Multivariate Functional Principal Component Analysis Research Articles

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.

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.

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.

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.

Multipopulation mortality modelling and forecasting: the weighted multivariate functional principal component approaches

ABSTRACT Human mortality patterns and trajectories in closely related populations are likely linked together and share similarities. It is always desirable to model them simultaneously while taking their heterogeneity into account. This article introduces two new models for joint mortality modelling and forecasting multiple subpopulations using the multivariate functional principal component analysis techniques. The first model extends the independent functional data model to a multipopulation modelling setting. In the second one, we propose a novel multivariate functional principal component method for coherent modelling. Its design primarily fulfils the idea that when several subpopulation groups have similar socio-economic conditions or common biological characteristics such close connections are expected to evolve in a non-diverging fashion. We demonstrate the proposed methods by using sex-specific mortality data. Their forecast performances are further compared with several existing models, including the independent functional data model and the Product-Ratio model, through comparisons with mortality data of ten developed countries. The numerical examples show that the first proposed model maintains a comparable forecast ability with the existing methods. In contrast, the second proposed model outperforms the first model as well as the existing models in terms of forecast accuracy.

Sparse and Robust Multivariate Functional Principal Component Analysis for Passenger Flow Pattern Discovery in Metro Systems

Modern metro systems in big cities have accumulated large amounts of passenger transit transaction data via the deployment of automatic fare collection (AFC) devices, which could facilitate a thorough analysis of passenger flow dynamics. To discover the underlying passenger flow patterns over all stations in an entire metro system, this paper proposes a multivariate functional principal component analysis (MFPCA) method. The functional integrity of a single daily passenger flow profile at each station is explicitly utilized, and the complex correlation among a multitude of daily passenger flow profiles from all stations is comprehensively modeled. Moreover, to simultaneously improve the interpretability of eigenvectors and mitigate the sensitivity to outliers, the MFPCA is formulated as a minimization problem with both a sparsity and a robustness penalty term. A computationally efficient algorithm is developed accordingly to obtain the eigenvectors. The superiority of our proposed sparse and robust MFPCA (SRMFPCA) is validated using a Hong Kong Mass Transit Railway (MTR) dataset. The derived sparse and smooth eigenvectors can be well interpreted as empirically meaningful passenger flow patterns. The results of our method can be further used as solid foundations for station clustering, correlation analysis and outlier identification.

Sparse multivariate functional principal component analysis

We introduce a sparse multivariate functional principal component analysis method by incorporating ideas from the group sparse maximum variance method to multivariate functional data. Our method can avoid the “curse of dimensionality” from a high‐dimensional dataset and enjoy interpretability at the same time. In particular, our unsupervised method can capture important latent factors to explain variability of the dataset, which can induce a clear distinction between important variables in the principal components and unnecessary features based on the sparseness structure. Furthermore, our method can be applied to functional data from a multidimensional domain that hinges on different intervals. In the numerical experiment, we show that our method works well in both low‐ and high‐dimensional multivariate functional data regardless of the number and the type of basis. We further apply our method to stock market data and electroencephalography data in an alcoholism study to demonstrate the theoretical result.

A robust functional partial least squares for scalar‐on‐multiple‐function regression

AbstractThe scalar‐on‐function regression model has become a popular analysis tool to explore the relationship between a scalar response and multiple functional predictors. Most of the existing approaches to estimate this model are based on the least‐squares estimator, which can be seriously affected by outliers in empirical datasets. When outliers are present in the data, it is known that the least‐squares‐based estimates may not be reliable. This paper proposes a robust functional partial least squares method, allowing a robust estimate of the regression coefficients in a scalar‐on‐multiple‐function regression model. In our method, the functional partial least squares components are computed via the partial robust M‐regression. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments and two chemometric datasets: glucose concentration spectrometric data and sugar process data. The results produced by the proposed method are compared favorably with some of the classical functional or multivariate partial least squares and functional principal component analysis methods.

Transfer learning of degradation modeling and prognosis based on multivariate functional analysis with heterogeneous sampling rates

Multivariate functional principal component analysis (MFPCA) is a widely used tool for modeling and prognosis of degradation signals. However, MFPCA usually assumes samples from each process share the same sampling rate. To relax this assumption and facilitate the information sharing among multiple degradation processes, we propose a transfer learning framework to conduct modeling and prognosis of degradation data with heterogeneous sampling rates. This framework features the flexible modeling capability and focuses on transferring degradation information from correlated processes to the prognosis of an interested (target) process. More specifically, we use kernel smoothing to handle the heterogeneous sampling rates so that the mean and covariance in each sample can be successfully estimated. Then, the conditional expectation is developed to estimate the MFPC scores, which contain correlation information among processes. These MFPC scores are then updated in a Bayesian way by incorporating the observed data from the in-service unit in the target process. The updated MFPC scores deliver a superior performance of prognosis. The proposed method is validated and compared with various benchmarks in extensive numerical studies and one case study. The results show the proposed method can successfully handle the issue of heterogeneous sampling rates and provide an improved prognosis result.

Multivariate functional additive mixed models

Multivariate functional data can be intrinsically multivariate like movement trajectories in 2D or complementary such as precipitation, temperature and wind speeds over time at a given weather station. We propose a multivariate functional additive mixed model (multiFAMM) and show its application to both data situations using examples from sports science (movement trajectories of snooker players) and phonetic science (acoustic signals and articulation of consonants). The approach includes linear and nonlinear covariate effects and models the dependency structure between the dimensions of the responses using multivariate functional principal component analysis. Multivariate functional random intercepts capture both the auto-correlation within a given function and cross-correlations between the multivariate functional dimensions. They also allow us to model between-function correlations as induced by, for example, repeated measurements or crossed study designs. Modelling the dependency structure between the dimensions can generate additional insight into the properties of the multivariate functional process, improves the estimation of random effects, and yields corrected confidence bands for covariate effects. Extensive simulation studies indicate that a multivariate modelling approach is more parsimonious than fitting independent univariate models to the data while maintaining or improving model fit.

Multidimensional economic indicators and multivariate functional principal component analysis (MFPCA) in a comparative study of countries’ competitiveness

Multidimensional economic indicators and multivariate functional principal component analysis (MFPCA) in a comparative study of countries’ competitiveness