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Medical and public health research increasingly involves the collection of complex and high dimensional data. In particular, functional data—where the unit of observation is a curve or set of curves that are finely sampled over a grid—is frequently obtained. Moreover, researchers often sample multiple curves per person resulting in repeated functional measures. A common question is how to analyze the relationship between two functional variables. We propose a general function-on-function regression model for repeatedly sampled functional data on a fine grid, presenting a simple model as well as a more extensive mixed model framework, and introducing various functional Bayesian inferential procedures that account for multiple testing. We examine these models via simulation and a data analysis with data from a study that used event-related potentials to examine how the brain processes various types of images.

The link between different psychophysiological measures during emotion episodes is not well understood. To analyse the functional relationship between electroencephalography and facial electromyography, we apply historical function-on-function regression models to electroencephalography and electromyography data that were simultaneously recorded from 24 participants while they were playing a computerized 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 componentwise gradient boosting algorithm, which scales well to large data sets and complex models.

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 article 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. Supplementary materials for this article are available online.

We discuss scalar-on-function regression models where all parameters of the assumed response distribution can be modelled depending on covariates. We thus combine signal regression models with generalized additive models for location, scale and shape. Our approach is motivated by 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 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 componentwise 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.

We develop a general theory and estimation methods for functional linear sufficient dimension reduction, where both the predictor and the response can be random functions, or even vectors of functions. Unlike the existing dimension reduction methods, our approach does not rely on the estimation of conditional mean and conditional variance. Instead, it is based on a new statistical construction—the weak conditional expectation, which is based on Carleman operators and their inducing functions. Weak conditional expectation is a generalization of conditional expectation. Its key advantage is to replace the projection on to an L 2-space—which defines conditional expectation—by projection on to an arbitrary Hilbert space, while still maintaining the unbiasedness of the related dimension reduction methods. This flexibility is particularly important for functional data, because attempting to estimate a full-fledged conditional mean or conditional variance by slicing or smoothing over the space of vector-valued functions may be inefficient due to the curse of dimensionality. We evaluated the performances of the our new methods by simulation and in several applied settings.

In recent years, research into the design, development and use of drones has been of great interest. In this paper, methods and equipment for testing electric motors in the operating regimes are presented. These types of brushless electric motors are mainly used in the equipment of rotary wing UAVs (unmanned aerial vehicles). Determining their performance curves is essential for correlating operating regime data with the response of the structure to the thrust force developed by the motor-propeller assembly. The mathematical analysis of the duty cycles data correlation with the rotor rotation speed involves the use of curve fitting and smoothing functions. To determine the thrust force developed by the motor-propeller assembly, a Wheatstone half-bridge was carried out, using strain gauges.

Two long-standing research problems of interest to sociologists are sources of variations in social inequalities and differential contributions of the temporal dimensions of age, time period, and cohort to variations in social phenomena. Recently, scholars have introduced a model called Variance Function Regression for the study of the former problem, and a model called Hierarchical Age-Period-Cohort regression has been developed for the study of the latter. This article presents an integration of these two models as a means to study the evolution of social inequalities along distinct temporal dimensions. We apply the integrated model to survey data on subjective health status. We find substantial age, period, and cohort effects, as well as gender differences, not only for the conditional mean of self-rated health (i.e., between-group disparities), but also for the variance in this mean (i.e., within-group disparities)—and it is detection of age, period, and cohort variations in the latter disparities that application of the integrated model permits. Net of effects of age and individual-level covariates, in recent decades, cohort differences in conditional means of self-rated health have been less important than period differences that cut across all cohorts. By contrast, cohort differences of variances in these conditional means have dominated period differences. In particular, post-baby boom birth cohorts show significant and increasing levels of within-group disparities. These findings illustrate how the integrated model provides a powerful framework through which to identify and study the evolution of variations in social inequalities across age, period, and cohort temporal dimensions. Accordingly, this model should be broadly applicable to the study of social inequality in many different substantive contexts.

Regression models with interaction effects have been widely used in multivariate analysis to improve model flexibility and prediction accuracy. In functional data analysis, however, due to the challenges of estimating three-dimensional coefficient functions, interaction effects have not been considered for function-on-function linear regression. In this article, we propose function-on-function regression models with interaction and quadratic effects. For a model with specified main and interaction effects, we propose an efficient estimation method that enjoys a minimum prediction error property and has good predictive performance in practice. Moreover, converting the estimation of three-dimensional coefficient functions of the interaction effects to the estimation of two- and one-dimensional functions separately, our method is computationally efficient. We also propose adaptive penalties to account for varying magnitudes and roughness levels of coefficient functions. In practice, the forms of the models are usually unspecified. We propose a stepwise procedure for model selection based on a predictive criterion. This method is implemented in our R package FRegSigComp. Supplemental materials are available online.

Smoothness penalties are efficient regularization and dimension reduction tools for functional regressions. However, for spiky functional data observed on a dense grid, the coefficient function in a functional regression can be spiky and, hence, the smoothness regularization is inefficient and leads to over-smoothing. We propose a novel approach to fit the function-on-function regression model by viewing the spiky coefficient functions as derivatives of smooth auxiliary functions. Compared with the smoothness regularization or sparsity regularization imposed directly on the spiky coefficient function in existing methods, imposing smoothness regularization on the smooth auxiliary functions can more efficiently reduce the dimension and improve the performance of the fitted model. Using the estimated smooth auxiliary functions and taking derivatives, we can fit the model and make predictions. Simulation studies and real-data applications show that compared with existing methods, the new method can greatly improve model performance when the coefficient function is spiky, and performs similarly well when the coefficient function is smooth.

Quantifying certain physiological traits under heat-stress is crucial for maximizing genetic gain for wheat yield and yield-related components. In-season estimation of different physiological traits related to heat stress tolerance can ensure the finding of germplasm, which could help in making effective genetic gains in yield. However, estimation of those complex traits is time- and labor-intensive. Unmanned aerial vehicle (UAV) based hyperspectral imaging could be a powerful tool to estimate indirectly in-season genetic variation for different complex physiological traits in plant breeding that could improve genetic gains for different important economic traits, like grain yield. This study aims to predict in-season genetic variations for cellular membrane thermostability (CMT), yield and yield related traits based on spectral data collected from UAVs; particularly, in cases where there is a small sample size to collect data from and a large range of features collected per sample. In these cases, traditional methods of yield-prediction modeling become less robust. To handle this, a functional regression approach was employed that addresses limitations of previous techniques to create a model for predicting CMT, grain yield and other traits in wheat under heat stress environmental conditions and when data availability is constrained. The results preliminarily indicate that the overall models of each trait studied presented a good accuracy compared to their data’s standard deviation. The yield prediction model presented an average error of 13.42%, showing the function-on-function algorithm chosen for the model as reliable for small datasets with high dimensionality.