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## Functional data analysis for sparse longitudinal data. (2005)

Venue: | Journal of the American Statistical Association |

Citations: | 123 - 24 self |

### Citations

1476 | Multivariate analysis - Mardia, Kent, et al. - 1979 |

1215 | Comprehensive identification of cell cycleregulated genes of the Yeast Saccharomyces Cerevisiae by microarray hybridization - Spellman, Sherlock, et al. - 1998 |

959 | Functional Data Analysis
- Ramsay, Silverman
- 2005
(Show Context)
Citation Context ...nt of Statistics, Colorado State University, Fort Collins, CO 80523 (E-mail: fyao@stat.colostate.edu). Hans-Georg Müller is Professor (E-mail: mueller@wald.ucdavis.edu) and Jane-Ling Wang is Professor (E-mail: wang@wald.ucdavis.edu), Department of Statistics, University of California, Davis, CA 95616. This research was supported in part by National Science Foundation grants DMS-98-03637, DMS-99-71602, DMS-02-04869, DMS-03-54448, and DMS-04-06430. The authors thank the associate editor and two referees for insightful comments on a previous version of this article that led to many improvements. Ramsay and Silverman 1997.) Staniswalis and Lee (1998) proposed kernel-based functional principal components analysis for repeated measurements with an irregular grid of time points. The case of irregular grids was also studied by Besse, Cardot, and Ferraty (1997) and Boularan, Ferré, and Vieu (1993). However, when the time points vary widely across subjects and are sparse, down to one or two measurements, the FPC scores defined through the Karhunen–Loève expansion are not well approximated by the usual integration method. Shi, Weiss, and Taylor (1996), Rice and Wu (2000), James, Hastie, and Sugar (2001), and James an... |

936 |
Local Polynomial Modelling and Its Applications
- Fan, Gijbels
- 1996
(Show Context)
Citation Context ...o be iid and independent of the random coefficients ξik, where i = 1, . . . ,n, j = 1, . . . ,Ni, k = 1,2, . . . . Then the model that we consider is Yij = Xi(Tij) + ij = µ(Tij) + ∞∑ k=1 ξikφk(Tij) + ij, Tij ∈ T , (1) where Eij = 0, var(ij) = σ 2, and the number of measurements Ni made on the ith subject is considered random, reflecting sparse and irregular designs. The random variables Ni are assumed to be iid and independent of all other random variables. 2.2 Estimation of the Model Components We assume that mean, covariance, and eigenfunctions are smooth. We use local linear smoothers (Fan and Gijbels 1996) for function and surface estimation, fitting local lines in one dimension and local planes in two dimensions by weighted least squares. In a first step, we estimate the mean function µ based on the pooled data from all individuals. The formula for this local linear smoother is in (A.1) in the Appendix. Data-adaptive methods for bandwidth choice are available (see Müller and Prewitt 1993 for surface smoothing and Rice and Silverman 1991 for one-curve-leave-out cross-validation); subjective choices are often adequate. (For issues of smoothing dependent data, see Lin and Carroll 2000.) Adapting ... |

735 | Analysis of longitudinal data - Diggle, Heagerty, et al. - 2002 |

226 |
Estimating the mean and covariance structure nonparametrically when the data are curves
- Rice, Silverman
- 1991
(Show Context)
Citation Context ...oose the number of eigenfunctions that provide a reasonable approximation to the infinite-dimensional process, one may use the cross-validation score based on the one-curveleave-out prediction error (=-=Rice and Silverman, 1991-=-). Let ˆµ (−i) and ˆ φ (−i) k be the estimated mean and eigenfunctions after removing the data for the ith subject. Then we choose K so as to minimize the cross-validation score based on the squared p... |

123 |
An optimal selection of regression variables
- Shibata
- 1984
(Show Context)
Citation Context ...lidation score based on the one-curveleave-out prediction error (Rice and Silverman 1991). Let µ(−i) and φ(−i)k be the estimated mean and eigenfunctions after removing the data for the ith subject. Then we choose K so as to minimize the cross-validation score based on the squared prediction error, CV(K) = n∑ i=1 Ni∑ j=1 { Yij − Y (−i)i (Tij) }2 , (10) where Y (−i)i is the predicted curve for the ith subject, computed after removing the data for this subject, that is, Y (−i)i (t) = µ(−i)(t) +∑Kk=1 ξ (−i)ik φ(−i)k (t), where ξik is obtained by (5). One can also adapt AIC-type criteria (Shibata 1981) to this situation. In simulations not reported here, we found that AIC is computationally more efficient while the results are similar to those obtained by cross-validation. A pseudo-Gaussian loglikelihood, summing the contributions from all subjects, conditional on the estimated FPC scores ξik (5), is given by L = n∑ i=1 { −Ni 2 log (2π) − Ni 2 log σ 2 − 1 2σ 2 ( Yi − µi − K∑ k=1 ξikφik )T × ( Yi − µi − K∑ k=1 ξikφik )} , (11) where we define AIC = −L + K. 3. ASYMPTOTIC PROPERTIES We derive consistency and distribution results demonstrating the consistency of the estimated FPC s... |

114 | Nonparametric mixed effects models for unequally sampled noisy curves.
- Rice, Wu
- 2001
(Show Context)
Citation Context ...≥ λ2 ≥ . . . . We consider an extended version of the model that incorporates uncorrelated measurement errors with mean zero and constant variance σ 2 to reflect additive measurement errors (see also =-=Rice and Wu, 2000-=-). Let Yij be the jth observation of the random function Xi(·), made at a random time Tij, and ɛij the additional measurement errors that are assumed to be i.i.d. and independent of the random coeffic... |

99 | Principal component models for sparse functional data
- James, Hastie, et al.
- 2000
(Show Context)
Citation Context ...ith an irregular grid of time points. The case of irregular grids was also studied by Besse, Cardot, and Ferraty (1997) and Boularan, Ferré, and Vieu (1993). However, when the time points vary widely across subjects and are sparse, down to one or two measurements, the FPC scores defined through the Karhunen–Loève expansion are not well approximated by the usual integration method. Shi, Weiss, and Taylor (1996), Rice and Wu (2000), James, Hastie, and Sugar (2001), and James and Sugar (2003) proposed B-splines to model the individual curves with random coefficients through mixed effects models. James et al. (2001) and James and Sugar (2003) emphasized the case of sparse data, postulating a reduced-rank mixed-effects model through a B-spline basis for the underlying random trajectories. In contrast, we represent the trajectories directly through the Karhunen–Loève expansion, determining the eigenfunctions from the data. Perhaps owing to the complexity of their modeling approach, James et al. (2001) did not investigate the asymptotic properties of the estimated components in relation to the true components, such as the behavior of the estimated covariance structure, eigenvalues, and eigenfunctions, espec... |

90 | Asymptotic theory for the principal component analysis of a vector random function: some applications to statistical inference. - Dauxois, Pousse, et al. - 1982 |

86 |
Smoothed functional principal components analysis by choice of norm. The Annals of Statistics
- Silverman
- 1996
(Show Context)
Citation Context ...gence of estimated eigenfunctions and eigenvalues. To our knowledge, there exist only few published asymptotic results for functional principal components (Dauxois, Pousse and Romain 1982; Bosq 1991; =-=Silverman 1996-=-), and none for functional data analysis in the sparse situation. Fourth, we derive the asymptotic distribution that is needed to obtain pointwise confidence intervals for individual trajectories, and... |

85 | Clustering for sparsely sampled functional data.
- James, Sugar
- 2003
(Show Context)
Citation Context ...man 1997.) Staniswalis and Lee (1998) proposed kernel-based functional principal components analysis for repeated measurements with an irregular grid of time points. The case of irregular grids was also studied by Besse, Cardot, and Ferraty (1997) and Boularan, Ferré, and Vieu (1993). However, when the time points vary widely across subjects and are sparse, down to one or two measurements, the FPC scores defined through the Karhunen–Loève expansion are not well approximated by the usual integration method. Shi, Weiss, and Taylor (1996), Rice and Wu (2000), James, Hastie, and Sugar (2001), and James and Sugar (2003) proposed B-splines to model the individual curves with random coefficients through mixed effects models. James et al. (2001) and James and Sugar (2003) emphasized the case of sparse data, postulating a reduced-rank mixed-effects model through a B-spline basis for the underlying random trajectories. In contrast, we represent the trajectories directly through the Karhunen–Loève expansion, determining the eigenfunctions from the data. Perhaps owing to the complexity of their modeling approach, James et al. (2001) did not investigate the asymptotic properties of the estimated components in relati... |

70 | Two-step estimation of functional linear models with applications to longitudinal data,”
- Fan, Zhang
- 2000
(Show Context)
Citation Context ...quite well suited for analysis by PACE is illustrated by Figure 2. As this figure shows, the assembled pairs (Tij,Tik) are sufficiently dense in the domain plane, and estimation of the covariance function (A.2) is feasible for these data. Further details Figure 2. Assembled Pairs (Tij ,Tik ) of All Subjects, i = 1, . . . , n, j, k = 1, . . . , Ni , for the CD4 Count Data. Although the data available per subject are sparse, the assembled data fill the domain of the covariance surface quite densely. about design, methods, and medical implications of the study were given by Kaslow et al. (1987). Fan and Zhang (2000) and Wu and Chiang (2000) analyzed these data with varying coefficient models adapted to longitudinal data, and Diggle, Liang, and Zeger (1994) discussed classical longitudinal approaches for these data. The objectives of our analysis are to estimate the overall trend over time, uncover subject-specific variation patterns, extract the dominant modes of variation, and recover individual trajectories from sparse measurements. This includes predicting the time course for an individual given only few observations, and constructing pointwise and simultaneous bands for an individual’s trajectory. Th... |

70 | Nonparametric Function Estimation for Clustered Data When the Predictor is Measured Without/With Error,”
- Lin, Carroll
- 2000
(Show Context)
Citation Context ...smoothers (Fan and Gijbels 1996) for function and surface estimation, fitting local lines in one dimension and local planes in two dimensions by weighted least squares. In a first step, we estimate the mean function µ based on the pooled data from all individuals. The formula for this local linear smoother is in (A.1) in the Appendix. Data-adaptive methods for bandwidth choice are available (see Müller and Prewitt 1993 for surface smoothing and Rice and Silverman 1991 for one-curve-leave-out cross-validation); subjective choices are often adequate. (For issues of smoothing dependent data, see Lin and Carroll 2000.) Adapting to estimated correlations when estimating the mean function did not lead to improvements (simulations not reported); therefore, we do not incorporate such adjustments. Note that in model (1), cov(Yij,Yil|Tij,Til) = cov(X(Tij), X(Til)) + σ 2δjl, where δjl is 1 if j = l and 0 otherwise. Let Gi(Tij,Til) = (Yij − µ(Tij))(Yil − µ(Til)) be the “raw” covariances, where µ(t) is the estimated mean function obtained from the previous step. It is easy to see that E[Gi(Tij,Til)|Tij,Til] ≈ cov(X(Tij),X(Til)) + σ 2δjl. Therefore, the diagonal of the raw Yao, Müller, and Wang: FDA for Sparse L... |

61 | Some statistical methods for comparison of growth curves. - Rao - 1958 |

57 |
Nonparametric regression analysis of longitudinal data
- Staniswalis, Lee
- 1998
(Show Context)
Citation Context ...Therefore the diagonal of the raw covariances should be removed, i.e., only Gi(Tij, Til), j = l, should be included as input data for the covariance surface smoothing step (as previously observed in =-=Staniswalis and Lee, 1998-=-). We use one-curve-leave-out cross-validation to choose the smoothing parameter for this surface smoothing step. The variance σ 2 of the measurement errors is of interest in model (1). Let G(s, t) ... |

49 | Principal components analysis of sampled functions. - Besse, Ramsay - 1986 |

46 | Principal modes of variation for processes with continuous sample curves - Castro, Lawton, et al. - 1986 |

45 |
Inference for Density Families Using Functional Principal Component Analysis,”
- Kneip, Utikal
- 2001
(Show Context)
Citation Context ...n. There exists an extensive literature on FPC analysis when individuals are measured at a dense grid of regularly spaced time points. The method was introduced by Rao (1958) for growth curves, and the basic principle has been studied by Besse and Ramsay (1986), Castro, Lawton, and Sylvestre (1986), and Berkey, Laird, Valadian, and Gardner (1991). Rice and Silverman (1991) discussed smoothing and smoothing parameter choice in this context, whereas Jones and Rice (1992) emphasized applications. Various theoretical properties have been studied by Silverman (1996), Boente and Fraiman (2000), and Kneip and Utikal (2001). (For an introduction and summary, see Fang Yao is Assistant Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523 (E-mail: fyao@stat.colostate.edu). Hans-Georg Müller is Professor (E-mail: mueller@wald.ucdavis.edu) and Jane-Ling Wang is Professor (E-mail: wang@wald.ucdavis.edu), Department of Statistics, University of California, Davis, CA 95616. This research was supported in part by National Science Foundation grants DMS-98-03637, DMS-99-71602, DMS-02-04869, DMS-03-54448, and DMS-04-06430. The authors thank the associate editor and two referees for insightf... |

42 |
Modelization, nonparametric estimation and prediction for continuous time processes, in: G. Roussas (Ed.), Nonparametric Functional Estimation and Related Topics,
- Bosq
- 1991
(Show Context)
Citation Context ...form convergence of estimated eigenfunctions and eigenvalues. To our knowledge, there exist only few published asymptotic results for functional principal components (Dauxois, Pousse and Romain 1982; =-=Bosq 1991-=-; Silverman 1996), and none for functional data analysis in the sparse situation. Fourth, we derive the asymptotic distribution that is needed to obtain pointwise confidence intervals for individual t... |

42 | Displaying the important features of large collections of similar curves - Jones, Rice - 1992 |

38 | Simultaneous nonparametric regressions of unbalanced longitudinal data - Besse, Cardot, et al. - 1997 |

37 | An Analysis of Paediatric CD4 Counts for Acquired Immune Deficiency Syndrome Using Flexible Random Curves,” - Shi, Weiss, et al. - 1996 |

37 | Shrinkage estimation for functional principal component scores with application to the population kinetics of plasma folate
- Yao, Miiller, et al.
(Show Context)
Citation Context ...m with independent measurements (Kneip 1994), are random vectors of large but fixed dimensions (Ferré 1995), or are random trajectories sampled on dense and regular grids (Cardot, Ferraty, and Sarda 1999). The contributions of this article are as follows. First, we provide a new technique, PACE, for longitudinal and functional data, a method designed to handle sparse and irregular longitudinal data for which the pooled time points are sufficiently dense. Second, the presence of additional measurement errors is taken into account, extending previous approaches of Staniswalis and Lee (1998) and Yao et al. (2003). Third, an emphasis is on the derivation of asymptotic consistency properties, by first establishing uniform convergence for smoothed estimates of the mean and covariance functions under mild assumptions. These uniform consistency results are developed for smoothers in the situation where repeated, and thus dependent, measurements are obtained for the same subject. Then we couple these results with the theory of eigenfunctions and eigenvalues of compact linear operators, to obtain uniform convergence of estimated eigenfunctions and eigenvalues. To our knowledge, there exist only few published... |

34 |
Nonparametric Estimation of Common Regressors for Similar Curve Data”,
- Kneip
- 1994
(Show Context)
Citation Context ... of the assumed random nature of the observed sample of trajectories, which sets our work apart from previous results where either the observed functions are non-random with independent measurements (=-=Kneip, 1994-=-), are random vectors of large but fixed dimensions (Ferré, 1995) or are random trajectories sampled on dense and regular grids (Cardot, Ferraty and Sarda, 1999). 3The contributions of this paper are... |

22 | Kernel Smoothing on Varying Coefficient Models With Longitudinal Dependent Variable,”
- Wu, Chiang
- 2000
(Show Context)
Citation Context ...lysis by PACE is illustrated by Figure 2. As this figure shows, the assembled pairs (Tij,Tik) are sufficiently dense in the domain plane, and estimation of the covariance function (A.2) is feasible for these data. Further details Figure 2. Assembled Pairs (Tij ,Tik ) of All Subjects, i = 1, . . . , n, j, k = 1, . . . , Ni , for the CD4 Count Data. Although the data available per subject are sparse, the assembled data fill the domain of the covariance surface quite densely. about design, methods, and medical implications of the study were given by Kaslow et al. (1987). Fan and Zhang (2000) and Wu and Chiang (2000) analyzed these data with varying coefficient models adapted to longitudinal data, and Diggle, Liang, and Zeger (1994) discussed classical longitudinal approaches for these data. The objectives of our analysis are to estimate the overall trend over time, uncover subject-specific variation patterns, extract the dominant modes of variation, and recover individual trajectories from sparse measurements. This includes predicting the time course for an individual given only few observations, and constructing pointwise and simultaneous bands for an individual’s trajectory. The estimate of the mean fu... |

16 |
An Accelerated-Time Model for Response Curves,”
- Capra, Müller
- 1997
(Show Context)
Citation Context ...(1998). Let |T |denote the length of T , and let T1 be the interval T1 = [inf{x : x ∈ T }+ |T |/4, sup{x : x ∈ T }− |T |/4]. The proposed estimate of σ 2 is σ 2 = 2|T | ∫ T1 {V(t) − G(t)}dt (2) if σ 2 > 0 and σ 2 = 0 otherwise. The estimates of eigenfunctions and eigenvalues correspond to the solutions φk and λk of the eigenequations, ∫ T G(s, t)φk(s)ds = λkφk(t), (3) where the φk are subject to ∫ T φk(t) 2 dt = 1 and ∫T φk(t) × φm(t)dt = 0 for m < k. We estimate the eigenfunctions by discretizing the smoothed covariance, as previously described by Rice and Silverman (1991) and Capra and Müller (1997). 2.3 Functional Principal Components Analysis Through Conditional Expectation The FPC scores ξik = ∫ (Xi(t) − µ(t))φk(t)dt have traditionally been estimated by numerical integration, which works well when the density of the grid of measurements for each subject is sufficiently large. Because in our model the Yij are available only at discrete random times Tij, reflecting the sparseness of the data, the integrals in the definition of the FPC scores ξik accordingly would be approximated by sums, substituting Yij as defined in (1) for Xi(Tij) and estimates µ(tij) for µ(tij) and φk(tij) for φk(... |

10 | Growth Curves: A Two-Stage Nonparametric Approach,” - Boularan, Ferré, et al. - 1993 |

6 | Asymptotics for nonparametric regression - Bhattacharya, K, et al. - 1993 |

5 |
Improvement of Some Multivariate Estimates by Reduction of
- Ferré
- 1995
(Show Context)
Citation Context ...ries, which sets our work apart from previous results where either the observed functions are non-random with independent measurements (Kneip, 1994), are random vectors of large but fixed dimensions (=-=Ferré, 1995-=-) or are random trajectories sampled on dense and regular grids (Cardot, Ferraty and Sarda, 1999). 3The contributions of this paper are as follows: First, we provide a new technique, Principal Compon... |

4 | Modeling Adolescent Blood Pressure Patterns and Their Prediction of Adult Pressures,” - Berkey, Laird, et al. - 1991 |

4 |
Multiparameter Bandwidth Processes and Adaptive Surface Smoothing,”
- Müller, Prewitt
- 1993
(Show Context)
Citation Context ...ables Ni are assumed to be iid and independent of all other random variables. 2.2 Estimation of the Model Components We assume that mean, covariance, and eigenfunctions are smooth. We use local linear smoothers (Fan and Gijbels 1996) for function and surface estimation, fitting local lines in one dimension and local planes in two dimensions by weighted least squares. In a first step, we estimate the mean function µ based on the pooled data from all individuals. The formula for this local linear smoother is in (A.1) in the Appendix. Data-adaptive methods for bandwidth choice are available (see Müller and Prewitt 1993 for surface smoothing and Rice and Silverman 1991 for one-curve-leave-out cross-validation); subjective choices are often adequate. (For issues of smoothing dependent data, see Lin and Carroll 2000.) Adapting to estimated correlations when estimating the mean function did not lead to improvements (simulations not reported); therefore, we do not incorporate such adjustments. Note that in model (1), cov(Yij,Yil|Tij,Til) = cov(X(Tij), X(Til)) + σ 2δjl, where δjl is 1 if j = l and 0 otherwise. Let Gi(Tij,Til) = (Yij − µ(Tij))(Yil − µ(Til)) be the “raw” covariances, where µ(t) is the estimated ... |

3 | Simultaneous Nonparametric Regression of - Besse, Cardot, et al. - 1997 |

2 |
Some Statistical Methods for
- Rao
- 1958
(Show Context)
Citation Context ...reduction is mandatory, and FPC analysis has become a common tool to achieve this, by reducing random trajectories to a set of FPC scores. However, this method encounters difficulties when applied to longitudinal data with only few repeated observations per subject. Beyond dimension reduction, FPC analysis attempts to characterize the dominant modes of variation of a sample of random trajectories around an overall mean trend function. There exists an extensive literature on FPC analysis when individuals are measured at a dense grid of regularly spaced time points. The method was introduced by Rao (1958) for growth curves, and the basic principle has been studied by Besse and Ramsay (1986), Castro, Lawton, and Sylvestre (1986), and Berkey, Laird, Valadian, and Gardner (1991). Rice and Silverman (1991) discussed smoothing and smoothing parameter choice in this context, whereas Jones and Rice (1992) emphasized applications. Various theoretical properties have been studied by Silverman (1996), Boente and Fraiman (2000), and Kneip and Utikal (2001). (For an introduction and summary, see Fang Yao is Assistant Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523 (E... |

1 |
Displaying the Important Features of
- Jones, Rice
- 1992
(Show Context)
Citation Context ...ction, FPC analysis attempts to characterize the dominant modes of variation of a sample of random trajectories around an overall mean trend function. There exists an extensive literature on FPC analysis when individuals are measured at a dense grid of regularly spaced time points. The method was introduced by Rao (1958) for growth curves, and the basic principle has been studied by Besse and Ramsay (1986), Castro, Lawton, and Sylvestre (1986), and Berkey, Laird, Valadian, and Gardner (1991). Rice and Silverman (1991) discussed smoothing and smoothing parameter choice in this context, whereas Jones and Rice (1992) emphasized applications. Various theoretical properties have been studied by Silverman (1996), Boente and Fraiman (2000), and Kneip and Utikal (2001). (For an introduction and summary, see Fang Yao is Assistant Professor, Department of Statistics, Colorado State University, Fort Collins, CO 80523 (E-mail: fyao@stat.colostate.edu). Hans-Georg Müller is Professor (E-mail: mueller@wald.ucdavis.edu) and Jane-Ling Wang is Professor (E-mail: wang@wald.ucdavis.edu), Department of Statistics, University of California, Davis, CA 95616. This research was supported in part by National Science Foundation... |