Multivariate functional principal component analysis for endogenously stratified data

Cardiovascular diseases (CVDs) are one of the primary reasons for death worldwide. These diseases often occur due to the occlusion of coronary arteries, thereby leading to insufficient blood and oxygen supply that damages cardiac muscle cells. Electrocardiogram (ECG) signals which reflect heart electrical activity are being used for diagnosing various cardiac diseases. Typically, a standard ECG consists of 12 channels referred to as leads which enable practitioners to monitor heartbeats through different channels where each heartbeat lasts approximately 600 ms. The majority of studies focus on the classification and early diagnosis of arrhythmias. Although the current studies on change-point methods have acquired massive accuracy in detecting potential shifts during a multi-channel process, they lack flexibility in manually assigning more weights to the channels, which are of more importance for experts. This could be addressed by implementing the weighted multivariate functional principal component analysis (WMFPCA). The objective of this study is to develop a novel change-point detection method to monitor long-term cardiovascular treatment. A third-order tensor structure was employed to represent the 12-lead ECG data in three dimensions (beats × samples × leads). Exploiting intra-beat, inter-beat, and inter-lead correlations along with channel significance in the third-order tensor, the WMFPCA is incorporated into Hotelling's statistic to construct monitoring schemes. Simulation results show that the proposed approach outperforms the existing methods in monitoring multi-channel processes. Finally, applying the suggested model on a real-world dataset containing Myocardial Infarction (MI) subjects verifies the model.

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.

The paper entitled \(\textit{Multivariate Functional Principal Component Analysis for Endogenously Stratified Data}\) has been published in the journal Afrika Statistika, Vol. 17 (4), 2023, pp. 3321–3366 (DOI: 10.16929/as/2022.3321.308). The authors have identified the need for substantial modifications, including additions, deletions, and revisions. This erratum is published to address these necessary corrections and updates.

In this paper, we propose a generalized functional linear regression model for a binary outcome indicating the presence/absence of a cardiac disease with multivariate functional data among the relevant predictors. In particular, the motivating aim is the analysis of electrocardiographic traces of patients whose pre-hospital electrocardiogram (ECG) has been sent to 118 Dispatch Center of Milan (the Italian free-toll number for emergencies) by life support personnel of the basic rescue units. The statistical analysis starts with a preprocessing of ECGs treated as multivariate functional data. The signals are reconstructed from noisy observations. The biological variability is then removed by a nonlinear registration procedure based on landmarks. Thus, in order to perform a data-driven dimensional reduction, a multivariate functional principal component analysis is carried out on the variance-covariance matrix of the reconstructed and registered ECGs and their first derivatives. We use the scores of the Principal Components decomposition as covariates in a generalized linear model to predict the presence of the disease in a new patient. Hence, a new semi-automatic diagnostic procedure is proposed to estimate the risk of infarction (in the case of interest, the probability of being affected by Left Bundle Brunch Block). The performance of this classification method is evaluated and compared with other methods proposed in literature. Finally, the robustness of the procedure is checked via leave-j-out techniques.

With the emergence of numerical sensors in many aspects of every- day life, there is an increasing need in analyzing multivariate functional data. This work focuses on the clustering of such functional data, in order to ease their modeling and understanding. To this end, a novel clustering technique for multivariate functional data is presented. This method is based on a func- tional latent mixture model which fits the data in group-specific functional subspaces through a multivariate functional principal component analysis. A family of parsimonious models is obtained by constraining model parameters within and between groups. An EM algorithm is proposed for model inference and the choice of hyper-parameters is addressed through model selection. Nu- merical experiments on simulated datasets highlight the good performance of the proposed methodology compared to existing works. This algorithm is then applied to the analysis of the pollution in French cities for one year.

SummaryThe transitional phase of menopause induces significant hormonal fluctuations, exerting a profound influence on the long-term well-being of women. In an extensive longitudinal investigation of women’s health during mid-life and beyond, known as the Study of Women’s Health Across the Nation (SWAN), hormonal biomarkers are repeatedly assessed, following an asynchronous schedule compared to other error-prone covariates, such as physical and cardiovascular measurements. We conduct a subgroup analysis of the SWAN data employing a semiparametric mixture regression model, which allows us to explore how the relationship between hormonal responses and other time-varying or time-invariant covariates varies across subgroups. To address the challenges posed by asynchronous scheduling and measurement errors, we model the time-varying covariate trajectories as functional data with reduced-rank Karhunen–Loéve expansions, where splines are employed to capture the mean and eigenfunctions. Treating the latent subgroup membership and the functional principal component (FPC) scores as missing data, we propose an Expectation-Maximization algorithm to effectively fit the joint model, combining the mixture regression for the hormonal response and the FPC model for the asynchronous, time-varying covariates. In addition, we explore data-driven methods to determine the optimal number of subgroups within the population. Through our comprehensive analysis of the SWAN data, we unveil a crucial subgroup structure within the aging female population, shedding light on important distinctions and patterns among women undergoing menopause.

The standard paradigm in Affective Computing involves acquiring one/several markers (e.g., physiological signals) of emotions and training models on these to predict emotions. However, due to the internal nature of emotions, labelling/annotation of emotional experience is done manually by humans using specially developed annotation tools. To effectively exploit the resulting subjective annotations for developing affective systems, their quality needs to be assessed. This entails, (i) evaluating the variations in annotations, across different subjects and emotional stimuli, to detect spurious/unexpected patterns; and (ii) developing strategies to effectively combine these subjective annotations into a ground truth annotation. This article builds on our previous work by presenting a novel Functional Data Analysis based approach to assess the quality of annotations. Specifically, the bivariate annotation time-series are transformed into functions, such that each resulting functional annotation then becomes a sample element for analysis like Multivariate Functional Principal Component Analysis (MFPCA) that evaluate variation across all annotations. The resulting scores from MFPCA provide interesting insights into annotation patterns and facilitate the use of multivariate statistical techniques to address both (i) and (ii). Given the presented efficacy of these methods, we believe they offer an exciting new approach to assessing the quality of annotations.

Postural synergies allow a rich set of hand configurations to be represented in lower dimension space compared to the original joint space. In our previous works, we have shown that this can be extended to trajectories thanks to the multivariate functional principal component analysis, obtaining a set of basis functions able to represent grasping movements learned from human demonstration. In this article, we introduce a human cognition-inspired approach for generalizing and improving robot grasping skills in this motion synergies subspace. The use of a reinforcement learning (RL) algorithm allows the robot to explore the surrounding space and improve its capability to reach and grasp objects. The learning method is the policy improvement with path integrals, running in the policy space. Bootstrapped with synergy coefficients obtained from neural networks, the policy reward is based on a force closure grasp quality index computed at the end of the task, measuring how a firm is the grip. We finally show that combining neural networks and RL allows the robot manipulator to have a good initial estimate of the grasping configuration and faster convergence to an optimal grasp with respect to a database approach, the latter a less general solution in presence of new objects.

Methods and tools for implementing regularized multivariate functional principal component analysis ('ReMFPCA') for multivariate functional data whose variables might be observed over different dimensional domains. 'ReMFPCA' is an object-oriented interface leveraging the extensibility and scalability of R6. It employs a parameter vector to control the smoothness of each functional variable. By incorporating smoothness constraints as penalty terms within a regularized optimization framework, 'ReMFPCA' generates smooth multivariate functional principal components, offering a concise and interpretable representation of the data. For detailed information on the methods and techniques used in 'ReMFPCA', please refer to Haghbin et al. (2023) &lt;<a href="https://doi.org/10.48550%2FarXiv.2306.13980" target="_top">doi:10.48550/arXiv.2306.13980</a>&gt;.

We consider modeling and analysis of functional data arising in method comparison studies. The observed data consist of repeated measurements of a continuous variable obtained using multiple methods of measurement on a sample of subjects. The data are treated as multivariate functional data that are observed with noise at a common set of discrete time points which may vary from subject to subject. The proposed methodology uses functional principal components analysis within the framework of a mixed-effects model to represent the observations in terms of a small number of method-specific principal components. Two approaches for estimating the unknowns in the model, both adaptations of general techniques developed for multivariate functional principal components analysis, are presented. Bootstrapping is employed to get estimates of bias and covariance matrix of model parameter estimates. These in turn are used to compute confidence intervals for parameters and functions thereof, such as the measures of similarity and agreement between the measurement methods, that are necessary for data analysis. The estimation approaches are evaluated using simulation. The methodology is illustrated by analyzing two datasets.

Data with spatio-temporal structure can arise in many contexts, therefore a considerable interest in modelling these data has been generated, but the complexity of spatio-temporal models, together with the size of the dataset, results in a challenging task. The modelization is even more complex in presence of multivariate data. Since some modelling problems are more natural to think through in functional terms, even if only a finite number of observations is available, treating the data as functional can be useful (Berrendero et al. in Comput Stat Data Anal 55:2619–2634, 2011). Although in Ramsay and Silverman (Functional data analysis, 2nd edn. Springer, New York, 2005) the case of multivariate functional data is also contemplated, they do not cope with more than one dimension (only time is considered as covariate) in the domain of the considered functions. In estimating functional data through smoothing methods, a proper framework for incorporating space-time structures can be found in the generalized additive models (GAM), while classical dimension reduction techniques for functional data lead to functional principal component analysis (FPCA). In a previous work Ruggieri et al. (J Appl Stat 40(4):795–807, 2013) extended temporal FPCA, that is FPCA on data modelled as functions of the one-dimensional time, to multivariate (more than one variable, in our case pollutants) context. In this paper the computational aspects of FPCA are extended to more than one dimension: space (long, lat) and/or space-time; moreover, multidimensional (spatial, spatio-temporal) FPCA is extended to multivariate case. In order to provide a generalization of FPCA to multidimensional (spatio-temporal) and simultaneously multivariate data, we link GAM models together with the approach proposed by Ramsay and Silverman (2005). The paper describes all the computational details useful to implement this approach, while its effectiveness will be shown by a multivariate spatio-temporal environmental dataset, whose structure is actually very common in literature.

Despite improved techniques, the determination of tumor origin in poorly differentiated adenocarcinomas still remains a challenge for the pathologist. Here we report the use of protein profiling combined with principal component analysis to improve diagnostic decision-making in tumor samples, in which standard pathologic investigations cannot present reliable results. A poorly differentiated adenocarcinoma of unknown origin located in the pelvis, infiltrating the sigmoid colon as well as the ovary, served as a model to evaluate our proteomic approach. Firstly, we characterized the protein expression profiles from eight advanced colon and seven ovarian adenocarcinomas using two-dimensional gel electrophoresis (2-DE). Qualitative and quantitative patterns were recorded and compared to the tumor of unknown origin. Based on these protein profiles, match sets from the different tumors were created. Finally, a multivariate principal component analysis was applied to the entire 2-DE data to disclose differences in protein patterns between the different tumors. Over 89% of the unknown tumor sample spots could be matched with the colon standard gel, whereas only 63% of the spots could be matched with the ovarian standard. In addition, principal component analysis impressively displayed the clustering of the unknown case within the colon cancer samples, whereas this case did not cluster at all within the group of ovarian adenocarcinomas. These results show that 2-DE protein expression profiling combined with principal component analysis is a sensitive method for diagnosing undifferentiated adenocarcinomas of unknown origin. The described approach can contribute greatly to diagnostic decision-making and, with further technical improvements and a higher throughput, become a powerful tool in the armentarium of the pathologist.

Phytoplankton community composition of Saginaw Bay, Lake Huron, during the zebra mussel ( Dreissena polymorpha) invasion: A multivariate analysis

The aim of present study was to measure of 15 element levels (Hg, Ca, As, Fe, Mg, Pb, Mn, Co, Al, Se, Cr, Ni, P, Zn and Cd) using ICP-OES (Inductively Coupled Plasma Optical Emission spectroscopy) in Persian tahini (Ardeh) samples. Further, the correlation between the type and amount of essential elements and toxic elements samples was evaluated with multivariate principal component analysis. The LOD, LOQ and recovery of methods were ranged from 0.43 to 2.166, 0.128 to 6.498 µg/kg and 95.3 to 104.7%, respectively. According to our results, the highest and lowest mean of toxic metals were related to Al 122.20 ± 56.70 and Cd 7.82 ± 1.40, respectively. Also, the highest mean of essential metals was related to the Mg 29,345.37 ± 4816.00 and the lowest mean of essential metals was related to the Se and P that were not detected in all samples. Finally, in all samples, level of all elements (toxic and essential) were lower than the standard levels of Iran and other countries, consequently, for the consumers of this product in Iran, there is no risk.

We present the application of a nonparametric method to performing functional principal component analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal component scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing of healthy adults with a small tracer dose of 14C-folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken. The proposed methodology, incorporating shrinkage and data-adaptive features, is seen to be well suited for describing population kinetics of 14C-folate-specific activity and random effects, and can also be applied to other functional data analysis problems.