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.

Smoothing splines approximation using Hilbert curve basis selection

Smoothing splines have been used pervasively in nonparametric regressions. However, the computational burden of smoothing splines is significant when the sample size n is large. When the number of predictors d ≥ 2 , the computational cost for smoothing splines is at the order of O(n 3) using the standard approach. Many methods have been developed to approximate smoothing spline estimators by using q basis functions instead of n ones, resulting in a computational cost of the order O(nq 2). These methods are called the basis selection methods. Despite algorithmic benefits, most of the basis selection methods require the assumption that the sample is uniformly-distributed on a hyper-cube. These methods may have deteriorating performance when such an assumption is not met. To overcome the obstacle, we develop an efficient algorithm that is adaptive to the unknown probability density function of the predictors. Theoretically, we show the proposed estimator has the same convergence rate as the full-basis estimator when q is roughly at the order of O[n 2d/{(pr+1)(d +2)}] , where p ∈[1, 2] and r ≈ 4 are some constants depend on the type of the spline. Numerical studies on various synthetic datasets demonstrate the superior performance of the proposed estimator in comparison with mainstream competitors.

Within-Day Time-Varying Associations Between Behavioral Cognitions and Physical Activity in Adults

This study used time-varying effect modeling to examine time-of-day differences in how behavioral cognitions predict subsequent physical activity (PA). Adults (N = 116) participated in three 4-day "bursts" of ecological momentary assessment (EMA). Participants were prompted with eight EMA questionnaires per day assessing behavioral cognitions (i.e., intentions, self-efficacy, outcome expectations) and wore an accelerometer during waking hours. Subsequent PA was operationalized as accelerometer-derived minutes of moderate- or vigorousintensity PA in the 2 hr following the EMA prompt. On weekdays, intentions positively predicted subsequent PA in the morning (9:25 a.m.-11:45 a.m.) and in the evening (8:15 p.m.-10:00 p.m.). Self-efficacy positively predicted subsequent PA on weekday evenings (7:35 p.m.-10:00 p.m.). Outcome expectations were unrelated to subsequent PA on weekdays. On weekend days, behavior cognitions and subsequent PA were unrelated regardless of time of day. This study identifies windows of opportunity and vulnerability for motivation-based PA interventions aiming to deliver intervention content within the context of adults' daily lives.