Structured Additive Regression Models for Functional Data Fabian Scheipl & Sonja Greven Institut für Statistik Ludwig-Maximilians-Universität München III International Workshop on Advances in FDA Castro Urdiales, June 2019
Joint work with: Alexander Bauer, Andreas Bender, Sarah Brockhaus, Jona Cederbaum, Ciprian Crainiceanu, Karen Fuchs, Antonio Gasparrini, Jan Gertheiss, Jonathan Gellar, Jeff Goldsmith, Wolfgang Hartl, Torsten Hothorn, Andrada Ivanescu, Helmut Küchenhoff, Matthew McLean, Friedrich Leisch, Viola Obermeier, David Rügamer, Ana-Maria Staicu, Simon N. Wood
Outline Introduction Functional data analysis Research program Structured additive functional regression Model Specification of Model Terms Estimation Penalized Likelihood Estimation Gradient Boosting Example Feeding Behavior of Pigs Summary & Outlook Wrap-Up
Outline Introduction Functional data analysis Research program Structured additive functional regression Model Specification of Model Terms Estimation Penalized Likelihood Estimation Gradient Boosting Example Feeding Behavior of Pigs Summary & Outlook Wrap-Up
Functional Data ◮ unit of observation is a curve over some interval (or 2D/3D) ◮ theoretical view: data are realizations of stochastic processes, curves live in a function space ◮ in practice: (densely/sparsely) sampled on a finite grid ◮ examples: spectroscopy, longitudinal blood marker profiles, medical imaging, accelerometers, .... 3500 0.5 2500 0.0 Total CD4 Cell Count NIR − 0.5 1500 − 1.0 500 − 1.5 0 1000 1500 2000 2500 -20 -10 0 10 20 30 40 Months since seroconversion wavelength [nm] 2 / 35
Functional Data ◮ unit of observation is a curve over some interval (or 2D/3D) ◮ theoretical view: data are realizations of stochastic processes, curves live in a function space ◮ in practice: (densely/sparsely) sampled on a finite grid ◮ examples: spectroscopy, longitudinal blood marker profiles, medical imaging, accelerometers, .... pig 57 100 60 day 20 02:00 07:00 12:00 17:00 22:00 2 / 35
Functional Data Analysis (FDA) FDA is statistics for functional data. Increasing interest in relating functional variables to other variables of interest: functional regression ◮ function-on-scalar regression ◮ scalar-on-function (also: “time-to-event”-on-function) ◮ function-on-function (Cuevas, 2014; Morris, 2015) 3 / 35
Outline Introduction Functional data analysis Research program Structured additive functional regression Model Specification of Model Terms Estimation Penalized Likelihood Estimation Gradient Boosting Example Feeding Behavior of Pigs Summary & Outlook Wrap-Up
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Aim A general framework for functional regression with choice of 1. scalar or functional response (observed on (un)equal grids) 2. modeled feature of the conditional response distribution, e.g. expectation, quantile or higher moments 3. linear, smooth and interaction effects of scalar and functional covariates, (functional) random effects 4. bases, e.g. splines, functional principal components 5. estimation method: gradient boosting or penalized likelihood Key idea: model observations within curves, shift functional structure to additive predictor. ⇒ penalized scalar regression using varying coefficient models = Implementation in R-packages FDboost and refund . 4 / 35
Outline Introduction Functional data analysis Research program Structured additive functional regression Model Specification of Model Terms Estimation Penalized Likelihood Estimation Gradient Boosting Example Feeding Behavior of Pigs Summary & Outlook Wrap-Up
Structured Additive Functional Models Observations ( Y i , X i ) , i = 1 , . . . , N , with ◮ Y i a functional (scalar) response over interval T = [ a , b ] , [ t , t ] ◮ X i a set of scalar and/or functional covariates Structured Additive Regression Model J � ξ ( Y i | X i = x i )( t ) = h ( x i )( t ) = h j ( x i , t ) , j = 1 ◮ ξ the modeled feature of the conditional response distribution, e.g. expectation (with link function), median, a quantile, .... ◮ partial effects h j ( x i , t ) are real valued functions over T depending on one or more covariates. Scheipl et al. (2015, 2016); Brockhaus et al. (2015); Greven & Scheipl (2017) 5 / 35
Outline Introduction Functional data analysis Research program Structured additive functional regression Model Specification of Model Terms Estimation Penalized Likelihood Estimation Gradient Boosting Example Feeding Behavior of Pigs Summary & Outlook Wrap-Up
Partial Effects h j ( x , t ) Model: ξ ( Y | X = x ) = h ( x )( t ) = � j h j ( x , t ) covariate(s) type of effect h j ( x , t ) (none) smooth intercept α ( t ) scalar covariate z linear effect z β ( t ) smooth effect γ ( z , t ) two scalars z 1 , z 2 linear interaction z 1 z 2 β ( t ) functional varying coefficient z 1 f ( z 2 , t ) smooth interaction f ( z 1 , z 2 , t ) functional covariate x ( s ) linear functional effect S x ( s ) β ( s , t ) ds � � u ( t ) historical effect ℓ ( t ) x ( s ) β ( s , t ) ds smooth functional effect F ( x ( s ) , s , t ) ds � scalar z , functional x ( s ) linear interaction z x ( s ) β ( s , t ) ds � smooth interaction x ( s ) β ( z , s , t ) ds � grouping variable g functional random intercept b g ( t ) grouping variable g and functional random slope zb g ( t ) scalar z curve indicator i curve-specific smooth residual e i ( t ) 6 / 35
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