Functional data analysis: Application to daily observation of COVID-19 prevalence in France

<abstract> <p>In this paper we use the technique of functional data analysis to model daily hospitalized, deceased, Intensive Care Unit (ICU) cases and return home patient numbers along the COVID-19 outbreak, considered as functional data across different departments in France while our response variables are numbers of vaccinations, deaths, infected, recovered and tests in France. These sets of data were considered before and after vaccination started in France. After smoothing our data set, analysis based on functional principal components method was performed. Then, a clustering using k-means techniques was done to understand the dynamics of the pandemic in different French departments according to their geographical location on France map. We also performed canonical correlations analysis between variables. Finally, we made some predictions to assess the accuracy of the method using functional linear regression models.</p> </abstract>

Degrees of Freedom in Functional Principal Components Analysis

This paper develops the analytical form of the degrees of freedom in functional principal components analysis. Under the framework of unbiased risk estimation, we derive an unbiased estimator with a clear analytical formula for the degrees of freedom in the one-way penalized functional principal components analysis paradigm. Specifically, a new analytical formula incorporating binary smoothing parameters is also derived based on the singular value decomposition and half-smoothed method regarding the two-way penalized functional principal components analysis framework. The performance of our procedures is demonstrated by simulation studies.

Functional Data Analysis: Transition from Daily Observation of COVID-19 Prevalence in France to Functional Curves

In this paper we use the technique of functional data analysis to model daily hospitalized, deceased, ICU cases and return home patient numbers along the COVID-19 outbreak, considered as functional data across different departments in France while our response variables are numbers of vaccinations, deaths, infected, recovered and tests in France. These sets of data were considered before and after vaccination started in France. We used some smoothing techniques to smooth our data set, then analysis based on functional principal components method was performed, clustering using k-means techniques was done to understand the dynamics of the pandemic in different French departments according to their geographical location on France map and we also performed canonical correlations analysis between variables. Finally, we made some predictions to assess the accuracy of the method using functional linear regression models.

Dynamic Functional Principal Components for Testing Causality

In this paper, we investigate the causality in the sense of Granger for functional time series. The concept of causality for functional time series is defined, and a statistical procedure of testing the hypothesis of non-causality is proposed. The procedure is based on projections on dynamic functional principal components and the use of a multivariate Granger test. A comparative study with existing procedures shows the good results of our test. An illustration on a real dataset is provided to attest the performance of the proposed procedure.

COVID-19 Data Imputation by Multiple Function-on-Function Principal Component Regression

The aim of this paper is the imputation of missing data of COVID-19 hospitalized and intensive care curves in several Spanish regions. Taking into account that the curves of cases, deceases and recovered people are completely observed, a function-on-function regression model is proposed to estimate the missing values of the functional responses associated with hospitalized and intensive care curves. The estimation of the functional coefficient model in terms of principal components’ regression with the completely observed data provides a prediction equation for the imputation of the unobserved data for the response. An application with data from the first wave of COVID-19 in Spain is developed after properly homogenizing, registering and smoothing the data in a common interval so that the observed curves become comparable. Finally, Canonical Correlation Analysis is performed on the functional principal components to interpret the relationship between hospital occupancy rate and illness response variables.

An Animated Functional Data Analysis Interface to Cluster Rapid Lung Function Decline and Enhance Center-Level Care in Cystic Fibrosis

Identifying disease progression through enhanced decision support tools is key to chronic management in cystic fibrosis at both the patient and care center level. Rapid decline in lung function relative to patient level and center norms is an important predictor of outcomes. Our objectives were to construct and utilize center-level classification of rapid decliners to develop an animated dashboard for comparisons within patients over time, multiple patients within centers, or between centers. A functional data analysis technique known as functional principal components analysis was applied to lung function trajectories from 18,387 patients across 247 accredited centers followed through the United States Cystic Fibrosis Foundation Patient Registry, in order to cluster patients into rapid decline phenotypes. Smaller centers (<30 patients) had older patients with lower baseline lung function and less severe rates of decline and had maximal decline later, compared to medium (30–150 patients) or large (>150 patients) centers. Small centers also had the lowest prevalence of early rapid decliners (17.7%, versus 24% and 25.7% for medium and large centers, resp.). The animated functional data analysis dashboard illustrated clustering and center-specific summaries of the rapid decline phenotypes. Clinical scenarios and utility of the center-level functional principal components analysis (FPCA) approach are considered and discussed.

State space functional principal component analysis to identify spatiotemporal patterns in remote sensing lake water quality

Satellite remote sensing can provide indicative measures of environmental variables that are crucial to understanding the environment. The spatial and temporal coverage of satellite images allows scientists to investigate the changes in environmental variables in an unprecedented scale. However, identifying spatiotemporal patterns from such images is challenging due to the complexity of the data, which can be large in volume yet sparse within individual images. This paper proposes a new approach, state space functional principal components analysis (SS-FPCA), to identify the spatiotemporal patterns in processed satellite retrievals and simultaneously reduce the dimensionality of the data, through the use of functional principal components. Furthermore our approach can be used to produce interpolations over the sparse areas. An algorithm based on the alternating expectation–conditional maximisation framework is proposed to estimate the model. The uncertainty of the estimated parameters is investigated through a parametric bootstrap procedure. Lake chlorophyll-a data hold key information on water quality status. Such information is usually only available from limited in situ sampling locations or not at all for remote inaccessible lakes. In this paper, the SS-FPCA is used to investigate the spatiotemporal patterns in chlorophyll-a data of Taruo Lake on the Tibetan Plateau, observed by the European Space Agency MEdium Resolution Imaging Spectrometer.

Identifying longitudinal child growth curve pattern using Functional Principal Component Analysis from birth cohort study in South India

Background: Human growth studies has long been of interest to researchers and health authorities. Predominantly, the study of physical growth in children is a challenging and complex issue. The goal of a variety of studies from across the world is to develop overall health and well-being in children. It is therefore important that we need to identify an accurate and reliable approach for characterising growth trajectories to distinguish between children who have healthy growth and those growth is poor. Many statistical approaches are available to assess the longitudinal growth data and which are difficult to recognize the pattern. The purpose of this study is to identify the longitudinal child growth trajectory pattern and factors association on the growth function using an advanced statistical technique. Methods: This longitudinal birth cohort study (n=290) was conducted in three nearby urban slums communities in Vellore, South India. Pregnant women were identified during a survey conducted in 2002 and infants were recruited from birth between the period of March 2002 and August 2003 following written informed consent from the mother. Growth outcomes of height and weight measurements were recorded for each child continuously in first 36 months. Functional Principal Component Analysis was used to classify the longitudinal child growth trajectory pattern. Functional linear model was used to assess the factors association with the growth functions. Results: We have obtained four functional principal components explained by 86.5%, 3.9%, 3.1% and 2.2% of the variation respectively for the height functions. 38% of the children’s had poor growth trajectories in height. Similarly, we have obtained three functional principal components explained 76.2%, 8.8%, and 4.7% respectively for the weight functions. 44% of the children’s had poor growth in their weight trajectories. The study shows that gender, socio-economic status, parent’s education, breast feeding, and gravida are associated and, influence the growth pattern in children. Conclusions: The advanced FPC approach deals with subjects’ dynamics of growth and not with specific values at given times. FPCA may be a better alternative approach in the sense of both dimension reduction and pattern detection. FPCA may be used to offer greater insight for classification.

New Modeling Approaches Based on Varimax Rotation of Functional Principal Components

Functional Principal Component Analysis (FPCA) is an important dimension reduction technique to interpret the main modes of functional data variation in terms of a small set of uncorrelated variables. The principal components can not always be simply interpreted and rotation is one of the main solutions to improve the interpretation. In this paper, two new functional Varimax rotation approaches are introduced. They are based on the equivalence between FPCA of basis expansion of the sample curves and Principal Component Analysis (PCA) of a transformation of the matrix of basis coefficients. The first approach consists of a rotation of the eigenvectors that preserves the orthogonality between the eigenfunctions but the rotated principal component scores are not uncorrelated. The second approach is based on rotation of the loadings of the standardized principal component scores that provides uncorrelated rotated scores but non-orthogonal eigenfunctions. A simulation study and an application with data from the curves of infections by COVID-19 pandemic in Spain are developed to study the performance of these methods by comparing the results with other existing approaches.