Classification Of Functional Data Research Articles

Spatial depth-based classification for functional data

We enlarge the number of available functional depths by introducing the kernelized functional spatial depth (KFSD). KFSD is a local-oriented and kernel-based version of the recently proposed functional spatial depth (FSD) that may be useful for studying functional samples that require an analysis at a local level. In addition, we consider supervised functional classification problems, focusing on cases in which the differences between groups are not extremely clear-cut or the data may contain outlying curves. We perform classification by means of some available robust methods that involve the use of a given functional depth, including FSD and KFSD, among others. We use the functional \textit{k}-nearest neighbor classifier as a benchmark procedure. The results of a simulation study indicate that the KFSD-based classification approach leads to good results. Finally, we consider two real classification problems, obtaining results that are consistent with the findings observed with simulated curves.

Mixtures of spatial spline regressions for clustering and classification

Classification and clustering of functional data arise in many areas of modern research. Currently, techniques for performing such tasks have concentrated on applications to univariate functions. Such techniques can be extended to the domain of classifying and clustering bivariate functions (i.e. surfaces) over rectangular domains. This is achieved by combining the current techniques in spatial spline regression (SSR) with finite mixture models and mixed-effects models. As a result, three novel techniques have been developed: spatial spline mixed models (SSMM) for fitting populations of surfaces, mixtures of SSR (MSSR) for clustering surfaces, and MSSR discriminant analysis (MSSRDA) for classification of surfaces. Through simulations and applications to problems in handwritten character recognition, it is shown that SSMM, MSSR, and MSSRDA are effective in performing their desired tasks. It is also shown that in the context of handwritten character recognition, MSSR and MSSRDA are comparable to established methods, and are able to outperform competing approaches in missing-data situations.

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.

A survey of functional principal component analysis

Advances in data collection and storage have tremendously increased the presence of functional data, whose graphical representations are curves, images or shapes. As a new area of statistics, functional data analysis extends existing methodologies and theories from the realms of functional analysis, generalized linear model, multivariate data analysis, nonparametric statistics, regression models and many others. From both methodological and practical viewpoints, this paper provides a review of functional principal component analysis, and its use in explanatory analysis, modeling and forecasting, and classification of functional data.

Model-based functional mixture discriminant analysis with hidden process regression for curve classification

In this paper, we study the modeling and the classification of functional data presenting regime changes over time. We propose a new model-based functional mixture discriminant analysis approach based on a specific hidden process regression model that governs the regime changes over time. Our approach is particularly adapted to handle the problem of complex-shaped classes of curves, where each class is potentially composed of several sub-classes, and to deal with the regime changes within each homogeneous sub-class. The proposed model explicitly integrates the heterogeneity of each class of curves via a mixture model formulation, and the regime changes within each sub-class through a hidden logistic process. Each class of complex-shaped curves is modeled by a finite number of homogeneous clusters, each of them being decomposed into several regimes. The model parameters of each class are learned by maximizing the observed-data log-likelihood by using a dedicated expectation–maximization (EM) algorithm. Comparisons are performed with alternative curve classification approaches, including functional linear discriminant analysis and functional mixture discriminant analysis with polynomial regression mixtures and spline regression mixtures. Results obtained on simulated data and real data show that the proposed approach outperforms the alternative approaches in terms of discrimination, and significantly improves the curves approximation.

Generative models for functional data using phase and amplitude separation

Constructing generative models for functional observations is an important task in statistical functional analysis. In general, functional data contains both phase (or x or horizontal) and amplitude (or y or vertical) variability. Traditional methods often ignore the phase variability and focus solely on the amplitude variation, using cross-sectional techniques such as fPCA for dimensional reduction and data modeling. Ignoring phase variability leads to a loss of structure in the data and inefficiency in data models. This paper presents an approach that relies on separating the phase (x-axis) and amplitude (y-axis), then modeling these components using joint distributions. This separation, in turn, is performed using a technique called elastic shape analysis of curves that involves a new mathematical representation of functional data. Then, using individual fPCAs, one each for phase and amplitude components, it imposes joint probability models on principal coefficients of these components while respecting the nonlinear geometry of the phase representation space. These ideas are demonstrated using random sampling, for models estimated from simulated and real datasets, and show their superiority over models that ignore phase-amplitude separation. Furthermore, the generative models are applied to classification of functional data and achieve high performance in applications involving SONAR signals of underwater objects, handwritten signatures, and periodic body movements recorded by smart phones.

Robust Classification of Functional and Quantitative Image Data Using Functional Mixed Models

This article 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.

Supervised classification for functional data: A weighted distance approach

A natural methodology for discriminating functional data is based on the distances from the observation or its derivatives to group representative functions (usually the mean) or their derivatives. It is proposed to use a combination of these distances for supervised classification. Simulation studies show that this procedure performs very well, resulting in smaller testing classification errors. Applications to real data show that this technique behaves as well as–and in some cases better than–existing supervised classification methods for functions.

Achieving near Perfect Classification for Functional Data

SummaryWe 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.

Using mathematical morphology for unsupervised classification of functional data

This paper is concerned with the unsupervised classification of functional data by using mathematical morphology. Different morphological operators are used to extract relevant structures of the functions (considered as sets through their subgraph representations). These operators can be considered as preprocessing tools whose outputs are also functional data. We explore some dissimilarity measures and clustering methods for the classification of the transformed data. Our approach is illustrated through a detailed analysis of two data sets. These techniques, which have mainly been used in image processing, provide a flexible and robust toolbox for improving the results in unsupervised functional data classification.

Supervised Classification for a Family of Gaussian Functional Models

Abstract. In the framework of supervised classification (discrimination) for functional data, it is shown that the optimal classification rule can be explicitly obtained for a class of Gaussian processes with ‘triangular’ covariance functions. This explicit knowledge has two practical consequences. First, the consistency of the well‐known nearest neighbours classifier (which is not guaranteed in the problems with functional data) is established for the indicated class of processes. Second, and more important, parametric and non‐parametric plug‐in classifiers can be obtained by estimating the unknown elements in the optimal rule. The performance of these new plug‐in classifiers is checked, with positive results, through a simulation study and a real data example.

A clustering algorithm for spatially dependent functional data

We propose a new algorithm for unsupervised classification of functional data – indexed by the sites of a spatial lattice – that exploits spatial dependence by repeatedly generating random connectivity maps and by clustering, at each iteration, local representatives of neighbouring functional data. The algorithm supports different implementations – both with respect to the treatment of spatial dependence and to the employed clustering technique – consistent with different contexts arising in applications. The performance of the algorithm is tested on synthetic data. We then illustrate a case study where irradiance data are analyzed to investigate the exploitability of solar energy in different areas of the planet.

Functional clustering and alignment methods with applications

We consider the issue of classification of functional data and, in particular, we deal with the problem of curve clustering when curves are misaligned. In the proposed setting, we aim at jointly aligning and clustering the curves, via the solution of an optimization problem. We describe an iterative procedure for the solution of the optimization problem, and we detail two alternative specifications of the procedure, a k-mean version and a k-medoid version. We illustrate via applications to real data the robustness of the alignment and clustering procedure under the different specifications.

A Bayesian Hierarchical Model for Classification with Selection of Functional Predictors

In functional data classification, functional observations are often contaminated by various systematic effects, such as random batch effects caused by device artifacts, or fixed effects caused by sample-related factors. These effects may lead to classification bias and thus should not be neglected. Another issue of concern is the selection of functions when predictors consist of multiple functions, some of which may be redundant. The above issues arise in a real data application where we use fluorescence spectroscopy to detect cervical precancer. In this article, we propose a Bayesian hierarchical model that takes into account random batch effects and selects effective functions among multiple functional predictors. Fixed effects or predictors in nonfunctional form are also included in the model. The dimension of the functional data is reduced through orthonormal basis expansion or functional principal components. For posterior sampling, we use a hybrid Metropolis-Hastings/Gibbs sampler, which suffers slow mixing. An evolutionary Monte Carlo algorithm is applied to improve the mixing. Simulation and real data application show that the proposed model provides accurate selection of functional predictors as well as good classification.

Barycentric Discriminant Analysis (BDA): a new pattern recognition classifier that identifies voxels and regions of interest relevant for classification of functional brain imaging data.

Barycentric Discriminant Analysis (BDA): a new pattern recognition classifier that identifies voxels and regions of interest relevant for classification of functional brain imaging data.

Penalized optimal scoring for the classification of multi-dimensional functional data

Many fields of research need to classify individual systems based on one or more data series, which are obtained by sampling an unknown continuous curve with noise. In other words, the underlying process is an unknown function which the observed variables represent only imperfectly. Although functional logistic regression has many attractive features for this classification problem, this method is applicable only when the number of individuals to be classified (or available to estimate the model) is large compared to the number of curves sampled per individual. To overcome this limitation, we use penalized optimal scoring to construct a new method for the classification of multi-dimensional functional data. The proposed method consists of two stages. First, the series of observed discrete values available for each individual are expressed as a set of continuous curves. Next, the penalized optimal scoring model is estimated on the basis of these curves. A similar penalized optimal scoring method was described in my previous work, but this model is not suitable for the analysis of continuous functions. In this paper we adopt a Gaussian kernel approach to extend the previous model. The high accuracy of the new method is demonstrated on Monte Carlo simulations, and used to predict defaulting firms on the Japanese Stock Exchange.

Information-theoretic feature selection for functional data classification

The classification of functional or high-dimensional data requires to select a reduced subset of features among the initial set, both to help fighting the curse of dimensionality and to help interpreting the problem and the model. The mutual information criterion may be used in that context, but it suffers from the difficulty of its estimation through a finite set of samples. Efficient estimators are not designed specifically to be applied in a classification context, and thus suffer from further drawbacks and difficulties. This paper presents an estimator of mutual information that is specifically designed for classification tasks, including multi-class ones. It is combined to a recently published stopping criterion in a traditional forward feature selection procedure. Experiments on both traditional benchmarks and on an industrial functional classification problem show the added value of this estimator.

On depth measures and dual statistics. A methodology for dealing with general data

A general depth measure, based on the use of one-dimensional linear continuous projections, is proposed. The applicability of this idea in different statistical setups (including inference in functional data analysis, image analysis and classification) is discussed. A special emphasis is made on the possible usefulness of this method in some statistical problems where the data are elements of a Banach space. The asymptotic properties of the empirical approximation of the proposed depth measure are investigated. In particular, its asymptotic distribution is obtained through U -statistics techniques. The practical aspects of these ideas are discussed through a small simulation study and a real-data example.

Classification of functional data: A segmentation approach

We suggest a classification and feature extraction method on functional data where the predictor variables are curves. The method, called functional segment discriminant analysis (FSDA), combines the classical linear discriminant analysis and support vector machine. FSDA is particularly useful for irregular functional data, characterized by spatial heterogeneity and local patterns like spikes. FSDA not only reduces the computation and storage burden by using a fraction of the spectrum, but also identifies important predictors and extracts features. FSDA is highly flexible, easy to incorporate information from other data sources and/or prior knowledge from the investigators. We apply FSDA to two public domain data sets and discuss the understanding developed from the study.

Theoretical, Statistical, and Practical Perspectives on Pattern-based Classification Approaches to the Analysis of Functional Neuroimaging Data

The goal of pattern-based classification of functional neuroimaging data is to link individual brain activation patterns to the experimental conditions experienced during the scans. These "brain-reading" analyses advance functional neuroimaging on three fronts. From a technical standpoint, pattern-based classifiers overcome fatal f laws in the status quo inferential and exploratory multivariate approaches by combining pattern-based analyses with a direct link to experimental variables. In theoretical terms, the results that emerge from pattern-based classifiers can offer insight into the nature of neural representations. This shifts the emphasis in functional neuroimaging studies away from localizing brain activity toward understanding how patterns of brain activity encode information. From a practical point of view, pattern-based classifiers are already well established and understood in many areas of cognitive science. These tools are familiar to many researchers and provide a quantitatively sound and qualitatively satisfying answer to most questions addressed in functional neuroimaging studies. Here, we examine the theoretical, statistical, and practical underpinnings of pattern-based classification approaches to functional neuroimaging analyses. Pattern-based classification analyses are well positioned to become the standard approach to analyzing functional neuroimaging data.