Functional Principal Component Research Articles

Filtrated common functional principal component analysis of multigroup functional data

Filtrated common functional principal component analysis of multigroup functional data

Dynamic Survival Prediction Using Sparse Longitudinal Images via Multi-Dimensional Functional Principal Component Analysis

Our work is motivated by predicting the progression of Alzheimer’s disease (AD) based on a series of longitudinally observed brain scan images. Existing works on dynamic prediction for AD focus primarily on extracting predictive information from multivariate longitudinal biomarker values or brain imaging data at the baseline; whereas in practice, the subject’s brain scan image represented by a multi-dimensional data matrix is collected at each follow-up visit. It is of great interest to predict the progression of AD directly from a series of longitudinally observed images. We propose a novel multi-dimensional functional principal component analysis based on alternating regression on tensor-product B-spline, which circumvents the computational difficulty of doing eigendecomposition, and offers the flexibility of accommodating sparsely and irregularly observed image series. We then use the functional principal component scores as features in the Cox proportional hazards model. We further develop a dynamic prediction framework to provide a personalized prediction that can be updated as new images are collected. Our method extracts visibly interpretable images of the functional principal components and offers an accurate prediction of the conversion to AD. We examine the effectiveness of our method via simulation studies and illustrate its application on the Alzheimer’s Disease Neuroimaging Initiative data. Supplementary materials for this article are available online.

Field-based high-throughput phenotyping enhances phenomic and genomic predictions for grain yield and plant height across years in maize.

Field-based phenomic prediction employs novel features, like vegetation indices (VIs) from drone images, to predict key agronomic traits in maize, despite challenges in matching biomarker measurement time points across years or environments. This study utilized functional principal component analysis (FPCA) to summarize the variation of temporal VIs, uniquely allowing the integration of this data into phenomic prediction models tested across multiple years (2018-2021) and environments. The models, which included 1 genomic, 2 phenomic, 2 multikernel, and 1 multitrait type, were evaluated in 4 prediction scenarios (CV2, CV1, CV0, and CV00), relevant for plant breeding programs, assessing both tested and untested genotypes in observed and unobserved environments. Two hybrid populations (415 and 220 hybrids) demonstrated the visible atmospherically resistant index's strong temporal correlation with grain yield (up to 0.59) and plant height. The first 2 FPCAs explained 59.3 ± 13.9% and 74.2 ± 9.0% of the temporal variation of temporal data of VIs, respectively, facilitating predictions where flight times varied. Phenomic data, particularly when combined with genomic data, often were comparable to or numerically exceeded the base genomic model in prediction accuracy, particularly for grain yield in untested hybrids, although no significant differences in these models' performance were consistently observed. Overall, this approach underscores the effectiveness of FPCA and combined models in enhancing the prediction of grain yield and plant height across environments and diverse agricultural settings.

Gradient‐based approach to sufficient dimension reduction with functional or longitudinal covariates

AbstractIn this paper, we focus on the sufficient dimension reduction problem in regression analysis with real‐valued response and functional or longitudinal covariates. We propose a new method based on gradients of the conditional distribution function to estimate the sufficient dimension reduction subspace. While existing inverse‐regression‐type methods relies on a linearity condition, our method is based on the gradient of conditional distribution function and its validity only requires smoothness conditions on the population parameters. Practically, the proposed estimator can be obtained by standard algorithm of functional principal component analysis. The proposed method is demonstrated through extensive simulations and two empirical examples.

Traffic Flow Prediction by an Ensemble Framework with Data Denoising and Functional Principal Components Analysis

Accurate traffic flow data are important for congestion identification and real-time traffic control. Raw traffic flow data may be disturbed by different noises during the acquisition process, which leads to the degradation of model prediction performance. To address this problems, this paper proposes a data denoising method based on the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) combined with wavelet threshold denoising (WTD) to suppress potential outliers in the data, and then introduces functional principal component analysis to accomplish the traffic forecasting task. We use the proposed framework for real traffic flow prediction and validate the prediction performance of the proposed framework using root mean square error and mean absolute error. In the prediction performance comparison, five denoising methods are considered: empirical mode decomposition (EMD), ensemble empirical mode decomposition, complete ensemble empirical mode decomposition with adaptive noise, ICEEMDAN, and WTD. The prediction results show that the model combined with the denoising method outperforms the model without the denoising method. ICEEMDAN combined with WTD’s denoising method improves prediction performance more than other methods. Furthermore, the WTD method with dmey type achieves higher accuracy than other types.

Regular variation in Hilbert spaces and principal component analysis for functional extremes

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm ‖X‖ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal ‘directions’ of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert–Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

High dimensional variable selection through group Lasso for multiple function‐on‐function linear regression: A case study in PM10 monitoring

SummaryAnalyzing the effect of chemical and local meteorological variables over the behaviour in concentrations in the Abruzzo region (Italy), with the objective of forecasting and controlling air quality, motivates the current work. Given that the available data are curves that represent the day‐to‐day variations, a multiple function‐on‐function linear regression (MFFLR) model is considered. By assuming the Karhunen‐Loève expansion, MFFLR model can be reduced to a classical linear regression model for each principal component of the functional response in terms of all principal components (PCs) of the functional predictors. In this sense, a regularization approach for functional principal component regression based on the merge of functional data analysis with group Lasso is proposed. This novel methodology allows to estimate the model and, simultaneously, select those relevant functional predictors with the functional response, where each functional independent variable is represented by a group of input variables derived by the PCs.

Repeated measures in functional logistic regression

We present a proposal to extend the functional logistic regression model –which models a binary scalar response variable from a functional predictor– to the case where the functional observations are not independent because the same functional variable is measured in the same individuals in different experimental conditions (repeated measures). The extension is addressed by including a random effect in the model. The functional approach of this model assumes that all functional objects are elements of the same finite-dimensional subspace of the space of square-integrable functions L2 in the same compact domain allowing the functions to be treated through the basis coefficients on the basis that generates the subspace to which functional objects belong (basis expansion). This methodology usually induces a multicollinearity problem in the multivariate model that emerges, which is solved with the use of the functional principal components of the functional predictor, resulting in a new functional principal component random effects model. The proposal is contextualized through a simulation study that contains three simulation scenarios for four different functional parameters and considering the lack of independence.

Machine learning of ECG waveforms and cardiac magnetic resonance for response and survival after cardiac resynchronization therapy

Cardiac resynchronization therapy (CRT) can lead to marked symptom reduction and improved survival in selected patients with heart failure with reduced ejection fraction (HFrEF); however, many candidates for CRT based on clinical guidelines do not have a favorable response. A better way to identify patients expected to benefit from CRT that applies machine learning to accessible and cost-effective diagnostic tools such as the 12-lead electrocardiogram (ECG) could have a major impact on clinical care in HFrEF by helping providers personalize treatment strategies and avoid delays in initiation of other potentially beneficial treatments. This study addresses this need by demonstrating that a novel approach to ECG waveform analysis using functional principal component decomposition (FPCD) performs better than measures that require manual ECG analysis with the human eye and also at least as well as a previously validated but more expensive approach based on cardiac magnetic resonance (CMR). Analyses are based on five-fold cross validation of areas under the curve (AUCs) for CRT response and survival time after the CRT implant using Cox proportional hazards regression with stratification of groups using a Gaussian mixture model approach. Furthermore, FPCD and CMR predictors are shown to be independent, which demonstrates that the FPCD electrical findings and the CMR mechanical findings together provide a synergistic model for response and survival after CRT. In summary, this study provides a highly effective approach to prognostication after CRT in HFrEF using an accessible and inexpensive diagnostic test with a major expected impact on personalization of therapies.

Degradation modeling and remaining useful lifetime prediction based on functional variance process

AbstractDynamic fluctuation is a common phenomenon in degradation processes. Hence, how to model it properly has a great impact on the degradation modeling as well as the remaining useful lifetime prediction. To capture the dynamic features and to avoid the risk of the model mis‐specification, a nonparametric degradation model based on functional variance process is proposed in this article. The model is composed of a unit‐specific mean trend and a degradation fluctuation which follows a stochastic process. The mean trend is estimated by the local smoother method, while the stochastic fluctuation is estimated by the functional principal component analysis method. The asymptotic properties of the estimators are proved. Also, the prediction for the remaining useful lifetime is discussed and the estimator is proved to converge in distribution. Moreover, a Bayesian scheme is developed to forecast the remaining useful lifetime for units with incomplete degradation observations. Simulation results show the superiority of the proposed method by comparing it with some existing methods. Finally, two real data sets are analyzed and used to illustrate the application of the method.

A sustainable model for the simultaneous determination of tadalafil and dapoxetine hydrochloride in their binary mixture: Overcoming the challenges of environmental, operational, and instrumental variability for spectral data

This study employed functional principal component analysis (FPCA) and covariate design of experiments (DoE) to mitigate the susceptibility of chemometric methods to unexpected variations stemming from operational and instrumental factors. A comparative analysis with partial least squares (PLS) revealed that our proposed approach effectively reduced variability across different analysts, days, and instruments. Specifically, FPCA was utilized to compress available spectral wavelength information, while covariate DoE aided in selecting an optimal training set within the experimental space. Subsequently, PLS was applied for the simultaneous determination of tadalafil (TD) and dapoxetine hydrochloride (DP) in their binary mixture. Validation of the proposed method through accuracy profiles demonstrated its reliability, paving the way for its application in pharmaceutical analysis.

Using functional principal component analysis (FPCA) to quantify sitting patterns derived from wearable sensors

BackgroundSedentary behavior (SB) is a recognized risk factor for many chronic diseases. ActiGraph and activPAL are two commonly used wearable accelerometers in SB research. The former measures body movement and the latter measures body posture. The goal of the current study is to quantify the pattern and variation of movement (by ActiGraph activity counts) during activPAL-identified sitting events, and examine associations between patterns and health-related outcomes, such as systolic and diastolic blood pressure (SBP and DBP).MethodsThe current study included 314 overweight postmenopausal women, who were instructed to wear an activPAL (at thigh) and ActiGraph (at waist) simultaneously for 24 hours a day for a week under free-living conditions. ActiGraph and activPAL data were processed to obtain minute-level time-series outputs. Multilevel functional principal component analysis (MFPCA) was applied to minute-level ActiGraph activity counts within activPAL-identified sitting bouts to investigate variation in movement while sitting across subjects and days. The multilevel approach accounted for the nesting of days within subjects.ResultsAt least 90% of the overall variation of activity counts was explained by two subject-level principal components (PC) and six day-level PCs, hence dramatically reducing the dimensions from the original minute-level scale. The first subject-level PC captured patterns of fluctuation in movement during sitting, whereas the second subject-level PC delineated variation in movement during different lengths of sitting bouts: shorter (< 30 minutes), medium (30 -39 minutes) or longer (> 39 minute). The first subject-level PC scores showed positive association with DBP (standardized β^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\\beta }$$\\end{document}: 2.041, standard error: 0.607, adjusted p = 0.007), which implied that lower activity counts (during sitting) were associated with higher DBP.ConclusionIn this work we implemented MFPCA to identify variation in movement patterns during sitting bouts, and showed that these patterns were associated with cardiovascular health. Unlike existing methods, MFPCA does not require pre-specified cut-points to define activity intensity, and thus offers a novel powerful statistical tool to elucidate variation in SB patterns and health.Trial registrationClinicalTrials.gov NCT03473145; Registered 22 March 2018; https://clinicaltrials.gov/ct2/show/NCT03473145; International Registered Report Identifier (IRRID): DERR1-10.2196/28684

Minimax estimation of functional principal components from noisy discretized functional data

AbstractFunctional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

Different PCA approaches for vector functional time series with applications to resistive switching processes

This paper is motivated by modeling the cycle-to-cycle variability associated with the resistive switching operation behind memristors. Although the data generated by this stochastic process are by nature current–voltage curves associated with the creation (set process) and destruction (reset process) of a conductive filament, the statistical analysis is usually based on analyzing only the scalar time series related to the reset and set voltages/currents in consecutive cycles. As the data are by nature curves, functional principal component analysis is a suitable candidate to explain the main modes of variability associated with these processes. Taking into account this data-driven motivation, in this paper we propose two new forecasting approaches based on studying the sequential cross-dependence between and within a multivariate functional time series in terms of vector autoregressive modeling of the most explicative functional principal component scores. The main difference between the two methods lies in whether a univariate or multivariate PCA is performed so that we have a different set of principal component scores for each functional time series or the same one for all of them. Finally, the sample performance of the proposed methodologies is illustrated by an application on a bivariate functional time series of reset/set curves.

Longitudinal study of paralytic shellfish toxins along Canada's coast

Paralytic shellfish toxins (PST) in shellfish products have led to severe risks to human health. To monitor the risk, the Canadian Shellfish Sanitation Program has been collecting longitudinal PST measurements in blue mussel (Mytilus edulis) and soft-shell clam (Mya arenaria) samples in six coastal provinces of Canada. The spatial distributions of major temporal variation patterns were studied via Functional Principal Component Analysis. Seasonal increases in PST contamination were found to vary the most in terms of magnitude along the coastlines, which provides support for location-specific management of the time-sensitive PST contamination. In British Columbia, the first functional principal component (FPC1) indicated the variance among the magnitudes, while FPC2 indicated the seasonality of the PST levels. The temporal variations tended to be positively correlated with the abundance of dianoflagellates Alexandrium spp., and negatively with precipitation and inorganic nutrients. These findings indicate the underlying mechanism of PST variation in various geographical settings. In New Brunswick, Prince Edward, and Nova Scotia, the top FPCs indicated that the PST contamination differed mostly in the seasonal increase of the PST level during summer.

Functional sufficient dimension reduction through information maximization with application to classification

Considering the case where the response variable is a categorical variable and the predictor is a random function, two novel functional sufficient dimensional reduction (FSDR) methods are proposed based on mutual information and square loss mutual information. Compared to the classical FSDR methods, such as functional sliced inverse regression and functional sliced average variance estimation, the proposed methods are appealing because they are capable of estimating multiple effective dimension reduction directions in the case of a relatively small number of categories, especially for the binary response. Moreover, the proposed methods do not require the restrictive linear conditional mean assumption and the constant covariance assumption. They avoid the inverse problem of the covariance operator which is often encountered in the functional sufficient dimension reduction. The functional principal component analysis with truncation be used as a regularization mechanism. Under some mild conditions, the statistical consistency of the proposed methods is established. Simulation studies and real data analyzes are used to evaluate the finite sample properties of our methods.

Language-Specific Prosody in Statements of Palenquero/Spanish Bilinguals

This study explores the extent to which Palenquero/Spanish bilinguals, a population that is said to have a residual high tone of African origin, keep their two languages temporally and intonationally distinct across statements. While creole languages that emerged from the contact of African and European languages, such as Palenquero, may develop hybrid prosodic systems with tones from substrate languages, and stress from the majority language, language-specific prosody might be expected to converge or simplify over the course of time. As prosodic convergence seems to be inescapable under Palenquero’s circumstances, which factors could support language-specific prosody in this population, if there are any? Two-hundred and thirty-four five-syllable statements were elicited through a discourse completion task, with the participation of ten Palenquero/Spanish bilinguals, in two unilingual sessions. Both phrase-final lengthening and F0 contours were assessed using linear mixed-effects models testing their association with final stress, language, and generation. F0 contours were dimensionally reduced using functional principal component analysis. Despite the strong similarities between the two languages, results indicate that both groups keep their two languages intonationally distinct using plateau-shaped contours in Palenquero initial rises followed by steeper declinations in Spanish. However, elderly bilinguals implement penultimate lengthening language-specifically, being more pronounced in Palenquero. Adults, in contrast, do not show this distinction. In addition to this, elderly speakers show hyperarticulation in Spanish intonation, increasing the difference between their languages. This leads us to believe that adults exhibit a more simplified prosodic system between their languages, relative to elderly bilinguals. In spite of such differences, both generations seem to have the same underlying process (perhaps a substrate effect) driving plateau-shaped intonation in Palenquero, which enhances language differentiation.

Impaired Neuromotor Control During Gait in Concussed Adolescents-A Frequency Analysis.

Disruptions in gait function are common after concussion in adolescents; however, the neuromotor control deficits driving these gait disruptions are not well known. Fifteen concussed (age mean [SD]): 17.4 [0.6], 13 females, days since injury: 26.3 [9.9]) and 17 uninjured (age: 18.0 [0.7], 10 females) adolescents completed 3 trials each of single-task gait and dual-task gait (DT). During DT, participants simultaneously walked while completing a serial subtraction task. Gait metrics and variability in instantaneous mean frequency in lower extremity muscles were captured by inertial sensors and surface electromyography, respectively. A 2-way analysis of covariance was used to compare gait metrics across groups and conditions. Functional principal components analysis was used to identify regions of variability in instantaneous mean frequency curves. Functional principal component scores were compared across groups using a Welch statistic. Both groups displayed worse performance on gait metrics during DT condition compared to single-task, with no differences between groups (P < .001). Concussed adolescents displayed significantly greater instantaneous mean frequency, indicated by functional principal component 1, in the tibialis anterior, biceps femoris, and semitendinosus (P < .05) during single-task and DT compared with uninjured adolescents. Our observations suggest that concussed adolescents display inefficient motor unit recruitment lasting longer than 2weeks following injury, regardless of the addition of a secondary task.

Quantifying Apathy in Late-Life Depression: Unraveling Neurobehavioral Links Through Daily Activity Patterns and Brain Connectivity Analysis

BackgroundBetter understanding apathy in late-life depression would help improve prediction of poor prognosis of diseases such as dementia. Actimetry provides an objective and ecological measure of apathy from patients’ daily motor activity. We aimed to determine whether patterns of motor activity were associated with apathy and brain connectivity in networks that underlie goal-directed behaviors. MethodsResting-state functional magnetic resonance imaging and diffusion magnetic resonance imaging were collected from 38 nondemented participants with late-life depression. Apathy was evaluated using the diagnostic criteria for apathy, the Apathy Evaluation Scale, and the Apathy Motivation Index. Functional principal components (fPCs) of motor activity were derived from actimetry recordings taken for 72 hours. Associations between fPCs and apathy were estimated by linear regression. Subnetworks whose connectivity was significantly associated with fPCs were identified via threshold-free network-based statistics. The relationship between apathy and microstructure metrics was estimated along fibers by diffusion tensor imaging and a multicompartment model called neurite orientation dispersion and density imaging via tractometry. ResultsWe found 2 fPCs associated with apathy: mean diurnal activity, negatively associated with Apathy Evaluation Scale scores, and an early chronotype, negatively associated with Apathy Motivation Index scores. Mean diurnal activity was associated with increased connectivity in the default mode, cingulo-opercular, and frontoparietal networks, while chronotype was associated with a more heterogeneous connectivity pattern in the same networks. We did not find significant associations between microstructural metrics and fPCs. ConclusionsOur findings suggest that mean diurnal activity and chronotype could provide indirect ambulatory measures of apathy in late-life depression, associated with modified functional connectivity of brain networks that underlie goal-directed behaviors.

Temporal and spatial variation of domoic acid along Canada's coast

Shellfish poisonings have posed severe risks to human health globally. The Canadian Shellfish Sanitation Program was established in 1948 to monitor the toxin levels at shellfish harvesting sites along the coast of six provinces in Canada. Domoic acid has been a causal toxin for amnesic shellfish poisoning, and a macro-scale analysis of the temporal and spatial variation of domoic acid along Canada's coast were conducted in this study. We aggregated the toxin levels by week in blue mussel (Mytilus edulis) and soft-shell clam (Mya arenaria) samples, respectively, over a one-year scale. The subsequent application of Functional Principal Component Analysis unveiled that magnitudes of seasonal variation and peaked DA levels around early summer, spring, or mid-fall formed the largest variation in the toxin levels in blue mussels along coastlines of British Columbia and Prince Edward Island and in soft-shell calms along those of New Brunswick and Nova Scotia. In Quebec, the DA levels were low and varied mostly in terms of the overall magnitude across spring to fall. Downstream correlation analyses in British Columbia further discovered that, at most sites, the strongest correlations were negative between precipitation as well as inorganic nutrients (including nitrate, nitrite, phosphate, and silicate) on one side and DA a few weeks afterward on the other. These findings indicated associations between amnesic shellfish poisoning and environmental stresses.