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We propose a flexible regression framework to model the conditional distribution of multilevel generalized multivariate functional data of potentially mixed type, e.g. binary and continuous data. We make pointwise parametric distributional assumptions for each dimension of the multivariate functional data and model each distributional parameter as an additive function of covariates. The dependency between the different outcomes and, for multilevel functional data, also between different functions within a level is modelled by shared latent multivariate Gaussian processes. For a parsimonious representation of the latent processes, (generalized) multivariate functional principal components are estimated from the data and used as an empirical basis for these latent processes in the regression framework. Our modular two-step approach is very general and can easily incorporate new developments in the estimation of functional principal components for all types of (generalized) functional data. Flexible additive covariate effects for scalar or even functional covariates are available and are estimated in a Bayesian framework. We provide an easy-to-use implementation in the accompanying R package 'gmfamm' on CRAN and conduct a simulation study to confirm the validity of our regression framework and estimation strategy. The proposed multivariate functional model is applied to four dimensional traffic data in Berlin, which consists of the hourly numbers and mean speed of cars and trucks at different locations.

Reliable estimation and approximation of probability density functions is fundamental for their further processing. However, their specific properties, i.e. scale invariance and relative scale, prevent the use of standard methods of spline approximation and have to be considered when building a suitable spline basis. Bayes Hilbert space methodology allows to account for these properties of densities and enables their conversion to a standard Lebesgue space of square integrable functions using the centered log-ratio transformation. As the transformed densities fulfill a zero integral constraint, the constraint should likewise be respected by any spline basis used. Bayes Hilbert space methodology also allows to decompose bivariate densities into their interactive and independent parts with univariate marginals. As this yields a useful framework for studying the dependence structure between random variables, a spline basis ideally should admit a corresponding decomposition. This paper proposes a new spline basis for (transformed) bivariate densities respecting the desired zero integral property. We show that there is a one-to-one correspondence of this basis to a corresponding basis in the Bayes Hilbert space of bivariate densities using tools of this methodology. Furthermore, the spline representation and the resulting decomposition into interactive and independent parts are derived. Finally, this novel spline representation is evaluated in a simulation study and applied to empirical geochemical data.

Functional data analysis (FDA) is a statistical framework that allows for the analysis of curves, images, or functions on higher dimensional domains. The goals of FDA, such as descriptive analyses, classification, and regression, are generally the same as for statistical analyses of scalar-valued or multivariate data, but FDA brings additional challenges due to the high- and infinite dimensionality of observations and parameters, respectively. This paper provides an introduction to FDA, including a description of the most common statistical analysis techniques, their respective software implementations, and some recent developments in the field. The paper covers fundamental concepts such as descriptives and outliers, smoothing, amplitude and phase variation, and functional principal component analysis. It also discusses functional regression, statistical inference with functional data, functional classification and clustering, and machine learning approaches for functional data analysis. The methods discussed in this paper are widely applicable in fields such as medicine, biophysics, neuroscience, and chemistry, and are increasingly relevant due to the widespread use of technologies that allow for the collection of functional data. Sparse functional data methods are also relevant for longitudinal data analysis. All presented methods are demonstrated using available software in R by analyzing a data set on human motion and motor control. To facilitate the understanding of the methods, their implementation, and hands-on application, the code for these practical examples is made available on Github: https://github.com/davidruegamer/FDA_tutorial .

The joint modeling of multiple longitudinal biomarkers together with a time-to-event outcome is a challenging modeling task of continued scientific interest. In particular, the computational complexity of high dimensional (generalized) mixed effects models often restricts the flexibility of shared parameter joint models, even when the subject-specific marker trajectories follow highly nonlinear courses. We propose a parsimonious multivariate functional principal components representation of the shared random effects. This allows better scalability, as the dimension of the random effects does not directly increase with the number of markers, only with the chosen number of principal component basis functions used in the approximation of the random effects. The functional principal component representation additionally allows to estimate highly flexible subject-specific random trajectories without parametric assumptions. The modeled trajectories can thus be distinctly different for each biomarker. We build on the framework of flexible Bayesian additive joint models implemented in the R-package 'bamlss', which also supports estimation of nonlinear covariate effects via Bayesian P-splines. The flexible yet parsimonious functional principal components basis used in the estimation of the joint model is first estimated in a preliminary step. We validate our approach in a simulation study and illustrate its advantages by analyzing a study on primary biliary cholangitis.

Geochemical mapping of risk element concentrations in soils is performed in countries around the world. It results in large datasets of high analytical quality, which can be used to identify soils that violate individual legislative limits for safe food production. However, there is a lack of advanced data mining tools that would be suitable for sensitive exploratory data analysis of big data while respecting the natural variability of soil composition. To distinguish anthropogenic contamination from natural variation, the analysis of the entire data distributions for smaller sub-areas is key. In this article, we propose a new data mining method for geochemical mapping data based on functional data analysis of probability density functions in the framework of Bayes spaces after post-stratification of a big dataset to smaller districts. Proposed tools allow us to analyse the entire distribution, going beyond a superficial detection of extreme concentration anomalies. We illustrate the proposed methodology on a dataset gathered according to the Czech national legislation (1990--2009). Taking into account specific properties of probability density functions and recent results for orthogonal decomposition of multivariate densities enabled us to reveal real contamination patterns that were so far only suspected in Czech agricultural soils. We process the above Czech soil composition dataset by first compartmentalising it into spatial units, in particular the districts, and by subsequently clustering these districts according to diagnostic features of their uni- and multivariate distributions at high concentration ends. Comparison between compartments is key to the reliable distinction of diffuse contamination. In this work, we used soil contamination by Cu-bearing pesticides as an example for empirical testing of the proposed data mining approach.

This paper presents a novel approach to functional principal component analysis (FPCA) in Bayes spaces in the setting where densities are the object of analysis, but only few individual samples from each density are observed. We use the observed data directly to account for all sources of uncertainty, instead of relying on prior estimation of the underlying densities in a two-step approach, which can be inaccurate if small or heterogeneous numbers of samples per density are available. To account for the constrained nature of densities, we base our approach on Bayes spaces, which extend the Aitchison geometry for compositional data to density functions. For modeling, we exploit the isometric isomorphism between the Bayes space and the $\mathbb{L}^2$ subspace $\mathbb{L}_0^2$ with integration-to-zero constraint through the centered log-ratio transformation. As only discrete draws from each density are observed, we treat the underlying functional densities as latent variables within a maximum likelihood framework and employ a Monte Carlo Expectation Maximization (MCEM) algorithm for model estimation. Resulting estimates are useful for exploratory analyses of density data, for dimension reduction in subsequent analyses, as well as for improved preprocessing of sparsely sampled density data compared to existing methods. The proposed method is applied to analyze the distribution of maximum daily temperatures in Berlin during the summer months for the last 70 years, as well as the distribution of rental prices in the districts of Munich.

We propose regression models for curve-valued responses in two or more dimensions, where only the image but not the parametrization of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of objects. In the square-root-velocity framework, a parametrization invariant distance for curves is obtained as the quotient space metric with respect to the action of re-parametrization, which is by isometries. With this special case in mind, we discuss the generalization of 'linear' regression to quotient metric spaces more generally, before illustrating the usefulness of our approach for curves modulo re-parametrization. We address the issue of sparsely or irregularly sampled curves by using splines for modeling smooth conditional mean curves. We test this model in simulations and apply it to human hippocampal outlines, obtained from Magnetic Resonance Imaging scans. Here we model how the shape of the irregularly sampled hippocampus is related to age, Alzheimer's disease and sex.

We propose new ensemble models for multivariate functional data classification as combinations of semi-metric-based weak learners. Our models extend current semi-metric-type methods from the univariate to the multivariate case, propose new semi-metrics to compute distances between functions, and consider more flexible options for combining weak learners using stacked generalisation methods. We apply these ensemble models to identify respondents' difficulty with survey questions, with the aim to improve survey data quality. As predictors of difficulty, we use mouse movement trajectories from the respondents' interaction with a web survey, in which several questions were manipulated to create two scenarios with different levels of difficulty., Comment: 24 pages, 3 tables, 0 figures

Determining the mean shape of a collection of curves is not a trivial task, in particular when curves are only irregularly/sparsely sampled at discrete points. We newly propose an elastic full Procrustes mean of shapes of (oriented) plane curves, which are considered equivalence classes of parameterized curves with respect to translation, rotation, scale, and re-parameterization (warping), based on the square-root-velocity (SRV) framework. Identifying the real plane with the complex numbers, we establish a connection to covariance estimation in irregular/sparse functional data analysis. We introduce Hermitian covariance smoothing and show how to employ this extension of existing covariance estimation methods for obtaining an estimator of the (in)elastic full Procrustes mean, also in the sparse case not yet covered by existing (intrinsic) elastic shape means. For this, we provide different groundwork results which are also of independent interest: we characterize (the decomposition of) the covariance structure of rotation-invariant bivariate stochastic processes using complex representations, and we identify sampling schemes that allow for exact observation of derivatives / SRV transforms of sparsely sampled curves. We demonstrate the performance of the approach in a phonetic study on tongue shapes and in different realistic simulation settings, inter alia based on handwriting data.

We extend the generalised functional additive mixed model to include (functional) compositional covariates carrying relative information of a whole. Relying on the isometric isomorphism of the Bayes Hilbert space of probability densities with a subspace of the $L^2$, we include functional compositions as transformed functional covariates with constrained effect function. The extended model allows for the estimation of linear, nonlinear and time-varying effects of scalar and functional covariates, as well as (correlated) functional random effects, in addition to the compositional effects. We use the model to estimate the effect of the age, sex and smoking (functional) composition of the population on regional Covid-19 incidence data for Spain, while accounting for climatological and socio-demographic covariate effects and spatial correlation., Comment: submitted for publication

Corporate carbon emissions data is disclosed by approximately 65% of large and mid-sized companies globally, despite being a key indicator of corporate climate performance. With investors increasingly looking to integrate climate risk into their investment strategies and risk reporting, this creates demand for robust prediction models that can generate reliable estimates for missing carbon disclosures. However, these estimates lack transparency and are frequently used in the investment decisions process with the same confidence as corporate reported data. As disclosures remain mostly voluntary and the propensity to disclose is shaped by several factors (e.g. size, sector, geography), missing emissions data should be assumed to be missing not at random (MNAR). However, widely used estimation methods (e.g. linear regression models) typically do not correct for MNAR bias and do not accurately reflect the uncertainty of estimated data. The objective of this paper is to address these issues: (1) account for the uncertainty of the missing data and thus obtain regression coefficients by multiple imputation (MI) (2) correct for potential bias by using MI algorithms based on Heckman's sample selection model introduced by Galimard et al. (3) estimate missing carbon disclosures with linear models based on MI and report on the uncertainty of predicted values, measured as the length of the prediction interval. In the simulation, our approach resulted in an accuracy gain based on root mean squared error of up to 30%, and up to a 40% higher coverage rate than the existing models. When applied to commercial data, the results suggested up to 20% higher coverage for proposed methods., Comment: This paper was selected for oral presentation at the Global Research Alliance for Sustainable Finance and Investment (GRASFI) Conference in Beijing, China (2021). The paper was also submitted for publication at Energy Economics and is currently under review

Motivated by research on gender identity norms and the distribution of the woman's share in a couple's total labor income, we consider functional additive regression models for probability density functions as responses with scalar covariates. To preserve nonnegativity and integration to one under vector space operations, we formulate the model for densities in a Bayes Hilbert space, which allows to not only consider continuous densities, but also, e.g., discrete or mixed densities. Mixed ones occur in our application, as the woman's income share is a continuous variable having discrete point masses at zero and one for single-earner couples. Estimation is based on a gradient boosting algorithm, allowing for potentially numerous flexible covariate effects and model selection. We develop properties of Bayes Hilbert spaces related to subcompositional coherence, yielding (odds-ratio) interpretation of effect functions and simplified estimation for mixed densities via an orthogonal decomposition. Applying our approach to data from the German Socio-Economic Panel Study (SOEP) shows a more symmetric distribution in East German than in West German couples after reunification and a smaller child penalty comparing couples with and without minor children. These West-East differences become smaller, but are persistent over time.

The "shape" of a planar curve and/or landmark configuration is considered its equivalence class under translation, rotation and scaling, its "form" its equivalence class under translation and rotation while scale is preserved. We extend generalized additive regression to models for such shapes/forms as responses respecting the resulting quotient geometry by employing the squared geodesic distance as loss function and a geodesic response function to map the additive predictor to the shape/form space. For fitting the model, we propose a Riemannian $L_2$-Boosting algorithm well suited for a potentially large number of possibly parameter-intensive model terms, which also yields automated model selection. We provide novel intuitively interpretable visualizations for (even non-linear) covariate effects in the shape/form space via suitable tensor-product factorization. The usefulness of the proposed framework is illustrated in an analysis of 1) astragalus shapes of wild and domesticated sheep and 2) cell forms generated in a biophysical model, as well as 3) in a realistic simulation study with response shapes and forms motivated from a dataset on bottle outlines.

We provide statistical analysis methods for samples of curves when the image but not the parametrisation of the curves is of interest. A parametrisation invariant analysis can be based on the elastic distance of the curves modulo warping, but existing methods have limitations in common realistic settings where curves are irregularly and potentially sparsely observed. We provide methods and algorithms to approximate the elastic distance for such curves via interpreting them as polygons. Moreover, we propose to use spline curves for modelling smooth or polygonal Fr\'echet means of open or closed curves with respect to the elastic distance and show identifiability of the spline model modulo warping. We illustrate the use of our methods for elastic mean and distance computation by application to two datasets. The first application clusters sparsely sampled GPS tracks based on the elastic distance and computes smooth means for each cluster to find new paths on Tempelhof field in Berlin. The second classifies irregularly sampled handwritten spirals of Parkinson's patients and controls based on the elastic distance to a mean spiral curve computed using our approach. All developed methods are implemented in the \texttt{R}-package \texttt{elasdics} and evaluated in simulations.

Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary like 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 e.g.\ repeated measurements or crossed study designs. Modeling 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 modeling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.

Tuning of model-based boosting algorithms relies mainly on the number of iterations, while the step-length is fixed at a predefined value. For complex models with several predictors such as Generalized Additive Models for Location, Scale and Shape (GAMLSS), imbalanced updates of predictors, where some distribution parameters are updated more frequently than others, can be a problem that prevents some submodels to be appropriately fitted within a limited number of boosting iterations. We propose an approach using adaptive step-length (ASL) determination within a non-cyclical boosting algorithm for GAMLSS to prevent such imbalance. Moreover, for the important special case of the Gaussian distribution, we discuss properties of the ASL and derive a semi-analytical form of the ASL that avoids manual selection of the search interval and numerical optimization to find the optimal step-length, and consequently improves computational efficiency. We show competitive behavior of the proposed approaches compared to penalized maximum likelihood and boosting with a fixed step-length for GAMLSS models in two simulations and two applications, in particular for cases of large variance and/or more variables than observations. In addition, the idea of the ASL is also applicable to other models with more than one predictor like zero-inflated count model, and brings up insights into the choice of the reasonable defaults for the step-length in simpler special case of (Gaussian) additive models.

A central goal of survey research is to collect robust and reliable data from respondents. However, despite researchers' best efforts in designing questionnaires, respondents may experience difficulty understanding questions' intent and therefore may struggle to respond appropriately. If it were possible to detect such difficulty, this knowledge could be used to inform real-time interventions through responsive questionnaire design, or to indicate and correct measurement error after the fact. Previous research in the context of web surveys has used paradata, specifically response times, to detect difficulties and to help improve user experience and data quality. However, richer data sources are now available, in the form of the movements respondents make with the mouse, as an additional and far more detailed indicator for the respondent-survey interaction. This paper uses machine learning techniques to explore the predictive value of mouse-tracking data with regard to respondents' difficulty. We use data from a survey on respondents' employment history and demographic information, in which we experimentally manipulate the difficulty of several questions. Using features derived from the cursor movements, we predict whether respondents answered the easy or difficult version of a question, using and comparing several state-of-the-art supervised learning methods. In addition, we develop a personalization method that adjusts for respondents' baseline mouse behavior and evaluate its performance. For all three manipulated survey questions, we find that including the full set of mouse movement features improved prediction performance over response-time-only models in nested cross-validation. Accounting for individual differences in mouse movements led to further improvements., Comment: 40 pages, 2 Figures, 3 Tables

This work addresses the problem of conducting valid inference for additive and linear mixed models after model selection. One possible solution to overcome overconfident inference results after model selection is selective inference, which constitutes a post-selection inference framework, yielding valid inference statements by conditioning on the selection event. We extend recent work on selective inference to the class of additive and linear mixed models for any type of model selection mechanism that can be expressed as a function of the outcome variable (and potentially on covariates on which it conditions). We investigate the properties of our proposal in simulation studies and apply the framework to a data set in monetary economics. Due to the generality of our proposed approach, the presented approach also works for non-standard selection procedures, which we demonstrate in our application. Here, the final additive mixed model is selected using a hierarchical selection procedure, which is based on the conditional Akaike information criterion and involves varying data set sizes.

Functional data typically contains amplitude and phase variation. In many data situations, phase variation is treated as a nuisance effect and is removed during preprocessing, although it may contain valuable information. In this note, we focus on joint principal component analysis (PCA) of amplitude and phase variation. As the space of warping functions has a complex geometric structure, one key element of the analysis is transforming the warping functions to $L^2(\mathcal{T})$. We present different transformation approaches and show how they fit into a general class of transformations. This allows us to compare their strengths and limitations. In the context of PCA, our results offer arguments in favour of the centered log-ratio transformation. We further embed existing approaches from Hadjipantelis et al. (2015) and Lee and Jung (2017) for joint PCA of amplitude and phase variation into the framework of multivariate functional PCA, where we study the properties of the estimators based on an appropriate metric. The approach is illustrated through an application from seismology., Comment: Contains Supplementary Files

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 formulations

We propose a statistical inference framework for the component-wise functional gradient descent algorithm (CFGD) under normality assumption for model errors, also known as $L_2$-Boosting. The CFGD is one of the most versatile tools to analyze data, because it scales well to high-dimensional data sets, allows for a very flexible definition of additive regression models and incorporates inbuilt variable selection. Due to the variable selection, we build on recent proposals for post-selection inference. However, the iterative nature of component-wise boosting, which can repeatedly select the same component to update, necessitates adaptations and extensions to existing approaches. We propose tests and confidence intervals for linear, grouped and penalized additive model components selected by $L_2$-Boosting. Our concepts also transfer to slow-learning algorithms more generally, and to other selection techniques which restrict the response space to more complex sets than polyhedra. We apply our framework to an additive model for sales prices of residential apartments and investigate the properties of our concepts in simulation studies.

Model selection in mixed models based on the conditional distribution is appropriate for many practical applications and has been a focus of recent statistical research. In this paper we introduce the R-package cAIC4 that allows for the computation of the conditional Akaike Information Criterion (cAIC). Computation of the conditional AIC needs to take into account the uncertainty of the random effects variance and is therefore not straightforward. We introduce a fast and stable implementation for the calculation of the cAIC for linear mixed models estimated with lme4 and additive mixed models estimated with gamm4 . Furthermore, cAIC4 offers a stepwise function that allows for a fully automated stepwise selection scheme for mixed models based on the conditional AIC. Examples of many possible applications are presented to illustrate the practical impact and easy handling of the package.

Joint models of longitudinal and survival data have become an important tool for modeling associations between longitudinal biomarkers and event processes. The association between marker and log-hazard is assumed to be linear in existing shared random effects models, with this assumption usually remaining unchecked. We present an extended framework of flexible additive joint models that allows the estimation of nonlinear, covariate specific associations by making use of Bayesian P-splines. Our joint models are estimated in a Bayesian framework using structured additive predictors for all model components, allowing for great flexibility in the specification of smooth nonlinear, time-varying and random effects terms for longitudinal submodel, survival submodel and their association. The ability to capture truly linear and nonlinear associations is assessed in simulations and illustrated on the widely studied biomedical data on the rare fatal liver disease primary biliary cirrhosis. All methods are implemented in the R package bamlss to facilitate the application of this flexible joint model in practice., Comment: Changes to initial commit: minor language editing, additional information in Section 4, formatting in Supplementary Information

Complex statistical models such as scalar-on-image regression often require strong assumptions to overcome the issue of non-identifiability. While in theory it is well understood that model assumptions can strongly influence the results, this seems to be underappreciated, or played down, in practice. The article gives a systematic overview of the main approaches for scalar-on-image regression with a special focus on their assumptions. We categorize the assumptions and develop measures to quantify the degree to which they are met. The impact of model assumptions and the practical usage of the proposed measures are illustrated in a simulation study and in an application to neuroimaging data. The results show that different assumptions indeed lead to quite different estimates with similar predictive ability, raising the question of their interpretability. We give recommendations for making modeling and interpretation decisions in practice, based on the new measures and simulations using hypothetic coefficient images and the observed data.

Statistical inference after model selection requires an inference framework that takes the selection into account in order to be valid. Following recent work on selective inference, we derive analytical expressions for inference after likelihood- or test-based model selection for linear models.

The R add-on package FDboost is a flexible toolbox for the estimation of functional regression models by model-based boosting. It provides the possibility to fit regression models for scalar and functional response with effects of scalar as well as functional covariates, i.e., scalar-on-function, function-on-scalar and function-on-function regression models. In addition to mean regression, quantile regression models as well as generalized additive models for location scale and shape can be fitted with FDboost. Furthermore, boosting can be used in high-dimensional data settings with more covariates than observations. We provide a hands-on tutorial on model fitting and tuning, including the visualization of results. The methods for scalar-on-function regression are illustrated with spectrometric data of fossil fuels and those for functional response regression with a data set including bioelectrical signals for emotional episodes., Comment: Revised version

The joint modeling of longitudinal and time-to-event data is an important tool of growing popularity to gain insights into the association between a biomarker and an event process. We develop a general framework of flexible additive joint models that allows the specification of a variety of effects, such as smooth nonlinear, time-varying and random effects, in the longitudinal and survival parts of the models. Our extensions are motivated by the investigation of the relationship between fluctuating disease-specific markers, in this case autoantibodies, and the progression to the autoimmune disease type 1 diabetes. By making use of Bayesian P-splines we are in particular able to capture highly nonlinear subject-specific marker trajectories as well as a time-varying association between the marker and the event process allowing new insights into disease progression. The model is estimated within a Bayesian framework and implemented in the R-package bamlss.

We propose a fast bivariate smoothing approach for symmetric surfaces that has a wide range of applications. We show how it can be applied to estimate the covariance function in longitudinal data as well as multiple additive covariances in functional data with complex correlation structures. Our symmetric smoother can handle (possibly noisy) data sampled on a common, dense grid as well as irregularly or sparsely sampled data. Estimation is based on bivariate penalized spline smoothing using a mixed model representation and the symmetry is used to reduce computation time compared to the usual non-symmetric smoothers. We outline the application of our approach in functional principal component analysis and demonstrate its practical value in two applications. The approach is evaluated in extensive simulations. We provide documented open source software implementing our fast symmetric bivariate smoother building on established algorithms for additive models.

The link between different psychophysiological measures during emotion episodes is not well understood. To analyse the functional relationship between electroencephalography (EEG) and facial electromyography (EMG), we apply historical function-on-function regression models to EEG and EMG data that were simultaneously recorded from 24 participants while they were playing a computerised gambling task. Given the complexity of the data structure for this application, we extend simple functional historical models to models including random historical effects, factor-specific historical effects, and factor-specific random historical effects. Estimation is conducted by a component-wise gradient boosting algorithm, which scales well to large data sets and complex models., Comment: Revised version

We discuss scalar-on-function regression models where all parameters of the assumed response distribution can be modeled depending on covariates. We thus combine signal regression models with generalized additive models for location, scale and shape (GAMLSS). We compare two fundamentally different methods for estimation, a gradient boosting and a penalized likelihood based approach, and address practically important points like identifiability and model choice. Estimation by a component-wise gradient boosting algorithm allows for high dimensional data settings and variable selection. Estimation by a penalized likelihood based approach has the advantage of directly provided statistical inference. The motivating application is a time series of stock returns where it is of interest to model both the expectation and the variance depending on lagged response values and functional liquidity curves.

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 conditional on estimated principal components. We provide R-code implementing our approach. The method is motivated by and applied to data from speech production research.

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 non-central $t$-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".

Regression models with functional responses and covariates constitute a powerful and increasingly important model class. However, regression with functional data poses well known and challenging problems of non-identifiability. This non-identifiability can manifest itself in arbitrarily large errors for coefficient surface estimates despite accurate predictions of the responses, thus invalidating substantial interpretations of the fitted models. We offer an accessible rephrasing of these identifiability issues in realistic applications of penalized linear function-on-function-regression and delimit the set of circumstances under which they are likely to occur in practice. Specifically, non-identifiability that persists under smoothness assumptions on the coefficient surface can occur if the functional covariate's empirical covariance has a kernel which overlaps that of the roughness penalty of the spline estimator. Extensive simulation studies validate the theoretical insights, explore the extent of the problem and allow us to evaluate their practical consequences under varying assumptions about the data generating processes. A case study illustrates the practical significance of the problem. Based on theoretical considerations and our empirical evaluation, we provide immediately applicable diagnostics for lack of identifiability and give recommendations for avoiding estimation artifacts in practice., Comment: 27 pages, 8 figures

We develop a flexible framework for modeling high-dimensional imaging data observed longitudinally. The approach decomposes the observed variability of repeatedly measured high-dimensional observations into three additive components: a subject-specific imaging random intercept that quantifies the cross-sectional variability, a subject-specific imaging slope that quantifies the dynamic irreversible deformation over multiple realizations, and a subject-visit-specific imaging deviation that quantifies exchangeable effects between visits. The proposed method is very fast, scalable to studies including ultrahigh-dimensional data, and can easily be adapted to and executed on modest computing infrastructures. The method is applied to the longitudinal analysis of diffusion tensor imaging (DTI) data of the corpus callosum of multiple sclerosis (MS) subjects. The study includes $176$ subjects observed at $466$ visits. For each subject and visit the study contains a registered DTI scan of the corpus callosum at roughly 30,000 voxels., Comment: Published in at http://dx.doi.org/10.1214/14-AOAS748 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We present a new statistical modelling approach where the response is a function of high frequency count data. Our application is about investigating the relationship between the health outcome fat mass and physical activity (PA) measured by accelerometer. The accelerometer quantifies the intensity of physical activity as counts per epoch over a given period of time. We use data from the Avon longitudinal study of parents and children (ALSPAC) where accelerometer data is available as a time series of accelerometer counts per minute over seven days for a subset of children. In order to compare accelerometer profiles between individuals and to reduce the high dimension a functional summary of the profiles is used. We use the histogram as a functional summary due to its simplicity, suitability and ease of interpretation. Our model is an extension of generalised regression of scalars on functions or signal regression. It allows also multi-dimensional functional predictors and additive non-linear predictors for metric covariates. The additive multidimensional functional predictors allow investigating specific questions about whether the effect of PA varies over its intensity, by gender, by time of day or by day of the week. The key feature of the model is that it utilises the full profile of measured PA without requiring cut-points defining intensity levels for light, moderate and vigorous activity. We show that the (not necessarily causal) effect of PA is not linear and not constant over the activity intensity. Also, there is little evidence to suggest that the effect of PA intensity varies by gender or whether it happens on weekdays or on weekends.

The functional generalized additive model (FGAM) was recently proposed in McLean et al. (2013) as a more flexible alternative to the common functional linear model (FLM) for regressing a scalar on functional covariates. In this paper, we develop a Bayesian version of FGAM for the case of Gaussian errors with identity link function. Our approach allows the functional covariates to be sparsely observed and measured with error, whereas the estimation procedure of McLean et al. (2013) required that they be noiselessly observed on a regular grid. We consider both Monte Carlo and variational Bayes methods for fitting the FGAM with sparsely observed covariates. Due to the complicated form of the model posterior distribution and full conditional distributions, standard Monte Carlo and variational Bayes algorithms cannot be used. The strategies we use to handle the updating of parameters without closed-form full conditionals should be of independent interest to applied Bayesian statisticians working with nonconjugate models. Our numerical studies demonstrate the benefits of our algorithms over a two-step approach of first recovering the complete trajectories using standard techniques and then fitting a functional regression model. In a real data analysis, our methods are applied to forecasting closing price for items up for auction on the online auction website eBay., Comment: substantial updates based on referee comments

Motivated by modern observational studies, we introduce a class of functional models that expands nested and crossed designs. These models account for the natural inheritance of correlation structure from sampling design in studies where the fundamental sampling unit is a function or image. Inference is based on functional quadratics and their relationship with the underlying covariance structure of the latent processes. A computationally fast and scalable estimation procedure is developed for ultra-high dimensional data. Methods are illustrated in three examples: high-frequency accelerometer data for daily activity, pitch linguistic data for phonetic analysis, and EEG data for studying electrical brain activity during sleep.

We propose an extensive framework for additive regression models for correlated 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. Additionally, our framework includes linear and nonlinear effects of functional and scalar covariates that may vary smoothly over the index of the functional response. It accommodates densely or sparsely observed functional responses and predictors which may be observed with additional error and includes both spline-based and functional principal component-based terms. Estimation and inference in this framework is based on standard additive mixed models, allowing us to take advantage of established methods and robust, flexible algorithms. We provide easy-to-use open source software in the pffr() function for the R-package refund. Simulations show that the proposed method recovers relevant effects reliably, handles small sample sizes well and also scales to larger data sets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach., Comment: 26 pages, 8 figures