Generalized functional additive mixed models

Abstract

We propose a comprehensive framework for additive regression models for non-Gaussian functional responses, allowing for multiple (partially) nested or crossed functional random effects with flexible correlation structures for, e.g., spatial, temporal, or longitudinal functional data as well as linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. Our implementation handles functional responses from any exponential family distribution as well as many others like Beta-or scaled and shifted t-distributions. Development is motivated by and evaluated on an application to large-scale longitudinal feeding records of pigs. Results in extensive simulation studies as well as replications of two previously published simulation studies for generalized functional mixed models demonstrate the good performance of our proposal. The approach is implemented in well-documented open source software in the pffr function in R-package refund. 0 arXiv:1506.05384v3 [stat.ME] 6 May 2016 Scheipl, Gertheiss, Greven/GFAMM

Signal Regression Models for Location, Scale and Shape with an Application to Stock Returns

“…In future research, we shall consider GAMLSSs for functional responses with scalar and/or functional covariates, with the aim of extending the flexible functional regression model framework of Scheipl et al . () and Brockhaus et al . (, ), estimated by penalized likelihood and boosting respectively, to simultaneous models for several response distribution parameters.…”

A note on modeling sparse exponential-family functional response curves

“…However, our proposed techniques are also applicable for densely observed data, and in that setting retain the benefits of parsimony and useful descriptions of the major patterns of variability in the population. That being said, dense data at the subject level allow the estimation of subject-specific deviations from the mean using, for example, a spline basis expansion [18] for each subject rather than an FPC expansion. Such an approach is expected to yield approximately unbiased estimates of the mean function for dense data, although it is not feasible for sparse data.…”

“…By using the estimated covariance matrix of the basis coefficients, one could rotate the initial basis to obtain the FPC to be used in the two-step model. Recently, Scheipl et al [18] considered ( i ) to model non-Gaussian functional data and found that doing so works well in terms of accurately predicting the subject-specific curves for densely observed data, but it may be problematic in the case of sparsely observed data. Specifically, using a relatively large number of basis functions, subject-specific basis coefficients are hard to estimate with a small number of irregularly sampled observations per subject.…”

“…Analyzing repeated exponential-family outcomes using functional data approaches is currently an area of intensive research [12, 13, 14, 15, 16, 17, 18]. Hall et al [12] extended model (1) to handle non-Gaussian functional data.…”

Comparison and contrast of two general functional regression modelling frameworks

“…In this article, Greven and Scheipl describe an impressively general framework for performing functional regression that encompasses all three of these areas, and can be described as containing many of the existing methods as special cases. The formulation and building up of this general framework has been a topic in this research group’s work over the past few years, introducing much of this structure for Gaussian functions largely represented by splines and fit using generalized additive model (GAM) software in Scheipl, Staicu and Greven (2015), incorporating functional principal components (fPC) to flexibly model sparse, irregularly sampled outcomes in Cederbaum, et al (2015), extending to generalized outcomes in Scheipl, Gertheiss, and Greven (2016), and introducing a new boosting-based fitting procedure that allows extension to robust functional regression and confers other benefits in Brockhaus, et al (2015). Other specific work has been done developing details for additive scalar-on function models (McLean et al 2014) and function-on-function regression (Ivanescu, et al 2015; Scheipl and Greven 2016; Brockhaus, et al 2016), and undoubtedly this productive group will continue to further develop this framework in the coming years.…”