CRAN Task View: Functional Data Analysis

Functional data analysis (FDA) deals with data that “provides information about curves, surfaces or anything else varying over a continuum.” This task view tries to provide an overview of available packages in this developing field.
In practice, there is substantial overlap between time series data and functional data, so many packages listed under the TimeSeries task view could be useful for functional data as well and vice versa.

General functional data analysis

The packages listed below provide “infrastructure” for representing and handling function-valued data and/or implement many widely applicable functional data methods:

• fda provides object-types for functional data with corresponding functions for smoothing, plotting and simple regression models, c.f. Ramsay et al. (2009, doi:10.1007/978-0-387-98185-7).
• fdapace provides functional principal component based methods for sparsely or densely sampled random trajectories and time courses for functional regression and correlation, for longitudinal data analysis, the analysis of stochastic processes from samples of realized trajectories, and for the analysis of underlying dynamics.
• fdasrvf performs alignment, PCA, and regression of multidimensional or unidimensional functions using the square-root velocity framework (Srivastava et al., 2011). This framework allows for elastic analysis of functional data through phase and amplitude separation.
• fda.usc provides routines for exploratory and descriptive analysis of functional data such as depth measurements, outlier detection, as well as unsupervised and supervised classification, (univariate, nonparametric) regression models with a functional covariate and functional analysis of variance.
• fds contains 19 data sets with functional data.
• funData provides S4 classes for univariate and multivariate functional and image data and utility functions.
• tf provides S3 vector data types for functional data with arithmetic and summary methods, derivation, integration and smoothing, plotting, data import and export, and data wrangling (subsetting, sub-assigning, zooming, extracting functional features) that allow including such vectors in data frames for joint analysis of functional and scalar variables.

Inference methods for functional data

General purpose regression and classification:

• FDboost implements flexible additive regression models and variable selection for scalar-on-function, function-on-scalar and function-on-function regression models that are fitted by a component-wise gradient boosting algorithm.
• mlr3fda extends ‘mlr3’ for functional data analysis, with support for irregular and regular functional data as defined in ‘tf’, providing ‘PipeOps’ for preprocessing functional columns and for extracting scalar features to be used in standard machine learning algorithms.
• refund provides spline-based methods for penalized function-on-scalar, scalar-on-function, and function-on-function regression as well as methods for functional PCA. Some of the functions are also applicable to image data.
• dbstats provides prediction methods where explanatory information is coded as a matrix of distances between individuals. It includes distance based versions of lm and glm, as well as nonparametric versions of both, based on local estimation. To apply these methods to functional data it is sufficient to calculate a distance matrix between the observed functional data.

Specialized models and applications:

• cfda implements analysis of categorical functional data, by computing an encoding (a real functional variable) for each state.
• denseFLMM and sparseFLMM estimate linear mixed models for densely and sparsely sampled functional responses, respectively, based on functional principal component analysis.
• fastFMM implements a fast univariate inference approach for fitting functional mixed models.
  
• fdaconcur implements concurrent and history index models for both dense and sparsely observed functional data
• fdaPDE implements statistical analysis of functional and spatial data over multidimensional complex domains, based on regression models with partial differential regularization, discretized through the finite element method.
• GPFDA uses functional regression as the mean structure and Gaussian processes as the covariance structure.
• growfunctions estimates a collection of time-indexed functions under either of Gaussian process (GP) or intrinsic Gaussian Markov random field (iGMRF) prior formulations where a Dirichlet process mixture allows sub-groupings of the functions to share the same covariance or precision parameters.
• multifamm implements multivariate functional additive mixed models (multiFAMM) based on univariate sparse functional regression models and multivariate functional principal component analysis, c.f. Volkmann et al. (2012, doi:10.48550/arXiv.2103.06606).
• robflreg implements robust methods for functional linear regression (scalar-on-function, function-on-function) based on robust partial least squares decompositions.
• slasso implements a smooth LASSO estimator for the function-on-function linear regression model, c.f. Centofanti et al. (2020, doi:10.48550/arXiv.2007.00529).
  
• splinetree implements regression trees and random forests for longitudinal or functional data using a spline projection method.
• adass implements the adaptive smoothing spline estimator for the function-on-function linear regression model described, c.f. Centofanti et al. (2023, doi:10.1007/s00180-022-01223-6).

Statistical tests:

• fdANOVA implements analysis of variance testing procedures for univariate and multivariate functional data.
• fdatest provides an implementation of the Interval Testing Procedure for functional data in different frameworks (i.e., one or two-population frameworks, functional linear models) by means of different basis expansions (i.e., B-spline, Fourier, and phase-amplitude Fourier).
• gmtFD implements general multiple tests for univariate and multivariate functional data.
• multiFANOVA implements multiple contrast tests for functional data for one-way and multi-way analysis of variance for functional data.
• rmfanova implements statistical tests for functional repeated measures analysis testing for equality or pairwise comparisons (post-hoc analysis) of mean functions in repeated experiments.
• rofanova implements robust functional analysis of variance, c.f. Centofanti et al. (2021, doi:10.48550/arXiv.2112.10643).
• SCBmeanfd provides methods for estimating and inferring the mean of functional data. The methods include simultaneous confidence bands, local polynomial fitting, bandwidth selection by plug-in and cross-validation, goodness-of-fit tests for parametric models, equality tests for two-sample problems, and plotting functions.

Clustering functional data

• fdacluster implements k-means, hierarchical agglomerative and DBSCAN clustering for functional data which allows for jointly aligning and clustering curves.
• funFEM ’s algorithm (Bouveyron et al., 2014) allows to cluster functional data by modeling the curves within a common and discriminative functional subspace.
• funLBM implements model-based co-clustering of functional data, i.e., simultaneously clustering the rows and the columns of a data matrix where each entry of the matrix is a function or a time series.
• sasfunclust implements the sparse and smooth functional clustering (SaS-Funclust) method, c.f. Centofanti et al. (2024, doi:10.1007/s00362-023-01408-1), that aims to classify a sample of curves into homogeneous groups while jointly detecting the most informative portions of domain.

Registering and aligning functional data

• elasdics provides functions to align 2D curves and to compute mean curves based on the elastic distance defined in the square-root-velocity framework, c.f. Steyer et al. (2021, doi:10.48550/arXiv.2104.11039).
• elastes provides functions for the computation of functional elastic shape means over sets of open planar curves, particularly for settings where these curves are only sparsely and irregularly observed, c.f. Stöcker et al. (2022, doi:10.48550/arXiv.2203.10522).
• fdasrvf performs alignment, PCA, and regression of multidimensional or unidimensional functions using the square-root velocity framework (Srivastava et al., 2011). This framework allows for elastic analysis of functional data through phase and amplitude separation.
• registr provides registration for (incomplete) non-Gaussian functional data, c.f Wrobel et al. (2019, doi:10.1111/biom.12963), Wrobel and Bauer (2021, doi:10.21105/joss.02964).
• warpMix implements warping (alignment) for functional data using B-spline based mixed effects models.

Time series of functional data

• ftsa provides functions for visualizing, modeling, forecasting and hypothesis testing of functional time series.
• ftsspec provides functions for estimating the spectral density operator of functional time series (FTS) and comparing the spectral density operator of two functional time series, in a way that allows detection of differences of the spectral density operator in frequencies and along the curve length.
• freqdom provides frequency domain methods for multivariate and functional time series and implements dynamic functional principal components and functional regression in the presence of temporal dependence.
• freqdom.fda provides a wrapper for functionality of freqdom for objects from fda
• pcdpca extends multivariate dynamic principal components to periodically correlated multivariate and functional<!– time series.
• fdaACF contains functions to quantify the serial correlation across lags of a given functional time series, see also github.com/GMestreM/fdaACF .

Visualization and Exploratory Data Analysis

• rainbow contains functional data sets and functions for functional data display, exploratory analysis and outlier detection.
• refund.shiny provides a function plot_shiny that uses S3 and generates a Shiny app of interactive visualizations for functional data analyses produce using the refund package, described in c.f. Wrobel et al. (2016, doi:10.1002/sta4.109)
• fdaoutlier provides a collection of functions for functional data outlier detection. Methods implemented include directional outlyingness, MS-plot, total variation depth, and sequential transformations among others.

Other

• conformalInference.fd computes full conformal, split conformal and multi split conformal prediction regions when the response has functional nature.
• ddalpha implements depth-based classification and calculation of data depth, also for functional data.
• face implements Fast Covariance Estimation for Sparse Functional Data, c.f. Xiao et al. (2018, doi:10.1007/s11222-017-9744-8).
• fdadensity implements Petersen and Müller (2016, doi:10.1214/15-AOS1363) for the analysis of samples of density functions via specialized Functional Principal Components Analysis.
• fdaPOIFD implements partially observed integrated functional depth with applications to visualization, outlier detection and classification, c.f. Elías et al. (2022, doi:10.1080/10618600.2022.2070171).
• frechet implements Fréchet regression for for non-Euclidean responses, e.g. distributions in L^2-Wasserstein space or covariance matrices, c.f. Petersen & Müller (2019, doi:10.1214/17-AOS1624).
• funcharts provides functional control charts for statistical process monitoring of functional data, c.f. Capezza et al (2023, doi:10.1080/00224065.2023.2219012)
  
• mfaces implements multivariate functional principal component analysis via fast covariance estimation for multivariate sparse functional data or longitudinal data, c.f Li, Xiao, and Luo (2020, doi:10.1002/sta4.245)
• MFPCA calculates multivariate FPCA for “multimodal” data observed on domains with different dimensionalities, c.f. Happ and Greven (2018, doi:10.1080/01621459.2016.1273115).

If you want to suggest additions or improvements to this list, please submit an issue or pull request in the GitHub repository.

CRAN packages

Related links

• Website of the canonical FDA book by Ramsay and Silverman
• PACE: collection of MATLAB scripts from UC Davis
• WFMM: powerful software for Bayesian wavelet-based functional mixed models (C++/Matlab)
• scikit-FDA: comprehensive Python package for FDA
• fdasrsf: comprehensive Python package for SRVF FDA

Other resources

• CRAN Task View: TimeSeries