Multivariate Functional Principal Component Research Articles

Functional Multivariable Logistic Regression With an Application to HIV Viral Suppression Prediction.

Motivated by improving the prediction of the human immunodeficiency virus (HIV) suppression status using electronic health records (EHR) data, we propose a functional multivariable logistic regression model, which accounts for the longitudinal binary process and continuous process simultaneously. Specifically, the longitudinal measurements for either binary or continuous variables are modeled by functional principal components analysis, and their corresponding functional principal component scores are used to build a logistic regression model for prediction. The longitudinal binary data are linked to underlying Gaussian processes. The estimation is done using penalized spline for the longitudinal continuous and binary data. Group-lasso is used to select longitudinal processes, and the multivariate functional principal components analysis is proposed to revise functional principal component scores with the correlation. The method is evaluated via comprehensive simulation studies and then applied to predict viral suppression using EHR data for people living with HIV in SouthCarolina.

A federated data fusion-based prognostic model for applications with multi-stream incomplete signals

Most prognostic methods require a decent amount of data for model training. In reality, however, the amount of historical data owned by a single organization might be small or not large enough to train a reliable prognostic model. To address this challenge, this article proposes a federated prognostic model that allows multiple users to jointly construct a failure time prediction model using their multi-stream, high-dimensional, and incomplete data while keeping each user’s data local and confidential. The prognostic model first employs multivariate functional principal component analysis to fuse the multi-stream degradation signals. Then, the fused features coupled with the times-to-failure are utilized to build a (log)-location-scale regression model for failure prediction. To estimate parameters using distributed datasets and keep the data privacy of all participants, we propose a new federated algorithm for feature extraction. Numerical studies indicate that the performance of the proposed model is the same as that of classic non-federated prognostic models and is better than that of the models constructed by each user itself.

Multivariate functional principal component analysis and k-means clustering to identify kinematic foot types during gait in children with cerebral palsy

BackgroundChildren with neuromuscular disorders, such as cerebral palsy, frequently develop foot deformities, such as equinopronovalgus and equinosupovarus, leading to walking difficulties and discomfort. Traditional assessment methods, including clinical measures and radiographs, often fail to capture the dynamic nature of these deformities, resulting in suboptimal treatment. 3D gait analysis using multisegment foot models offers a more detailed understanding of these deformities. Research questionTo determine whether the combination of multisegment foot models, multivariate functional principal component analysis, and k-means cluster analyses could identify distinct, clinically relevant foot types in a large pediatric cohort with cerebral palsy. MethodsThis was a retrospective analysis of 3D gait data from 197 patients with cerebral palsy collected using a multisegment foot model. Multivariate functional principal component analysis was used to reduce these data prior to using k-means clustering to identify foot posture clusters. Further analyses, including ANOVA and Fisher's Exact tests, were used to evaluate demographic, radiographic, and gait characteristics to explain the clinical relevance of each cluster. ResultsAnalysis of kinematic data from 371 feet revealed six clinically significant clusters, with a low misclassification rate of 2 %. One-factor ANOVAs demonstrated significant differences across clusters for all MPCs, whereas no significant differences were noted in basic anthropometric variables. Significant variations were observed in radiographic and gait function variables, and a strong association between GMFCS levels and cluster categorization was identified. SignificanceThe novel approach of integrating multivariate functional principal component analysis and k-means clustering identified a spectrum of foot deformities in children with CP, ranging from equinosupovarus to marked equinopronovalgus. This methodology provides an objective classification based on kinematic data and can facilitate improved diagnosis and treatment of cerebral palsy-related foot deformities.

Evaluation and Selection of Multi-Spectral Indices to Classify Vegetation Using Multivariate Functional Principal Component Analysis

The identification, classification and mapping of different plant communities and habitats is of fundamental importance for defining biodiversity monitoring and conservation strategies. Today, the availability of high temporal, spatial and spectral data from remote sensing platforms provides dense time series over different spectral bands. In the case of supervised mapping, time series based on classical vegetation indices (e.g., NDVI, GNDVI, …) are usually input characteristics, but the selection of the best index or set of indices (which guarantees the best performance) is still based on human experience and is also influenced by the study area. In this work, several different time series, based on Sentinel-2 images, were created exploring new combinations of bands that extend the classic basic formulas as the normalized difference index. Multivariate Functional Principal Component Analysis (MFPCA) was used to contemporarily decompose the multiple time series. The principal multivariate seasonal spectral variations identified (MFPCA scores) were classified by using a Random Forest (RF) model. The MFPCA and RF classifications were nested into a forward selection strategy to identify the proper and minimum set of indices’ (dense) time series that produced the most accurate supervised classification of plant communities and habitat. The results we obtained can be summarized as follows: (i) the selection of the best set of time series is specific to the study area and the habitats involved; (ii) well-known and widely used indices such as the NDVI are not selected as the indices with the best performance; instead, time series based on original indices (in terms of formula or combination of bands) or underused indices (such as those derivable with the visible bands) are selected; (iii) MFPCA efficiently reduces the dimensionality of the data (multiple dense time series) providing ecologically interpretable results representing an important tool for habitat modelling outperforming conventional approaches that consider only discrete time series.

Demonstrating the relevance of spatial-functional statistical analysis in marine ecological studies: The case of environmental variations in micronektonic layers

In this study, we conducted an analysis of a multifrequency acoustics dataset acquired from scientific echosounders in the West African water. Our objective was to explore the spatial arrangement of marine organism aggregations. We investigated various attributes of these intricate biological entities, such as thickness, relative density, and depth, in relation to their surroundings. These environmental conditions were represented at a fine scale using a towed multiparameter system. This study is closely intertwined with two key domains: Fisheries acoustics techniques and functional data analysis. Fisheries acoustics techniques facilitate the collection of high-resolution spatial and temporal data concerning marine organisms at various depths and spatial scales, all without causing any disturbance. On the other hand, spatial-functional data analysis is a statistical approach for examining data characterised by functional attributes distributed across a spatial domain. This analysis encompasses dimension reduction techniques, as well as supervised and unsupervised methods, which take into consideration spatial dependencies within extensive datasets.We began by applying multivariate statistical techniques and subsequently employed Functional Data Analysis (FDA). In the modeling section, we introduced the spatial dimension with the spatial coordinates as covariates in the General Additive Model (GAM) and Functional Generalized Spectral Additive Model (FGSAM) models, aiming to underscore its relevance in those contexts. In an exploratory phase, Multivariate Functional Principal Component Analysis provided detailed insights into the variations of parameters at different depths, a capability not offered by traditional Principal Component Analysis. When it came to regression tasks, we explored the interactions between descriptors of Sound Scattering Layers and key environmental variables, both with and without considering spatial dimensions. Our findings revealed significant distinctions between northern and southern Sound Scattering Layers, as well as between coastal and high-sea regions. The use of the spatial locations enhanced the performance of GAM and FGSAM, particularly in the case of salinity, reflecting the influence of water mixing and seawater temperature. The multifaceted effects of environmental variations on Sound Scattering Layers underscore the importance of spatial-functional statistical analysis in ecological studies involving complex, spatially functional objects. Beyond the scope of this specific case study, the application of functional data analysis shows promise for a wide array of ecological studies dealing with extensive spatial datasets.

Projection-based two-sample inference for sparsely observed multivariate functional data.

Modern longitudinal studies collect multiple outcomes as the primary endpoints to understand the complex dynamics of the diseases. Oftentimes, especially in clinical trials, the joint variation among the multidimensional responses plays a significant role in assessing the differential characteristics between two or more groups, rather than drawing inferences based on a single outcome. We develop a projection-based two-sample significance test to identify the population-level difference between the multivariate profiles observed under a sparse longitudinal design. The methodology is built upon widely adopted multivariate functional principal component analysis to reduce the dimension of the infinite-dimensional multi-modal functions while preserving the dynamic correlation between the components. The test applies to a wide class of (non-stationary) covariance structures of the response, and it detects a significant group difference based on a single p-value, thereby overcoming the issue of adjusting for multiple p-values that arise due to comparing the means in each of components separately. Finite-sample numerical studies demonstrate that the test maintains the type-I error, and is powerful to detect significant group differences, compared to the state-of-the-art testing procedures. The test is carried out on two significant longitudinal studies for Alzheimer's disease and Parkinson's disease (PD) patients, namely, TOMMORROW study of individuals at high risk of mild cognitive impairment to detect differences in the cognitive test scores between the pioglitazone and the placebo groups, and Azillect study to assess the efficacy of rasagiline as a potential treatment to slow down the progression of PD.

Dynamic prediction of survival using multivariate functional principal component analysis: A strict landmarking approach.

Dynamically predicting patient survival probabilities using longitudinal measurements has become of great importance with routine data collection becoming more common. Many existing models utilize a multi-step landmarking approach for this problem, mostly due to its ease of use and versatility but unfortunately most fail to do so appropriately. In this article we make use of multivariate functional principal component analysis to summarize the available longitudinal information, and employ a Cox proportional hazards model for prediction. Additionally, we consider a centred functional principal component analysis procedure in an attempt to remove the natural variation incurred by the difference in age of the considered subjects. We formalize the difference between a 'relaxed' landmarking approach where only validation data is landmarked and a 'strict' landmarking approach where both the training and validation data are landmarked. We show that a relaxed landmarking approach fails to effectively use the information contained in the longitudinal outcomes, thereby producing substantially worse prediction accuracy than a strict landmarking approach.

Decomposing acoustic signal reveals the pelagic response to a frontal system

The pelagic zone is home to a large diversity of organisms such as macrozooplankton and micronekton (MM), connecting the surface productive waters to the mesopelagic layers (200-1000 m) through diel vertical migrations (DVM). Active acoustics complement net sampling observations by detecting sound-scattering layers (SL) of organisms, allowing to monitor the MM dynamics with a high spatio-temporal resolution. Multi-frequency analyses are a pertinent approach to better integrate the rich diversity of organisms composing SLs and their respective dynamics. However, analysing simultaneously emitted acoustic signals with distinct depth ranges and separating spatial from temporal variability is challenging and needs adapted tools to be fully exploited. This study examines the pelagic realm in a transition zone between the Southern Ocean and the subtropical Indian Ocean, crossing the Saint-Paul and Amsterdam islands’ natural reserve. We extended a Multivariate Functional Principal Component Analysis (mfPCA) to analyse the joint vertical variation of five frequencies from two oceanographic cruises (2016 and 2022), allowing the decomposition of the acoustic dataset into orthogonal vertical modes (VM) of variability. We found the first VM to be linked to the temporal variability due to DVM, while the following majorly depict patterns in spatial distribution. Overall, from the subantarctic to the subtropical zones, we observed (i) enrichment of densities in the surface layer (0–100 m), (ii) a decrease in densities in the intermediate layer during the daytime (100–300 m) and (iii) the apparition of an intensive deep scattering layer on the 38 kHz. We explored VMs’ connection with in-situ environmental conditions by clustering our observations into three distinct environmental-acoustic regions. These regions were compared with vertically integrated nautical area scattering coefficient distribution, a proxy for marine organisms’ biomass. Additionally, we analysed species assemblage changes from complementary cruises to further elucidate the observed acoustic distribution. We show that the mfPCA method is promising to better integrate the pelagic horizontal, vertical and temporal dimensions which is a step towards further investigating the control of the environment on the distribution and structuring of pelagic communities.

Constrained Multivariate Functional Principal Components Analysis for Novel Outcomes in Eye-Tracking Experiments

Constrained Multivariate Functional Principal Components Analysis for Novel Outcomes in Eye-Tracking Experiments

Adaptive sampling and monitoring of partially observed images

Image-based monitoring techniques have achieved great success in many engineering applications. However, most existing image monitoring methods require fully observed images to implement modeling/monitoring. This requirement limits the application of these techniques for images with large size and/or cameras with limited scanning area. In this article, we propose to solve this issue by splitting the large image into multiple sub-images and using partially observed sub-images to implement monitoring of the whole large image. More specifically, the two-dimensional multivariate functional principal component analysis (2D-MFPCA) is proposed to model the cross-and-within correlation among sub-images, which facilitates the information augmentation from partially observed sub-images. The asymptotic properties of the 2D-MFPCA estimators are investigated to justify the use of MFPC scores as the monitoring statistics, which are fed into a multivariate CUSUM control chart to implement adaptive sampling and monitoring of image data. The developed chart can dynamically select locations of partially observed sub-images and adaptively focus on the most suspicious sub-images. The proposed method is validated and compared with various benchmark methods in both numerical and case studies. The results demonstrate the proposed method can achieve superior monitoring performance for large images when only partially observed sub-images are available.

Multi-feature clustering of step data using multivariate functional principal component analysis

Multi-feature clustering of step data using multivariate functional principal component analysis

Multivariate spatiotemporal functional principal component analysis for modeling hospitalization and mortality rates in the dialysis population.

Dialysis patients experience frequent hospitalizations and a higher mortality rate compared to other Medicare populations, in whom hospitalizations are a major contributor to morbidity, mortality, and healthcare costs. Patients also typically remain on dialysis for the duration of their lives or until kidney transplantation. Hence, there is growing interest in studying the spatiotemporal trends in the correlated outcomes of hospitalization and mortality among dialysis patients as a function of time starting from transition to dialysis across the United States Utilizing national data from the United States Renal Data System (USRDS), we propose a novel multivariate spatiotemporal functional principal component analysis model to study the joint spatiotemporal patterns of hospitalization and mortality rates among dialysis patients. The proposal is based on a multivariate Karhunen-Loéve expansion that describes leading directions of variation across time and induces spatial correlations among region-specific scores. An efficient estimation procedure is proposed using only univariate principal components decompositions and a Markov Chain Monte Carlo framework for targeting the spatial correlations. The finite sample performance of the proposed method is studied through simulations. Novel applications to the USRDS data highlight hot spots across the United States with higher hospitalization and/or mortality rates and time periods of elevated risk.

Statistical Learning for Nonlinear Dynamical Systems with Applications to Aircraft-UAV Collisions

ABSTRACT This article investigates a physics-informed statistical approach capable of (i) learning nonlinear system dynamics by using data generated from a nonlinear system as well as the underlying governing physics, and (ii) predicting system dynamics with reasonable accuracy and a computational speed much faster than numerical methods. The proposed approach obtains the reduced-order model from the full-order governing equations. A function-to-function regression, based on multivariate Functional Principal Component Analysis, establishes the mapping between external forcing and system dynamics, while a multivariate Gaussian Process is used to capture the relationship between parameters and external forcing. In the application, the proposed approach is applied to predict aircraft nose skin deformation after Unmanned Aerial Vehicle (UAV) collisions at different impact attitudes (i.e., pitch, yaw and roll degrees). We show that the proposed physics-informed statistical model can achieve a 12% out-of-sample mean relative error, and is more than 103 times faster than Finite Element Analysis (FEA). Computer code and sample data are available on GitHub.

Multivariate functional data clustering using adaptive density peak detection.

Clustering for multivariate functional data is a challenging problem since the data are represented by a set of curves and functions belonging to an infinite-dimensional space. In this article, we propose a novel clustering method for multivariate functional data usingan adaptive density peak detection technique. It is a quick cluster center identification algorithm based on the two measures of each functional data observation: the functional density estimate and the distance to the closest observation with a higher functional density. We suggest two types of functional density estimators for multivariate functional data. The first one is a functional -nearest neighbor density estimator based on (a) an L2 distance between raw functional curves, or (b) a semimetric of multivariate functional principal components. The second one is a -nearest neighbor density estimator based on multivariate functional principal scores. Our clustering method is computationally fast since it does not need an iterative process. The flexibility and advantages of the method are examined by comparing it with other existing clustering methods in simulation studies. A user-friendly R package FADPclust is developed for public use. Finally, our method is applied to a real case study in lung cancer research.

Adaptive Sampling for Monitoring Multi-Profile Data with Within-and-between Profile Correlation

Multi-profile data can provide within-and-between profile information for efficiently modeling and monitoring system status. In practice, however, acquisition of such data requires large number of sensors, which raises various concerns and difficulties, for example, cost, energy, and data transmission bandwidth, in accessing the full data from each sensor. In this article, we propose an adaptive sampling strategy for multi-profile monitoring by using limited portion of data. The proposed sampling and monitoring scheme incorporates the within-and-between profile correlation and features the balance between random search and greedy search in identifying the most informative profiles. More specifically, the multivariate functional principal component analysis (MFPCA) is used to capture the within-and-between profile correlation, and the MFPC scores are augmented for unobservable profiles to feed into a multivariate CUSUM chart. Two properties of the proposed method for allocating sampling resources among sensors are investigated. Numerical and case studies are conducted under various scenarios to demonstrate the effectiveness of the method.

Bayesian multivariate sparse functional principal components analysis with application to longitudinal microbiome multiomics data

Microbiome researchers often need to model the temporal dynamics of multiple complex, nonlinear outcome trajectories simultaneously. This motivates our development of multivariate Sparse Functional Principal Components Analysis (mSFPCA), extending existing SFPCA methods to simultaneously characterize multiple temporal trajectories and their interrelationships. As with existing SFPCA methods, the mSFPCA algorithm characterizes each trajectory as a smooth mean plus a weighted combination of the smooth major modes of variation about the mean, where the weights are given by the component scores for each subject. Unlike existing SFPCA methods, the mSFPCA algorithm allows estimation of multiple trajectories simultaneously, such that the component scores, which are constrained to be independent within a particular outcome for identifiability, may be arbitrarily correlated with component scores for other outcomes. A Cholesky decomposition is used to estimate the component score covariance matrix efficiently and guarantee positive semidefiniteness given these constraints. Mutual information is used to assess the strength of marginal and conditional temporal associations across outcome trajectories. Importantly, we implement mSFPCA as a Bayesian algorithm using R and stan, enabling easy use of packages such as PSIS-LOO for model selection and graphical posterior predictive checks to assess the validity of mSFPCA models. Although we focus on application of mSFPCA to microbiome data in this paper, the mSFPCA model is of general utility and can be used in a wide range of real-world applications.

Detecting differences in gait initiation between older adult fallers and non-fallers through multivariate functional principal component analysis

Gait initiation (GI) is an important locomotor transition task that includes anticipatory postural adjustments and the joint propulsion necessary for the first step of walking. Discrete variable analysis between GI of fallers and non-fallers has shown important between-group differences. More complex time series analysis, such as functional principal component analysis (FPCA) may highlight group differences not detectable using discrete comparisons alone. This study aims to characterize the differences between fallers and non-fallers by examining the kinematics and kinetics of gait initiation using multivariate FPCA (mFPCA). A sample of 56 community-dwelling older adults completed five walking trials where GI was measured by force platforms. mFPCA of center of pressure kinematics and kinetics was conducted and functional principal component scores were compared between groups. Overall mFPCA provided a comprehensive assessment of GI that supports and enhances previous findings with respect to differences between faller and non-faller cohorts. During weight transfer and forward progress, fallers demonstrate a greater range of mediolateral movement and lower lateral force than non-fallers. During the first step, fallers have a more gradual rise in vertical force, as well as a greater lateral movement toward the edge of their base of support. Fallers also demonstrate a shorter step length, indicating an altered approach to GI, where mediolateral and anteroposterior stability may be prioritized over forward advancement.

Multivariate functional principal component analysis for endogenously stratified data

We address the problem of performing dimension reduction on multivariate functional data observed on different domains in an endogenously stratified sampling context. The aim is to propose a new multivariate functional principal component analysis \(MFPCA) approach for data sampled by a stratification of a population according to a binary variable of interest. This estimation strategy is derived from a direct relationship between univariate and multivariate FPCA for finite Karhunen-Loève decompositions. The proposed methodology yields encouraging results and can be applied to data with measurement errors. Computational results on simulated data highlight the good performance of the proposed methodology compared to the classical MFPCA, which ignores the type of data sampling. A real-life application considering breast cancer cells data is also presented.

An Inverse Reinforcement Learning Approach for Customizing Automated Lane Change Systems

Vehicle automation seeks to enhance road safety and improve the driving experience. However, a standard system does not account for variations in users and driving conditions. Customizing vehicle automation based on users’ preferences aims to improve the user experience and adoption of the technologies. This study introduces a systematic paradigm that starts with naturalistic driving data to identify the driving behaviors and styles for a customized automated lane change system. The driving behaviors are first extracted using Multivariate Functional Principal Component Analysis (MFPCA) with minimum prior expert knowledge. The driving styles are identified by clustering the extracted driving behaviors. An Inverse Reinforcement Learning (IRL) algorithm is then used to train the automated lane change system from grouped demonstrations of the identified driving styles to capture the preferences of a group of drivers with a similar driving style. The performance of the proposed customized automated lane change system is compared to (1) a non-customized system trained on all the sample trips, (2) customized systems built on expert-coded reward functions, and (3) customized systems trained using a Generative Adversarial Imitation Learning (GAIL) algorithm. The results show that our method outperforms all the other systems with respect to the prediction accuracy of the lane change actions. Additionally, our method gains insights on the representative behaviors of different driving styles to enable customization of automated lane change systems.

Co-clustering of multivariate functional data for the analysis of air pollution in the South of France

Nowadays, air pollution is a major threat for public health with clear relationships with many diseases, especially cardiovascular ones. The spatiotemporal study of pollution is of great interest for governments and local authorities when deciding for public alerts or new city policies against pollution increase. The aim of this work is to study spatiotemporal profiles of environmental data collected in the south of France (Région Sud) by the public agency AtmoSud. The idea is to better understand the exposition to pollutants of inhabitants on a large territory with important differences in term of geography and urbanism. The data gather the recording of daily measurements of five environmental variables, namely, three pollutants (PM10, NO2, O3) and two meteorological factors (pressure and temperature) over six years. Those data can be seen as multivariate functional data: quantitative entities evolving along time for which there is a growing need of methods to summarize and understand them. For this purpose a novel co-clustering model for multivariate functional data is defined. The model is based on a functional latent block model which assumes for each co-cluster a probabilistic distribution for multivariate functional principal component scores. A stochastic EM algorithm, embedding a Gibbs sampler, is proposed for model inference as well as a model selection criteria for choosing the number of co-clusters. The application of the proposed co-clustering algorithm on environmental data of the Région Sud allowed to divide the region, composed by 357 zones, into six macroareas with common exposure to pollution. We showed that pollution profiles vary accordingly to the seasons, and the patterns are similar during the six years studied. These results can be used by local authorities to develop specific programs to reduce pollution at the macroarea level and to identify specific periods of the year with high pollution peaks in order to set up specific health prevention programs. Overall, the proposed co-clustering approach is a powerful resource to analyse multivariate functional data in order to identify intrinsic data structure and to summarize variables profiles over long periods of time.