1
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Luo R, Qi X. Nonlinear function-on-scalar regression via functional universal approximation. Biometrics 2023; 79:3319-3331. [PMID: 36799710 DOI: 10.1111/biom.13838] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/06/2022] [Accepted: 01/23/2023] [Indexed: 02/18/2023]
Abstract
We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and real datasets.
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Affiliation(s)
- Ruiyan Luo
- Department of Population Health Sciences, Georgia State University, Atlanta, Georgia, USA
| | - Xin Qi
- Department of Population Health Sciences, Georgia State University, Atlanta, Georgia, USA
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2
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Park Y, Han K, Simpson DG. Testing Linear Operator Constraints in Functional Response Regression with Incomplete Response Functions. Electron J Stat 2023; 17:3143-3180. [PMID: 39105139 PMCID: PMC11299897 DOI: 10.1214/23-ejs2177] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 08/07/2024]
Abstract
Hypothesis testing procedures are developed to assess linear operator constraints in function-on-scalar regression when incomplete functional responses are observed. The approach enables statistical inferences about the shape and other aspects of the functional regression coefficients within a unified framework encompassing three incomplete sampling scenarios: (i) partially observed response functions as curve segments over random sub-intervals of the domain; (ii) discretely observed functional responses with additive measurement errors; and (iii) the composition of former two scenarios, where partially observed response segments are observed discretely with measurement error. The latter scenario has been little explored to date, although such structured data is increasingly common in applications. For statistical inference, deviations from the constraint space are measured via integratedL 2 -distance between the model estimates from the constrained and unconstrained model spaces. Large sample properties of the proposed test procedure are established, including the consistency, asymptotic distribution and local power of the test statistic. Finite sample power and level of the proposed test are investigated in a simulation study covering a variety of scenarios. The proposed methodologies are illustrated by applications to U.S. obesity prevalence data, analyzing the functional shape of its trends over time, and motion analysis in a study of automotive ergonomics.
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Affiliation(s)
- Yeonjoo Park
- Department of Management Science and Statistics, University of Texas at San Antonio, San Antonio, TX
| | - Kyunghee Han
- Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL
| | - Douglas G Simpson
- Department of Statistics, University of Illinois at Urbana-Champaign, Urbana-Champaign, IL
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3
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Zhang D, Li L, Sripada C, Kang J. Image response regression via deep neural networks. J R Stat Soc Series B Stat Methodol 2023; 85:1589-1614. [PMID: 38584801 PMCID: PMC10994199 DOI: 10.1093/jrsssb/qkad073] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2022] [Revised: 06/22/2023] [Accepted: 06/28/2023] [Indexed: 04/09/2024]
Abstract
Delineating associations between images and covariates is a central aim of imaging studies. To tackle this problem, we propose a novel non-parametric approach in the framework of spatially varying coefficient models, where the spatially varying functions are estimated through deep neural networks. Our method incorporates spatial smoothness, handles subject heterogeneity, and provides straightforward interpretations. It is also highly flexible and accurate, making it ideal for capturing complex association patterns. We establish estimation and selection consistency and derive asymptotic error bounds. We demonstrate the method's advantages through intensive simulations and analyses of two functional magnetic resonance imaging data sets.
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Affiliation(s)
- Daiwei Zhang
- Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania, Philadelphia, PA, USA
| | - Lexin Li
- Department of Biostatistics and Epidemiology, University of California, Berkeley, CA, USA
| | - Chandra Sripada
- Department of Psychiatry, University of Michigan, Ann Arbor, MI, USA
- Department of Philosophy, University of Michigan, Ann Arbor, MI, USA
| | - Jian Kang
- Department of Biostatistics, University of Michigan, Ann Arbor, MI, USA
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Zhang C, Lin H, Liu L, Liu J, Li Y. Functional data analysis with covariate-dependent mean and covariance structures. Biometrics 2023; 79:2232-2245. [PMID: 36065564 DOI: 10.1111/biom.13744] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/29/2021] [Accepted: 08/11/2022] [Indexed: 11/27/2022]
Abstract
Functional data analysis has emerged as a powerful tool in response to the ever-increasing resources and efforts devoted to collecting information about response curves or anything that varies over a continuum. However, limited progress has been made with regard to linking the covariance structures of response curves to external covariates, as most functional models assume a common covariance structure. We propose a new functional regression model with covariate-dependent mean and covariance structures. Particularly, by allowing variances of random scores to be covariate-dependent, we identify eigenfunctions for each individual from the set of eigenfunctions that govern the variation patterns across all individuals, resulting in high interpretability and prediction power. We further propose a new penalized quasi-likelihood procedure that combines regularization and B-spline smoothing for model selection and estimation and establish the convergence rate and asymptotic normality of the proposed estimators. The utility of the developed method is demonstrated via simulations, as well as an analysis of the Avon Longitudinal Study of Parents and Children concerning parental effects on the growth curves of their offspring, which yields biologically interesting results.
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Affiliation(s)
- Chenlin Zhang
- Center of Statistical Research and School of Statistics, Southwestern University of Finance and Economics, Chengdu, China
| | - Huazhen Lin
- Center of Statistical Research and School of Statistics, Southwestern University of Finance and Economics, Chengdu, China
| | - Li Liu
- School of Mathematics and Statistics, Wuhan University, Wuhan, China
| | - Jin Liu
- Centre for Quantitative Medicine, Program in Health Services & Systems Research, Duke-NUS Medical School, Singapore
| | - Yi Li
- Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, USA
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5
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Zhu H, Zhang Y, Li Y, Lian H. Semiparametric function-on-function quantile regression model with dynamic single-index interactions. Comput Stat Data Anal 2023; 182:107727. [PMID: 39044771 PMCID: PMC11264192 DOI: 10.1016/j.csda.2023.107727] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/15/2023]
Abstract
In this paper we propose a new semiparametric function-on-function quantile regression model with time-dynamic single-index interactions. Our model is very flexible in taking into account of the nonlinear time-dynamic interaction effects of the multivariate longitudinal/functional covariates on the longitudinal response, that most existing quantile regression models for longitudinal data are special cases of our proposed model. We propose to approximate the bivariate nonparametric coefficient functions by tensor product B-splines, and employ a check loss minimization approach to estimate the bivariate coefficient functions and the index parameter vector. Under some mild conditions, we establish the asymptotic normality of the estimated single-index coefficients using projection orthogonalization technique, and obtain the convergence rates of the estimated bivariate coefficient functions. Furthermore, we propose a score test to examine whether there exist interaction effects between the covariates. The finite sample performance of the proposed method is illustrated by Monte Carlo simulations and an empirical data analysis.
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Affiliation(s)
- Hanbing Zhu
- School of Big Data and Statistics, Anhui University, Hefei 230601, China
| | - Yuanyuan Zhang
- School of Mathematical Sciences, Soochow University, Suzhou 215006, China
| | - Yehua Li
- Department of Statistics, University of California, Riverside, CA 92521, USA
| | - Heng Lian
- Department of Mathematics, City University of Hong Kong, Hong Kong, China
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6
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Zhu W, Xu S, Liu C, Li Y. Minimax Powerful Functional Analysis of Covariance Tests: with Application to Longitudinal Genome-Wide Association Studies. Scand Stat Theory Appl 2023; 50:266-295. [PMID: 39076352 PMCID: PMC11286231 DOI: 10.1111/sjos.12583] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/07/2021] [Accepted: 01/07/2022] [Indexed: 11/27/2022]
Abstract
We model the Alzheimer's Disease-related phenotype response variables observed on irregular time points in longitudinal Genome-Wide Association Studies as sparse functional data and propose nonparametric test procedures to detect functional genotype effects while controlling the confounding effects of environmental covariates. Our new functional analysis of covariance tests are based on a seemingly unrelated kernel smoother, which takes into account the within-subject temporal correlations, and thus enjoy improved power over existing functional tests. We show that the proposed test combined with a uniformly consistent nonparametric covariance function estimator enjoys the Wilks phenomenon and is minimax most powerful. Data used in the preparation of this article were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database, where an application of the proposed test lead to the discovery of new genes that may be related to Alzheimer's Disease.
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Affiliation(s)
| | - Sheng Xu
- Global Statistics and Data Science, BeiGene Co., Ltd., China
| | - Catherine Liu
- Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, China
| | - Yehua Li
- Department of Statistics, University of California, Riverside, CA, 92521, USA
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7
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Sang P, Kashlak AB, Kong L. A reproducing kernel Hilbert space framework for functional classification. J Comput Graph Stat 2022. [DOI: 10.1080/10618600.2022.2138407] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/31/2022]
Affiliation(s)
- Peijun Sang
- Department of Statistics and Actuarial Science, University of Waterloo
| | - Adam B Kashlak
- Department of Mathematical and Statistical Sciences, University of Alberta
| | - Linglong Kong
- Department of Mathematical and Statistical Sciences, University of Alberta
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Abstract
Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.
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Cai X, Xue L, Cao J. Robust estimation and variable selection for function‐on‐scalar regression. CAN J STAT 2021. [DOI: 10.1002/cjs.11661] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/10/2022]
Affiliation(s)
- Xiong Cai
- School of Statistics and Mathematics Nanjing Audit University Nanjing 211815 China
| | - Liugen Xue
- College of Statistics and Data Science, Faculty of Science Beijing University of Technology Beijing 100124 China
| | - Jiguo Cao
- Department of Statistics and Actuarial Science Simon Fraser University Burnaby V5A1S6 BC Canada
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Wang Y, Liu X. Extreme quantile regression for tail single-index varying-coefficient models. COMMUN STAT-THEOR M 2021. [DOI: 10.1080/03610926.2021.1961154] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
Affiliation(s)
- Yingjie Wang
- State Key Laboratory of Mechanics and Control of Mechanical Structures, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China
| | - Xinsheng Liu
- State Key Laboratory of Mechanics and Control of Mechanical Structures, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, China
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Affiliation(s)
- Hua Liu
- School of Economics and Finance, Xi’an Jiaotong University, Xi’an, Shaanxi, China
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Jinhong You
- Institute of Data Science and Interdisciplinary Studies, Shanghai Lixin University of Accounting and Finance, Shanghai, China
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Jiguo Cao
- Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, British Columbia, Canada
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12
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Huang Q, You J, Zhang L. Efficient inference of longitudinal/functional data models with time‐varying additive structure. Scand Stat Theory Appl 2021. [DOI: 10.1111/sjos.12540] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
Affiliation(s)
- Qian Huang
- School of Statistics and Management Shanghai University of Finance and Economics Shanghai China
| | - Jinhong You
- School of Statistics and Management Shanghai University of Finance and Economics Shanghai China
| | - Liwen Zhang
- School of Statistics and Management Shanghai University of Finance and Economics Shanghai China
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13
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Huang Z, Sun X, Zhang R. Estimation for partially varying-coefficient single-index models with distorted measurement errors. METRIKA 2021. [DOI: 10.1007/s00184-021-00823-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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14
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Yan X, Pu X, Xun X, Zhou Y. A New Functional Estimation Procedure for Varying Coefficient Models. COMMUN STAT-THEOR M 2021. [DOI: 10.1080/03610926.2019.1646767] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
Affiliation(s)
- Xingyu Yan
- Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai, P.R. China
| | - Xiaolong Pu
- Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai, P.R. China
| | - Xiaolei Xun
- School of Data Science, Fudan University, Shanghai, P.R. China
| | - Yingchun Zhou
- Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE, School of Statistics, East China Normal University, Shanghai, P.R. China
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15
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Li T, Li T, Zhu Z, Zhu H. Regression Analysis of Asynchronous Longitudinal Functional and Scalar Data. J Am Stat Assoc 2020. [DOI: 10.1080/01621459.2020.1844211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Affiliation(s)
- Ting Li
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, China
| | - Tengfei Li
- Department of Radiology and Biomedical Research Imaging Center (BRIC), University of North Carolina at Chapel Hill, Chapel Hill, NC
| | - Zhongyi Zhu
- Department of Statistics, Fudan University, Shanghai, China
| | - Hongtu Zhu
- Department of Biostatistics, University of North Carolina at Chapel Hill, Chapel Hill, NC
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16
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Ye Z, Hooker G, Ellner SP. The Jensen effect and functional single index models: Estimating the ecological implications of nonlinear reaction norms. Ann Appl Stat 2020. [DOI: 10.1214/20-aoas1349] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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17
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Abstract
Covariance estimation is essential yet underdeveloped for analyzing multivariate functional data. We propose a fast covariance estimation method for multivariate sparse functional data using bivariate penalized splines. The tensor-product B-spline formulation of the proposed method enables a simple spectral decomposition of the associated covariance operator and explicit expressions of the resulting eigenfunctions as linear combinations of B-spline bases, thereby dramatically facilitating subsequent principal component analysis. We derive a fast algorithm for selecting the smoothing parameters in covariance smoothing using leave-one-subject-out cross-validation. The method is evaluated with extensive numerical studies and applied to an Alzheimer's disease study with multiple longitudinal outcomes.
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Affiliation(s)
- Cai Li
- Department of Statistics, North Carolina State Univerisy, NC, USA
| | - Luo Xiao
- Department of Statistics, North Carolina State Univerisy, NC, USA
| | - Sheng Luo
- Department of Biostatistics and Bioinformatics, Duke Universitye, NC, USA
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18
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Abstract
AbstractIn this article, we focus on the estimation of varying-coefficient mixed effects models for longitudinal and sparse functional response data, by using the generalized least squares method coupling a modified local kernel smoothing technique. This approach provides a useful framework that simultaneously takes into account the within-subject covariance and all observation information in the estimation to improve efficiency. We establish both uniform consistency and pointwise asymptotic normality for the proposed estimators of varying-coefficient functions. Numerical studies are carried out to illustrate the finite sample performance of the proposed procedure. An application to the white matter tract dataset obtained from Alzheimer’s Disease Neuroimaging Initiative study is also provided.
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19
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Empirical likelihood for partially linear single-index models with missing observations. Comput Stat Data Anal 2020. [DOI: 10.1016/j.csda.2019.106877] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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20
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Chen F, Jiang Q, Feng Z, Zhu L. Model checks for functional linear regression models based on projected empirical processes. Comput Stat Data Anal 2020. [DOI: 10.1016/j.csda.2019.106897] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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21
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Li J, Yue M, Zhang W. Subgroup identification via homogeneity pursuit for dense longitudinal/spatial data. Stat Med 2019; 38:3256-3271. [PMID: 31066095 DOI: 10.1002/sim.8192] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2018] [Revised: 04/08/2019] [Accepted: 04/12/2019] [Indexed: 12/23/2022]
Abstract
In the clinical trial community, it is usually not easy to find a treatment that benefits all patients since the reaction to treatment may differ substantially across different patient subgroups. The heterogeneity of treatment effect plays an essential role in personalized medicine. To facilitate the development of tailored therapies and improve the treatment efficacy, it is important to identify subgroups that exhibit different treatment effects. We consider a very general framework for subgroup identification via the homogeneity pursuit methods usually employed in econometric time series analysis. The change point detection algorithm in our procedure is most suitable for analyzing dense longitudinal or spatial data which are quite common for biomedical studies these days. We demonstrate that our proposed method is fast and accurate through extensive numerical studies. In particular, our method is illustrated by analyzing a diffusion tensor imaging data set.
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Affiliation(s)
- Jialiang Li
- Department of Statistics and Applied Probability, National University of Singapore, Singapore.,Duke-NUS Medical School, Singapore.,Singapore Eye Research Institute, Singapore
| | - Mu Yue
- School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, China
| | - Wenyang Zhang
- Department of Mathematics, University of York, York, UK
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22
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Affiliation(s)
- Tao Huang
- School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, People's Republic of China
| | - Jialiang Li
- Department of Statistics & Applied Probability, National University of Singapore, Duke-NUS Graduate Medical School, Singapore Eye Research Institute, Singapore, Singapore
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