A general framework for functional regression modelling

Researchers are increasingly interested in regression models for functional data. This article discusses a comprehensive framework for additive (mixed) models for functional responses and/or functional covariates based on the guiding principle of reframing functional regression in terms of corresponding models for scalar data, allowing the adaptation of a large body of existing methods for these novel tasks. The framework encompasses many existing as well as new models. It includes regression for generalized' functional data, mean regression, quantile regression as well as generalized additive models for location, shape and scale (GAMLSS) for functional data. It admits many flexible linear, smooth or interaction terms of scalar and functional covariates as well as (functional) random effects and allows flexible choices of basesparticularly splines and functional principal componentsand corresponding penalties for each term. It covers functional data observed on common (dense) or curve-specific (sparse) grids. Penalized-likelihood-based and gradient-boosting-based inference for these models are implemented in R packages refund and FDboost, respectively. We also discuss identifiability and computational complexity for the functional regression models covered. A running example on a longitudinal multiple sclerosis imaging study serves to illustrate the flexibility and utility of the proposed model class. Reproducible code for this case study is made available online

Boosting Functional Response Models for Location, Scale and Shape with an Application to Bacterial Competition

We extend Generalized Additive Models for Location, Scale, and Shape (GAMLSS) to regression with functional response. This allows us to simultaneously model point-wise mean curves, variances and other distributional parameters of the response in dependence of various scalar and functional covariate effects. In addition, the scope of distributions is extended beyond exponential families. The model is fitted via gradient boosting, which offers inherent model selection and is shown to be suitable for both complex model structures and highly auto-correlated response curves. This enables us to analyze bacterial growth in \textit{Escherichia coli} in a complex interaction scenario, fruitfully extending usual growth models.Comment: bootstrap confidence interval type uncertainty bounds added; minor changes in formulation

Regression modelling with I-priors

We introduce the I-prior methodology as a unifying framework for estimating a variety of regression models, including varying coefficient, multilevel, longitudinal models, and models with functional covariates and responses. It can also be used for multi-class classification, with low or high dimensional covariates. The I-prior is generally defined as a maximum entropy prior. For a regression function, the I-prior is Gaussian with covariance kernel proportional to the Fisher information on the regression function, which is estimated by its posterior distribution under the I-prior. The I-prior has the intuitively appealing property that the more information is available on a linear functional of the regression function, the larger the prior variance, and the smaller the influence of the prior mean on the posterior distribution. Advantages compared to competing methods, such as Gaussian process regression or Tikhonov regularization, are ease of estimation and model comparison. In particular, we develop an EM algorithm with a simple E and M step for estimating hyperparameters, facilitating estimation for complex models. We also propose a novel parsimonious model formulation, requiring a single scale parameter for each (possibly multidimensional) covariate and no further parameters for interaction effects. This simplifies estimation because fewer hyperparameters need to be estimated, and also simplifies model comparison of models with the same covariates but different interaction effects; in this case, the model with the highest estimated likelihood can be selected. Using a number of widely analyzed real data sets we show that predictive performance of our methodology is competitive. An R-package implementing the methodology is available (Jamil, 2019)

What's the Risk? Older Women Report Fewer Symptoms for Suspected Acute Coronary Syndrome than Younger Women.

The purpose of the study was to determine whether older (≥65 years) and younger (<65 years) women presenting to the emergency department (ED) with symptoms suggestive of acute coronary syndrome (ACS) varied on risk factors, comorbid conditions, functional status, and symptoms that have implications for emergent cardiac care. Women admitted to five EDs were enrolled. The ACS Symptom Checklist was used to measure symptoms. Comorbid conditions and functional status were measured with the Charlson Comorbidity Index and Duke Activity Status Index. Logistic regression models were used to evaluate symptom differences in older and younger women adjusting for ACS diagnosis, functional status, body mass index (BMI), and comorbid conditions. Analyses were stratified by age, and interaction of symptom by age was tested. Four hundred women were enrolled. Mean age was 61.3 years (range 21-98). Older women (n = 163) were more likely to have hypertension, hypercholesterolemia, never smoked, lower BMI, more comorbid conditions, and lower functional status. Younger women (n = 237) were more likely to be members of minority groups, be college-educated, and have a non-ACS discharge diagnosis. Younger women had higher odds of experiencing chest discomfort, chest pain, chest pressure, shortness of breath, nausea, sweating, and palpitations. Lack of chest symptoms and shortness of breath (key symptoms triggering a decision to seek emergency care) may cause older women to delay seeking treatment, placing them at risk for poorer outcomes. Younger African American women may require more comprehensive risk reduction strategies and symptom management

Model-based boosting in high dimensions

Summary: The R add-on package mboost implements functional gradient descent algorithms (boosting) for optimizing general loss functions utilizing componentwise least squares, either of parametric linear form or smoothing splines, or regression trees as base learners for fitting generalized linear, additive and interaction models to potentially high-dimensional data. Availability: Package mboost is available from the Comprehensive R Archive Network () under the terms of the General Public Licence (GPL). Contact: [email protected]

Image-based remapping of spatially-varying material appearance

BRDF models are ubiquitous tools for the representation of material appearance. However, there is now an astonishingly large number of different models in practical use. Both a lack of BRDF model standardisation across implementations found in different renderers, as well as the often semantically different capabilities of various models, have grown to be a major hindrance to the interchange of production assets between different rendering systems. Current attempts to solve this problem rely on manually finding visual similarities between models, or mathematical ones between their functional shapes, which requires access to the shader implementation, usually unavailable in commercial renderers. We present a method for automatic translation of material appearance between different BRDF models, which uses an image-based metric for appearance comparison, and that delegates the interaction with the model to the renderer. We analyse the performance of the method, both with respect to robustness and visual differences of the fits for multiple combinations of BRDF models. While it is effective for individual BRDFs, the computational cost does not scale well for spatially-varying BRDFs. Therefore, we further present a parametric regression scheme that approximates the shape of the transformation function and generates a reduced representation which evaluates instantly and without further interaction with the renderer. We present respective visual comparisons of the remapped SVBRDF models for commonly used renderers and shading models, and show that our approach is able to extrapolate transformed BRDF parameters better than other complex regression schemes

Flexible Regression Models for Estimating Interactions between a Treatment and Scalar/Functional Predictors

In this dissertation, we develop regression models for estimating interactions between a treatment variable and a set of baseline predictors in their eect on the outcome in a randomized trial, without restriction to a linear relationship. The proposed semiparametric/nonparametric regression approaches for representing interactions generalize the notion of an interaction between a categorical treatment variable and a set of predictors on the outcome, from a linear model context. In Chapter 2, we develop a model for determining a composite predictor from a set of baseline predictors that can have a nonlinear interaction with the treatment indicator, implying that the treatment efficacy can vary across values of such a predictor without a linearity restriction. We introduce a parsimonious generalization of the single-index models that targets the eect of the interaction between the treatment conditions and the vector of predictors on the outcome. A common approach to interrogate such treatment-by-predictor interaction is to t a regression curve as a function of the predictors separately for each treatment group. For parsimony and insight, we propose a single-index model with multiple-links that estimates a single linear combination of the predictors (i.e., a single-index), with treatment-specic nonparametrically-dened link functions. The approach emphasizes a focus on the treatment-by-predictors interaction eects on the treatment outcome that are relevant for making optimal treatment decisions. Asymptotic results for estimator are obtained under possible model misspecication. A treatment decision rule based on the derived single-index is dened, and it is compared to other methods for estimating optimal treatment decision rules. An application to a clinical trial for the treatment of depression is presented to illustrate the proposed approach for deriving treatment decision rules. In Chapter 3, we allow the proposed single-index model with multiple-links to have an unspecified main effect of the predictors on the outcome. This extension greatly increases the utility of the proposed regression approach for estimating the treatment-by-predictors interactions. By obviating the need to model the main eect, the proposed method extends the modied covariate approach of [Tian et al., 2014] into a semiparametric regression framework. Also, the approach extends [Tian et al., 2014] into general K treatment arms. In Chapter 4, we introduce a regularization method to deal with the potential high dimensionality of the predictor space and to simultaneously select relevant treatment effect modiers exhibiting possibly nonlinear associations with the outcome. We present a set of extensive simulations to illustrate the performance of the treatment decision rules estimated from the proposed method. An application to a clinical trial for the treatment of depression is presented to illustrate the proposed approach for deriving treatment decision rules. In Chapter 5, we develop a novel additive regression model for estimating interactions between a treatment and a potentially large number of functional/scalar predictor. If the main effect of baseline predictors is misspecied or high-dimensional (or, innite dimensional), any standard nonparametric or semiparametric approach for estimating the treatment-bypredictors interactions tends to be not satisfactory because it is prone to (possibly severe) inconsistency and poor approximation to the true treatment-by-predictors interaction effect. To deal with this problem, we impose a constraint on the model space, giving the orthogonality between the main and the interaction effects. This modeling method is particularly appealing in the functional regression context, since a functional predictor, due to its infinite dimensional nature, must go through some sort of dimension reduction, which essentially involves a main effect model misspecication. The main effect and the interaction effect can be estimated separately due to the orthogonality between the two effects, which side-steps the issue of misspecication of the main effect. The proposed approach extends the modied covariate approach of [Tian et al., 2014] into an additive regression model framework. We impose a concave penalty in estimation, and the method simultaneously selects functional/scalar treatment effect modifiers that exhibit possibly nonlinear interaction effects with the treatment indicator. The dissertation concludes in Chapter 6