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%%% Keywords:	Autocovariance Operator, Clustering, Covariance Surface, Eigenfunction, Infinite-dimensional Data, Karhunen-Lo\`eve Representation, Longitudinal Data, Nonparametrics, Panel Data, Principal Component, Registration, Regression, Smoothing, Square Integrable Function, Stochastic Process, Time Course, Tracking, Registration, Warping
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\begin{document}
\begin{center}{\bf Functional Data Analysis}\bigskip

Hans-Georg M\"uller\\
Department of Statistics\\ University of California, Davis\\
One
Shields
Ave., Davis, CA 95616, USA.\\
e-mail: mueller@wald.ucdavis.edu\\
\end{center}

KEY WORDS: Autocovariance Operator, Clustering, Covariance Surface, Eigenfunction, Infinite-dimensional Data, Karhunen-Lo\`eve Representation, Longitudinal Data, Nonparametrics, Panel Data, Principal Component, Registration, Regression,
Smoothing, Square Integrable Function, Stochastic Process, Time Course, Tracking, Warping.\\


\noindent{\bf 1. Overview}\\

Functional data analysis (FDA) refers to the statistical analysis of
data samples consisting of random functions or surfaces, where each
function is viewed as one sample element. Typically, the random
functions contained in the sample are considered to be independent
and to correspond to smooth realizations of an underlying stochastic process.
FDA methodology then provides a statistical approach to the analysis of
repeatedly observed stochastic processes or data generated by such processes.
FDA differs from time series approaches, as the sampling design is very flexible, stationarity of the
underlying process is not needed, and autoregressive-moving average models or similar time regression
models
play no role, except where the elements of such models are functions themselves.

FDA also differs from multivariate analysis, the area of
statistics that deals with finite-dimensional random vectors, as functional data are inherently
infinite-dimensional and smoothness often is a central assumption. Smoothness has no meaning for multivariate data
analysis, which in contrast to FDA is permutation invariant.
Even sparsely and irregularly observed longitudinal data
can be analyzed with FDA methodology. FDA thus is useful for the
analysis of longitudinal or otherwise sparsely sampled data. It is also a key methodology for
the analysis of time course, image and
tracking data.

The approaches and models of FDA
are essentially nonparametric, allowing for flexible modeling. The statistical tools of FDA include
smoothing, e.g., based on series expansions, penalized splines, or local polynomial smoothing, and functional principal
component analysis. A distinction between smoothing methods and FDA is that smoothing
is typically used in situations where one wishes to obtain an estimate for one non-random
object (where objects here are functions or surfaces) from noisy observations, while FDA aims
at the analysis of a sample of random objects,
which may be assumed to be completely observed without noise or to be
sparsely observed with noise; many scenarios of interest
fall in between these extremes.

An important special situation arises when the underlying
random processes generating the data are Gaussian processes,
an assumption that is often invoked
to justify linear procedures and to simplify methodology and theory.
Functional data are ubiquitous and may for example involve samples of density
functions \cp{knei:01}, hazard functions, or behavioral tracking data.
Application areas that have been emphasized in the statistical literature include growth curves \cp{rao:58,
mull:84:2}, econometrics and e-commerce \cp{rams:02:2,jank:06:2},
evolutionary
biology \cp{kirk:89,izem:05}, and genetics and genomics \cp{opge:06,mull:08:3}.
FDA also applies to panel data as considered in
economics and other social sciences.\\

\noindent{\bf 2. Methodology}\\

Key FDA methods include functional principal component analysis
\cp{cast:86,rice:91}, warping and curve registration \cp{gerv:04} and
functional regression \cp{rams:91}. Theoretical foundations and
asymptotic methods of FDA are closely tied to perturbation theory
of linear operators in Hilbert space \cp{daux:82,bosq:00,mas:03}; a reproducing kernel
Hilbert space approach has also been proposed \cp{euba:08}, as well as
Bayesian approaches \cp{tele:08}. Finite sample
implementations typically require to address ill-posed problems, emplying
suitable regularization, which is often implemented by penalized least squares or penalized
likelihood and by truncated
series expansions. A broad overview of methods and applied aspects of FDA
can be found in the textbook \ci{rams:05} and some additional reviews are in
\ci{rice:04,zhao:04,mull:08:7}.

The basic statistical
methodologies of ANOVA, regression, correlation, classification and
clustering that are available for scalar and vector data have
spurred analogous developments for functional data. An additional
aspect is that the time axis itself may be subject to random
distortions and adequate functional models sometimes need to reflect
such time-warping (also referred to as alignment or registration).

Another issue is that often the random
trajectories are not directly observed. Instead, for each sample
function one has available measurements on a time grid that may
range from very dense to extremely sparse. Sparse and randomly
distributed measurement times are frequently encountered in
longitudinal studies. Additional contamination of the measurements
of the trajectory levels by errors is also common. These situations
require careful modeling of the relationship between the recorded
observations and the assumed underlying functional trajectories
\cp{rice:01, jame:03, mull:05:4}.

Initial analysis of functional
data includes exploratory plotting of the observed functions in a
``spaghetti plot'' to obtain an initial idea of functional shapes,
to check for outliers and to identify potential ``landmarks''. Preprocessing may
include outlier removal and registration to adjust
for time-warping \cp{gass:95,gerv:04, mull:04:4, jame:07,knei:08}.\\

\noindent{\bf 3. Functional Principal Components}\\

Basic objects in FDA are the mean function $\mu$ and the covariance
function $G$. For square integrable random functions $X(t)$, \begin{eqnarray}
\mu(t)=E(Y(t)), \quad G(s,t)&=&\cov\left\{X(s),X(t)\right\},\quad
s,t \in \T, \end{eqnarray} with auto-covariance operator $(A f)(t) =
\int_{\T}\, f(s) G(s,t)\, ds.$ This linear operator of
Hilbert-Schmidt type has orthonormal eigenfunctions $\pk,\,
k=1,2,\ldots,$ with associated ordered eigenvalues $\lambda_{1} \ge
\lambda_{2} \ge \ldots$, such that $A\, \pk = \lambda_k \, \pk.$
The foundation for functional principal component analysis is the
Karhunen-Lo\`eve representation of random functions \cp{karh:46,gren:50,gikh:69}
$X(t)=\mu(t)+\sum\limits_{k=1}^{\infty} \xk\,\pk(t),$ where
$\xk=\int_{\T} (Y(t)-\mu(t))\pk(t)\,dt$ are uncorrelated centered
random variables with $\var(\xk)=\lambda_{k}$, referred to as functional
principal components (FPCs).

Estimation of eigenfunctions, eigenvalues and of FPCs is a core objective
of FDA. Various smoothing-based methods and applications for various sampling designs
have been considered \cp{jone:92, silv:96, stan:98, card:00:1,jame:00, paul:09:1}.
Estimators employing smoothing methods (local least squares or
splines) have been developed for various sampling schemes (sparse,
dense, with errors) to obtain a data-based version of the
eigen-representation, where one regularizes by truncating at a finite
number $K$ of included components. The idea is to borrow strength
from the entire sample of functions, rather than estimating each
function separately. The functional data are then represented by the
subject-specific vectors of score estimates $\hxk,\, k=1,\ldots, K$,
which can be used to represent individual trajectories and for
subsequent statistical analysis \cp{mull:05:4}.

More adequate representations of functional data are
sometimes obtained by fitting pre-specified fixed
basis functions with random coefficients. In particular,
B-splines \cp{sy:97}, P-splines \cp{yao:06} and wavelets \cp{morr:06} have been
successfully applied. A general relation between mixed linear models and fitting
functional models with basis expansion coefficients
can be used to advantage for modeling and implementation of these approaches.
In the theoretical analysis, one may distinguish between an essentially multivariate
analysis, which results from assuming that the number of series terms is actually finite, leading to
parametric rates of convergence, and an essentially functional approach. In the latter, the number
of components is assumed to increase with sample size and this leads to ``functional'' rates
of convergence that depend on the properties of underlying processes, such as decay and spacing of
the eigenvalues of the autocovariance operator.\\

\noindent{\bf 4. Functional Regression and Related Models}\\

Functional regression models may include one or several functions
among the predictors, responses, or both. For pairs $(X,Y)$ with
centered random predictor functions $X$ and scalar responses $Y$,
the linear model is $$E(Y|X)=\int_{\T} (X(s)-\mu(s))\beta(s)\,ds.$$ The
regression parameter function $\beta$ can be represented in a
suitable basis, for example the eigenbasis, with coefficient
estimates determined by least squares or similar criteria. The functional linear model
has been thoroughly studied, including optimal rates of convergence
\cp{card:03:1,card:03:2,mull:05:5,cai:06,hall:07:1,li:07,mas:09}.

The class of useful functional regression models is large, due to the
infinite-dimensional nature of the functional predictors. The case
where the response is functional \cp{rams:91} also is of interest.
Flexible extensions of
the functional linear model for example include nonparametric
approaches \cp{ferr:06},
where unfavorable small ball probabilities and the non-existence of a density in general
random function
space impose limits on convergence \cp{mull:09:5},
and multiple index models \cp{jame:05}. Another extension is the
functional additive model \cp{mull:08:2}. For functional
predictors $X=\mu + \sum_{k=1}^\infty A_k \pk$ and scalar responses $Y$, this model is
given by $$E(Y|X)=\sum_{k=1}^\infty f_k(A_k) \pk$$ for smooth
functions $f_k$ with $E(f_k(A_k))=0$.


Another
variant of the functional linear model, which is also applicable for classification purposes, is
the generalized functional linear model $E(Y|X)=g\{\mu + \int_{\T} \,
X(s)\beta(s)\,ds\}$ with link function $g$ \cp{jame:02,esca:04, card:05:1, mull:05:1}. The link function (and
an additional
variance function if applicable) is adapted to the
(often discrete) distribution of $Y$; the components of the
model can be estimated by quasi-likelihood.
Besides discriminant analysis via the binomial functional generalized linear model,
various other methods have been studied for
functional clustering and discriminant analysis \cp{jame:03,chio:07,chio:08}.

Of practical relevance are extensions towards
polynomial functional regression models \cp{mull:10:1}, hierarchical functional
models \cp{crai:09:1},
models with varying domains, models with more than one predictor function,
and functional (autoregressive) time series models, among others. In addition
to the functional trajectories themselves, derivatives are of
interest to study the dynamics of the underlying processes \cp{rams:05}.
Software for functional data analysis evolves rapidly and is available from various sources. Freely
available software includes
for example
the fda package (R and matlab), at \newline
{\tt http://www.psych.mcgill.ca/misc/fda/software.html},
and the PACE package (matlab), at {\tt http://anson.ucdavis.edu/~mueller/data/pace.html}.\\


\noindent{\bf Acknowledgments}\\

Research supported in part by NSF Grant DMS-0806199.
Based on an article from Lovric, Miodrag (2011), International
Encyclopedia of Statistical Science. Heidelberg: Springer Science
+Business Media, LLC.


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\end{document}
