this paper we present a proof of this conjecture in our proof we use an inequality for spectral radii of nonnegative matrices

on meet and join matrices associated with incidence functions we study recently meet matrices on meet semilattices as an abstract generalization of greatest common divisor gcd matrices analogously in this paper we consider join matrices on lattices as an abstract generalization of least common multiple lcm matrices a formula for the determinant of join matrices on join closed sets bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join closed sets are given the concept of a semi multiplicative function gives us formulae for meet matrices on join closed sets and join matrices on meet closed sets finally we show what new the study of meet and join matrices contributes to the usual gcd and lcm matrices

defining probability density for a distribution of random functions the notion of probability density for a random function is not as straightforward as in finite dimensional cases while a probability density function generally does not exist for functional data we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis this leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension this density approximation is estimable readily from data it accurately represents in a monotone way key features of small ball approximations to density our results on estimators of the densities of principal component scores are also of independent interest they reveal interesting shape differences that have not previously been considered the statistical implications of these results and properties are identified and discussed and practical ramifications are illustrated in numerical work

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

on greatest common divisor matrices and their applications abstract let s x 1 x 2 x n be a set of distinct positive integers the n n matrix s s ij where s ij x i x j the greatest common divisor of x i and x j is called the greatest common divisor gcd matrix on s we study the structure of a gcd matrix and obtain interesting relations between its determinant euler s totient function and moebius function we also determine some arithmetic progressions related to gcd matrices then we generalize the results to general partially ordered sets and show a variety of applications

total nonnegativity of gcd matrices and kernels abstract let x x 1 x n be a vector of distinct positive integers the n n matrix s s x gcd x i x j i j 1 n where gcd x i x j denotes the greatest common divisor of x i and x j is called the greatest common divisor gcd matrix on x by a surprising result of beslin and ligh 1989 7 all gcd matrices are positive definite in this paper we completely characterize the gcd matrices satisfying the stronger property of being totally nonnegative tn or totally positive tp as we show a gcd matrix is never tp when n 3 and is tn if and only if it is t n 2 i e all its 2 2 minors are nonnegative we next demonstrate that a gcd matrix is tn 2 if and only if the exponents of each prime divisor in the prime factorization of the x i s form a monotonic sequence reformulated in the language of kernels our results characterize the subsets of integers over which the kernel k x y gcd x y is totally nonnegative the proofs of our characterizations depend on gantmacher and krein s notion of a green s matrix we conclude by showing that a gcd matrix is tn if and only if it is a green s matrix as a consequence we obtain explicit formulas for all the minors and for the inverse of totally nonnegative gcd matrices

1 f α noise from self organized critical models with uniform driving using the well known olami feder christensen model as our paradigm we show how to modify uniform driven self organized critical models to generate 1 f ensuremath alpha noise this model can reproduce all the main features of 1 f ensuremath alpha noise 1 ensuremath alpha is close to one and does not depend on the dimension of the system 2 the 1 f ensuremath alpha behavior is found for very low frequencies 3 the spatial correlations do not obey a power law that proves that spatially extended systems based on activation deactivation processes do not have to be point driven to produce 1 f ensuremath alpha noise the essential ingredient is a local memory of the activation deactivation process

most predictive design points for functional data predictors we suggest a way of reducing the very high dimension of a functional predictor x to a low number of dimensions chosen so as to give the best predictive performance specifically if x is observed on a fine grid of design points t 1 t r we propose a method for choosing a small subset of these say t i1 t ik to optimize the prediction of a response variable y the values t ij are referred to as the most predictive design points or covariates for a given value of k and are computed using information contained in a set of independent observations x i y i of x y the algorithm is based on local linear regression and calculations can be accelerated using linear regression to preselect the design points boosting can be employed to further improve the predictive performance we illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings

on the use of reproducing kernel hilbert spaces in functional classification the hajek feldman dichotomy establishes that two gaussian measures are either mutually absolutely continuous with respect to each other and hence there is a radon nikodym density for each measure

linear processes that produce 1 f or flicker noise 1 f noise and flicker noises i e the class of 1 mathit f mathrm ensuremath alpha noises with 0 51 5 are as ubiquitous as they are mysterious several physical mechanisms to generate 1 f noise have been devised and most of them try to obtain a broad nearly flat distribution of relaxation times which would then yield a 1 f spectrum however they are all very specialized and none of them addresses the question of the apparent universality of this noise while they all fail in some respect i show here that the power spectral density of a relaxing linear system driven by white noise is determined by the eigenvalue density of the linear operator associated with the system i also show that the eigenvalue densities of linear operators that describe diffusion and transport lead to 1 f or flicker noise using the concepts developed in the paper and a rough approximation of transport in a resistor i derive the hooge formula for the spectrum of conductance fluctuations

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

multiscale clustering of nonparametric regression curves abstract in a wide range of modern applications one observes a large number of time series rather than only a single one it is often natural to suppose that there is some group structure in the observed time series when each time series is modeled by a nonparametric regression equation one may in particular assume that the observed time series can be partitioned into a small number of groups whose members share the same nonparametric regression function we develop a bandwidth free clustering method to estimate the unknown group structure from the data more precisely speaking we construct multiscale estimators of the unknown groups and their unknown number which are free of classical bandwidth or smoothing parameters in the theoretical part of the paper we analyze the statistical properties of our estimators our theoretical results are derived under general conditions which allow the data to be dependent both in time series direction and across different time series the technical analysis of the paper is complemented by simulated and real data examples

generalized eigenvalue problems for meet and join matrices on semilattices abstract we study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices we provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions the applied methodology is flexible and it is shown in the case of gcd and lcm matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice these results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices which we briefly consider as well we investigate the effectiveness of the obtained bounds for select examples involving number theoretical lattices

joint modeling and clustering paired generalized longitudinal trajectories with application to cocaine abuse treatment data in a cocaine dependence treatment study we have paired binary longitudinal trajectories that record the cocaine use patterns of each patient before and after a treatment to better understand the drug using behaviors among the patients we propose a general framework based on functional data analysis to jointly model and cluster these paired non gaussian longitudinal trajectories our approach assumes that the response variables follow distributions from the exponential family with the canonical parameters determined by some latent gaussian processes to reduce the dimensionality of the latent processes we express them by a truncated karhunen loeve kl expansion allowing the mean and covariance functions to be different across clusters we further represent the mean and eigenfunctions functions by flexible spline bases and determine the orders of the truncated kl expansions using data driven methods by treating the cluster membership as a missing value we cluster the cocaine use trajectories by a l

passive motion in dynamical disorder as a model for stock market prices abstract a new model for stock price fluctuations is proposed based upon an analogy with the motion of tracers in gaussian random fields as used in turbulent dispersion models and in studies of transport in dynamically disordered media analytical and numerical results for this model in a special limiting case of a single scale field show characteristics similar to those found in empirical studies of stock market data specifically short term returns have a non gaussian distribution with super diffusive volatility assuming a power law decay of the time correlation of the disorder the returns correlation decays rapidly but the correlation function of the absolute returns exhibits a slow power law decay the returns distribution converges towards gaussian over long times some important characteristics of empirical data are not however reproduced by the model notably the scaling of tails of the cumulative distribution function of returns implied volatilities for options pricing are found by numerical simulation

shape descriptors for classification of functional data curve discrimination is an important task in engineering and other sciences we propose several shape descriptors for classifying functional data inspired by form analysis from the image analysis field statistical moments coefficients of the components of independent component analysis and two mathematical morphology descriptors morphological covariance and spatial size distributions these are applied to three problems an artificial problem a speech recognition problem and a biomechanical application shape descriptors are compared with other methods in the literature with better or similar performance obtaining

strong consistency of k means clustering a random sample is divided into the k clusters that minimise the within cluster sum of squares conditions are found that ensure the almost sure convergence as the sample size increases of the set of means of the k clusters the result is proved for a more general clustering criterion

functional classification in hilbert spaces let x be a random variable taking values in a separable hilbert space x with label y spl isin 0 1 we establish universal weak consistency of a nearest neighbor type classifier based on n independent copies x sub i y sub i of the pair x y extending the classical result of stone to infinite dimensional hilbert spaces under a mild condition on the distribution of x we also prove strong consistency we reduce the infinite dimension of x by considering only the first d coefficients of a fourier series expansion of each x sub i and then we perform k nearest neighbor classification in spl ropf sup d both the dimension and the number of neighbors are automatically selected from the data using a simple data splitting device an application of this technique to a signal discrimination problem involving speech recordings is presented

componentwise classification and clustering of functional data the infinite dimension of functional data can challenge conventional methods for classification and clustering a variety of techniques have been introduced to address this problem particularly in the case of prediction but the structural models that they involve can be too inaccurate or too abstract or too difficult to interpret for practitioners in this paper we develop approaches to adaptively choose components enabling classification and clustering to be reduced to finite dimensional problems we explore and discuss properties of these methodologies our techniques involve methods for estimating classifier error rate and cluster tightness and for choosing both the number of components and their locations to optimize these quantities a major attraction of this approach is that it allows identification of parts of the function domain that convey important information for classification and clustering it also permits us to determine regions that are relevant to one of these analyses but not the other copyright 2012 oxford university press

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

transport in randomly fluctuating spatially periodic potentials abstract the motion of overdamped particles in a one dimensional spatially periodic potential is considered the potential is also randomly fluctuating in time due to multiplicative colored noise terms and has a deterministic tilt numerical simulations show two distinct parameter regimes corresponding to free running near deterministic particles and particles which are trapped in local minima of the potential with intermittent escape flights perturbation and asymptotic methods are developed to understand the drift velocity and diffusion coefficient in each parameter regime

deconvolution when classifying noisy data involving transformations in the present study we consider the problem of classifying spatial data distorted by a linear transformation or convolution and contaminated by additive random noise in this setting we show that classifier performance can be improved if we carefully invert the data before the classifier is applied however the inverse transformation is not constructed so as to recover the original signal and in fact we show that taking the latter approach is generally inadvisable we introduce a fully data driven procedure based on cross validation and use several classifiers to illustrate numerical properties of our approach theoretical arguments are given in support of our claims our procedure is applied to data generated by light detection and ranging lidar technology where we improve on earlier approaches to classifying aerosols this article has supplementary materials online

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

clustering for sparsely sampled functional data we develop a flexible model based procedure for clustering functional data the technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points in addition to producing final cluster assignments the procedure generates predictions and confidence intervals for missing portions of curves our approach also provides many useful tools for evaluating the resulting models clustering can be assessed visually via low dimensional representations of the curves and the regions of greatest separation between clusters can be determined using a discriminant function finally we extend the model to handle multiple functional and finite dimensional covariates and show how it can be applied to standard finite dimensional clustering problems involving missing data

exactly solvable model of continuous stationary 1 f noise an exactly solvable model generating a continuous random process with a 1 f power spectrum is presented examples of such processes include the angular phase speed of trajectories near stable equilibrium points in two dimensional dynamical systems perturbed by colored gaussian noise an exact formula giving the correlation function of the 1 f noise in terms of the correlation of the perturbing colored noises is derived and used to show that the 1 f spectrum is found in a wide variety of cases the 1 f noise is non gaussian as demonstrated by calculating its one time probability distribution function numerical simulations confirm and extend the theoretical results

on meet matrices on posets abstract we consider meet matrices on posets as an abstract generalization of greatest common divisor gcd matrices some of the most important properties of gcd matrices are presented in terms of meet matrices

comparing the shapes of regression functions summary does a regression function follow a specified shape do two regression functions have the same shape how can regression functions be grouped based on shape these questions can occur when investigating monotonicity when counting local maxima or when studying variation in families of curves one can address these questions by considering the rank correlation coefficient between two functions this correlation is a generalisation of the rank correlation between two finite sets of numbers and is equal to one if and only if the two functions have the same shape a sample rank correlation based on smoothed estimates of the regression functions consistently estimates the true correlation this sample rank correlation can be used as a measure of similarity between functions in cluster analysis and as a measure of monotonicity or modality

exact solution for single scale gaussian random transport a quadrature expression is derived for the probability density function of passive tracers advected from a point by a one dimensional single scale gaussian velocity field the effect of trapping on the tracer moments and the lagrangian velocity variance is explicitly demonstrated

optimal designs for longitudinal and functional data summary we propose novel optimal designs for longitudinal data for the common situation where the resources for longitudinal data collection are limited by determining the optimal locations in time where measurements should be taken as for all optimal designs some prior information is needed to implement the optimal designs proposed we demonstrate that this prior information may come from a pilot longitudinal study that has irregularly measured and noisy measurements where for each subject one has available a small random number of repeated measurements that are randomly located on the domain a second possibility of interest is that a pilot study consists of densely measured functional data and one intends to take only a few measurements at strategically placed locations in the domain for the future collection of similar data we construct optimal designs by targeting two criteria optimal designs to recover the unknown underlying smooth random trajectory for each subject from a few optimally placed measurements such that squared prediction errors are minimized optimal designs that minimize prediction errors for functional linear regression with functional or longitudinal predictors and scalar responses again from a few optimally placed measurements the optimal designs proposed address the need for sparse data collection when planning longitudinal studies by taking advantage of the close connections between longitudinal and functional data analysis we demonstrate in simulations that the designs perform considerably better than randomly chosen design points and include a motivating data example from the baltimore longitudinal study of aging the designs are shown to have an asymptotic optimality property

greatest common divisor matrices abstract let s x 1 x 2 x n be a set of distinct positive integers then n n matrix s s ij where s ij x i x j the greatest common divisor of x i and x j is call the greatest common divisor gcd matrix on s we initiate the study of gcd matrices in the direction of their structure determinant and arithmetic in z n several open problems are posed

functional linear discriminant analysis for irregularly sampled curves we introduce a technique for extending the classical method of linear discriminant analysis lda to data sets where the predictor variables are curves or functions this procedure which we call functional linear discriminant analysis flda is particularly useful when only fragments of the curves are observed all the techniques associated with lda can be extended for use with flda in particular flda can be used to produce classifications on new test curves give an estimate of the discriminant function between classes and provide a one or two dimensional pictorial representation of a set of curves we also extend this procedure to provide generalizations of quadratic and regularized discriminant analysis

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

analysis of long period variable stars with nonparametric tests for trend detection in astronomy the study of variable stars that is stars characterized by showing significant variation in their brightness over time has made crucial contributions to our understanding of many phenomena from stellar birth and evolution to the calibration of the extragalactic distance scale in this article we develop a method for analyzing multiple pseudo periodic time series with the goal of detecting temporal trends in their periods we allow for nonstationary noise and for clustering among the various time series we apply this method to the long standing astronomical problem of identifying variable stars whose regular brightness fluctuations have periods that change over time the results of our analysis show that such changes can be substantial raising the possibility that astronomers estimates of galactic distances can be refined two significant contributions of our approach relative to existing methods for this problem are as follows 1 the method is nonparametric making minimal assumpt

on eigenvalues of meet and join matrices associated with incidence functions abstract let p be a locally finite meet semilattice let s x 1 x 2 x n x i x j i j be a finite subset of p and let f be a complex valued function on p then the n n matrix s f where s f ij f x i x j is called the meet matrix on s with respect to f the join matrix on s with respect to f is defined dually on a locally finite join semilattice in this paper we give lower bounds for the smallest eigenvalues of certain positive definite meet matrices with respect to f on any set s we also estimate eigenvalues of meet matrices respect to any f on meet closed set s and with respect to semimultiplicative f on join closed set s the same is carried out dually for join matrices

clustering high dimensional time series based on parallelism this article considers the problem of clustering high dimensional time series based on trend parallelism the underlying process is modeled as a nonparametric trend function contaminated by locally stationary errors a special class of nonstationary processes for each group where the parallelism holds i semiparametrically estimate its representative trend function and vertical shifts of group members and establish their central limit theorems an information criterion consisting of in group similarities and number of groups is then proposed for the purpose of clustering i prove its theoretical consistency and propose a splitting coalescence algorithm to reduce the computational burden in practice the method is illustrated by both simulation and a real data example

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

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

cats clustering after transformation and smoothing cats clustering after transformation and smoothing is a technique for nonparametrically estimating and clustering a large number of curves our motivating example is a genetic microarray experiment but the method is very general the method includes transformation and smoothing multiple curves multiple nonparametric testing for screening out flat curves clustering curves with similar shape and nonparametrically inferring the clustering estimation error rate

multilevel functional clustering analysis in this article we investigate clustering methods for multilevel functional data which consist of repeated random functions observed for a large number of units e g genes at multiple subunits e g bacteria types to describe the within and between variability induced by the hierarchical structure in the data we take a multilevel functional principal component analysis mfpca approach we develop and compare a hard clustering method applied to the scores derived from the mfpca and a soft clustering method using an mfpca decomposition in a simulation study we assess the estimation accuracy of the clustering membership and the cluster patterns under a series of settings small versus moderate number of time points various noise levels and varying number of subunits per unit we demonstrate the applicability of the clustering analysis to a real data set consisting of expression profiles from genes activated by immunity system cells prevalent response patterns are identified by clustering the expression profiles using our multilevel clustering analysis

achieving near perfect classification for functional data summary we show that in functional data classification problems perfect asymptotic classification is often possible making use of the intrinsic very high dimensional nature of functional data this performance is often achieved by linear methods which are optimal in important cases these results point to a marked contrast between classification for functional data and its counterpart in conventional multivariate analysis where the dimension is kept fixed as the sample size diverges in the latter setting linear methods can sometimes be quite inefficient and there are no prospects for asymptotically perfect classification except in pathological cases where for example a variance vanishes by way of contrast in finite samples of functional data good performance can be achieved by truncated versions of linear methods truncation can be implemented by partial least squares or projection onto a finite number of principal components using in both cases cross validation to determine the truncation point we establish consistency of the cross validation procedure

variable selection for the prediction of c 0 1 valued autoregressive processes using reproducing kernel hilbert spaces a model for the prediction of functional time series is introduced where observations are assumed to be continuous random functions we model the dependence of the data with a nonstandard autoregr

classification using censored functional data we consider classification of functional data when the training curves are not observed on the same interval different types of classifier are suggested one of which involves a new curve extension procedure our approach enables us to exploit the information contained in the endpoints of these intervals by incorporating it in an explicit but flexible way we study asymptotic properties of our classifiers and show that in a variety of settings they can even produce asymptotically perfect classification the performance of our techniques is illustrated in applications to real and simulated data supplementary materials for this article are available online

functional clustering and identifying substructures of longitudinal data summary a functional clustering fc method k centres fc for longitudinal data is proposed the k centres fc approach accounts for both the means and the modes of variation differentials between clusters by predicting cluster membership with a reclassification step the cluster membership predictions are based on a non parametric random effect model of the truncated karhunen loeve expansion coupled with a non parametric iterative mean and covariance updating scheme we show that under the identifiability conditions derived the k centres fc method proposed can greatly improve cluster quality as compared with conventional clustering algorithms moreover by exploring the mean and covariance functions of each cluster thek centres fc method provides an additional insight into cluster structures which facilitates functional cluster analysis practical performance of the k centres fc method is demonstrated through simulation studies and data applications including growth curve and gene expression profile data

modeling and forecasting daily electricity load curves a hybrid approach we propose a hybrid approach for the modeling and the short term forecasting of electricity loads two building blocks of our approach are 1 modeling the overall trend and seasonality by fitting a generalized additive model to the weekly averages of the load and 2 modeling the dependence structure across consecutive daily loads via curve linear regression for the latter a new methodology is proposed for linear regression with both curve response and curve regressors the key idea behind the proposed methodology is dimension reduction based on a singular value decomposition in a hilbert space which reduces the curve regression problem to several ordinary i e scalar linear regression problems we illustrate the hybrid method using french electricity loads between 1996 and 2009 on which we also compare our method with other available models including the electricite de france operational model supplementary materials for this article are available online

clustering multivariate functional data with phase variation when functional data come as multiple curves per subject characterizing the source of variations is not a trivial problem the complexity of the problem goes deeper when there is phase variation in addition to amplitude variation we consider clustering problem with multivariate functional data that have phase variations among the functional variables we propose a conditional subject specific warping framework in order to extract relevant features for clustering using multivariate growth curves of various parts of the body as a motivating example we demonstrate the effectiveness of the proposed approach the found clusters have individuals who show different relative growth patterns among different parts of the body

bayesian curve classification using wavelets we propose classification models for binary and multicategory data where the predictor is a random function we use bayesian modeling with wavelet basis functions that have nice approximation properties over a large class of functional spaces and can accommodate a wide variety of functional forms observed in real life applications we develop an unified hierarchical model to encompass both the adaptive wavelet based function estimation model and the logistic classification model we couple together these two models are to borrow strengths from each other in a unified hierarchical framework the use of gibbs sampling with conjugate priors for posterior inference makes the method computationally feasible we compare the performance of the proposed model with other classification methods such as the existing naive plug in methods by analyzing simulated and real data sets

on the use of fuzzy constraints in semisupervised clustering this paper introduces fuzzy hss a semisupervised hierarchical clustering approach that uses fuzzy instance level constraints these constraints are external information on the shape of fuzzy must link and fuzzy cannot link restrictions they allow uncertainty when indicating whether two instances of a dataset belong to the same group fuzzy must link constraints give a degree of belief of two instances belonging to the same group analogously fuzzy cannot link constraints indicate the degree of belief of two instances not belonging to the same group these constraints have been introduced in a hierarchical clustering process allowing us to obtain the optimal number of groups in a dendrogram when the number of clusters is not known the optimal amount of constraints needed in the process is determined by means of fuzzy entropy an extensive experimental study is provided by comparing this fuzzy semisupervised approach with classic unsupervised methods as well as a crisp semisupervised alternative