Reduced Rank Mixed Effects Models for Spatially Correlated Hierarchical Functional Data
Lan ZhouJianhua Z. HuangJosue G. MartinezArnab MaityVeerabhadran BaladandayuthapaniRaymond J. Carroll
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Abstract:
Hierarchical functional data are widely seen in complex studies where subunits are nested within units, which in turn are nested within treatment groups. We propose a general framework of functional mixed effects model for such data: within-unit and within-subunit variations are modeled through two separate sets of principal components; the subunit level functions are allowed to be correlated. Penalized splines are used to model both the mean functions and the principal components functions, where roughness penalties are used to regularize the spline fit. An expectation–maximization (EM) algorithm is developed to fit the model, while the specific covariance structure of the model is utilized for computational efficiency to avoid storage and inversion of large matrices. Our dimension reduction with principal components provides an effective solution to the difficult tasks of modeling the covariance kernel of a random function and modeling the correlation between functions. The proposed methodology is illustrated using simulations and an empirical dataset from a colon carcinogenesis study. Supplemental materials are available online.Keywords:
Mixed model
Rank (graph theory)
Kernel (algebra)
본 연구는 함수적 데이터 분석의 몇 가지 이론에 대해 소개하고 분석 기법을 실제 자료에 적용하는 내용을 다루었다. 함수적 데이터 분석의 이론적 내용으로 기저를 이용해 자료를 함수적 데이터로 표현하는 방법, 그리고 함수적 데이터의 변동성을 조사하는 주성분분석, 선형모형 등에 대해 살펴보았다. 그리고 우리나라 기온 데이터와 강수량 데이터를 대상으로 각각 함수적 데이터 분석 기법을 적용해 보았다. 또한, 기온과 강수량 데이터에 대해 함수적 회귀모형을 적합시켜 두 변수간의 함수관계를 살펴보았다. In this paper we review some methods for analyzing functional data and illustrate real application of functional data analysis. Representing methods for functional data by using basis function, analyzing functional variation by functional principal component analysis and functional linear models are reviewed. For a real application, we use temperature and precipitation data measured in Korea from the January of 1970 to the May of 2004. We apply functional principal component analysis for each data and test the significance of regional division done by using shining hours. We also estimate functional regression model for temperature and precipitation.
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This dissertation is concerned with functional data analysis. Functional data consists of a collection of curves or functions defined on an interval. These curves can be obtained by splitting a continuous time record such as temperature into daily or annual curves. Functional data is also obtained when an experimenter records a curve of data from each subject in a sample, e.g., a growth trajectory of an animal or plant. Several examples of different models for functional data are given. We use the method of principle component analysis to obtain the necessary regularization in each model. Functional principal component analysis is summarized as a natural extension of the traditional vector principal component analysis. The first functional model is concerned with inference based on the mean function of a functional time series. We develop and asymptotically justify a testing procedure for the equality of means in two functional samples exhibiting temporal dependence. As a second example, we consider a quadratic functional regression model in which a scalar response depends on a functional predictor. We develop a test of the significance of the nonlinear term in the model. The asymptotic behavior of our testing procedure is established. In the third model, we observe two sequences of curves which are connected via an integral operator. This model includes linear models as well as autoregressive models in Hilbert spaces. We develop a procedure to test the stability of the model. In the fourth model, we propose a functional version of the popular ARCH model. We establish conditions for the existence of a strictly stationary solution, derive weak dependence and moment conditions, show consistency of the estimators, and perform an empirical study demonstrating how our model matches with real data.
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We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Lo\`eve expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.
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Representation
Component (thermodynamics)
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본 연구는 함수적 데이터 분석의 몇 가지 이론에 대해 소개하고 분석 기법을 실제 자료에 적용하는 내용을 다루었다. 함수적 데이터 분석의 이론적 내용으로 기저를 이용해 자료를 함수적 데이터로 표현하는 방법, 그리고 함수적 데이터의 변동성을 조사하는 주성분분석, 선형모형 등에 대해 살펴보았다. 그리고 우리나라 기온 데이터와 강수량 데이터를 대상으로 각각 함수적 데이터 분석 기법을 적용해 보았다. 또한, 기온과 강수량 데이터에 대해 함수적 회귀모형을 적합시켜 두 변수간의 함수관계를 살펴보았다. 【In this paper we review some methods for analyzing functional data and illustrate real application of functional data analysis. Representing methods for functional data by using basis function, analyzing functional variation by functional principal component analysis and functional linear models are reviewed. For a real application, we use temperature and precipitation data measured in Korea from the January of 1970 to the May of 2004. We apply functional principal component analysis for each data and test the significance of regional division done by using shining hours. We also estimate functional regression model for temperature and precipitation.】
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The functional linear model is a popular tool to investigate the relationship between a scalar/functional response variable and a scalar/functional covariate. We generalize this model to a functional linear mixed-effects model when repeated measurements are available on multiple subjects. Each subject has an individual intercept and slope function, while shares common population intercept and slope function. This model is flexible in the sense of allowing the slope random effects to change with the time. We propose a penalized spline smoothing method to estimate the population and random slope functions. A REML-based EM algorithm is developed to estimate the variance parameters for the random effects and the data noise. Simulation studies show that our estimation method provides an accurate estimate for the functional linear mixed-effects model with the finite samples. The functional linear mixed-effects model is demonstrated by investigating the effect of the 24-hour nitrogen dioxide on the daily maximum ozone concentrations and also studying the effect of the daily temperature on the annual precipitation.
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Population model
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This paper is devoted to the R package fda.usc which includes some utilities for functional data analysis. This package carries out exploratory and descriptive analysis of functional data analyzing its most important features such as depth measurements or functional outliers detection, among others. The R package fda.usc also includes functions to compute functional regression models, with a scalar response and a functional explanatory data via non-parametric functional regression, basis representation or functional principal components analysis. There are natural extensions such as functional linear models and semi-functional partial linear models, which allow non-functional covariates and factors and make predictions. The functions of this package complement and incorporate the two main references of functional data analysis: The R package fda and the functions implemented by Ferraty and Vieu (2006).
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Abstract In commonly used functional regression models, the regression of a scalar or functional response on the functional predictor is assumed to be linear. This means that the response is a linear function of the functional principal component scores of the predictor process. We relax the linearity assumption and propose to replace it by an additive structure, leading to a more widely applicable and much more flexible framework for functional regression models. The proposed functional additive regression models are suitable for both scalar and functional responses. The regularization needed for effective estimation of the regression parameter function is implemented through a projection on the eigenbasis of the covariance operator of the functional components in the model. The use of functional principal components in an additive rather than linear way leads to substantial broadening of the scope of functional regression models and emerges as a natural approach, because the uncorrelatedness of the functional principal components is shown to lead to a straightforward implementation of the functional additive model, based solely on a sequence of one-dimensional smoothing steps and without the need for backfitting. This facilitates the theoretical analysis, and we establish the asymptotic consistency of the estimates of the components of the functional additive model. We illustrate the empirical performance of the proposed modeling framework and estimation methods through simulation studies and in applications to gene expression time course data. Keywords: : Additive modelAsymptoticsFunctional data analysisFunctional regressionLinear modelPrincipal componentsSmoothingStochastic processes
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Linear form
Principal component regression
Generalized additive model
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We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Lo\`eve expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.
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Data in many experiments arise as curves and therefore it is natural to use a curve as a basic unit in the analysis, which is termed functional data analysis (FDA). In longitudinal studies, recent developments in FDA have extended classical linear models and linear mixed effects models to functional linear models (also termed varying-coefficient models) and functional mixed effects models. In this paper we focus our review on the functional mixed effects models using smoothing splines, because functional linear models are special cases of this more general framework. Due to the connection between smoothing splines and linear mixed effects models, functional mixed effects models can be fitted using existing software such as SAS Proc Mixed. A case study is presented as an illustration.
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We propose an estimation approach to analyse correlated functional data, which are observed on unequal grids or even sparsely. The model we use is a functional linear mixed model, a functional analogue of the linear mixed model. Estimation is based on dimension reduction via functional principal component analysis and on mixed model methodology. Our procedure allows the decomposition of the variability in the data as well as the estimation of mean effects of interest, and borrows strength across curves. Confidence bands for mean effects can be constructed conditionally on estimated principal components. We provide R -code implementing our approach in an online appendix. The method is motivated by and applied to data from speech production research.
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