Penalized function-on-function regression

Effect of Digoxin vs Beta-Blockers on Left Atrial Strain for Heart Rate-Controlled Atrial Fibrillation: The DIGOBET-AF Randomized Clinical Trial

BACKGROUND AND OBJECTIVE: Left atrial strain (LAS) has prognostic value in patients with atrial fibrillation (AF). Consequently, therapies that improve LAS may help reduce AF-related adverse cardiac events. We aimed to compare how digoxin and bisoprolol modulate LAS in patients with AF being treated with rate control.

METHODS

This was a bicentric randomized controlled trial. Patients with AF, naïve to beta-blockers and digoxin, and scheduled for treatment with a rate control strategy were randomized to receive oral bisoprolol 5-10 mg daily or digoxin 0.25 mg daily. The primary aim was to compare the change in peak LAS before and after 30 days of treatment between the two groups.

RESULTS

A total of 60 patients, equally distributed between the two groups, completed the trial. By day 30, there was no significant difference in global peak LAS between the groups. However, when analyzed separately, the two-chamber view showed a significantly higher peak LAS in the digoxin group than in the BB group (mean 7.5 ± standard deviation 3.2% vs. 5.9 ± 3.4%; p = 0.004). Similarly, the four-chamber view also showed a higher peak LAS in the digoxin group (7.2 ± 3.6% vs. 6.4 ± 3.8%; p = 0.047). Considering the entire LAS curve rather than solely the peak value, digoxin significantly increased all LAS curves. In the global and four-chamber view, the digoxin maximum effect occurred significantly earlier than the peak of the LAS curve (p < 0.001). This effect remained constant over the cardiac cycle in the two-chamber curve (p < 0.001).

CONCLUSION

Our findings suggest that, in patients with rate-controlled AF, digoxin positively modulates LAS when compared with bisoprolol.

CLINICAL TRIALS REGISTRATION NUMBER

NCT05540600, https://clinicaltrials.gov .

Functional regression clustering with multiple functional gene expressions

Gene expression data is often collected in time series experiments, under different experimental conditions. There may be genes that have very different gene expression profiles over time, but that adjust their gene expression patterns in the same way under experimental conditions. Our aim is to develop a method that finds clusters of genes in which the relationship between these temporal gene expression profiles are similar to one another, even if the individual temporal gene expression profiles differ. We propose a K-means-type algorithm in which each cluster is defined by a function-on-function regression model, which, inter alia, allows for multiple functional explanatory variables. We validate this novel approach through extensive simulations and then apply it to identify groups of genes whose diurnal expression pattern is perturbed by the season in a similar way. Our clusters are enriched for genes with similar biological functions, including one cluster enriched in both photosynthesis-related functions and polysomal ribosomes, which shows that our method provides useful and novel biological insights.

Functional Neural Networks for High-Dimensional Genetic Data Analysis

Artificial intelligence (AI) is a thriving research field with many successful applications in areas such as computer vision and speech recognition. Machine learning methods, such as artificial neural networks (ANN), play a central role in modern AI technology. While ANN also holds great promise for human genetic research, the high-dimensional genetic data and complex genetic structure bring tremendous challenges. The vast majority of genetic variants on the genome have small or no effects on diseases, and fitting ANN on a large number of variants without considering the underlying genetic structure (e.g., linkage disequilibrium) could bring a serious overfitting issue. Furthermore, while a single disease phenotype is often studied in a classic genetic study, in emerging research fields (e.g., imaging genetics), researchers need to deal with different types of disease phenotypes. To address these challenges, we propose a functional neural networks (FNN) method. FNN uses a series of basis functions to model high-dimensional genetic data and a variety of phenotype data and further builds a multi-layer functional neural network to capture the complex relationships between genetic variants and disease phenotypes. Through simulations, we demonstrate the advantages of FNN for high-dimensional genetic data analysis in terms of robustness and accuracy. The real data applications also showed that FNN attained higher accuracy than the existing methods.

A tensor based varying-coefficient model for multi-modal neuroimaging data analysis

All neuroimaging modalities have their own strengths and limitations. A current trend is toward interdisciplinary approaches that use multiple imaging methods to overcome limitations of each method in isolation. At the same time neuroimaging data is increasingly being combined with other non-imaging modalities, such as behavioral and genetic data. The data structure of many of these modalities can be expressed as time-varying multidimensional arrays (tensors), collected at different time-points on multiple subjects. Here, we consider a new approach for the study of neural correlates in the presence of tensor-valued brain images and tensor-valued covariates, where both data types are collected over the same set of time points. We propose a time-varying tensor regression model with an inherent structural composition of responses and covariates. Regression coefficients are expressed using the B-spline technique, and the basis function coefficients are estimated using CP-decomposition by minimizing a penalized loss function. We develop a varying-coefficient model for the tensor-valued regression model, where both covariates and responses are modeled as tensors. This development is a non-trivial extension of function-on-function concurrent linear models for complex and large structural data, where the inherent structures are preserved. In addition to the methodological and theoretical development, the efficacy of the proposed method based on both simulated and real data analysis (e.g., the combination of eye-tracking data and functional magnetic resonance imaging (fMRI) data) is also discussed.

Nonparametric predictive model for sparse and irregular longitudinal data

We propose a kernel-based estimator to predict the mean response trajectory for sparse and irregularly measured longitudinal data. The kernel estimator is constructed by imposing weights based on the subject-wise similarity on L2 metric space between predictor trajectories, where we assume that an analogous fashion in predictor trajectories over time would result in a similar trend in the response trajectory among subjects. In order to deal with the curse of dimensionality caused by the multiple predictors, we propose an appealing multiplicative model with multivariate Gaussian kernels. This model is capable of achieving dimension reduction as well as selecting functional covariates with predictive significance. The asymptotic properties of the proposed nonparametric estimator are investigated under mild regularity conditions. We illustrate the robustness and flexibility of our proposed method via extensive simulation studies and an application to the Framingham Heart Study.

SpaceANOVA: Spatial co-occurrence analysis of cell types in multiplex imaging data using point process and functional ANOVA

Motivation

Multiplex imaging platforms have enabled the identification of the spatial organization of different types of cells in complex tissue or tumor microenvironment (TME). Exploring the potential variations in the spatial co-occurrence or co-localization of different cell types across distinct tissue or disease classes can provide significant pathological insights, paving the way for intervention strategies. However, the existing methods in this context either rely on stringent statistical assumptions or suffer from a lack of generalizability.

Results

We present a highly powerful method to study differential spatial co-occurrence of cell types across multiple tissue or disease groups, based on the theories of the Poisson point process (PPP) and functional analysis of variance (FANOVA). Notably, the method accommodates multiple images per subject and addresses the problem of missing tissue regions, commonly encountered in such a context due to the complex nature of the data-collection procedure. We demonstrate the superior statistical power and robustness of the method in comparison to existing approaches through realistic simulation studies. Furthermore, we apply the method to three real datasets on different diseases collected using different imaging platforms. In particular, one of these datasets reveals novel insights into the spatial characteristics of various types of precursor lesions associated with colorectal cancer.

Availability

The associated R package can be found here, https://github.com/sealx017/SpaceANOVA.

Instrumental variable estimation for functional concurrent regression models

In this work we propose a functional concurrent regression model to estimate labor supply elasticities over the years 1988 through 2014 using Current Population Survey data. Assuming, as is common, that individuals' wages are endogenous, we introduce instrumental variables in a two-stage least squares approach to estimate the desired labor supply elasticities. Furthermore, we tailor our estimation method to sparse functional data. Though recent work has incorporated instrumental variables into other functional regression models, to our knowledge this has not yet been done in the functional concurrent regression model, and most existing literature is not suited for sparse functional data. We show through simulations that this two-stage least squares approach greatly eliminates the bias introduced by a naive model (i.e. one that does not acknowledge endogeneity) and produces accurate coefficient estimates for moderate sample sizes.

Robust scalar-on-function partial quantile regression

Compared with the conditional mean regression-based scalar-on-function regression model, the scalar-on-function quantile regression is robust to outliers in the response variable. However, it is susceptible to outliers in the functional predictor (called leverage points). This is because the influence function of the regression quantiles is bounded in the response variable but unbounded in the predictor space. The leverage points may alter the eigenstructure of the predictor matrix, leading to poor estimation and prediction results. This study proposes a robust procedure to estimate the model parameters in the scalar-on-function quantile regression method and produce reliable predictions in the presence of both outliers and leverage points. The proposed method is based on a functional partial quantile regression procedure. We propose a weighted partial quantile covariance to obtain functional partial quantile components of the scalar-on-function quantile regression model. After the decomposition, the model parameters are estimated via a weighted loss function, where the robustness is obtained by iteratively reweighting the partial quantile components. The estimation and prediction performance of the proposed method is evaluated by a series of Monte-Carlo experiments and an empirical data example. The results are compared favorably with several existing methods. The method is implemented in an R package robfpqr.

Sex and age variations in the impact of puberty on cortical thickness and associations with internalizing symptoms and suicidal ideation in early adolescence

PURPOSE

The childhood-to-adolescence transition is a notable period of change including pubertal development, neurodevelopment, and psychopathology onset, that occurs in divergent patterns between sexes. This study examined the effects of sex and puberty on cortical thickness (CT) in children and explored whether CT changes over time related to emergence of psychopathology in early adolescence.

METHODS

We used longitudinal data (baseline ages 9-10 and Year 2 [Y2] ages 11-12) from the ABCD Study (n = 9985). Linear and penalized function-on-function regressions modeled the impact of puberty, as it interacts with sex, on CT. Focusing on regions that showed sex differences, linear and logistic regressions modeled associations between change in CT and internalizing problems and suicide ideation.

RESULTS

We identified significant sex differences in the inverse relation between puberty and CT in fifteen primarily posterior brain regions. Nonlinear pubertal effects across age were identified in the fusiform, isthmus cingulate, paracentral, and precuneus. All effects were stronger for females relative to males during this developmental window. We did not identify associations between CT change and early adolescent clinical outcomes.

CONCLUSION

During this age range, puberty is most strongly associated with regional changes in CT in females, which may have implications for the later emergence of psychopathology.

Fixed-effects inference and tests of correlation for longitudinal functional data

We propose an inferential framework for fixed effects in longitudinal functional models and introduce tests for the correlation structures induced by the longitudinal sampling procedure. The framework provides a natural extension of standard longitudinal correlation models for scalar observations to functional observations. Using simulation studies, we compare fixed effects estimation under correctly and incorrectly specified correlation structures and also test the longitudinal correlation structure. Finally, we apply the proposed methods to a longitudinal functional dataset on physical activity. The computer code for the proposed method is available at https://github.com/rli20ST758/FILF.

Inference in Functional Linear Quantile Regression

In this paper, we study statistical inference in functional quantile regression for scalar response and a functional covariate. Specifically, we consider a functional linear quantile regression model where the effect of the covariate on the quantile of the response is modeled through the inner product between the functional covariate and an unknown smooth regression parameter function that varies with the level of quantile. The objective is to test that the regression parameter is constant across several quantile levels of interest. The parameter function is estimated by combining ideas from functional principal component analysis and quantile regression. An adjusted Wald testing procedure is proposed for this hypothesis of interest, and its chi-square asymptotic null distribution is derived. The testing procedure is investigated numerically in simulations involving sparse and noisy functional covariates and in a capital bike share data application. The proposed approach is easy to implement and the R code is published online at https://github.com/xylimeng/fQR-testing.

Adaptive smoothing spline estimator for the function-on-function linear regression model

In this paper, we propose an adaptive smoothing spline (AdaSS) estimator for the function-on-function linear regression model where each value of the response, at any domain point, depends on the full trajectory of the predictor. The AdaSS estimator is obtained by the optimization of an objective function with two spatially adaptive penalties, based on initial estimates of the partial derivatives of the regression coefficient function. This allows the proposed estimator to adapt more easily to the true coefficient function over regions of large curvature and not to be undersmoothed over the remaining part of the domain. A novel evolutionary algorithm is developed ad hoc to obtain the optimization tuning parameters. Extensive Monte Carlo simulations have been carried out to compare the AdaSS estimator with competitors that have already appeared in the literature before. The results show that our proposal mostly outperforms the competitor in terms of estimation and prediction accuracy. Lastly, those advantages are illustrated also in two real-data benchmark examples. The AdaSS estimator is implemented in the package , openly available online on CRAN.

Effect of legroom proportions and individual factors for sitting with crossed legs: implications on the interior design of automated driving vehicles

Sitting with crossed legs is a commonly adopted sitting posture in everyday situations. Yet, little is known about suitable design criteria to facilitate such a position inside a vehicle. This study is aimed at determining how much space is necessary for crossing the legs while considering legroom restrictions, anthropometric measures, and individual flexibility. More specifically, 3 D-kinematics of an ankle-on-knee leg-crossing task and the easiness to move ratings of 30 participants were assessed with restrictions of the legroom (2 heights × 3 distances) as well as without restrictions. Functional regression models revealed adaptations to a legroom restriction in the execution of movement, which occurred mainly in the knee joint and increased with more restricted legroom proportions. Therefore, the present study suggests a distance of 120% of the buttock-knee length between the dashboard and the occupant, as it requires only moderate adaptations and does not affect the perceived easiness of move. Practitioner Summary: This research investigated how much space is needed to cross the legs while sitting in a vehicle, finding that the movement execution is affected by legroom proportions, as well as individual anthropometry and flexibility. The study further presents the use of predicted motion traces to determine spatial requirements of movements. Abbreviations: BKL: buttock-knee length; H-point: hip point.

Elucidating age and sex-dependent association between frontal EEG asymmetry and depression: An application of multiple imputation in functional regression

Frontal power asymmetry (FA), a measure of brain function derived from electroencephalography, is a potential biomarker for major depressive disorder (MDD). Though FA is functional in nature, it is typically reduced to a scalar value prior to analysis, possibly obscuring its relationship with MDD and leading to a number of studies that have provided contradictory results. To overcome this issue, we sought to fit a functional regression model to characterize the association between FA and MDD status, adjusting for age, sex, cognitive ability, and handedness using data from a large clinical study that included both MDD and healthy control (HC) subjects. Since nearly 40% of the observations are missing data on either FA or cognitive ability, we propose an extension of multiple imputation (MI) by chained equations that allows for the imputation of both scalar and functional data. We also propose an extension of Rubin's Rules for conducting valid inference in this setting. The proposed methods are evaluated in a simulation and applied to our FA data. For our FA data, a pooled analysis from the imputed data sets yielded similar results to those of the complete case analysis. We found that, among young females, HCs tended to have higher FA over the θ, α, and β frequency bands, but that the difference between HC and MDD subjects diminishes and ultimately reverses with age. For males, HCs tended to have higher FA in the β frequency band, regardless of age. Young male HCs had higher FA in the θ and α bands, but this difference diminishes with increasing age in the α band and ultimately reverses with increasing age in the θ band.

A Bayesian multiple imputation approach to bivariate functional data with missing components

Existing missing data methods for functional data mainly focus on reconstructing missing measurements along a single function-a univariate functional data setting. Motivated by a renal study, we focus on a bivariate functional data setting, where each sampling unit is a collection of two distinct component functions, one of which may be missing. Specifically, we propose a Bayesian multiple imputation approach based on a bivariate functional latent factor model that exploits the joint changing patterns of the component functions to allow accurate and stable imputation of one component given the other. We further extend the framework to address multilevel bivariate functional data with missing components by modeling and exploiting inter-component and intra-subject correlations. We develop a Gibbs sampling algorithm that simultaneously generates multiple imputations of missing component functions and posterior samples of model parameters. For multilevel bivariate functional data, a partially collapsed Gibbs sampler is implemented to improve computational efficiency. Our simulation study demonstrates that our methods outperform other competing methods for imputing missing components of bivariate functional data under various designs and missingness rates. The motivating renal study aims to investigate the distribution and pharmacokinetic properties of baseline and post-furosemide renogram curves that provide further insights into the underlying mechanism of renal obstruction, with post-furosemide renogram curves missing for some subjects. We apply the proposed methods to impute missing post-furosemide renogram curves and obtain more refined insights.

High-Dimensional Spatial Quantile Function-on-Scalar Regression

This article develops a novel spatial quantile function-on-scalar regression model, which studies the conditional spatial distribution of a high-dimensional functional response given scalar predictors. With the strength of both quantile regression and copula modeling, we are able to explicitly characterize the conditional distribution of the functional or image response on the whole spatial domain. Our method provides a comprehensive understanding of the effect of scalar covariates on functional responses across different quantile levels and also gives a practical way to generate new images for given covariate values. Theoretically, we establish the minimax rates of convergence for estimating coefficient functions under both fixed and random designs. We further develop an efficient primal-dual algorithm to handle high-dimensional image data. Simulations and real data analysis are conducted to examine the finite-sample performance.

Bayesian Wavelet-packet Historical Functional Linear Models

Historical Functional Linear Models (HFLM) quantify associations between a functional predictor and functional outcome where the predictor is an exposure variable that occurs before, or at least concurrently with, the outcome. Prior work on the HFLM has largely focused on estimation of a surface that represents a time-varying association between the functional outcome and the functional exposure. This existing work has employed frequentist and spline-based estimation methods, with little attention paid to formal inference or adjustment for multiple testing and no approaches that implement wavelet-bases. In this work, we propose a new functional regression model that estimates the time-varying, lagged association between a functional outcome and a functional exposure. Building off of recently developed function-on-function regression methods, the model employs a novel use the wavelet-packet decomposition of the exposure and outcome functions that allows us to strictly enforce the temporal ordering of exposure and outcome, which is not possible with existing wavelet-based functional models. Using a fully Bayesian approach, we conduct formal inference on the time-varying lagged association, while adjusting for multiple testing. We investigate the operating characteristics of our wavelet-packet HFLM and compare them to those of two existing estimation procedures in simulation. We also assess several inference techniques and use the model to analyze data on the impact of lagged exposure to particulate matter finer than 2.5μg, or PM_2.5, on heart rate variability in a cohort of journeyman boilermakers during the morning of a typical day's shift.

Longitudinal dynamic functional regression

The paper develops a parsimonious modelling framework to study the time-varying association between scalar outcomes and functional predictors observed at many instances, in longitudinal studies. The methods enable us to reconstruct the full trajectory of the response and are applicable to Gaussian and non-Gaussian responses. The idea is to model the time-varying functional predictors by using orthogonal basis functions and to expand the time-varying regression coefficient by using the same basis. Numerical investigation through simulation studies and data analysis show excellent performance in terms of accurate prediction and efficient computations, when compared with existing alternatives. The methods are inspired and applied to an animal science application, where of interest is to study the association between the feed intake of lactating sows and the minute-by-minute temperature throughout the 21 days of their lactation period. R code and an R illustration are provided.

Conditional Analysis for Mixed Covariates, with Application to Feed Intake of Lactating Sows

We propose a novel modeling framework to study the effect of covariates of various types on the conditional distribution of the response. The methodology accommodates flexible model structure, allows for joint estimation of the quantiles at all levels, and provides a computationally efficient estimation algorithm. Extensive numerical investigation confirms good performance of the proposed method. The methodology is motivated by and applied to a lactating sow study, where the primary interest is to understand how the dynamic change of minute-by-minute temperature in the farrowing rooms within a day (functional covariate) is associated with low quantiles of feed intake of lactating sows, while accounting for other sow-specific information (vector covariate).

Optimal Penalized Function-on-Function Regression under a Reproducing Kernel Hilbert Space Framework

Many scientific studies collect data where the response and predictor variables are both functions of time, location, or some other covariate. Understanding the relationship between these functional variables is a common goal in these studies. Motivated from two real-life examples, we present in this paper a function-on-function regression model that can be used to analyze such kind of functional data. Our estimator of the 2D coefficient function is the optimizer of a form of penalized least squares where the penalty enforces a certain level of smoothness on the estimator. Our first result is the Representer Theorem which states that the exact optimizer of the penalized least squares actually resides in a data-adaptive finite dimensional subspace although the optimization problem is defined on a function space of infinite dimensions. This theorem then allows us an easy incorporation of the Gaussian quadrature into the optimization of the penalized least squares, which can be carried out through standard numerical procedures. We also show that our estimator achieves the minimax convergence rate in mean prediction under the framework of function-on-function regression. Extensive simulation studies demonstrate the numerical advantages of our method over the existing ones, where a sparse functional data extension is also introduced. The proposed method is then applied to our motivating examples of the benchmark Canadian weather data and a histone regulation study.

Dynamic prediction in functional concurrent regression with an application to child growth

In many studies, it is of interest to predict the future trajectory of subjects based on their historical data, referred to as dynamic prediction. Mixed effects models have traditionally been used for dynamic prediction. However, the commonly used random intercept and slope model is often not sufficiently flexible for modeling subject-specific trajectories. In addition, there may be useful exposures/predictors of interest that are measured concurrently with the outcome, complicating dynamic prediction. To address these problems, we propose a dynamic functional concurrent regression model to handle the case where both the functional response and the functional predictors are irregularly measured. Currently, such a model cannot be fit by existing software. We apply the model to dynamically predict children's length conditional on prior length, weight, and baseline covariates. Inference on model parameters and subject-specific trajectories is conducted using the mixed effects representation of the proposed model. An extensive simulation study shows that the dynamic functional regression model provides more accurate estimation and inference than existing methods. Methods are supported by fast, flexible, open source software that uses heavily tested smoothing techniques.

Simple fixed-effects inference for complex functional models

We propose simple inferential approaches for the fixed effects in complex functional mixed effects models. We estimate the fixed effects under the independence of functional residuals assumption and then bootstrap independent units (e.g. subjects) to conduct inference on the fixed effects parameters. Simulations show excellent coverage probability of the confidence intervals and size of tests for the fixed effects model parameters. Methods are motivated by and applied to the Baltimore Longitudinal Study of Aging, though they are applicable to other studies that collect correlated functional data.

Dynamic child growth prediction: A comparative methods approach

Abstract:

We introduce a class of dynamic regression models designed to predict the future of growth curves based on their historical dynamics. This class of models incorporates both baseline and time-dependent covariates, start with simple regression models and build up to dynamic function-on-function regressions. We compare the performance of the dynamic prediction models in a variety of signal-to-noise scenarios and provide practical solutions for model selection. We conclude that (a) prediction performance increases substantially when using the entire growth history relative to using only the last and first observation; (b) smoothing incorporated using functional regression approaches increases prediction performance; and (c) the interpretation of model parameters is substantially improved using functional regression approaches. Because many growth curve datasets exhibit missing and noisy data, we propose a bootstrap of subjects approach to account for the variability associated with the missing data imputation and smoothing. Methods are motivated by and applied to the CONTENT dataset, a study that collected monthly child growth data on 197 children from birth until month 15. R code describing the fitting approaches is provided in a supplementary file.

Discussion of the paper ‘A general framework for functional regression modelling’

This discussion provides our reaction to the article by Greven and Scheipl. It contains an overview of their article and a description of the many areas of research that remain open and could benefit from further methodological and computational development.