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Review of Functional Data Analysis

Jane-Ling Wang, Jeng-Min Chiou, Hans-Georg Mueller

arXiv:1507.05135v1stat.ME

TL;DR

Functional data are intrinsically infinite dimensional, and FDA methodology and theory vary with the measurement schedule. This paper reviews foundational and advanced FDA methods while highlighting challenges for sparse and irregular sampling and unresolved unification questions.

  • Problem

    Functional data are intrinsically infinite dimensional, posing challenges for their analysis; unifying methods and theory remains an issue.

  • Method

    The paper reviews first-generation functional data methods, including foundational FDA concepts, nonlinear manifold-based models, and measurement-error accommodation.

  • Results

    FDA methodology and theory vary with the time-grid sampling plan, while sparse and irregular sampling requires more theoretical and methodological effort than dense sampling.

  • Takeaways & Limitations

    Sampling design is a foundational consideration because FDA methodology, theory, and convergence rates depend on the measurement schedule.

  • Takeaways & Limitations

    Some estimators are not efficient, so their resulting confidence intervals require cautious interpretation.

Abstract

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With the advance of modern technology, more and more data are being recorded continuously during a time interval or intermittently at several discrete time points. They are both examples of "functional data", which have become a prevailing type of data. Functional Data Analysis (FDA) encompasses the statistical methodology for such data. Broadly interpreted, FDA deals with the analysis and theory of data that are in the form of functions. This paper provides an overview of FDA, starting with simple statistical notions such as mean and covariance functions, then covering some core techniques, the most popular of which is Functional Principal Component Analysis (FPCA). FPCA is an important dimension reduction tool and in sparse data situations can be used to impute functional data that are sparsely observed. Other dimension reduction approaches are also discussed. In addition, we review another core technique, functional linear regression, as well as clustering and classification of functional data. Beyond linear and single or multiple index methods we touch upon a few nonlinear approaches that are promising for certain applications. They include additive and other nonlinear functional regression models, such as time warping, manifold learning, and dynamic modeling with empirical differential equations. The paper concludes with a brief discussion of future directions.

1 Introduction

Functional Data Analysis studies data represented as functions, whose intrinsic high dimensionality and varied sampling schedules create distinctive modeling challenges. This review focuses mainly on first-generation functional data and surveys dimension reduction, regression, clustering, classification, and emerging nonlinear methods.

  • 1 Introduction: Functional data are function-valued observations, often latent stochastic-process trajectories recorded discretely on dense, sparse, or subject-specific time grids.Measurement schedules may be fixed or random, equally spaced or irregular, and can vary across subjects.
  • 1 Introduction: High intrinsic dimensionality motivates dimension reduction, while smoothness lets neighboring measurements be pooled for regularization.The review highlights principal components and regression-oriented dimension reduction as key tools.
  • 1 Introduction: The review covers first-generation functional data while briefly introducing next-generation data involving multivariate, correlated, image, shape, or other complex objects.It is a subjective selection rather than an objective or comprehensive review, with possible omissions in this fast-moving field.
  • 1 Introduction: FDA accommodates measurement error because each subject contributes repeated measurements, supporting analysis of noisy functional trajectories.Measurement errors are commonly treated as random fluctuations around a smooth trajectory or as errors in observed measurements.
  • 1 Introduction: Sampling design changes FDA methodology and theory: sparse and irregular longitudinal data typically require more effort than densely sampled data.The review discusses comparisons and a unified treatment across sampling plans.
  • 1 Introduction: Beyond predominantly linear methods, the review surveys nonlinear approaches motivated by functional data with inherent nonlinear features.Examples include nonlinear regression, time warping, manifold learning, and dynamic modeling.

2 Mean and Covariance Function, and Functional Principal Component Analysis

Mean and covariance estimation for functional data depends on the sampling schedule, with smoothing methods accommodating irregular observations and measurement error. Sampling density determines convergence behavior, weighting efficiency, and the appropriate construction of simultaneous confidence bands.

  • Estimation of Mean and Covariance Functions: Sampling design determines estimation strategy: common grids permit empirical mean and covariance estimates, whereas differing schedules require smoothing across observations and subjects.For irregular designs, local polynomial smoothing estimates the mean from pooled observations and two-dimensional smoothing estimates covariance from off-diagonal raw covariances.
  • Estimation of Mean and Covariance Functions: Irregular-design covariance smoothing excludes diagonal raw covariances because measurement-error variance contaminates those terms.After estimating the covariance, measurement-error variance can be estimated by smoothing squared residuals minus the estimated covariance on the diagonal.
  • Sampling Designs and Weighting: Equal observation weighting favors subjects with more repeated measurements, while equal subject weighting changes efficiency according to the sampling plan.Equal subject weighting is generally more efficient for ultra-dense or some dense designs, whereas equal observation weighting is preferred for many other plans, including sparse data.
  • Sampling Designs and Weighting: Sparse sampling is a non-dense regime with the slowest convergence rates, while dense and ultra-dense designs differ by whether asymptotic bias remains.The sampling categories also affect the construction of simultaneous confidence bands.
  • Hypothesis Testing and Simultaneous Confidence Bands: Simultaneous confidence bands require different methods for ultra-dense, dense, and sparse functional data, with sparse data lacking tightness and requiring a rescaling approach.Nonparametric smoothing can introduce asymptotic bias, so confidence intervals and bands may require care when targeting the true function.
  • Hypothesis Testing and Simultaneous Confidence Bands: Eigenfunction estimates can attain a one-dimensional convergence rate when the covariance-surface estimate is undersmoothed.This result avoids the inverse-problem difficulty associated with functional data estimation under the stated undersmoothing choice.

3 Correlation and Regression: Inverse Problems and Dimension Reduction for Functional Data

Functional correlation and regression are fundamentally inverse problems because covariance operators are compact and their inverses may be unbounded. The review presents dimension reduction and regularization strategies, while noting that functional canonical correlation remains difficult for sparse data.

  • Inverse problems: Compact covariance operators can have unbounded inverses, so regression and correlation procedures involving Σ−1 require regularization.When the covariance operator has finitely many positive eigenvalues, the problem reduces to finite-dimensional multivariate analysis; otherwise its inverse is not defined on all of L2.
  • Functional correlation: Functional canonical correlation finds projections of X and Y whose inner products are maximally correlated, extending multivariate canonical correlation.Successive canonical pairs are constrained to be uncorrelated with earlier pairs.
  • Functional correlation: FCCA is ill-posed because infinite-dimensional weight functions can overfit when the sample of curves is small.Its first canonical correlation can also depend strongly on the regularization parameter.
  • Functional correlation: Sparse functional data remain an open challenge for FCCA because imputation before correlation introduces prediction error and potentially biased correlations.Existing FCCA formulations generally require densely recorded data for accurate inner-product evaluation.
  • Alternative measures: Alternative functional correlations avoid the inverse problem by using singular covariance expansions or cosine-based alignment of centered functional shapes.The singular approach replaces correlation with covariance, while cosine-based measures remove the static integral component before comparing dynamic parts.
  • Functional regression: Functional regression convergence depends on eigenvalue decay and regularity, with √n rates attainable for suitable smooth predictors and well-behaved eigenvalues.The cited results distinguish completely observed or dense data from other functional-data designs.

4 Clustering and classification of functional data

Functional clustering groups trajectories by similarity, whereas classification assigns new observations to predefined labeled classes. The review covers k-means, basis-expansion, FPCA, subspace, hierarchical, and model-based approaches for functional data.

  • Overview: Functional clustering is unsupervised grouping by similarity, while functional classification is supervised assignment to predetermined classes using labeled observations.Classification uses a discriminant function or classifier, whereas clustering seeks groups through a clustering criterion.
  • K-means clustering: Functional k-means minimizes within-cluster squared distances to mean-function centers, extending multivariate k-means to trajectories.Observations are partitioned into L groups using a suitable functional distance, often the L2 distance.
  • Practical considerations: Discrete, noisy, sparse, or irregular sampling motivates projecting infinite-dimensional data into a low-dimensional basis space before clustering.This projection makes functional distances and clustering objectives more practical to compute.
  • Basis expansion: Basis-expansion methods project trajectories onto shared basis functions and cluster their coefficient vectors as multivariate proxies.Common implementations use B-splines, Fourier bases, or wavelets, followed by k-means or related algorithms.
  • Subspace and model-based methods: Subspace-projected clustering defines centers through mean-and-eigenfunction subspaces, capturing stochastic structure beyond cluster mean functions.Model-based approaches can instead use statistical models such as mixture models as cluster centers.
  • Theoretical and applied scope: The reviewed functional clustering literature includes a strong consistency result for a basis-based clustering method.The cited method has been implemented with several basis choices, including P-splines, Gaussian orthonormalized bases, and wavelets.
  • FPCA clustering: FPCA-based clustering uses data-adaptive covariance eigenfunctions, with FPC-score distributions indicating cluster patterns while the overall mean does not affect clustering.This approach avoids choosing a fixed external basis and can use distances adapted to sparse functional data.

5 Nonlinear Methods for Functional Data

The review examines nonlinear functional-data methods motivated by the inadequacy of linear models for time variation and infinite-dimensional predictors. It covers structured nonlinear regression, additive models, continuously additive models, time warping, and manifold learning.

  • Motivation: Linear functional models can be inadequate when observation times are randomly distorted, making time variation a primary source of functional-data variation.Time warping motivates nonlinear methods that explicitly represent distorted observation time.
  • Structured nonlinear regression: Nonparametric regression with unrestricted functional predictors faces a curse of dimensionality because predictors are inherently infinite-dimensional.Lower-dimensional structure or manifolds can counteract this curse by yielding lower-dimensional convergence rates.
  • Structured nonlinear regression: Structured extensions, including single-index, additive, polynomial, and functional nonlinear models, balance flexibility with structural stability and polynomial convergence rates.These models extend classical linear regression while retaining interpretable structure.
  • Additive functional regression: Functional additive models use FPCA scores as predictors and smooth component functions, enabling one-dimensional smoothing when the principal components are independent.The same independence yields a decomposition of functional linear regression into infinitely many simple linear regressions.
  • Additive functional regression: Continuously additive models resolve the infeasibility of unrestricted time-additive regression by taking limits over increasingly dense grids and representing g with a bivariate spline.The construction assumes smooth bivariate dependence on time and the covariate.
  • Time warping and manifold learning: Time warping, manifold learning, and related nonlinear representations provide flexible low-dimensional descriptions of functional variation, while FPCA offers finite-dimensional reduction.Landmark methods use shared features such as peak locations, and manifold representations can differ from Karhunen–Loève expansions.

6 Outlook and Future Perspectives

Future FDA research is driven by increasingly complex, high-dimensional, dependent, spatial, and multivariate functional data. The review highlights dimension-reduction interfaces, unresolved methodological choices, and areas not covered comprehensively.

  • High-dimensional and functional data: Stringing converts high-dimensional, strongly correlated predictors into random functions whose FPC scores provide effective dimension reduction without relying on sparsity.Predictor locations are ordered to match correlation-based distances before values are smoothed and summarized.
  • Open problems: Open methodological problems include selecting the number of Karhunen–Loève components, choosing smoothing parameters, detecting outliers, and developing robust FDA methods.Sparse functional-data methods still lag behind methods for dense functional data.
  • Scope and omissions: The review does not comprehensively cover functional designs, domain selection, dependent functional data, or many recent developments.It explicitly focuses on concepts rather than applications.
  • Emerging data structures: FDA increasingly encompasses longitudinal, multivariate, spatially indexed, and other dependent functional data that pose novel analytical challenges.These settings include repeatedly observed functions for each subject and functional time series.
  • Applications and outlook: New applications in health monitoring, gene expression, genomics, finance, and neuroimaging are expanding the need for methodology for next-generation functional data.These data contain more complex features than the first-generation functional data emphasized in the review.
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