discriminant analysis by gaussian mixtures fisher rao linear discriminant analysis lda is a valuable tool for multigroup classification lda is equivalent to maximum likelihood classification assuming gaussian distributions for each class in this paper we fit gaussian mixtures to each class to facilitate effective classification in non normal settings especially when the classes are clustered low dimensional views are an important by product of lda our new techniques inherit this feature we can control the within class spread of the subclass centres relative to the between class spread our technique for fitting these models permits a natural blend with nonparametric versions of lda

fuzzy clustering for data time arrays with inlier and outlier time trajectories in many knowledge discovery and data mining tasks fuzzy clustering is one of the most common tools for data partitioning in this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories time series sequences are developed in particular by adopting an exploratory approach based on a geometric algebraic formulation of the data time array different kinds of dynamic fuzzy clustering models based on cross sectional and longitudinal aspects are suggested furthermore a modified version of the previous clustering models that can be seen as a generalization of these models is proposed by utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories trajectories with anomalous positions and slopes are present in the dataset in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure s domains and are characterized by strange slopes in these models the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given furthermore some remarks on dynamic three way extensions of a few robust fuzzy clustering models for two way data are suggested demonstrative examples are shown and a comparison assessment based on artificial multivariate time varying data is carried out

functional adaptive model estimation in this article we are interested in modeling the relationship between a scalar y and a functional predictor x t we introduce a highly flexible approach called functional adaptive model estimation fame which extends generalized linear models glms generalized additive models gams and projection pursuit regression ppr to handle functional predictors the fame approach can model any of the standard exponential family of response distributions that are assumed for glm or gam while maintaining the flexibility of ppr for example standard linear or logistic regression with functional predictors as well as far more complicated models can easily be applied using this approach we use a functional principal components decomposition of the predictor functions to aid visualization of the relationship between x t and y we also show how the fame procedure can be extended to deal with multiple functional and standard finite dimensional predictors possibly with missing data we illustrate the fame

efficient estimation of the nonparametric mean and covariance functions for longitudinal and sparse functional data abstractwe consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi likelihood coupling a modification of the local kernel smooth

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

the special modular transformation for polycnoidal waves of the korteweg de vries equation the modular transformation of the riemann theta function is used to show that the implicit dispersion relation for the n polycnoidal waves of the korteweg de vries equation has a countable infinity of branches for n 2 although the transformation also implies that each branch or mode can be written in a countable infinity of ways it is also shown that there is a unique physical representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude unfortunately the small amplitude physical representation is different by a modular transformation from the large amplitude physical representation for a given mode but this difference explains an apparent paradox as described in the text the general modular transformation expresses the theta function in terms of complex wavenumbers phase speeds and coordinates that have no physical relevance to the korteweg de vries equation but it is shown that for n 2 there is a subgroup here dubbed the special modular transformation which gives a real result this subgroup is explicitly constructed for general n and presented as a table for n 2

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

functional linear regression that s interpretable regression models to relate a scalar y to a functional predictor x t are becoming increasingly common work in this area has concentrated on estimating a coefficient function t with y related to x t through f t x t dt regions where t 0 correspond to places where there is a relationship between x t and y alternatively points where t 0 indicate no relationship hence for interpretation purposes it is desirable for a regression procedure to be capable of producing estimates of t that are exactly zero over regions with no apparent relationship and have simple structures over the remaining regions unfortunately most fitting procedures result in an estimate for t that is rarely exactly zero and has unnatural wiggles making the curve hard to interpret in this article we introduce a new approach which uses variable selection ideas applied to various derivatives of t to produce estimates that are both interpretable flexible and accurate we call our method functional linear regression that s interpretable flirti and demonstrate it on simulated and real world data sets in addition non asymptotic theoretical bounds on the estimation error are presented the bounds provide strong theoretical motivation for our approach

generalized linear models with functional predictors summary we present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function the technique is particularly useful when only fragments of each curve have been observed we demonstrate on both simulated and real data sets how this approach can be used to perform linear logistic and censored regression with functional predictors in addition we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors finally we extend the methodology to apply generalized linear models and principal components to standard missing data problems

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

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

estimating the covariance of fragmented and other related types of functional data we consider the problem of estimating the covariance function of functional data which are only observed on a subset of their domain such as fragments observed on small intervals or related types

choosing trajectory and data type when classifying functional data in some problems involving functional data it is desired to undertake prediction or classification before the full trajectory of a function is observed in such cases it is often preferable to suffer somewhat greater error in return for making a decision relatively early the prediction and classification problems can be treated similarly using mean squared prediction error or classification error respectively as the means for quantifying performance so in this paper we focus principally on classification we introduce a method for determining when an early decision can reasonably be made using only part of the trajectory and we show how to use the method to choose among data types our approach is fully nonparametric and no specific model is required properties of error rate are studied as functions of time and data type the effectiveness of the proposed method is illustrated in both theoretical and numerical terms the classification referred to in this paper would be termed supervised classification in machine learning to distinguish it from unsupervised classification or clustering copyright 2012 oxford university press

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

robust classification of functional and quantitative image data using functional mixed models this paper introduces new methods for performing classification of complex high dimensional functional data using the functional mixed model fmm framework the fmm relates a functional response to a set of predictors through functional fixed and random effects which allows it to account for various factors and between function correlations the methods include training and prediction steps in the training steps we train the fmm model by treating class designation as one of the fixed effects and in the prediction steps we classify the new objects using posterior predictive probabilities of class through a bayesian scheme we are able to adjust for factors affecting both the functions and the class designations while the methods can be used in any fmm framework we provide details for two specific bayesian approaches the gaussian wavelet based fmm g wfmm and the robust wavelet based fmm r wfmm both methods perform modeling in the wavelet space which yields parsimonious representations for the functions and can naturally adapt to local features and complex nonstationarities in the functions the r wfmm allows potentially heavier tails for features of the functions indexed by particular wavelet coefficients leading to a down weighting of outliers that makes the method robust to outlying functions or regions of functions the models are applied to a pancreatic cancer mass spectroscopy data set and compared with other recently developed functional classification methods

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

the mahalanobis distance for functional data with applications to classification this article presents a new semidistance for functional observations that generalizes the mahalanobis distance for multivariate datasets the main characteristics of the functional mahalanobis semidistance are shown to illustrate the applicability of this measure of proximity between functional observations new versions of several well known functional classification procedures are developed using the functional mahalanobis semidistance a monte carlo study and the analysis of two real examples indicate that the classification methods used in conjunction with the functional mahalanobis semidistance give better results than other well known functional classification procedures this article has supplementary material online

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

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

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

penalized discriminant analysis fisher s linear discriminant analysis lda is a popular data analytic tool for studying the relationship between a set of predictors and a categorical response in this paper we describe a penalized version of lda it is designed for situations in which there are many highly correlated predictors such as those obtained by discretizing a function or the grey scale values of the pixels in a series of images in cases such as these it is natural efficient and sometimes essential to impose a spatial smoothness constraint on the coefficients both for improved prediction performance and interpretability we cast the classification problem into a regression framework via optimal scoring using this our proposal facilitates the use of any penalized regression technique in the classification setting the technique is illustrated with examples in speech recognition and handwritten character recognition

components and completion of partially observed functional data functional data are traditionally assumed to be observed on the same domain motivated by a data set of heart rate temporal profiles we develop methodology for the analysis of incomplete functional samples where each curve may be observed on a subset of the domain and unobserved elsewhere we formalize this observation regime and develop the fundamental procedures of functional data analysis for this framework estimation of parameters mean and covariance operator and principal component analysis principal scores of a partially observed function cannot be computed directly and we solve this challenging issue by estimating their best predictions as linear functionals of the observed part of the trajectory next we propose a functional completion procedure that recovers the missing part by using the observed part of the curve we construct prediction intervals for principal scores and bands for missing parts of trajectories the prediction problems are seen to be ill posed inverse problems regularization techniques are used to obtain a stable solution a simulation study shows the good performance of our methods we illustrate the methods on the heart rate data and provide practical computational algorithms and theoretical arguments and proofs of all results

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

nonparametric mixed effects models for unequally sampled noisy curves we propose a method of analyzing collections of related curves in which the individual curves are modeled as spline functions with random coefficients the method is applicable when the individual curves are sampled at variable and irregularly spaced points this produces a low rank low frequency approximation to the covariance structure which can be estimated naturally by the em algorithm smooth curves for individual trajectories are constructed as best linear unbiased predictor blup estimates combining data from that individual and the entire collection this framework leads naturally to methods for examining the effects of covariates on the shapes of the curves we use model selection techniques akaike information criterion aic bayesian information criterion bic and cross validation to select the number of breakpoints for the spline approximation we believe that the methodology we propose provides a simple flexible and computationally efficient means of functional data analysis