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The estimation of large functional and longitudinal data, which refers to the estimation of mean function, estimation of covariance function, and prediction of individual trajectory, is one of the most challenging problems in the field of high-dimensional statistics. Functional Principal Components Analysis (FPCA) and Functional Linear Mixed Model (FLMM) are two major statistical tools used to address the estimation of large functional and longitudinal data; however, the former suffers from a dramatically increasing computational burden while the latter does not have clear asymptotic properties. In this paper, we propose a computationally effective estimator of large functional and longitudinal data within the framework of FLMM, in which all the parameters can be automatically estimated. Under certain regularity assumptions, we prove that the mean function estimation and individual trajectory prediction reach the minimax lower bounds of all nonparametric estimations. Through numerous simulations and real data analysis, we show that our new estimator outperforms the traditional FPCA in terms of mean function estimation, individual trajectory prediction, variance estimation, covariance function estimation, and computational effectiveness.

Individuals with autism spectrum disorder (ASD) tend to experience greater difficulties with social communication and sensory information processing. Of particular interest in ASD biomarker research is the study of visual attention, effectively quantified in eye tracking (ET) experiments. Eye tracking offers a powerful, safe, and feasible platform for gaining insights into attentional processes by measuring moment-by-moment gaze patterns in response to stimuli. Even though recording is done with millisecond granularity, analyses commonly collapse data across trials into variables such as proportion time spent looking at a region of interest (ROI). In addition, looking times in different ROIs are typically analyzed separately. We propose a novel multivariate functional outcome that carries proportion looking time information from multiple regions of interest jointly as a function of trial type, along with a novel constrained multivariate functional principal components analysis procedure to capture the variation in this outcome. The method incorporates the natural constraint that the proportion looking times from the multiple regions of interest must sum up to one. Our approach is motivated by the Activity Monitoring task, a social-attentional assay within the ET battery of the Autism Biomarkers Consortium for Clinical Trials (ABC-CT). Application of our methods to the ABC-CT data yields new insights into dominant modes of variation of proportion looking times from multiple regions of interest for school-age children with ASD and their typically developing (TD) peers, as well as richer analysis of diagnostic group differences in social attention.

Eye tracking (ET) experiments commonly record the continuous trajectory of a subject’s gaze on a two-dimensional screen throughout repeated presentations of stimuli (referred to as trials). Even though the continuous path of gaze is recorded during each trial, commonly derived outcomes for analysis collapse the data into simple summaries, such as looking times in regions of interest, latency to looking at stimuli, number of stimuli viewed, number of fixations, or fixation length. In order to retain information in trial time, we utilize functional data analysis (FDA) for the first time in literature in the analysis of ET data. More specifically, novel functional outcomes for ET data, referred to as viewing profiles, are introduced that capture the common gazing trends across trial time which are lost in traditional data summaries. Mean and variation of the proposed functional outcomes across subjects are then modeled using functional principal component analysis. Applications to data from a visual exploration paradigm conducted by the Autism Biomarkers Consortium for Clinical Trials showcase the novel insights gained from the proposed FDA approach, including significant group differences between children diagnosed with autism and their typically developing peers in their consistency of looking at faces early on in trial time.

This study examines the runoff prediction of each hydrometric station and each month in the mainstream of the Yellow River in China. From the perspective of functional data, the monthly runoff of each hydrometric station can be regarded as a function of both time and space. A sequence of such functions is formed by collecting the data over the years. We propose a new approach by combining the two-dimensional functional principal component analysis (FPCA) and time series analysis methods to predict the runoff. In the simulation, we compared the proposed method with two others: one based on one-dimensional FPCA and the seasonal auto-regressive integrated moving average (SARIMA) method. The method combining standard two-dimensional FPCA and time series analysis outperforms others in most cases, and is used to predict the runoff of each hydrometric station and each month in the Yellow River in 2018.

Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered, for example, as movement trajectories on the surface of the earth are an important special case. We consider an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the Fréchet mean function, and then performing a classical functional principal component analysis (FPCA) on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained with exponential maps. The tangent-space approximation yields upper bounds to residual variances if the Riemannian manifold has nonnegative curvature. We derive a central limit theorem for the mean function, as well as root-n uniform convergence rates for other model components. Our applications include a novel framework for the analysis of longitudinal compositional data, achieved by mapping longitudinal compositional data to trajectories on the sphere, illustrated with longitudinal fruit fly behavior patterns. RFPCA is shown to outperform an unrestricted FPCA in terms of trajectory recovery and prediction in applications and simulations.

Process monitoring and fault diagnosis using profile data remains an important and challenging problem in statistical process control (SPC). Although the analysis of profile data has been extensively studied in the SPC literature, the challenges associated with monitoring and diagnosis of multichannel (multiple) nonlinear profiles are yet to be addressed. Motivated by an application in multioperation forging processes, we propose a new modeling, monitoring, and diagnosis framework for phase-I analysis of multichannel profiles. The proposed framework is developed under the assumption that different profile channels have similar structure so that we can gain strength by borrowing information from all channels. The multidimensional functional principal component analysis is incorporated into change-point models to construct monitoring statistics. Simulation results show that the proposed approach has good performance in identifying change-points in various situations compared with some existing methods. The codes for implementing the proposed procedure are available in the supplementary material.

We consider a functional linear Cox regression model for characterizing the association between time-to-event data and a set of functional and scalar predictors. The functional linear Cox regression model incorporates a functional principal component analysis for modeling the functional predictors and a high-dimensional Cox regression model to characterize the joint effects of both functional and scalar predictors on the time-to-event data. We develop an algorithm to calculate the maximum approximate partial likelihood estimates of unknown finite and infinite dimensional parameters. We also systematically investigate the rate of convergence of the maximum approximate partial likelihood estimates and a score test statistic for testing the nullity of the slope function associated with the functional predictors. We demonstrate our estimation and testing procedures by using simulations and the analysis of the Alzheimer's Disease Neuroimaging Initiative (ADNI) data. Our real data analyses show that high-dimensional hippocampus surface data may be an important marker for predicting time to conversion to Alzheimer's disease. Data used in the preparation of this article were obtained from the ADNI database (adni.loni.usc.edu).

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, for example, 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 datasets. Applications with spatially and longitudinally observed functional data demonstrate the flexibility in modeling and interpretability of results of our approach.

Futbol Club Barcelona operates the largest stadium in Europe (with a seating capacity of almost one hundred thousand people) and manages recurring sports events. These are influenced by multiple conditions (time and day of the week, weather, adversary) and affect city dynamics—e.g., peak demand for related services like public transport and stores. We study fine grain audience entrances at the stadium segregated by visitor type and gate to gain insights and predict the arrival behavior of future games, with a direct impact on the organizational performance and productivity of the business. We can forecast the timeline of arrivals at gate level 72 h prior to kickoff, facilitating operational and organizational decision-making by anticipating potential agglomerations and audience behavior. Furthermore, we can identify patterns for different types of visitors and understand how relevant factors affect them. These findings directly impact commercial and business interests and can alter operational logistics, venue management, and safety.

Abstract Context Many female survivors of adolescent and young adult cancers (AYA survivors) have shortened reproductive lifespans. However, the timing and duration of ovarian function after cancer treatment are largely unknown. Objective To model the trajectory of ovarian function over two decades following cancer treatment and evaluate how trajectories vary by treatment gonadotoxicity and age. Design In a prospective cohort, AYA survivors aged 18-39 at variable times since cancer treatment completion provided dried blood spots (DBS) every 6 months for up to 18 months. Anti-Müllerian hormone (AMH) levels were measured using the Ansh DBS AMH enzyme-linked immunosorbent assay. The mean AMH trajectory was modeled for the entire cohort and separately by treatment gonadotoxicity and age using functional principal components analysis. Results 763 participants, mean (standard deviation) enrollment age 33.3 (4.7) and age at cancer diagnosis 25.9 (5.7) years, contributed 1905 DBS samples. The most common cancers were breast (26.9%), lymphoma (24.8%), and thyroid (18.0%). AMH trajectories differed among survivors by treatment gonadotoxicity (low, moderate, or high) (P < 0.001). Following low or moderately gonadotoxic treatments, AMH levels increased over 2-3 years and plateaued over 10-15 years before declining. In contrast, following highly gonadotoxic treatment, AMH levels were lower overall and declined shortly after peak at 2-3 years. Younger age at treatment was associated with higher trajectories, but a protective effect of younger age was not observed in survivors exposed to highly gonadotoxic treatments (Pinteraction < 0.001). Conclusions In this large AYA survivor cohort, timing and duration of ovarian function strongly depended on treatment gonadotoxicity and age at treatment. The findings provide novel, more precise information to guide reproductive decision-making.