Generalized Functional Concurrent Model · Summary: We propose a ﬂexible regression model to study the association between a functional response and a functional covariate that

Biometrics , 1–25

2015

Generalized Functional Concurrent Model

Janet S. Kim∗, Arnab Maity∗∗, and Ana-Maria Staicu∗∗∗

Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695, U.S.A.

*email: [email protected]

**email: [email protected]

***email: ana-maria [email protected]

Summary: We propose a flexible regression model to study the association between a functional response and a

functional covariate that are observed on the same domain. Our modeling describes the relationship between the

mean current response and the covariate by a smooth unknown bi-variate function that depends on both the current

value of the covariate and the time point itself. We develop estimation methodology that accommodates realistic

scenarios where the covariates are sampled with or without error on a sparse and/or irregular design, and prediction

that accounts for unknown model correlation structure. In this framework we also discuss the problem of testing

the null hypothesis that the covariate has no association with the response. The proposed methods are evaluated

numerically through simulations and two real data applications.

Key words: Functional data analysis; F-test; Generalized Functional Concurrent Model; Penalized B-splines;

Prediction.

This paper has been submitted for consideration for publication in Biometrics

Generalized Functional Concurrent Model 1

1. Introduction

Regression problems where both the response and the covariate are functional have become

more and more common in many scientific fields including medicine, finance, agriculture

to name a few. Such regression problems are often called function-on-function regression,

and their primary goal is to investigate the association between the response and predictor

functional variables. In this article, we propose a flexible concurrent model that relate the

current response to the predictor using a smooth unknown bivariate function that depends

on the current predictor and the specific time-point. We develop estimation, inference and

testing procedures for the functional covariate effect. The performance of the methods is

evaluated numerically in finite samples and two data applications, gait study (Olshen et al.,

1989; Ramsay and Silverman, 2005) and dietary calcium absorption study (Davis, 2002;

Senturk and Nguyen, 2011).

Functional regression models where both the response and the covariate are functional

have been long researched in the literature. Functional linear models rely on the assumption

that the current response depends on full trajectory of the covariate. The dependence is

modeled through a weighted integral of the full covariate trajectory through an unknown

bivariate coefficient surface as weight. Estimation procedures for this model have been

discussed in Ramsay and Silverman (2005), Yao, Muller, and Wang (2005b) and Wu, Fan,

and Muller (2010), among others. The crucial dependence assumption for this type of models

may be impractical in many real data situations. To bypass this limitation, one might use

the functional historical models (see e.g., Malfait and Ramsay (2003)), where the current

response is modeled using only the past observations of the covariate. Such models quantify

the relation between the response and the functional covariate/s using a linear relationship

via an unknown bivariate coefficient function. Another alternative is to assume a concurrent

relationship, where the current response is modeled based on only the current value of the

2 Biometrics, 2015

covariate function. Functional linear concurrent models assume a linear relationship; they

can be thought of a series of linear regressions for each time point, with the constraint that

the regression coefficient is a univariate smooth function of time. Estimation procedure for

such models have been described in Ramsay and Silverman (2005) and Senturk and Nguyen

(2011). While the linear approach provides easy interpretation for the estimated coefficient

function, it may not capture all the variability of the response in practical situations where

the underlying relationship is complex.

In this article, we consider the functional concurrent model where we allow the relationship

between the response and the covariate function to nonlinear. Specifically, we propose a

model where the value of the response variable at a certain time point depends on both

the time point and the covariate value at that time point through a smooth bivariate

unknown function. Such formulation allows us to capture potential complex relationships

between response and predictor functions as well as better capture out of sample prediction

performance, as we will observe in our numerical investigation. Our model contains the

standard linear concurrent model as a special case. We will show through numerical study

that when the true underlying relationship is linear (that is, the linear concurrent model is

in fact optimal), fitting our proposed model does not result in loss of prediction accuracy. On

the other hand, when the true relationship is nonlinear, fitting the linear concurrent model

results in high bias and loss of prediction accuracy.

In this paper, we make two main contribution. First, we propose a general nonlinear func-

tional concurrent model to describe association between two functional variables measured

on the same domain, and develop prediction of a new response when only a new covariate

and its evaluation points are available. We model the unknown regression function using

tensor products of B-spline basis functions and develop a penalized least squares estimation

procedure in conjunction with difference penalties (Eilers and Marx, 1996; Li and Ruppert,


2008). General implementation methods of the tensor product splines can be found in

Marx and Eilers (2005), Wood (2006a) and Wood (2006b). An important innovation of our

methodology is that it accounts for correlated residual structure. Specifically, the estimators

are calculated based on a working independence assumption, and the analytical form of

their standard error, involving the covariance of the error process, is derived. The unknown

covariance is then estimated by employing standard functional principal component analysis

(FPCA) based methods (see e.g., Yao et al. (2005b); Di et al. (2009); Goldsmith, Greven,

and Crainiceanu (2013)) to the resulted residuals. The final standard error bands are based

on the analytical expressions with plug-in estimator of the model covariance. We also develop

point-wise prediction intervals for new response curves.

Second, we develop a testing procedure for evaluating the global effect of the functional

covariate, that is, to test the null hypothesis that the functional covariate has no association

with the response variable. We propose a F ratio type test statistic (see e.g., Shen and

Faraway (2004); Xu et al. (2011)), and propose a resampling based algorithm to construct

the null distribution of the test statistics. Our resampling procedure also takes into account

the correlated error structure, and thus maintains nominal type I errors.

The paper is organized as follows. In Section 2 we introduce the generalized functional

concurrent model (GFCM) and its estimation procedure. Section 3 discusses prediction of

new response trajectories. Section 4 deals with resampling based hypothesis testing. Section 5

presents how to implement the proposed model and various extensions. In Section 6, we

investigate the empirical performance of our method through a simulation experiments and

real data applications. Section 7 provides a brief discussion.

4 Biometrics, 2015

2. Generalized Functional Concurrent Model

2.1 Modeling Framework

Suppose for i = 1, 2, . . . , n, we observe {(Wij, tij) : j = 1, 2, . . . , mX,i} and {(Yik, tik) : k =

1, 2, . . . , mY,i} where Wij’s and Yik’s denote the covariate and response, respectively, observed

at points tij and tik. It is assumed that tij, tik ∈ T for all i, j and k, and Wij = Xi(tij) + δij

where Xi(·) is a real-valued, continuous, square-integrable, random curve defined on the

compact interval T ; for convenience we take T = [0, 1] throughout the paper. Here δij is the

measurement error and it is assumed that δij are independent and identically distributed

with mean zero and variance τ 2. To illustrate ideas, we first consider Wij = Xi(tij), which

is equivalent to τ 2 = 0 and furthermore that tij = tj, tik = tk and furthermore that mX,i =

mY,i = m. We treat Yik = Yi(tik) to emphasize the dependence on the time points tik.

Adaptation of our methods to more realistic scenarios where τ 2 > 0 or the different sampling

design for Xi’s and Yi’s are discussed in Section 5.2. The main objective of the paper is

to provide flexible association models that relate the response at current time point to

the current predictor, and to develop a testing procedure to assess whether there is any

association between the predictor and response.

We introduce the functional generalized concurrent model

Yi(t) = F{Xi(t), t}+ εi(t), (1)

where F (·, ·) is an unknown smooth function defined on R×T , and εi(·) is a mean-zero error

process uncorrelated with the predictor Xi(·). The standard functional linear concurrent

model (Ramsay and Silverman, 2005; Ramsay, Hooker, and Graves, 2009) is a special case

of model in (1) with F (x, t) = β0(t) + xβ1(t), where β0(·) and β1(·) are unknown coefficient

functions. We introduce two main innovations in (1). First, the general bivariate function

F (·, ·) allows us to model a potentially complicated relationship between Yi(·) and Xi(·)

and extends the effect of the covariate beyond linearity. Second, we also allow the error


process εi(·) to have its own covariance structure unlike the standard concurrent models

where it is typically assumed that the errors are independent realizations from a white noise

process. To be specific, we assume that cov{εi(s), εi(t)} = G(s, t) where G(·, ·) is a smooth

unknown covariance function. The model dependence F (Xi(t), t) is inspired from McLean

et al. (2014) who used it to describe flexible association models between a scalar response

and one or multiple dense functional covariates. Here we consider its application in cases

when the response is functional, with unknown covariance structure, and the covariate is

also functional data observed densely as well as sparsely with or without error.

In the following we will develop an estimation procedure for the model components F (·, ·)

and G(·, ·), discuss prediction of a new trajectory, and develop a testing procedure to formally

assess the association between the response Yi(·) and the true latent predictor Xi(·) in this

general framework. The inferential procedures account for the nontrivial dependence within

the subject. Our model and methodology are presented for the setting involving a single

functional covariate; nevertheless they can be extended to incorporate other vector-valued

covariates via a linear or smooth effect.

2.2 Estimation

We model F (·, ·) using bivariate basis expansion using tensor product of B-splines as basis

functions (see e.g., Marx and Eilers (2005); McLean et al. (2014)). Specifically we write

F (x, t) =∑Kx

k=1

∑Ktl=1θk,lBX,k(x)BT,l(t),

where {BX,k(x) : k = 1, 2, . . . , Kx} and {BT,l(t) : l, 2, . . . , Kt} are B-splines defined on [0,1],

and θk,l are unknown parameters. Then, model (1) can be written as

Yi(t) =∑Kx

k=1

∑Ktl=1θk,lZi,k,l(t) + εi(t), (2)

where Zi,k,l(t) = BX,k{Xi(t)}BT,l(t), and Kx and Kt are the number of basis functions used.

A larger number of basis functions would result in a better but rougher fit, while a small

6 Biometrics, 2015

number of basis functions results in overly smooth estimate. As is typical in the literature,

we use rich bases to fully capture the complexity of the function and penalize coefficients

to ensure smoothness of the resulting fit. Such strategies are discussed in Marx and Eilers

(2005), Wood (2006a), Wood (2006b) and Goldsmith et al. (2012).

Define the KxKt-vector Zi(t) = [Zi,k,l(t)]l=1,...,Kt

k=1,...,Kx. Similarly, define the KxKt-vector of

unknown coefficients Θ = [θ1,1, . . . , θ1,Kt , . . . , θKx,1, . . . , θKx,Kt ]T . We can rewrite (2) as

Yi(t) = Zi(t)Θ + εi(t).

This model can be viewed as a linear model with unknown parameter vector Θ, except that

Yi(·) is defined as a smooth function, and that the error process εi(·) is a mean-zero process

with covariance G(·, ·).

To prevent overfitting, we propose to estimate the unknown parameter Θ by a penalized

criterion

∑ni=1||Yi(·)−Zi(·)Θ||2 + ΘT PΘ,

where ||·||2 is defined as common L2-norm corresponding to the inner product < f, g >=∫

fg,

and P is a penalty matrix containing penalty parameters that regularize the trade-off between

the goodness of fit and the smoothness of fit. In practice, we observe Yi(·) and Xi(·) on only

a finite and dense grid of points t1, . . . , tm. We can approximate the sum of squares as

PENSS(Θ) =∑n

i=1

∑mj=1{Yi(tj)−Zi(tj)Θ}2/m + ΘT PΘ. (3)

The case where the grid of points is irregular and sparse is presented in the Web Appendix.

The penalty matrix in (3) is defined as P = λxPTx Px +λtP

Tt Pt and depends on two penalty

parameters λx and λt. The parameters λx and λt control the smoothness in the direction of

x and t, respectively. The row penalty Px = Dx⊗

IKt and column penalty Pt = IKx

⊗Dt as

defined as in Eilers and Marx (1996), Marx and Eilers (2005) and McLean et al. (2014). Here


⊗stands for the Kronecker product, IK is the identity matrix with dimension K, and Dx

and Dt are matrices representing the row and column of second order difference penalties.

An explicit form of the estimator Θ is readily available for fixed values of the penalty param-

eters. Define the m-dimensional vectors of response and covariate Y i = [Yi(t1), Yi(t2), . . . , Yi(tm)]T

and X i = [Xi(t1), Xi(t2), . . . , Xi(tm)]T . Similarly, define the m-dimensional vectors of errors

Ei = [εi(t1), εi(t2), . . . , εi(tm)]T . Define matrices Zi such that the j-th row of Zi is Zi(tj).

The estimator Θ can be computed as

Θ = {∑ni=1ZT

i Zi + P}−1{∑ni=1ZT

i Y i}. (4)

In practice, the penalty parameters λx and λt can be chosen based on some appropriate

criteria. For example one can use generalized cross validation (GCV), restricted maximum

likelihood (REML), and maximum likelihood (ML) to estimate the smoothing parameters.

Estimation under (3) can be fully implemented in R using functions of the mgcv package

(Wood, 2011).

2.3 Variance Estimation

The penalized criterion (3) does not account for the possibly correlated error process. While

one can use standard software for solving the penalized least square problem, the default

variance estimates, based on working independent error structure, may lead to incorrect

inference. The variance of the parameter estimate Θ can be found as

var(Θ) = H{n∑

i=1

ZTi GZi}HT

where H = {∑ni=1ZT

i Zi+P}−1 and G = cov(Ei) = [G(tj, tk)]16j,k6m is the m×m covariance

matrix evaluated at the observed time points.

To address this problem, we assume that the error process has the form ε(t) = r(t) + e(t),

where r(t) is a zero-mean smooth stochastic process and e(t) is a zero-mean white noise

measurement error with variance σ2. Let Σ(s, t) be the autocovariance function of r(t). It

8 Biometrics, 2015

follows that the autocovariance of the random deviation ε(t), G(s, t) = Σ(s, t) + σ2I(s = t)

where I(·) is the indicator function, is unknown and needs to be estimated. To this end, we

assume that Σ admits a spectral decomposition Σ(s, t) =∑

k>1 φk(s)φk(t)λk where {φk(·), λk}

are the eigencomponents. We first compute the residuals εij = Yi(tj)−Zi(tj)Θ from the model

fit, and employ FPCA methods to estimate φk(·), λk, and σ2. Then we estimate G by

G(s, t) ≈ ∑Kk=1λkφk(s)φk(t) + σ2I(s = t),

for any s, t ∈ [0, 1], where K is the number of principal components. This truncation point is

typically chosen by setting the percent of variance explained by the first few eigencomponents

to pre-specified value such as 90% or 95%.

3. Prediction

A main focus in this paper is prediction of unobserved response when a new covariate and

its evaluation points are given. Suppose that we wish to predict new, unknown, response

Y0,i′(tk) for k = 1, 2, . . . , m when new observations X0,i′(tj) (j = 1, 2, . . . , m) are given. We

typically index subjects from new data by i′ ∈ {1, 2, . . . , n′} to differentiate from the index

i ∈ {1, 2, . . . , n} of original observed data. Then we assume that the model

Y0,i′(t) = F{X0,i′(t), t}+ ε0,i′(t) (5)

still holds for the new data, where the error process ε0,i′(t) has the same distributional

assumption as εi(t) in (1) and is uncorrelated with the new covariate X0,i′(t). For each

subject i′ = 1, . . . , n′, we predict the new response Y0,i′ by

Y0,i′(t) =∑Kx

k=1

∑Ktl=1θk,lZ0,i′,k,l(t),

where Z0,i′,k,l(t) = BX,k{X0,i′(t)}BT,l(t), and θk,l are estimated based on (4).

Uncertainty in the prediction depends on how small the difference is between the predicted

response Y0,i′(t) and the true response Y0,i′(t). We follow an approach similar to Ruppert,


Wand, and Carroll (2003) to estimate the prediction variance. Specifically, conditional on

the new covariate, we have

var{Y0,i′(t)− Y0,i′(t)} = var{ε0,i′(t)}+ var{Y0,i′(t)}.

Note that ε0,i′(t) in (5) is unobserved error process. However we can calculate the variance of

the ε0,i′(t) relying on the assumption that the error process in the prediction set have the same

covariance structure as in the original observed data set. Define Z0,i′ = [Z0,i′,k,l(t)]l=1,...,Kt

k=1,...,Kx.

Then the prediction variance becomes

var{Y 0,i′ − Y 0,i′} = G + Z0,i′H{∑ni=1ZT

i GZi}HTZT0,i′ , (6)

where the j-th row of Z0,i′ is Z0,i′(tj), G is the m ×m covariance matrix evaluated at the

observed time points. Note that this calculation involves both indices, i′ of the new data and

i of the original data. The prediction variance can be estimated by plugging-in the sample

estimate of G(·, ·) in (6). One can further define a 100(1−α)% point-wise prediction interval

for the new observation Y0,i′(t) by

C1−α,i′(t) = Y0,i′(t)± Φ−1(1− α/2)[var{Y0,i′(t)− Y0,i′(t)}]1/2

where Φ(·) is the standard Gaussian cumulative distribution function (cdf).

In the more general case when the new covariates X0,i′(tj) are only observed on a sparsely

sampled grid, we first compute the FPCA scores for the new covariates via the conditional

expectation formula (see e.g., Yao et al. (2005a)) with the estimated eigenfunctions from

the training data, and construct smooth versions of the new covariates. Then the procedure

described above can be readily applied for prediction.

4. Hypothesis Testing

In many situations, testing for association between the response and predictor variables is as

important, if not more, as it is to estimate the model components. Often before performing

10 Biometrics, 2015

any estimation, it is preferred to test for association first to determine whether there is

association to begin with and then a more in-depth analysis is done to determine the form

of the relationship. In this section we consider the problem of testing whether the functional

predictor variable is associated with the response. Specifically, we want to test

H0 : F{X(t), t} = F0(t) versus H1 : F{X(t), t} = F1{X(t), t},

where F0(t) is a univariate function of t, and F1(x, t) is the bivariate function of x and t.

Our testing procedure is based on first modelling the null effect F0(t) and then the full

model F1(x, t) using basis function expressions in a manner that ensures that the null

model is nested within the full model. Specifically, we propose to use BT = {BT,0(t) =

1, BT,l(t), l > 1} to model F0(·) under the null model, where BT,l(t) (l > 1) are the B-

splines evaluated at time point t. To model F1(·, ·) under the full model, we use the same

set of basis functions BT for t, but BX = {BX,0(x) = 1, BX,l(x), l > 1} for x, where

BX,l(x) (l > 1) are the B-splines evaluated at x. Then under the full model, we can write

F1(x, t) = F0(t)+∑∞

k=1

∑∞l=0 BX,k(x)BT,l(t)θk,l. We propose to compare the two models using

the F ratio

Tn =(RSS0 −RSS1)/(df1 − df0)

RSS1/(N − df1), (7)

where RSSi and dfi (i = 0, 1) are the residual sums of squares and the effective degrees-of-

freedom (Wood, 2006a), respectively, under the null and the full models. Here N denotes the

total number of observed data points. In this case, we have n subjects and m observations per

subject, and thus, the total number of observed data becomes N = nm. This can be easily

generalized when each subject have different number of observations. However, traditional

F tests only make sense when asymptotics of test statistics are evident. In our cases, it is

difficult to derive asymptotics of proposed test statistic in (7) due to potential uncertainty

caused by applying smoothing techniques and penalization. Mostly, the degrees-of-freedom


in test statistics will be affected by the penalization, and one might not achieve good level

and power of a test by simply using the critical values from an F distribution.

We instead propose to use a resampling based algorithm to construct an empirical sampling

distribution for the test statistic; the steps are as follows.

(1) Fit the full model F1(x, t) based on the estimation procedure of the GFCM, and obtain

residuals {εi(tj)}ni=1 by calculating Yi(tj)− Yi(tj) for each i and j.

(2) Fit the null model F0(t), and obtain its estimate F0(t).

(3) Calculate the value of the test statistic in (7) based on the null and the full model fits;

call this value Tn,obs.

(4) Resample B sets of bootstrap residuals E∗b(t) = {ε∗b,i(t)}n

i=1 (b = 1, 2, . . . , B) with

replacement from the residuals {εi(t)}ni=1 obtained in the first step.

(5) For b = 1, 2, . . . , B, generate response curves under the null model as Y ∗b,i = F0(t) +

ε∗b,i(t). This provides B bootstrap data sets {Xi(t), Y∗1,i(t)}n

i=1, {Xi(t), Y∗2,i(t)}n

i=1, . . .,

{Xi(t), Y∗B,i(t)}n

i=1.

(6) Given the B bootstrap data sets, fit the null and the full models, and evaluate the test

statistic in (7) for each data set, {T ∗b }B

b=1. These can be viewed as realizations from the

distribution of Tn under the assumption that H0 is true.

(7) Compute the p-value by p = B−1 ∑Bb=1 I{T ∗

b > Tn,obs}.

Our proposed resampling algorithm has two advantages. First, the exact form of the null

distribution of the test statistic Tn is not required; the resampled version of the test statistic

approximates the null distribution automatically. Second, our algorithm allows us to account

for correlated error process ε(·). This is automatically done by sampling the entire residual

vectors for each subject; and thus preserving the correlation structure within the residuals.

We observed in our numerical study (results not shown) that preserving such correlation

12 Biometrics, 2015

structure is of particular importance, as ignoring the correlation results in severely inflated

type I error.

5. Model Implementation and Extensions

5.1 Transformation of Functional Covariates

One challenge with our estimation approach is that some B-splines might not have observed

data on its support; this issue has been also emphasized in McLean et al. (2014). This problem

typically arises when the realizations of the functional covariate Xi(tj) are not dense over R.

To bypass this difficulty, we propose to first apply point-wise center/scaling transformation

of the covariates; it is worthwhile to note that McLean et al. (2014) used a different approach

to address this problem. In the generalized concurrent model framework, we define point-wise

center/scaling transformation of X(t) by

X∗(t) = {X(t)− µX(t)}/σX(t)

where µX(t) and σX(t) are mean and standard deviation of X(t). One can interpret the

transformed covariate X∗(t) as the amount of standard deviation X(t) is away from the

mean at time t. In practice, we estimate the mean and the standard deviation by the sample

mean µX(t) and the sample standard deviation σX(t), respectively, of the covariates. Thus

for a fixed point tj we will obtain realizations of the transformed covariates {X∗i (tj)}n

i=1 based

on the sample mean µX(tj) and the sample standard deviation σX(tj) at the same point.

The GFCM based on the transformed covariate X∗(t) can be written as

Yi(t) = F ∗{X∗i (t), t}+ εi(t) =

∑∞k=1

∑∞l=1θ

∗k,lZ

∗i,k,l(t) + εi(t), (8)

where Z∗i,k,l(t) = BX,k{X∗

i (t)}BT,l(t) and θ∗k,l are the parameters to be estimated. Note that

Z∗i,k,l(t) and θ∗k,l are different from the previous quantities denoted by Zi,k,l(t) and θk,l in

model (2). To emphasize the difference, we introduce a new smooth function F ∗ in (8). Let


Z∗i (t) = [Z∗

i,k,l(t)]l=1,...,Kt

k=1,...,Kxand Θ∗ = [θ∗1,1, . . . , θ

∗1,Kt

, . . . , θ∗Kx,1, . . . , θ∗Kx,Kt

]T be the updated

notations for the model components. Then the parameter estimates Θ∗ can be found as

Θ∗ = {∑ni=1Z∗T

i Z∗i + P}−1{∑n

i=1Z∗T

i Y i} (9)

where the j-th row of Z∗i is Z∗

i (tj). Let G∗(·, ·) = cov{X∗(s), X∗(t)}. We compute the

variance of Θ∗ as var(Θ∗) = H∗{∑ni=1Z∗T

i G∗Z∗i }H∗T

, where H∗ = {∑ni=1Z∗T

i Z∗i + P}−1,

and G∗ is the m×m covariance matrix of the process X∗(·) evaluated at the observed time

points t1, . . . , tm.

The prediction procedure described in Section 3 proceeds as before with the understanding

that one now uses the transformed version of the new covariates. The dependence structure

between Yi(t) and Xi(t) is, in general, not the same as the one between Yi(t) and X∗i (t).

Hence, the model fit F ∗(x, t) of F ∗(x, t) cannot be used directly to investigate the relationship

between the response Yi(t) and the covariate Xi(t). Instead, we will compare the response

Yi(t) and its estimator Yi(t) because the estimator will be consistent without being affected

by the transformation of the covariates. If the true relationship between Yi(t) and Xi(t) is

well captured by the model fit, the difference between Yi(t) and Yi(t) will be very small.

5.2 Extensions

This section discusses modifications of the methodology that are required by realistic situa-

tions. In particular, we consider the case when the functional covariate is observed densely

with error, or sparsely with or without noise, as well as when the sparseness of the covariate

is different from that of the response.

Assume first that functional covariate is observed on a dense and regular grid of points

but with error; i.e. the observed predictors Wij’s are such that Wij = Xi(tj) + δij and the

deviation δij has variance τ 2 > 0. Several approaches have been proposed to adjust for the

measurement errors; Zhang and Chen (2007) proposed to first smooth each noisy trajectory

using local polynomial kernel smoothing and then estimate the mean and standard deviation

14 Biometrics, 2015

of the covariate Xi(tij) by their sample estimators. The recovered trajectories, say Xi(·) will

estimate the latent ones Xi(·) with negligible error. The methodology described in Section 2.2

can be applied with Xi(·) in place of Xi(·). Numerical investigation of this approach is

included in the simulation section.

Then consider the case that the functional covariate is observed on a sparse and irregular

grid of points with measurement error, i.e. Wij = Xi(tij) + δij. The common assumption

made for this setting is that the number of observations mX,i for each subject is small, but

⋃ni=1{tij}mX,i

j=1 is dense in T = [0, 1]. Reconstructing the latent trajectories Xi(·) is based on

employing FPCA for sparse design to the observed Wij’s. Yao, Muller, and Wang (2005a)

proposed to use local linear smoothers to estimate mean and covariance functions. The

latent trajectories are predicted using a finite Karhunen-Loeve truncation using estimated

eigenvalues/eigenfunctions from the spectral decomposition of the estimated covariance and

predicted scores using conditional expectation. This method may be further applied to the

response variable when the sampling design of the response is sparse as well. An alternative

for the latter situation, is to use the standardized prediction of the covariate at the time points

tij at which the response is observed, X∗i (tij) and then continue the estimation using the

data {Yij, X∗i (tij) : j}n

i=1. In preliminary investigation, the former method yielded slightly

increased type I error rates; the results included in the simulation section use the latter

method, which shows good performance in both estimation and testing evaluation.

6. Numerical Study

In this section, we investigate the finite sample performance of our proposed methodology.

We investigate the prediction accuracy of our method in Section 6.1. The performance of

our proposed test is investigated in Section 6.2. Finally in Section 6.3, we assess practical

behavior of our method by applying it to the gait data (Olshen et al., 1989; Ramsay and


Silverman, 2005; Crainiceanu et al., 2014) and the dietary calcium absorption data (Davis,

2002; Senturk and Nguyen, 2011) sets.

6.1 Prediction Performance

6.1.1 Simulation Design. We aim to investigate the prediction accuracy of our method for

both within sample and out of sample prediction. To achieve this, we construct training and

test data sets assuming both are independent. We generate the covariate Xi(t) of training

data set by a nonlinear random process a0i+a1i

√2 sin(πt)+a2i

√2 cos(πt) where a0i ∼ N(0, 1),

a1i ∼ N(0, 0.852) and a2i ∼ N(0, 0.702) for i = 1, 2, . . . , n. Assuming that the test data

follows the same distribution as the training data, we generate the covariate Xi′(t) of test

data set for i′ = 1, 2, . . . , n′. Throughout the study, it is assumed that the covariate Xi(t)

of the training data are not observed directly. Instead we observe Wij = Xi(t)+WN(0,τ 2),

where τ = 0.59. The response Yi(·) is generated using the model (1) using two choices for

function F (·, ·): a non-linear version FNL(x, t) = 1 + x + t + 2x2t and a linear counterpart

F L(x, t) = 1 + x + t. We also consider three different types of the error process for Ei: (1)

E1i ∼ N(0, σ2

ε Imi); (2) E2

i ∼ N(0, σ2ε Σ) + N(0, σ2

ε Imi) where Σ has ARρ(1) structure, and

(3) E3i ∼ ξi1

√2 cos(πt) + ξi2

√2 sin(πt) + N(0, σ2

ε Imi). We set σ2

ε = 0.8 and ρ = 0.2. The

random variables ξi1 and ξi2 are uncorrelated and are generated from N(0,2) and N(0, 0.752),

respectively.

For the training data set, we considered three different sample sizes n = 50, 100 and 300.

For each sample size setting, we consider two different sampling designs, dense and sparse. In

the dense design, we sample m = 81 equally spaced time points for all i. The sparse design

assumes that tij = tik for all i and the number of observations mi is randomly selected from

{20, . . . , 31} for each i, preserving 24.7% ∼ 38.3% of the data. The time points tij for

j = 1, 2, . . . , mi are uniformly generated from the dense set {tj = (j − 1)/80}81j=1. In the

16 Biometrics, 2015

Web Appendix, we further present the cases with reduced sparseness. For the test set, we

use n′ = 100 and m = 81.

For each of these choices we fit both the standard linear functional concurrent model

(FCM) and our proposed generalized functional concurrent model (GFCM). When the true

model is nonlinear, i.e. F (x, t) = FNL(x, t), one would expect the GFCM to perform better

than FCM. On the other hand, if the true model is in fact linear, i.e. F (x, t) = F L(x, t) and

FCM is indeed optimal in this case, such a comparison corresponds to whether the GFCM

captures the relationship between the covariate and the response as well as FCM.

6.1.2 Evaluation Criteria. We perform N = 1000 Monte-Carlo simulations. Our per-

formance measures are the root mean squared prediction error (RMSPE), the integrated

coverage probability (ICP) and the integrated length (IL) of prediction bands. Define the

in-sample RMSPE by

RMSPEin =∑N

r=1[MSPE(r)]12 /N,

where MSPE(r) = (nm)−1 ∑ni=1

∑mj=1{Y (r)

i (tij) − Y(r)i (tij)}2, and Y

(r)i (tij) and its estimate

Y(r)i (tij) are from the r-th Monte Carlo simulation. The out-of-sample RMSPE, denoted by

RMSPEout, is defined similarly.

We approximate (1−α) level point-wise prediction intervals to observe coverage probabil-

ities at the nominal level. The ICP at the (1− α) level is given by

ICP(1− α) =∑N

r=1

∑n′i′=1

∫T I{Y0,i′(t) ∈ C

(r)1−α,i′(t)}dt/(Nn′),

where C(r)1−α,i′(t) are the point-wise prediction intervals from the r-th Monte Carlo simulation

and I(·) is the indicator function. We observe particular features of the prediction intervals

by measuring their length. The IL of the (1− α) level prediction intervals is defined by

IL =∑N

r=1

∑n′i′=1

∫T 2MOE

(r)1−α,i′(t)dt/(Nn′)


with MOE(r)1−α,i′(t) = Φ−1(1−α/2)× sd{Y0,i′(t)− Y

(r)0,i′ (t)}. Although the IL of the prediction

intervals remains small, the band might fluctuate dramatically at some time point, and this

will demonstrate a poor performance of variance estimation. Hence, we examine the range

of the estimated standard errors (SE). For the prediction intervals, we define the minimum

SE by

min(SE) =∑N

r=1

∑n′i′=1 min

t{2MOE

(r)1−α,i′(t)}/(Nn′).

We define the maximum SE, max(SE), similarly. Then, R(SE)= [min(SE), max(SE)] provides

the range of SE at the (1 − α) level. For the prediction intervals, the nominal significance

level is set as 0.95, 0.90, and 0.85.

6.1.3 Simulation Results. We obtain both the GFCM and the linear FCM fits using

Kx = Kt = 7 cubic B-splines for x and t. When we apply FPCA, we set 99% of variance to be

explained by the first few eigencomponents. Table 1 summarizes the predictive performance

of the proposed procedure for different simulation scenarios.

We first discuss the case when the true function is nonlinear (the top two panels in Table 1).

We observe that RMSPEin and RMSPEout from fitting the GFCM are smaller than those

from the linear FCM in all scenarios. The ICPs from the GFCM and the linear FCM are fairly

close to the nominal levels of 0.95, 0.90, and 0.85. However, on an average the linear FCM

produces larger intervals, indicated by larger IL values, and the wider R(SE) compared to

the GFCM. Such patterns confirm that the variability in prediction is not properly captured

by the linear FCM when the true model is nonlinear. For less complicated error patterns

(e.g., independent versus AR(1)), both models produce smaller prediction errors. However,

GFCM still produces smaller errors compared to linear FCM. Also it is evident from Table 1

that the conclusions made above are true for different sample sizes as well as for dense and

sparse sampling designs.

We now explore the true linear model case (the bottom two panels in Table 1). When

18 Biometrics, 2015

the true model is in fact linear, we would expect that the linear FCM would provide the

optimal results as it is the correct model to fit in this situation. From the results in Table 1),

we observe that the RMSPEin and the RMSPEout from the GFCM and the linear FCM are

almost same in all scenarios considered here. We observe a similar pattern when we compare

other measures such as IL, R(SE) and ICP as well. This clearly implies that even when the

true model is linear and one fits the more complicated model using the GFCM, prediction

accuracy and coverage is not lost. Again, the number of subjects, sparseness of the sampling,

and the error structure also slightly affect to the numerical result. Nevertheless, when the

true model is linear, the overall predictive performance of the GFCM and the linear FCM is

almost identical.

To better visualize the results, Figure 1 displays prediction bands for the true nonlinear

model (top panel) and for the true linear model (bottom panel) for a a single data set

observed densely. In the top panel, the prediction bands for the linear FCM (dashed line)

are much wider compared to the bands for the GFCM (solid line). This indicates that variance

estimation from the linear FCM is less accurate. In the bottom panel, the prediction bands of

the GFCM (grey solid line) and the linear FCM (dashed line) are almost identical indicating

a similar performance in terms of the variance estimation when the true model is linear.

In summary, from our numerical investigation we observe that when the true model is

linear, fitting the GFCM does not result in any substantial loss in prediction accuracy over

the FCM. However, when the true model is in fact nonlinear, fitting GFCM results is a

significant gain in prediction accuracy over the standard FCM.

6.2 Testing Performance

We assess the performance of resampling based tests using the same settings in the previous

experiment. The true null model is defined as FBS(x, t, d) = 1+2t+ t2 +d(xt/8). If d is zero,

the true model is a univariate function of time point t; otherwise, the true model depends


on both x and t. Type I error of the test is investigated by setting d = 0, and power of the

test correspond to nonzero values of d. We report the performance of the algorithm based

on 1000 Monte Carlo simulations and B = 200 bootstraps for each simulation.

Given a null model F0(t) = FBS(x, t, 0), we first compare the actual test levels to prescribed

nominal levels α = 5% and 10%. In Web Table 1, we provide the rejection probabilities

averaged over 1000 replications. It is evident from our results that the actual test levels get

closer to the nominal level as the sample size increases. With correlated error structures, the

test levels still maintain close to the nominal level both in the dense and sparse designs.

We now calculate powers of our proposed test at nominal level α = 5%. We consider true

null models with d = 0.1, 1, 2, 3, 4, 5 and 6, and results are presented in Figure 2. It is evident

from the figure that for each case, the power increases with respect to the increasing values

of d. We also notice that the tests based on the densely sampled design are more powerful

than the sparsely sampled design, as one would expect.

Nevertheless, one might encounter an important issue when implementing our resampling

based algorithm. When calculating Tn,obs or T ∗b , we occasionally observe negative RSS0 −

RSS1 both in the densely and the sparsely sampled designs. For those cases, we set RSS0−

RSS1 = 0. We also recognize negative df0 − df1 rarely in the sparse design. Such cases are

excluded when calculating the actual test level and the power in all cases.

6.3 Applications

6.3.1 Gait Data. In a study of gait deficiency, it is important to identify how the joints

in hip and knee interact during a gait cycle (Theologis, 2009). Typically, one represents the

timing of events occurring during a gait cycle as a percentage of this cycle, where the initial

contact of a foot is recorded as 0% and the second contact of the same foot as 100%. As

an illustrative example, we analyze longitudinal measurements of hip and knee angles taken

from 39 children as they walk through a single gait cycle (Olshen et al., 1989; Ramsay and

20 Biometrics, 2015

Silverman, 2005; Crainiceanu et al., 2014). The hip and knee angles are measured at 20

evaluation points {tj}20j=1 in [0,1], which are translated from percent values of the cycle. In

Web Figure 1, we display the observed individual trajectories of the hip and knee angles.

In the analysis, the hip and knee angles are considered as densely observed functional

covariate and functional response, respectively. We first employ our resampling based test

to investigate whether the hip angles are associated with the knee angles. We select Kx =

Kt = 11 cubic B-splines for x and t to fit the GFCM, and B = 250 bootstrap replications

are used. The bootstrap p-value is computed to be less than 0.004, thus we conclude that

the hip angle measured at a specific time point has a strong effect on the knee angle at the

same time point.

To assess how the hip and knee angles are related to each other we fit our proposed GFCM

as well as linear FCM. We assess predictive accuracy by splitting the data into training and

test sets of size 30 and 9. Assuming that the hip angles are observed with measurement

errors, we smooth the covariate by FPCA and then apply the center/scaling transformation.

We compare prediction errors obtained by fitting both the GFCM and the linear FCM. Also,

as a benchmark model we further fit a linear mixed effect (LME) model

Knee angleij = (β0 + b0i) + (β1 + b1i)Hip angleij + εij

where (b0i, b1i)T are the random coefficients from N(0, R) with some covariance matrix R,

and εij are the errors from N(0, σ2ε ). In this format, (b0i, b1i)

T and εij are assumed to be

independent. For the GFCM and the linear FCM, we report the RMSPE, the ICP, the IL

and the R(SE). For the LME model, we report similar measures, but we take average over

the repeated measurements instead of integrating over the time domain. The results are

collected in Table 2 (top panel).

We observe that the LME model provides a poor predictive performance compared to the

others, implying that models in the framework of concurrent regression model are obviously


better. The GFCM presents slightly better predictive performances than the linear FCM,

but the difference is negligible. For instance, prediction errors from the GFCM are smaller

than the linear FCM. The R(SE) obtained from the GFCM are narrower at all significance

levels. Figure 3, top panel, shows prediction bands obtained from the test data sets, and we

only detect a small difference between the GFCM (solid line) and the linear FCM (dashed

line). The bottom panel in Figure 3 displays a heat map of predicted surface in the GFCM. It

is evident that the relationship between the hip angles and the knee angels is simply linear.

6.3.2 Dietary Calcium Absorption Data. As a sparse data example, we present an appli-

cation to dietary calcium absorption data set (Davis, 2002; Senturk and Nguyen, 2011). In

a group of 188 patients, dietary and bone measurement tests are conducted approximately

every five year, and calcium intake and calcium absorption are measured with respect to

their ages. Patients are aged around 35 and 45 in the beginning of the study, and 1 to 4

repeated measurements are collected for each subject after then. Overall ages are ranging

approximately from 35 to 64. In the analysis, ages are transformed into the values in [0,1],

and the results are considered as evaluation points of the functions. There are dropouts

and missed visits among the patients, yielding a sparse and irregular design of sampling. We

investigate a relationship between the calcium intake and the calcium absorption measured at

the same time point. We applied both the GFCM and the linear FCM by considering that the

calcium intake and the calcium absorption are observed functional covariate and functional

response, respectively. In Web Figure 2, we display the observed individual trajectories of

the calcium intake and the calcium absorption along the patient’s age at the visit.

To begin, we employ our resampling based testing procedure to test the null hypothesis

of no association. We select Kx = Kt = 11 cubic B-splines for x and t to fit the GFCM,

and B = 250 bootstrap replications are used. The p-value of our test is obtained to be less

22 Biometrics, 2015

than 0.004, indicating that the calcium intake measured at a certain time point are strongly

associated with the calcium absorption at the same time point.

We next the analyze the predictive performance of the GFCM and FCM by forming the

training and test sets of size 148 and 40, respectively. For the test data set, we smooth

the individual trajectories to recover missed observations, thus the resulting covariate and

response are smooth and have all observations on its support. Assuming that the calcium

intake has a potential measurement errors, we smooth the sparse and noisy covariate of the

training data and then apply the center/scaling transformation. To fit the GFCM, we again

select Kx = Kt = 11 cubic B-splines for x and t. For the linear FCM, we use 11 cubic B-

splines. Results of predictive accuracy are presented in Table 2 (bottom panel). In the table,

the GFCM and the linear FCM are showing similar RMSPEin and RMSPEout, indicating a

simple linear association between the calcium intake and absorption. In Figure 4, we display

point-wise prediction intervals/bands for three subjects randomly selected from the test data

set. In the figure, there are negligible differences between the GFCM (grey solid lines) and

the linear FCM (dashed lines), further demonstrating linearity between the variables.

7. Discussion

This article proposes a generalized functional concurrent model (GFCM) when both the

response and the covariate are functional data observed on the same domain. Estimation and

inference are discussed in realistic scenarios involving settings such as complex correlated

error structures as well as sparse and/or irregular design, when the covariate is possibly

contaminated with noise. The model and estimation framework can be easily extended to

accommodate additional covariates via linear or smooth effects. In this flexible framework,

the article considers testing for no covariate effect using F test. Bootstrap is used to estimate

the null distribution of the test. The proposed methods are investigated numerically through

simulations mimicking realistic settings. The results show that the new framework captures


non-linear relationships between covariate and response, while it performs similarly to the

common functional concurrent model, when the relationship is linear. Formally testing for

linearity of the unknown relationship is also an interesting problem for future research.

Acknowledgements

Maity’s research was supported by National Institute of Health grant R00 ES017744 and

an North Carolina State University faculty Research and Professional Development grant.

Staicu’s research was supported by National Institute of Health grant RO1 NS085211.

Supplementary Materials

Web Appendices, Tables, and Figures referenced in Section 2 and Section 6 are available with

this paper at the Biometrics website on Wiley Online Library. The computer code implement-

ing our procedure is available with this paper at http://www4.stat.ncsu.edu/~maity/

under the software section.

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[Table 1 about here.]

[Table 2 about here.]

[Figure 1 about here.]




26 Biometrics, 2015

+++++++

+++

+++++

+++++++++++++

+++++++++++

+++++++++++++++

++++++++++

++++++

+++++++

++++

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

15

20

25

i’=27

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

+++++

+++++++++

+++++

++++++++

++++++

+++++++++++++++

+++++++++

++++++

++++++

+

+++++++++++

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

15

20

25

i’=37

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

+++

+++++

++++++

+++++++++

+++++++

+++++

++++++++++

+++++++++++

+++++

+++

++++

++++++++

+++++

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

15

20

25

i’=57

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

++

+++++++

+

+++

++

+++

++++++++++

++++++

++

++

+

+++++++

++

+

+

++++

++++++++++

+

+++++

+++++++

++++

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

i’=27

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

+++++

++++

+

++++

+++++

+

+++++++++++++

+++++

+++++++++

++++++++++

++++++

++++++

+

+++++

+++++

+

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

i’=37

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

++

+

++++

+

+

+

+++++++++++++

++++

++++++++

+++++++++++

+

+

+++++++

+

+++++++

+

+

+++

++++++++

++

+

++

0.0 0.2 0.4 0.6 0.8 1.0

−5

05

10

i’=57

t

Pre

dic

ted

re

sp

on

se

True RespGFCMFCM

Figure 1. Comparison of 95% prediction bands constructed for subjects i′ = 27, 37, and57 in the test data set. The case corresponds to a dense design with 100 subjects and E3

i

error structure. “+” are the response Y0,i′(t) in the test data, and dotted (“•”) lines are thetrue response without measurement errors. Solid and dashed lines are the prediction bandsobtained by fitting the GFCM and the linear FCM, respectively. The top (bottom) panelcorresponds to the case when the true function F is nonlinear (linear).


0 1 2 3 4 5 6

02

04

06

08

01

00

Independent Error Structure

d values

Pow

er

(%)

n=50n=100n=300

0 1 2 3 4 5 6

02

04

06

08

01

00

AR(1) Error Structure

d values

Pow

er

(%)

n=50n=100n=300

0 1 2 3 4 5 6

02

04

06

08

01

00

KL+WN

d valuesP

ow

er

(%)

n=50n=100n=300

0 1 2 3 4 5 6

02

04

06

08

01

00

Independent Error Structure

d values

Pow

er

(%)

n=50n=100n=300

0 1 2 3 4 5 6

02

04

06

08

01

00

AR(1) Error Structure

d values

Pow

er

(%)

n=50n=100n=300

0 1 2 3 4 5 6

02

04

06

08

01

00

KL+WN

d values

Pow

er

(%)

n=50n=100n=300

Figure 2. Powers (×100) of the tests at significance level α = 5%. The top (bottom) paneldisplays the result from the densely (sparsely) sampled. The error process in the left, middleand right panels is assumed to be E1

i , E2i and E3

i , respectively.

28 Biometrics, 2015

0.0 0.2 0.4 0.6 0.8 1.0

020

4060

80

i’=3

t

Pre

dict

ed k

nee

angl

e

0.0 0.2 0.4 0.6 0.8 1.0

020

4060

80

i’=5

t

Pre

dict

ed k

nee

angl

e

0.0 0.2 0.4 0.6 0.8 1.0

020

4060

80

i’=8

t

Pre

dict

ed k

nee

angl

e

t

Hip

ang

le

2

4

6

8

5 10 15 20

0 10 20 30 40 50 60 70 80

Figure 3. Results from the gait data analysis as described in Section 6.3. The top paneldisplays the comparison of 95% prediction bands obtained by fitting the GFCM (grey solidlines) and the linear FCM (dashed lines). The subjects i′ = 3, 5, and 8 in the test data set arerandomly selected. “•” represents the knee angles in the test data. The bottom panel showsthe heat map of F ∗{X∗

0,i′(t), t} obtained from the test data set of the gait data example.


0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

i’=18

t

Pred

icte

d ca

lciu

m a

bsor

ptio

n

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

i’=30

t

Pred

icte

d ca

lciu

m a

bsor

ptio

n

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

i’=40

t

Pred

icte

d ca

lciu

m a

bsor

ptio

n

Figure 4. Results from the calcium absorption data analysis as described in Section 6.3.Displayed is the comparison of 95% prediction bands obtained by fitting the GFCM (greysolid lines) and the linear FCM (dashed lines). The subjects i′ = 18, 30, and 40 in the testdata set are randomly selected. Thin solid line represents the curve of calcium absorptionobtained by smoothing the original sparse test data (“•”).

30 Biometrics, 2015

Table

1Res

ults

from

sim

ula

tion

study

as

dec

crib

edin

Sec

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6.D

ispla

yed

are

the

sum

mari

esofR

MSPE

in,R

MSPE

out,IC

P,IL

,and

R(S

E)

for

diff

eren

tsi

mula

tion

settin

gs.

1−

α=

0.9

51−

α=

0.9

01−

α=

0.8

5R

MSP

Ein

RM

SP

Eout

ICP

ILR

(SE

)IC

PIL

ICP

ILn

Ei

GLF

GLF

GLF

GLF

GLF

GLF

GLF

GLF

GLF

True

modelis

F(x

,t)=

FN

L(x

,t),

Dense

lySam

ple

dD

esi

gn

E1 i

0.9

63.3

90.9

63.0

40.9

52

0.9

53

3.8

012.1

8[3

.53,4.9

6]

[3.6

6,21.3

7]

0.9

05

0.9

27

3.1

910.2

20.8

57

0.9

01

2.7

98.9

450

E2 i

1.3

13.5

11.3

03.1

70.9

55

0.9

56

5.1

812.8

4[4

.95,6.1

6]

[5.0

5,21.7

0]

0.9

09

0.9

29

4.3

510.7

70.8

60

0.9

01

3.8

09.4

3E

3 i1.7

93.7

52.0

43.5

60.9

32

0.9

50

7.2

414.1

0[5

.45,9.8

0]

[8.1

9,22.7

8]

0.8

76

0.9

15

6.0

811.8

30.8

24

0.8

79

5.3

210.3

6

E1 i

0.9

73.5

60.9

42.9

40.9

56

0.9

61

3.8

112.6

7[3

.54,5.0

6]

[3.6

5,21.8

9]

0.9

10

0.9

37

3.2

010.6

30.8

64

0.9

14

2.8

09.3

0100

E2 i

1.3

23.6

71.2

83.0

70.9

58

0.9

63

5.1

913.3

0[4

.95,6.2

4]

[5.0

3,22.2

2]

0.9

13

0.9

39

4.3

511.1

60.8

64

0.9

14

3.8

19.7

7E

3 i1.8

43.9

01.9

43.4

60.9

41

0.9

59

7.2

514.5

5[5

.50,9.5

2]

[8.3

7,23.2

7]

0.8

87

0.9

27

6.0

812.2

10.8

37

0.8

95

5.3

210.6

9

E1 i

0.9

83.6

50.9

32.8

60.9

59

0.9

66

3.8

312.9

4[3

.54,5.1

8]

[3.6

3,22.0

7]

0.9

13

0.9

44

3.2

210.8

60.8

69

0.9

21

2.8

29.5

0300

E2 i

1.3

33.7

51.2

83.0

00.9

60

0.9

68

5.2

113.5

6[4

.96,6.3

6]

[5.0

2,22.3

9]

0.9

15

0.9

46

4.3

711.3

80.8

67

0.9

22

3.8

29.9

6E

3 i1.8

73.9

91.9

13.3

90.9

46

0.9

64

7.2

714.7

9[5

.57,9.4

3]

[8.4

3,23.5

2]

0.8

93

0.9

34

6.1

012.4

10.8

45

0.9

04

5.3

410.8

6

True

modelis

F(x

,t)=

FN

L(x

,t),

Sparse

lySam

ple

dD

esi

gn

E1 i

1.1

73.3

81.2

03.0

60.9

57

0.9

41

4.5

712.1

4[3

.75,7.3

7]

[3.6

2,22.9

0]

0.9

17

0.9

14

3.8

310.1

90.8

76

0.8

89

3.3

68.9

250

E2 i

1.4

73.5

01.4

83.1

90.9

57

0.9

45

5.7

912.7

4[5

.11,8.2

2]

[4.9

3,23.1

9]

0.9

15

0.9

18

4.8

610.6

90.8

72

0.8

89

4.2

59.3

5E

3 i1.9

13.7

32.1

33.5

80.9

34

0.9

47

7.6

414.0

4[5

.70,11.5

0]

[7.9

4,24.2

3]

0.8

81

0.9

11

6.4

111.7

80.8

30

0.8

75

5.6

110.3

1

E1 i

1.1

63.5

51.0

52.9

50.9

66

0.9

46

4.4

812.5

9[3

.71,7.0

6]

[3.3

1,23.4

5]

0.9

30

0.9

22

3.7

610.5

70.8

90

0.8

98

3.2

99.2

5100

E2 i

1.4

63.6

61.3

63.0

80.9

64

0.9

52

5.7

213.1

7[5

.08,7.9

5]

[4.6

9,23.7

2]

0.9

25

0.9

26

4.8

011.0

50.8

82

0.9

00

4.2

09.6

7E

3 i1.9

43.8

91.9

93.4

70.9

47

0.9

56

7.6

314.5

0[5

.79,10.9

1]

[8.1

3,24.6

6]

0.8

96

0.9

24

6.4

012.1

70.8

48

0.8

90

5.6

010.6

5

E1 i

1.1

53.6

50.9

62.8

70.9

73

0.9

54

4.4

112.9

0[3

.67,6.8

6]

[3.2

3,23.2

4]

0.9

39

0.9

31

3.7

010.8

20.9

02

0.9

09

3.2

49.4

7300

E2 i

1.4

63.7

61.2

93.0

10.9

70

0.9

61

5.6

513.5

0[5

.06,7.7

6]

[4.7

0,23.5

0]

0.9

32

0.9

38

4.7

411.3

30.8

90

0.9

13

4.1

59.9

1E

3 i1.9

63.9

91.9

13.4

00.9

55

0.9

63

7.6

114.8

0[5

.85,10.5

6]

[8.2

8,24.5

5]

0.9

05

0.9

33

6.3

912.4

20.8

59

0.9

02

5.5

910.8

7

True

modelis

F(x

,t)

=F

L(x

,t),

Dense

lySam

ple

dD

esi

gn

E1 i

0.9

00.9

00.9

20.9

20.9

44

0.9

44

3.5

33.5

3[3

.51,3.6

0]

[3.5

1,3.6

0]

0.8

93

0.8

93

2.9

62.9

70.8

42

0.8

43

2.5

92.6

050

E2 i

1.2

71.2

71.2

71.2

70.9

52

0.9

52

4.9

84.9

8[4

.93,5.1

6]

[4.9

3,5.1

6]

0.9

05

0.9

05

4.1

84.1

80.8

53

0.8

54

3.6

63.6

6E

3 i1.7

61.8

11.9

91.9

10.9

30

0.9

37

7.0

87.1

0[5

.34,9.1

7]

[5.3

7,8.8

6]

0.8

73

0.8

83

5.9

45.9

60.8

20

0.8

32

5.2

05.2

2

E1 i

0.9

00.9

00.9

20.9

20.9

44

0.9

44

3.5

33.5

4[3

.52,3.5

9]

[3.5

2,3.5

9]

0.8

94

0.8

94

2.9

72.9

70.8

43

0.8

43

2.6

02.6

0100

E2 i

1.2

71.2

71.2

61.2

60.9

53

0.9

53

4.9

84.9

8[4

.93,5.1

5]

[4.9

3,5.1

5]

0.9

05

0.9

05

4.1

84.1

80.8

54

0.8

54

3.6

63.6

6E

3 i1.8

01.8

31.9

21.9

00.9

38

0.9

41

7.1

07.1

3[5

.39,8.8

7]

[5.4

0,8.8

0]

0.8

83

0.8

87

5.9

65.9

80.8

31

0.8

37

5.2

15.2

3

E1 i

0.9

00.9

00.9

10.9

10.9

44

0.9

44

3.5

33.5

3[3

.52,3.5

8]

[3.5

2,3.5

8]

0.8

94

0.8

94

2.9

72.9

70.8

44

0.8

44

2.6

02.6

0300

E2 i

1.2

71.2

71.2

61.2

60.9

53

0.9

53

4.9

84.9

8[4

.93,5.1

5]

[4.9

3,5.1

5]

0.9

06

0.9

06

4.1

84.1

80.8

54

0.8

54

3.6

63.6

6E

3 i1.8

31.8

41.8

91.8

90.9

42

0.9

44

7.1

17.1

3[5

.43,8.7

0]

[5.4

4,8.7

1]

0.8

88

0.8

90

5.9

75.9

80.8

38

0.8

39

5.2

25.2

4

True

modelis

F(x

,t)

=F

L(x

,t),

Sparse

lySam

ple

dD

esi

gn

E1 i

0.9

20.9

20.9

20.9

20.9

48

0.9

49

3.6

33.6

3[3

.55,3.8

4]

[3.5

5,3.8

4]

0.8

98

0.8

99

3.0

43.0

40.8

49

0.8

50

2.6

62.6

650

E2 i

1.2

81.2

81.2

71.2

70.9

53

0.9

53

5.0

45.0

4[4

.95,5.3

1]

[4.9

5,5.3

2]

0.9

06

0.9

06

4.2

34.2

30.8

56

0.8

56

3.7

03.7

1E

3 i1.7

71.8

21.9

61.9

20.9

29

0.9

36

7.0

77.1

3[5

.27,9.7

9]

[5.2

9,9.7

6]

0.8

72

0.8

81

5.9

35.9

90.8

20

0.8

31

5.1

95.2

4

E1 i

0.9

20.9

20.9

20.9

20.9

48

0.9

49

3.6

13.6

1[3

.55,3.7

7]

[3.5

5,3.7

7]

0.8

99

0.8

99

3.0

33.0

30.8

49

0.8

50

2.6

52.6

5100

E2 i

1.2

81.2

81.2

71.2

70.9

54

0.9

54

5.0

35.0

4[4

.96,5.2

6]

[4.9

6,5.2

6]

0.9

07

0.9

07

4.2

34.2

30.8

57

0.8

57

3.7

03.7

0E

3 i1.8

11.8

41.9

21.9

00.9

38

0.9

41

7.1

27.1

6[5

.36,9.5

1]

[5.3

7,9.5

3]

0.8

83

0.8

88

5.9

76.0

10.8

32

0.8

38

5.2

35.2

6

E1 i

0.9

20.9

20.9

20.9

20.9

48

0.9

48

3.6

03.6

0[3

.55,3.7

2]

[3.5

5,3.7

2]

0.8

99

0.8

99

3.0

23.0

20.8

50

0.8

50

2.6

42.6

4300

E2 i

1.2

81.2

81.2

61.2

60.9

54

0.9

54

5.0

35.0

3[4

.97,5.1

9]

[4.9

7,5.1

9]

0.9

08

0.9

08

4.2

24.2

20.8

57

0.8

57

3.6

93.6

9E

3 i1.8

41.8

51.8

91.8

90.9

44

0.9

45

7.1

57.1

7[5

.44,9.1

7]

[5.4

5,9.1

9]

0.8

90

0.8

91

6.0

06.0

20.8

40

0.8

41

5.2

55.2

7


Table 2Summary of RMSPEin, RMSPEout, ICP, IL, and R(SE) in gait data example.

Gait data

1− α = 0.95 1− α = 0.90 1− α = 0.85RMSPEin RMSPEout ICP IL R(SE) ICP IL R(SE) ICP IL R(SE)

G 5.26 5.55 0.956 20.37 [15.78, 36.54] 0.900 17.10 [13.25, 30.66] 0.844 14.96 [11.59, 26.83]LF 5.37 5.59 0.956 20.65 [16.08, 36.44] 0.883 17.33 [13.50, 30.58] 0.828 15.17 [11.81, 26.76]

LME 20.49 22.46 0.994 116.62 [97.26, 154.12] 0.950 97.87 [81.62, 129.34] 0.906 85.65 [71.44, 113.20]

Calcium data

1− α = 0.95 1− α = 0.90 1− α = 0.85RMSPEin RMSPEout ICP IL R(SE) ICP IL R(SE) ICP IL R(SE)

G 0.079 0.091 0.990 0.34 [0.31, 0.50] 0.974 0.29 [0.26, 0.42] 0.954 0.25 [0.23, 0.37]LF 0.080 0.092 0.986 0.34 [0.31, 0.49] 0.972 0.29 [0.26, 0.41] 0.954 0.25 [0.23, 0.36]

Documents

Generalized Functional Concurrent Model · Summary: We propose a ﬂexible regression model to study the association between a functional response and a functional covariate that