Functional and Longitudinal Data Analysis:Perspectives on Smoothing

John A. Rice

Department of StatisticsUniversity of California, Berkeley

March 17, 2003

AbstractThe perspectives and methods of functional data analysis and lon-

gitudinal data analysis are contrasted and compared. Topics includekernel methods and random effects models for smoothing, basis func-tion methods, and examination of the relation of covariates to curveshapes. Some directions in which methodology might advance areidentified.

1 Introduction

Until recently, functional data analysis (FDA) and longitudinal data analysis(LDA) have been rather disjoint enterprises. I will attempt to compare andcontrast their perspectives and methods for this volume of Statistica Sinica,based on the stimulating conference, Emerging Issues in Longitudinal DataAnalysis, which brought the two communities together. Themes I pursueinclude considering how strength is borrowed in FDA and LDA, how theycan borrow strength from each other, the relationships between stochasticmodels and smoothing algorithms, and directions in which current method-ology might advance. This paper thus constitutes a comparison and a viewtoward future directions, from my personal perspective, and is not intendedto be a complete, all-encompassing review.

The remainder of the paper is structured as follows: In Section 2, kernelmethods are considered from both the FDA and LDA perspectives. Basis

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function methods are considered in Section 3. Linear and non-linear methodsof interpolation and extrapolation are treated in Section 4. Section 5 isconcerned with assessing the influence of covariates on curve shapes.

2 Local Kernel Smoothing and Random Ef-

fects Models

Kernel smoothing (linear averaging with smooth weights) arises from dif-ferent perspectives. As a simple example, consider the Nadaraya-Watsonestimate

f(x) =

∑ni=1 yiw(x− xi)∑ni=1 w(x− xi)

. (1)

From the FDA or non-parametric function estimation perspective, this esti-mate is appealing prima facie for purposes of estimating a broad variety offunctions f(x) under weak assumptions. In the LDA literature, smooth-ing procedures more typically arise from stochastic models. For a fixedvalue of x, the estimate (1) would correspond to a model yi = f(x) + εi(x),Var(εi(x)) ∝ (w(x− xi))

−1. It has the property that as |x− xi| increases asx is moved away from the data, f (x) tends to a constant that depends uponthe particular kernel used and the data (yi, xi). For example, David Ruppertpointed out that if x1 < x2 < · · · < xn and the kernel is Gaussian, then as xdecreases, f (x) → y1.

From an LDA perspective, this rather strange behavior could be avoidedby formulating a Gaussian random effects model, f(x) ∼ N(0, τ 2), leadingto the estimate

E[f(x)|y1, · · · , yn] =

∑ni=1 yiw(x− xi)

τ−2 +∑n

i=1w(x− xi)(2)

The FDA community would recognize this as a ridge estimate, as proposedin [17] to ameliorate fluctuations of local polynomial estimates in regions ofsparse data. This estimate extrapolates in a much more natural way thandoes (1), tending to 0 (or to the prior mean) as x moves away from the xi.

In the LDA literature, smoothers and extrapolators typically arise fromstochastic models, in particular from classical time-series models [9]. Anindividual trajectory is modeled as a signal, a Gaussian process u(t) withmean µ(t) and covariance function K(s, t), which is observed at times ti

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with noise, y(ti) = u(ti) + εi. The smoothed/extrapolated estimate of u(t)is then u(t) = E(u(t)|y) which is a weighted linear average of the observeddata. A kernel is thus implicitly determined. The parameters of the kernel,most importantly the bandwidth, are determined as maximum likelihoodestimates of the parameters of the covariance function. In practice, the co-variance function is usually chosen for computational convenience rather thanon principled grounds, and depends upon a parameter governing the corre-lation decay rate, i.e. a bandwidth parameter. Since according to the folkwisdom of kernel estimation, the shape of the kernel matters far less thanthe bandwidth, other aspects of the specification of the covariance functionmay not be of primary importance.

The correspondence of smoothing and stochastic models was pointed outin the case of smoothing splines in [10] and the implicitly determined ker-nel was studied in [19]. The relationship of smoothing splines to stochasticrandom effects models was exploited in [1] and [21].

Let us take a look at how estimation proceeds in such models with aview of learning what may be useful when viewed from the FDA perspective.Consider the simplest case, involving no covariates other than time, a meanfunction, depending upon linear parameters, individual random curves de-pending on random linear parameters, and a given covariance function. Themodel is

y(tij) = µ(tij) + fi(tij) + εij (3)

where tij, i = 1, . . . , n, j = 1, · · · , ni is the time of the jth measurementof the ith individual. The EM algorithm [11] proceeds iteratively. Givenestimates at stage k,

1. f(k+1i (t) = weighted linear combination of {y(tij) − µ(k)(tij)}

2. µ(k+1)(t) = weighted linear combination of the averages of {y(tij)−f (k)i (tij)}

3. Parameters of the covariance function are estimated by maximum like-lihood based on the expected values of the sufficient statistics.

This suggests an FDA analogue. As an illustration, I will use data onchildrens’ gaits ([13], [16]), the angles formed by each child’s ankle during asingle phase of a gait cycle as shown in figure 1.

A two-pass procedure was used, mimicking the first two EM steps above:the individual curves were first smoothed, the average of the smooths wasfound, this average was subtracted from the individual data points, and

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Figure 1: The angles formed by the ankles of 39 children during a singlephase of a gait cycle.

the results were smoothed again. (For a more elaborate related proceduresee [23].) A local linear smoother with a fixed bandwidth equal to 0.2 wasused. (No attempt to adaptively determine the bandwidth was made. Ananalogue would be to use cross validation in place of the maximum likelihoodestimation above, but I have not investigated this further.) The resultingestimate for a single child is shown in figure 2 and compared to a one-passsmooth with the same bandwidth. One usually thinks of borrowing strengthto reduce variance, but here by using all the curves in conjunction, the biasin estimating individual curves is substantially decreased!

Other possibilities for smoothing functional data are suggested by repre-senting the first two EM steps as

1. f(k+1)i = Smooth (Yi − µ(k))

2. µ(k+1) = Smooth (Average{Yi − f(k)i })

The smoothing operator could be nonlinear, for example robust or con-strained to be monotone. It could be based on local likelihoods. Similarly,averaging could be robust. The centering (subtraction of the mean) in thefirst step could be with respect to a registration template, with respect to asubset of the data such as a set of nearest neighbors, or with respect to the

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Figure 2: Comparison of one-pass smoothing (dashed line) and two-passsmoothing (dotted line) on ankle angle measurements (circles).

conditional expectation given a covariate. Looking ahead to the next section,the centering could be formed by a projection onto a set of basis functions.

3 Basis Function Methods

Both the FDA and LDA literature have a history of using expansions in basisfunctions, but the points of view have been rather different. The typical FDAapproach is to fit each curve individually by ordinary least squares to dataon a regular grid as

fi(x) =K∑

k=1

Ukiψk(x), (4)

where the functions {ψk(x)} form a basis and the Uki are the (random) co-efficients of curve i. As the dimension K of the subspace grows, increasinglyoscillatory functions are admitted. The choice of the truncation point isfrequently made by some sort of cross-validation procedure. This finite di-mensional representation is avowedly an approximation and the basis func-tions are chosen for good approximation properties. In FDA, the covariancestructure is of direct interest rather than viewed as a nuisance parameter

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complicating inferences about fixed effects. It can be estimated from theobservations y(tij) if they are on a regular grid.

The basis formed by the eigenfunctions of the covariance function isstressed in FDA [16]. Because of well known extremal properties of theeigenfunctions and eigenvalues, it is optimal in some senses. It affords acompact representation in which the leading functions often provide an in-terpretable description of the major modes of variation in the collection ofcurves. The underlying notion is that although the curves are evaluated at alarge number of points, there are not a large number of important “degreesof freedom” in the ensemble.

In the LDA tradition the design points would be sparse and perhapsirregularly spaced. The data vector of subject i might be modeled as

Yi(tij) = µi(tij) +

K∑

k=1

Ukiψk(tij)) + εi(tij). (5)

Here µi(t) is a parametric fixed-effects function, perhaps involving covariates;these fixed effects are of primary interest. The functions ψk(t) are typicallypolynomials andK is small, so that perhaps only linear or quadratic functionsare included. The observation noise process εi(t) is either specified to bewhite noise or a convenient continuous time stationary process; the Ornstein-Uhlenbeck process is very popular. The covariance structure of the randomtrajectories Yi(t) is thus determined by the functions ψk(t), the covarianceof the coefficients Uki and the covariance function of ε(t). This covariancestructure is not of primary interest and is largely specified a priori. Forlinear, Gaussian models, the smoothed observations are

Yi(t) = E[(µi(t) +

K∑

k=1

Ukiψk(tij))|Yi)] = WiYi (6)

where Wi is a matrix of weights derived from the covariance structure and Yi

is the vector of observations on subject i. Dropping the Gaussian assumption,the estimate is the best linear approximation to the Bayes estimate, theBLUP; “best” given the model, of course.

It is interesting to contrast the estimates of the scores of individual curveswith respect to a given eigenfunction as they would typically be constructedby FDA and LDA. For simplicity of notation, suppose that the data are on aregular grid. Let Y be the observations of a curve, which are assumed to be

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corrupted by white noise with variance σ2. The FDA-style estimate of thescore on the k-th eigenfunction ψk with eigenvalue λk would be Zk = 〈Y, ψk〉whereas the LDA-style estimate would be Zk = (λk/(λk + σ2))〈Y, ψk〉. Thelatter damps the contributions from the eigenfunctions with small eigenval-ues, which are typically rough, and hence smooths more. (Neither of theseestimates takes explicitly into account that the eigenfunctions and eigenval-ues are in fact estimated from the data.) This approach is explored in [25].

A hybrid approach was proposed in [18] and [15]. Basis functions wereused as an approximating sieve as in FDA, but fit to the irregular and sparsedata typical of LDA, using standard algorithms for Gaussian mixed models.Both papers used splines, but other basis functions could be used as well.Adaptive, data-driven methods were considered for choosing the truncationpoint K. The estimate of the covariance function of the underlying randomcurves was thus determined by the estimated covariance function of the ran-dom coefficients Uki and the basis functions ψk, from which eigenfuntionscould be estimated if desired. (An alternative approach of estimating theeigenfunctions and eigenvalues directly from sparse irregular data is givenin [8].) The procedure thus attempts to estimate a low frequency approxi-mation to the covariance function directly, without assuming it to be of anarbitrarily chosen parametric form, such as an Ornstein-Uhlenbeck process.

4 Interpolation and Extrapolation

Specification of the mean and covariance function, as in framework of theprevious section, allows the interpolation of missing data via (6).(Since thecomplete data are curves, almost all the data is missing.) Alternatively, themean and covariance function can be estimated directly by smoothing scat-terplots, [20], [5]. It appears that there has been little work on analyzing theconsequences of the nature and degree of smoothing done in estimating thecovariance function on the bias and variance of the “BLUP.” (The presenceof the B and the U in this context is not really appropriate, nor is the L,since the covariance function is estimated, leaving a rather uninformativeacronymn.)

The apparent effectiveness of these linear methods is quite surprisinggiven that the data are sparse and irregular and perhaps quite non-Gaussian.A typical task is to estimateE(Y (t0)) given Y (t1), Y (t2), Y (t3), say, and theremay be no instances of data for which Y is observed at these four points.

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Perhaps the key to the effectiveness of the nonparametric BLUP is that theestimate is additive, i.e. that the estimate is a weighted average of simpleunivariate linear predictors.

Multivariate normality, which only requires information on pairs to con-struct a full distribution, must be a quite dubious model in many longitudinaldata sets. One would think that in many applications the points on the ran-dom curves would be highly constrained in a non-linear way and that thedistributions would be concentrated on non-linear manifolds. Indeed thiscan be seen quite dramatically in the gait data. To empirically examine thejoint distribution of the ankle trajectories, I interpolated the data linearlyat 20 equispaced points and then used xgobi1 to search for non-Gaussianprojections of the 39 20-vectors. Two interesting projections are shown infigure 3.

Figure 3: Two highly non-Gaussian projections of ankle-angle trajectoriesobtained by projection pursuit.

Thus one would expect that modern nonparametric regression procedureswould be useful, but there has been little work on how to incorporate them inthe context of functional data analysis (for example, [14] is largely devoted tothe extension of classical Gaussian multivariate analysis to the functional set-ting). It would certainly be worthwhile to examine the feasibility of suitablenonparametric additive models [6]. More precisely, a family of smoothly re-lated additive models would be required to handle the various configurationsof observed data.

1http://www.research.att.com/areas/stat/xgobi/

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The results of a simple experiment illustrate the potential for using non-linear methods for estimating missing data. Data points with t < 0.3 weredeleted from an ankle-angle curve, a smooth curve was then fit to the re-maining points, and the seven nearest neighbors among the smoothed curvesfor the other subjects were found and averaged. This average alone wouldnot be a suitable estimate of the missing curve, because there would be adiscontinuity where it joined the observed curve. Continuity of the curve andits derivative were enforced in the following way: to extrapolate at a pointt < 0.3, a weighted linear combination of the nearest neighbor average and alinear extrapolation from the observed curve was computed, with the weightsdecaying linearly from w(0) = 1 to w(.3) = 0. Since the point is to illustratethe plausibility of nonlinear methods in general and of a nearest-neighborbased method in particular, no attempt was made to fine-tune the proce-dure; for example a weighted average of nearest neighbors could be found,the number of neighbors varied, and the extrapolation weights varied. Figure4 shows the results of performing this on four randomly selected curves, withvarying degrees of success.

5 Covariates

Linear models for functional and scalar covariates have been developed in theFDA literature, e.g [14],[4]. For the most part, though, this nonparametricpoint of view has not permeated the LDA literature, and has not addressedcommon problems such as data sparsity, irregularity, censoring and trunca-tion. Within the framework of basis function modeling the individual randomcurves are reduced to finite vectors of coefficients, making it possible to ex-amine the dependence of the curve shapes on covariates, via classical linearor modern nonparametric techniques. Nonparametric methods of clusteringand classification can be based on the coefficients.

To illustrate some possibilities for going beyond classical methods, andusing nonparametric smoothing techniques for informal data analysis, I willuse some data kindly provided by Hans Muller and Jane-Ling Wang on dailyfertility of a cohort of 1000 medflies ([3], [2]). Some examples of individualfertility curves are shown in figure 5. There are large daily fluctuations foreach fly and large variation from fly to fly.

Let Yi(t) denote the number of eggs laid on day t by fly i and let Li denoteits lifetime. We wish to examine the relationship between lifetime and the

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Figure 4: Four examples of using nearest neighbors to fill in missing data fort < 0.3. The original data are shown as points, the smooths using all thedata are solid lines, and the filled in data are shown by dashed lines.

function Y (t) (one point that is noteworthy is that Y (t) is only defined for1 ≤ t ≤ L). This endeavor is often referred to as “joint modeling” in the LDAliterature ([24], [22], [7]). The raw data are shown in figure 6, in which theegglaying profiles are ordered by longevity. Even if the figure is interactivelyviewed from many angles, the large variability makes it very hard to get asense of underlying patterns.

We thus consider smoothing, to form a nonparametric estimate of E(Y (t)|L),for t = 1, . . . , L. We construct this estimate in two stages. An initial estimateof E(Y (t)|L), for each fixed t, is constructed by local polynomial smoothingwith respect to L, using all the cases for which the lifetime is greater thanor equal to t. The results of of this are shown in the left panel of figure 7 (Asimilar plot appears in [2]). Next, fixing L = `, the function E(Y (t)|L = `)

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Figure 5: Fertility curves of nine medflies. Number of eggs produced is shownon the vertical axis and time in days is shown on the horizontal axis.

is estimated by smoothing the previous result with respect to t, yielding theresults shown in the left panel of the figure. In the second stage smoothing,I preserved the initial run of 0’s for small t that was produced in the firststage. There is an interesting tradeoff between smoothing with respect to Land with respect to t; in this case it seems advantageous to smooth with re-spect to L first, since the subsequent smoothing with respect to t, if needed,can be quite light. The figures show that the initial period of latency in-creases with the lifetime of the fly and that the time of peak egg laying shiftslater in time with increased lifetime. The contours slope gradually away fromthe vertical axis and there is an interesting valley starting around day 60.The longer lived flies exhibit a second fertility peak at about 35 days.

The conditional expectation of Y (t) given total egg production was esti-mated in a similar way. The results are shown in figure 8. In the left panel,

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Figure 6: Egg laying profiles ordered by lifetime

there is little sign of the rotation of the contours seen in figure 7, althougha hint of second peak production period appears for the most prolific flies.The right panel shows the result of normalizing each egg laying profile by thetotal number of eggs laid. With the exception of some deviation for smalltimes and some broadening as production increases, the curves exhibit littlechange with increasing total fertility. This is consistent with the multiplica-tive effects model of [4], which used functional principal components andcovariates.

Figure 7 suggests that lifetimes may be somewhat predictable from theegg production profiles, although the figure shows the conditional expectationof the latter given the former, rather than visa-versa. To explore this possi-bility, I conducted a simple illustrative experiment. The egg laying profilesfor the first 30 days of medflies which lived that long were used to attempt

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Figure 7: Left panel: smoothing with respect to lifetime. Right panel:smoothing the results shown in the left panel with respect to time.

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to predict the corresponding lifetimes. There were 667 such. The individualprofiles were smoothed with a local linear smoother, k(x) ∝ (1 − x2)2 witha bandwidth of 15 days. The lifetime of an individual medfly was then pre-dicted as the mean lifetime of its 25 nearest neighbors, using as a distancefunction the l1 norm. The resulting correlation was 0.27, so there was some,but not much, predictability by this method. (In general, correlations aresmall in this data, for example the correlation between lifetime and cumula-tive production for the first 15 days is 0.18 and the correlation between thetotal number of eggs in the first 30 days and the lifetime beyond 30 days isless than 0.1). A little exploration of the effects of modifying the weightsin the norm and the weighting of nearest neighbors made little appreciable

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difference. The point of this example is to illustrate the possibility of usingnonparametric regression techniques in such contexts, not to find the optimalsuch; for example, it would be natural to try tree-based regression methodsbased on features of the curves, such as the latency time noted above. Ananalysis of a very different spirit, based on a Cox model with a time-varyingcovariate, is presented in [12].

6 Final Remarks

One of the themes of this paper is that there are opportunities for a largerrole to be played by modern nonparametric methods in longitudinal dataanalysis ([26] is an interesting example). I have illustrated possibilities byusing nearest neighbor methods. These may not ultimately be the mosteffective, but they do have the advantage of applicability in non-linear, non-Gaussian situations; if the curves do tend to lie on low dimensional manifolds,this structure is tracked implicitly by nearest neighbor methods without thenecessity of explicit construction.

7 Acknowledgements

This work was supported by a grant from the National Science Foundation.I also wish to thank Hans Muller and Jane-Ling Wang for sharing the medflydata, for several very helpful conversations, and for comments on an earlierdraft of this manuscript. More generally, I would like to thank the partici-pants in the conference for sharing many interesting ideas.

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