Classification Of Functional Data Research Articles

Interpretable discriminant analysis for functional data supported on random nonlinear domains with an application to Alzheimer’s disease

Abstract We introduce a novel framework for the classification of functional data supported on nonlinear, and possibly random, manifold domains. The motivating application is the identification of subjects with Alzheimer’s disease from their cortical surface geometry and associated cortical thickness map. The proposed model is based upon a reformulation of the classification problem as a regularized multivariate functional linear regression model. This allows us to adopt a direct approach to the estimation of the most discriminant direction while controlling for its complexity with appropriate differential regularization. Our approach does not require prior estimation of the covariance structure of the functional predictors, which is computationally prohibitive in our application setting. We provide a theoretical analysis of the out-of-sample prediction error of the proposed model and explore the finite sample performance in a simulation setting. We apply the proposed method to a pooled dataset from Alzheimer’s Disease Neuroimaging Initiative and Parkinson’s Progression Markers Initiative. Through this application, we identify discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer’s disease that are consistent with the existing neuroscience literature.

Multiclass classification for multidimensional functional data through deep neural networks

Multiclass classification for multidimensional functional data through deep neural networks

Optimal Classification for Functional Data

Optimal Classification for Functional Data

Fast Convergence on Perfect Classification for Functional Data

We investigate the availability of approaching perfect classification on functional data with finite samples. The seminal work (Delaigle and Hall (2012)) showed that perfect classification for functional data is easier to achieve than for finite-dimensional data. This result is based on their finding that a sufficient condition for the existence of a perfect classifier, named a Delaigle--Hall condition, is only available for functional data. However, there is a danger that a large sample size is required to achieve the perfect classification even though the Delaigle--Hall condition holds, because a minimax convergence rate of errors with functional data has a logarithm order in sample size. This study solves this complication by proving that the Delaigle--Hall condition also achieves fast convergence of the misclassification error in sample size, under the bounded entropy condition on functional data. We study a reproducing kernel Hilbert space-based classifier under the Delaigle--Hall condition, and show that a convergence rate of its misclassification error has an exponential order in sample size. Technically, our proof is based on (i) connecting the Delaigle--Hall condition and a margin of classifiers, and (ii) handling metric entropy of functional data. Our experiments support our result, and also illustrate that some other classifiers for functional data have a similar property.

Review on functional data classification

AbstractA fundamental problem in functional data analysis is to classify a functional observation based on training data. The application of functional data classification has gained immense popularity and utility across a wide array of disciplines, encompassing biology, engineering, environmental science, medical science, neurology, social science, and beyond. The phenomenal growth of the application of functional data classification indicates the urgent need for a systematic approach to develop efficient classification methods and scalable algorithmic implementations. Therefore, we here conduct a comprehensive review of classification methods for functional data. The review aims to bridge the gap between the functional data analysis community and the machine learning community, and to intrigue new principles for functional data classification.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Clustering and Classification Statistical Models > Classification Models Data: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Some results on statistics and functional data analysis

This paper presents some recent results on regression and classification in Functional Data Analysis setting. The first work decated to Multiple kernel SVM for classifying functional data in Sobolev spaces focus on optimization of the information from each derivatives of the original functions, the second work devoted to An input/output-based aggregation rule for functional data classification when the last is interested with the problem of regression if the functional data lives in a finite dimensionnal submanifold.

Classification of multivariate functional data on different domains with Partial Least Squares approaches

Classification of multivariate functional data on different domains with Partial Least Squares approaches

A Functional Data Classification Framework Based on Functional Sufficient Dimension Reduction and Model Averaging

With the ever-evolving advancements in data collection and storage technologies, high-frequency data recorded in patterns that change over time have become increasingly common. In many application scenarios for this type of data, functional data classification has emerged as a prominent issue in the field of statistics. In light of this, this paper proposes a functional data classification model based on the functional sufficient dimensionality reduction method and the idea of model averaging. The proposed method utilizes techniques such as Functional Slice Inverse Regression (FSIR) and Functional Average Variance Estimation (FSAVE) to project an infinite-dimensional random function onto a function space spanned by a finite wiki function, ensuring that the original data's effective information for categorical variables is not lost. Additionally, the Bagging algorithm effectively addresses the overfitting and underfitting problems that arise in single predictive models, while employing model averaging instead of model selection to adaptively select the sufficient dimensionality reduction sub-directions. Notably, in the prediction stage, the number and types of base models are flexible. Empirical analysis demonstrates that the proposed method outperforms some comparative methods in terms of prediction accuracy and robustness.

Unbiased recursive decision tree for supervised functional data classification with applying on electrocardiogram signals

Unbiased recursive decision tree for supervised functional data classification with applying on electrocardiogram signals

A graph-based spectral classification of Type II supernovae

Given the ever-increasing number of time-domain astronomical surveys, employing robust, interpretative, and automated data-driven classification schemes is pivotal. Based on graph theory, we present new data-driven classification heuristics for spectral data. A spectral classification scheme of Type II supernovae (SNe II) is proposed based on the phase relative to the maximum light in the V band and the end of the plateau phase. We utilize a compiled optical data set that comprises 145 SNe and 1595 optical spectra in 4000–9000 Å. Our classification method naturally identifies outliers and arranges the different SNe in terms of their major spectral features. We compare our approach to the off-the-shelf umap manifold learning and show that both strategies are consistent with a continuous variation of spectral types rather than discrete families. The automated classification naturally reflects the fast evolution of Type II SNe around the maximum light while showcasing their homogeneity close to the end of the plateau phase. The scheme we develop could be more widely applicable to unsupervised time series classification or characterization of other functional data.

Functional data classification using the rotation forest method combined with the patch selection

Functional data classification is one of the major challenges in the field of functional data analysis. In this article, we propose the Rotation Forest method combined with the patch selection for functional data classification. The Rotation Forest method first divides the feature matrix into disjoint subsets. Feature extraction is performed on each subset by principal component analysis. All the coefficients of the principal components are retained and reordered to form a new feature matrix to do feature extraction for the training dataset. The patch selection is to connect the pixels in each patch in an orderly manner to form a sub-vector, then each sub-vector for each patch is connected according to the order of the patches to form a feature vector for each subject. This way, the patterns contained in the two-dimensional patches are transformed into information embedded in the one-dimensional feature vectors to be analyzed more efficiently. The new method combines the advantages of both the Rotation Forest method and the patch selection. It generates competitive prediction errors for new observations. Extensive simulations and a real dataset consisting of 372 brain image datasets suggest the superiority of this approach over existing methods.

Optimal Linear Discriminant Analysis for High-Dimensional Functional Data

Most of existing methods of functional data classification deal with one or a few processes. In this work we tackle classification of high-dimensional functional data, in which each observation is potentially associated with a large number of functional processes, p, which is comparable to or even much larger than the sample size n. The challenge arises from the complex inter-correlation structures among multiple functional processes, instead of a diagonal correlation for a single process. Since truncation is often needed for approximation in functional data, another difficulty stems from the fact that the discriminant set of the infinite-dimensional optimal classifier may be different from that of the truncated optimal classifier, when multiple (especially a large number of) processes are involved. We bridge the gap by proposing a penalized classifier that achieves both near-perfect classification that is unique to functional data, and discriminant set inclusion consistency in the sense that the classification-responsible functional predictors include those of the underlying optimal classifier. Simulation study and real data application are carried out to demonstrate its favorable performance. Supplementary materials for this article are available online.

Distance‐weighted discrimination for functional data

The main contribution of the paper is the development of a new margin‐based classifier called distance‐weighted discrimination (DWD) for functional data classification. The proposed classifier employs functional principal component analysis (FPCA) to reduce the dimensionality of the functional data and is free of the restrictive assumptions imposed by Bayes classifiers in terms of mean and covariance functions. Theoretical results show that the proposed classifier is Bayes risk consistent under mild assumptions. Simulation studies and real data examples demonstrate that the DWD classifier outperforms several conventional classifiers in terms of prediction accuracy. Overall, the paper provides a new approach for functional data classification with good empirical performance.

Joint classification and prediction of random curves using heavy‐tailed process functional regression

We propose a heavy-tailed process functional regression to jointly perform classification and prediction of time-varying functional data. We use two independent scale mixtures of Gaussian Processes to respectively model random effects and random errors, yielding robust inferences against both magnitude and shape outliers. We classify random curves by posterior predictive probabilities of class labels and offer a weighted prediction of future curve trends. A Bayesian estimation procedure is implemented through an MCMC sampling algorithm. The performance of classification and prediction of the proposed model is evaluated using simulated studies and some real data sets.

Variation pattern classification of functional data

AbstractA new classification method for functional data is proposed in this article. This work is motivated by the need to identify features that discriminate between neurological conditions on which local field potentials (LFPs) were recorded. Regardless of the condition, these LFPs have zero mean, and thus the first moments of these random processes do not have discriminating power. We propose the variation pattern classification (VPC) method which employs the second‐moment structure as the discriminating feature and uses the Hilbert–Schmidt norm to measure the discrepancy between the second‐moment structure of different groups. The proposed VPC method is demonstrated to be sensitive to the discrepancy, potentially leading to a higher rate of classification. One important innovation lies in the dimension reduction where the VPC method adaptively determines the basis functions (discriminative feature functions) that account for the major discrepancy. In addition, the selected discriminative feature functions provide insights into the discrepancy between different groups because they reveal the features of variation pattern that differentiate groups. Consistency properties are established and, furthermore, simulation studies and the analysis of rat brain LFP trajectories empirically demonstrate the advantages and effectiveness of the proposed method.

A Reproducing Kernel Hilbert Space Framework for Functional Classification

The intrinsic infinite-dimensional nature of functional data creates a bottleneck in the application of traditional classifiers to functional settings. These classifiers are generally either unable to generalize to infinite dimensions or have poor performance due to the curse of dimensionality. To address this concern, we propose building a distance-weighted discrimination (DWD) classifier on scores obtained by projecting data onto one specific direction. We choose this direction by minimizing, over a reproducing kernel Hilbert space, an empirical risk function containing the DWD classifier loss function. Our proposed classifier avoids overfitting and enjoys the appealing properties of DWD classifiers. We further extend this framework to accommodate functional data classification problems where scalar covariates are involved. In contrast to previous work, we establish a nonasymptotic estimation error bound on the relative misclassification rate. Through simulation studies and a real-world application, we demonstrate that the proposed classifier performs favorably relative to other commonly used functional classifiers in terms of prediction accuracy in finite-sample settings. Supplementary materials for this article are available online.

Pooling random forest and functional data analysis for biomedical signals supervised classification: Theory and application to electrocardiogram data.

Scientific progress has contributed to creating many devices to gather vast amounts of biomedical data over time. The goal of these devices is generally to monitor people's health conditions, diagnose, and prevent patients' diseases, for example, to discover cardiovascular disorders or predict epileptic seizures. A common way of investigating these data is classification, but these instruments generate signals often characterized by high dimensionality. Learning from these data is definitely a challenging task due to many issues, for example, the trade‐off between complexity and accuracy and the course of dimensionality. This study proposes a supervised classification method based on the joint use of functional data analysis, classification trees, and random forest to deal with massive biomedical data recorded over time. For this purpose, this research suggests different original tools to extract features and train functional classifiers, interpret the classification rules, assess leaves' quality and composition, avoid the classical drawbacks due to the COD, and improve the accuracy of the functional classifiers. Focusing on ECG data as a possible example, the final purpose of this study is to offer an original approach to identify and classify patients at risk using different types of biomedical signals. The results confirm that this line of research is exciting; indeed, the interpretative tools show evidence to be very useful for understanding classification rules. Furthermore, the performance of the proposed functional classifier, in terms of accuracy, is excellent because the latter breaks the previous classification record regarding a well‐known ECG dataset.

Model-based joint curve registration and classification

In this paper, we consider the problem of classification of misaligned multivariate functional data. We propose to use a model-based approach for the joint registration and classification of such data. The observed functional inputs are modeled as a functional nonlinear mixed effects model containing a nonlinear functional fixed effect constructed upon warping functions to account for curve alignment, and a nonlinear functional random effects component to address the variability among subjects. The warping functions are also modeled to accommodate common effect within groups and the variability between subjects. Then, a functional logistic regression model defined upon the representation of the aligned curves and scalar inputs is used to account for curve classification. EM-based algorithms are developed to perform maximum likelihood inference of the proposed models. The identifiability of the registration model and the asymptotical properties of the proposed method are established. The performance of the proposed procedure is illustrated via simulation studies and an analysis of a hyoid bone movement data application. The statistical developments proposed in this paper were motivated by the hyoid bone movement study, the methodology is designed and presented generality and can be applied to numerous areas of scientific research.

Measure inducing classification and regression trees for functional data

AbstractWe propose a tree‐based algorithm (μCART) for classification and regression problems in the context of functional data analysis, which allows to leverage measure learning and multiple splitting rules at the node level, with the objective of reducing error while retaining the interpretability of a tree. For each internal node, our main contribution is the idea of learning a weighted functional space by means of constrained convex optimization, which is then used to extract multiple weighted integral features from the functional predictors, in order to determine the binary split. The approach is designed to manage multiple functional predictors and/or responses, by defining suitable splitting rules and loss functions that can depend on the specific problem and can also be combined with additional scalar and categorical predictors, as the tree is grown with the original greedy CART algorithm. We focus on the case of scalar‐valued functional predictors defined on unidimensional domains and illustrate the effectiveness of our method in both classification and regression tasks, through a simulation study and four real‐world applications.

Diagnosis of Schizophrenia Based on Deep Learning Using fMRI.

Schizophrenia is a brain disease that frequently occurs in young people. Early diagnosis and treatment can reduce family burdens and reduce social costs. There is no objective evaluation index for schizophrenia. In order to improve the classification effect of traditional classification methods on magnetic resonance data, a method of classification of functional magnetic resonance imaging data is proposed in conjunction with the convolutional neural network algorithm. We take functional magnetic resonance imaging (fMRI) data for schizophrenia as an example, to extract effective time series from preprocessed fMRI data, and perform correlation analysis on regions of interest, using transfer learning and VGG16 net, and the functional connection between schizophrenia and healthy controls is classified. Experimental results show that the classification accuracy of fMRI based on VGG16 is up to 84.3%. On the one hand, it can improve the early diagnosis of schizophrenia, and on the other hand, it can solve the classification problem of small samples and high-dimensional data and effectively improve the generalization ability of deep learning models.