Functional Data Analysis Techniques Research Articles

Comprehensive functional data analysis of China’s dynamic energy security index

In the transition towards a cleaner and more sustainable global energy production and consumption, the energy security index (ESI) is an integral deciding factor for many countries. In this paper, the dynamics of China’s energy security index is constructed and analyzed using comprehensive functional data analysis (FDA) techniques. The energy security indices of 30 provinces in China between 2004 and 2017 are considered. The effect of investment (in energy sector) on energy security of the provinces is also checked using functional regression models. The average ESI for Shanxi (0.64737), Inner Mongolia (0.64540), and Shaanxi (0.51039), and the energy investment in Inner Mongolia, Shandong, and Shanxi form the top 10% of all the provinces considered. Exploratory analysis shows that there was an overall decrease in ESI in most of the provinces between 2004 and 2017, despite an increase in the investment in energy sectors of these provinces in the same period. The functional regression models show that investment in energy sector achieved its highest positive effect on energy security between year 2010 and 2014.

Cross-Validation, Information Theory, or Maximum Likelihood? A Comparison of Tuning Methods for Penalized Splines

Functional data analysis techniques, such as penalized splines, have become common tools used in a variety of applied research settings. Penalized spline estimators are frequently used in applied research to estimate unknown functions from noisy data. The success of these estimators depends on choosing a tuning parameter that provides the correct balance between fitting and smoothing the data. Several different smoothing parameter selection methods have been proposed for choosing a reasonable tuning parameter. The proposed methods generally fall into one of three categories: cross-validation methods, information theoretic methods, or maximum likelihood methods. Despite the well-known importance of selecting an ideal smoothing parameter, there is little agreement in the literature regarding which method(s) should be considered when analyzing real data. In this paper, we address this issue by exploring the practical performance of six popular tuning methods under a variety of simulated and real data situations. Our results reveal that maximum likelihood methods outperform the popular cross-validation methods in most situations—especially in the presence of correlated errors. Furthermore, our results reveal that the maximum likelihood methods perform well even when the errors are non-Gaussian and/or heteroscedastic. For real data applications, we recommend comparing results using cross-validation and maximum likelihood tuning methods, given that these methods tend to perform similarly (differently) when the model is correctly (incorrectly) specified.

Functional data analysis characterizes the shapes of the\xa0first COVID-19 epidemic\xa0wave in Italy

We investigate patterns of COVID-19 mortality across 20 Italian regions and their association with mobility, positivity, and socio-demographic, infrastructural and environmental covariates. Notwithstanding limitations in accuracy and resolution of the data available from public sources, we pinpoint significant trends exploiting information in curves and shapes with Functional Data Analysis techniques. These depict two starkly different epidemics; an “exponential” one unfolding in Lombardia and the worst hit areas of the north, and a milder, “flat(tened)” one in the rest of the country—including Veneto, where cases appeared concurrently with Lombardia but aggressive testing was implemented early on. We find that mobility and positivity can predict COVID-19 mortality, also when controlling for relevant covariates. Among the latter, primary care appears to mitigate mortality, and contacts in hospitals, schools and workplaces to aggravate it. The techniques we describe could capture additional and potentially sharper signals if applied to richer data.

Transmission dynamics and forecasts of the COVID-19 pandemic in Mexico, March-December 2020.

Mexico has experienced one of the highest COVID-19 mortality rates in the world. A delayed implementation of social distancing interventions in late March 2020 and a phased reopening of the country in June 2020 has facilitated sustained disease transmission in the region. In this study we systematically generate and compare 30-day ahead forecasts using previously validated growth models based on mortality trends from the Institute for Health Metrics and Evaluation for Mexico and Mexico City in near real-time. Moreover, we estimate reproduction numbers for SARS-CoV-2 based on the methods that rely on genomic data as well as case incidence data. Subsequently, functional data analysis techniques are utilized to analyze the shapes of COVID-19 growth rate curves at the state level to characterize the spatiotemporal transmission patterns of SARS-CoV-2. The early estimates of the reproduction number for Mexico were estimated between Rt ~1.1–1.3 from the genomic and case incidence data. Moreover, the mean estimate of Rt has fluctuated around ~1.0 from late July till end of September 2020. The spatial analysis characterizes the state-level dynamics of COVID-19 into four groups with distinct epidemic trajectories based on epidemic growth rates. Our results show that the sequential mortality forecasts from the GLM and Richards model predict a downward trend in the number of deaths for all thirteen forecast periods for Mexico and Mexico City. However, the sub-epidemic and IHME models perform better predicting a more realistic stable trajectory of COVID-19 mortality trends for the last three forecast periods (09/21-10/21, 09/28-10/27, 09/28-10/27) for Mexico and Mexico City. Our findings indicate that phenomenological models are useful tools for short-term epidemic forecasting albeit forecasts need to be interpreted with caution given the dynamic implementation and lifting of social distancing measures.

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.

On the intraday return curves of Bitcoin: Predictability and trading opportunities

Motivated by the potential inferences from intraday price data in the controversial Bitcoin market, we apply functional data analysis techniques to study cumulative intraday return (CIDR) curves. First, we indicate that Bitcoin CIDR curves are stationary, non-normal, uncorrelated, but exhibit conditional heteroscedastic, although we find that the projection scores of CIDR curves could be serially correlated during some certain periods. Second, we show the possibility of predicting the CIDR curves of Bitcoins based on the projection scores and then assess the forecasting performance. Finally, we utilize the functional forecasting methods to explore the intraday trading opportunities of Bitcoins and the results provide evidence of profitable trading opportunities based on intraday trading strategies, which confronts the efficient market hypothesis.

Associations between Google Search Trends for Symptoms and COVID-19 Confirmed and Death Cases in the United States.

We utilize functional data analysis techniques to investigate patterns of COVID-19 positivity and mortality in the US and their associations with Google search trends for COVID-19-related symptoms. Specifically, we represent state-level time series data for COVID-19 and Google search trends for symptoms as smoothed functional curves. Given these functional data, we explore the modes of variation in the data using functional principal component analysis (FPCA). We also apply functional clustering analysis to identify patterns of COVID-19 confirmed case and death trajectories across the US. Moreover, we quantify the associations between Google COVID-19 search trends for symptoms and COVID-19 confirmed case and death trajectories using dynamic correlation. Finally, we examine the dynamics of correlations for the top nine Google search trends of symptoms commonly associated with COVID-19 confirmed case and death trajectories. Our results reveal and characterize distinct patterns for COVID-19 spread and mortality across the US. The dynamics of these correlations suggest the feasibility of using Google queries to forecast COVID-19 cases and mortality for up to three weeks in advance. Our results and analysis framework set the stage for the development of predictive models for forecasting COVID-19 confirmed cases and deaths using historical data and Google search trends for nine symptoms associated with both outcomes.

Effects of acute wearable resistance loading on overground running lower body kinematics.

Field-based sports require athletes to run sub-maximally over significant distances, often while contending with dynamic perturbations to preferred coordination patterns. The ability to adapt movement to maintain performance under such perturbations appears to be trainable through exposure to task variability, which encourages movement variability. The aim of the present study was to investigate the extent to which various wearable resistance loading magnitudes alter coordination and induce movement variability during running. To investigate this, 14 participants (three female and 11 male) performed 10 sub-maximal velocity shuttle runs with either no weight, 1%, 3%, or 5% of body weight attached to the lower limbs. Sagittal plane lower limb joint kinematics from one complete stride cycle in each run were assessed using functional data analysis techniques, both across the participant group and within-individuals. At the group-level, decreases in ankle plantarflexion following toe-off were evident in the 3% and 5% conditions, while increased knee flexion occurred during weight acceptance in the 5% condition compared with unloaded running. At the individual-level, between-run joint angle profiles varied, with six participants exhibiting increased joint angle variability in one or more loading conditions compared with unloaded running. Loading of 5% decreased between-run ankle joint variability among two individuals, likely in accordance with the need to manage increased system load or the novelty of the task. In terms of joint coordination, the most considerable alterations to coordination occurred in the 5% loading condition at the hip-knee joint pair, however, only a minority of participants exhibited this tendency. Coaches should prescribe wearable resistance individually to perturb preferred coordination patterns and encourage movement variability without loading to the extent that movement options become limited.

A Bayesian Functional Methodology for Dengue Risk Mapping in Latin America and the Caribbean

Dengue fever has become one of the most outstanding infectious diseases in the world. Besides, the incidence and prevalence of dengue are increasing in the endemic areas of the tropical and subtropical regions. Space and time disease mapping models are common instruments to explain the patterns of disease counts, where hierarchical Bayesian models constitute a suitable framework for their formulation. These random events reflect interactions between nearby geographic locations, as well as correlations between close temporary instants. Functional data analysis techniques can better describe the evolution of disease mapping.In this paper, the risk of dengue in Mexico, Central and South America is studied from a Functional approach through a Bayesian estimation model focused on Hilbert-valued autoregressive processes combined with the Kalman filtering algorithm. Thus, the temporal functional evolution of spatial geographic patterns of incidence risk in disease mapping during 1998-2018 is approximated. Applying this methodology, the excess of smoothing that occurs with traditional models is avoided and the heterogeneity is conserved across the years. It improves the number of false positives created by noise and the number of false negatives as well. The results obtained with the application of this model are compared with those of previous models, corroborating the preceding statements and obtaining better results in the relative risk estimates, providing greater robustness and stability of disease risk estimates.

A comparison of parameter estimation in function-on-function regression

Recent technological developments have enabled us to collect complex and high-dimensional data in many scientific fields, such as population health, meteorology, econometrics, geology, and psychology. It is common to encounter such datasets collected repeatedly over a continuum. Functional data, whose sample elements are functions in the graphical forms of curves, images, and shapes, characterize these data types. Functional data analysis techniques reduce the complex structure of these data and focus on the dependences within and (possibly) between the curves. A common research question is to investigate the relationships in regression models that involve at least one functional variable. However, the performance of functional regression models depends on several factors, such as the smoothing technique, the number of basis functions, and the estimation method. This article provides a selective comparison for function-on-function regression models where both the response and predictor(s) are functions, to determine the optimal choice of basis function from a set of model evaluation criteria. We also propose a bootstrap method to construct a confidence interval for the response function. The numerical comparisons are implemented through Monte Carlo simulations and two real data examples.

Trajectories of the heart rate characteristics index, a physiomarker of sepsis in premature infants, predict Neonatal ICU mortality.

ObjectiveTrajectories of physiomarkers over time can be useful to define phenotypes of disease progression and as predictors of clinical outcomes. The aim of this study was to identify phenotypes of the time course of late-onset sepsis in premature infants in Neonatal Intensive Care Units.MethodsWe examined the trajectories of a validated continuous physiomarker, abnormal heart rate characteristics, using functional data analysis and clustering techniques.ParticipantsWe analyzed continuous heart rate characteristics data from 2989 very low birth weight infants (<1500 grams) from nine NICUs from 2004–2010.ResultDespite the relative homogeneity of the patients, we found extreme variability in the physiomarker trajectories. We identified phenotypes that were indicative of seven and 30 day mortality beyond that predicted by individual heart rate characteristics values or baseline demographic information.ConclusionTime courses of a heart rate characteristics physiomarker reveal snapshots of illness patterns, some of which were more deadly than others.

Clarifying and quantifying the geometric correlation for probability distributions of inter-sensor monitoring data: A functional data analytic methodology

In structural health monitoring (SHM), revealing the underlying correlations of monitoring data is of considerable significance, both theoretically and practically. In contrast to the traditional correlation analysis for numerical data, this study seeks to analyse the correlation of probability distributions of inter-sensor monitoring data. Due to induced by some commonly shared random excitations, many structural responses measured at different locations are usually correlated in distributions. Clarifying and quantifying such distributional correlations not only enables a more comprehensive understanding of the essential dependence properties of SHM data, but also has appealing application values; however, statistical methods pertinent to this topic are rare. To this end, this article proposes a novel approach using functional data analysis techniques. The monitoring data collected by each sensor are divided into time segments and later summarized by the corresponding probability density functions (PDFs). The geometric relations of the PDFs in terms of their shape mappings between sensors are first characterized by warping functions, and they are subsequently decomposed into finite functional principal components (FPCs); one FPC of the warping functions characterizes one deformation pattern in the transformation of the shapes of the PDFs from one sensor to another. Using this principle, the inter-sensor geometric correlation patterns of PDFs can be clarified by analysing the correlation of the FPC scores of warping functions to the PDFs from one sensor. To overcome the challenge of correlation quantification for real-valued samples (FPC scores) coupled with their functional counterparts (PDFs), a novel nonparametric functional regression (NFR)-based correlation coefficient is defined. Both simulation and real data studies are conducted to illustrate and validate the proposed method.

Intraday forecasts of a volatility index: functional time series methods with dynamic updating

As a forward-looking measure of future equity market volatility, the VIX index has gained immense popularity in recent years to become a key measure of risk for market analysts and academics. We consider discrete reported intraday VIX tick values as realisations of a collection of curves observed sequentially on equally spaced and dense grids over time and utilise functional data analysis techniques to produce 1-day-ahead forecasts of these curves. The proposed method facilitates the investigation of dynamic changes in the index over very short time intervals as showcased using the 15-s high-frequency VIX index values. With the help of dynamic updating techniques, our point and interval forecasts are shown to enjoy improved accuracy over conventional time series models.

Child Weight Gain Trajectories Linked To Oral Microbiota Composition

Gut and oral microbiota perturbations have been observed in obese adults and adolescents; less is known about their influence on weight gain in young children. Here we analyzed the gut and oral microbiota of 226 two-year-olds with 16S rRNA gene sequencing. Weight and length were measured at seven time points and used to identify children with rapid infant weight gain (a strong risk factor for childhood obesity), and to derive growth curves with innovative Functional Data Analysis (FDA) techniques. We showed that growth curves were associated negatively with diversity, and positively with the Firmicutes-to-Bacteroidetes ratio, of the oral microbiota. We also demonstrated an association between the gut microbiota and child growth, even after controlling for the effect of diet on the microbiota. Lastly, we identified several bacterial genera that were associated with child growth patterns. These results suggest that by the age of two, the oral microbiota of children with rapid infant weight gain may have already begun to establish patterns often seen in obese adults. They also suggest that the gut microbiota at age two, while strongly influenced by diet, does not harbor obesity signatures many researchers identified in later life stages.

"Dynamical correlation: A new method for quantifying synchrony with multivariate intensive longitudinal data": Correction to Liu et al. (2016).

Reports an error in "Dynamical correlation: A new method for quantifying synchrony with multivariate intensive longitudinal data" by Siwei Liu, Yang Zhou, Richard Palumbo and Jane-Ling Wang (Psychological Methods, 2016[Sep], Vol 21[3], 291-308). In the article, there were errors in the R script of Appendix B which could lead to incorrect significance testing results for dynamical correlation. We created an updated R script with corrections. In the updated R script, argument "na" from function "ind_DC" and argument "ms" from function "boot_test_DC" were removed. Codes to check if there is any missing value in the data, and to compute proportion of missing values in the data were added. A warning was added when too many missing values are present. In addition, argument 't' is now correctly labeled "a vector of time points where x,y are observed". The updated R script with corrections can be downloaded from the first author's personal website: https://siweiliu.weebly.com/publications.html. (The following abstract of the original article appeared in record 2016-07276-001.) In this article, we introduce dynamical correlation, a new method for quantifying synchrony between 2 variables with intensive longitudinal data. Dynamical correlation is a functional data analysis technique developed to measure the similarity of 2 curves. It has advantages over existing methods for studying synchrony, such as multilevel modeling. In particular, it is a nonparametric approach that does not require a prespecified functional form, and it places no assumption on homogeneity of the sample. Dynamical correlation can be easily estimated with irregularly spaced observations and tested to draw population-level inferences. We illustrate this flexible statistical technique with a simulation example and empirical data from an experiment examining interpersonal physiological synchrony between romantic partners. We discuss the advantages and limitations of the method, and how it can be extended and applied in psychological research. We also provide a set of R code for other researchers to estimate and test for dynamical correlation. (PsycINFO Database Record

IWTomics: testing high-resolution sequence-based 'Omics' data at multiple locations and scales.

With increased generation of high-resolution sequence-based 'Omics' data, detecting statistically significant effects at different genomic locations and scales has become key to addressing several scientific questions. IWTomics is an R/Bioconductor package (integrated in Galaxy) that, exploiting sophisticated Functional Data Analysis techniques (i.e. statistical techniques that deal with the analysis of curves), allows users to pre-process, visualize and test these data at multiple locations and scales. The package provides a friendly, flexible and complete workflow that can be employed in many genomic and epigenomic applications. IWTomics is freely available at the Bioconductor website (http://bioconductor.org/packages/IWTomics) and on the main Galaxy instance (https://usegalaxy.org/). Supplementary data are available at Bioinformatics online.

Force coordination strategies in on-water single sculling: Are asymmetries related to better rowing performance?

Asymmetries of the rowing stroke cycle have been assessed with reference to kinematics and foot-force measures in laboratory testing environments. It remains unclear how asymmetries in propulsive kinetic measures are related to on-water rowing performance. A new approach for the evaluation of both global and local asymmetries across the entire movement was used to assess the continuous role of asymmetries and whether these change according to the level of competitive representation. Twenty-seven highly skilled female rowers (national and international competition level), rowing at 32 strokes per minute in a single scull boat, were evaluated. A symmetry index (SI) and functional data analysis (FDA) techniques were applied to a continuous difference time-series, which described fluctuating asymmetry in propulsive pin forces for each rower. Univariate ANOVAs revealed that differences in asymmetries were present as a factor of competition level for the SI and results of FDA. International athletes were more likely to utilize an asymmetry strategy with increased stroke-side (port-side) force early in the drive phase and increased bow-side (starboard) force through the second half of the drive. This was likely the result of international performers customizing their movement strategies relative to known boat mechanical offsets. The first half of the drive phase was also found to be an adaptive part of the rowing stroke cycle, suggesting asymmetries may have a functional role in successful execution of movements during the rowing stroke.

Nonparametric Forecasting of Multivariate Probability Density Functions

The study of dependence between random variables is the core of theoretical and applied statistics. Static and dynamic copula models are useful for describing the dependence structure, which is fully encrypted in the copula probability density function. However, these models are not always able to describe the temporal change of the dependence patterns, which is a key characteristic of financial data. We propose a novel nonparametric framework for modelling a time series of copula probability density functions, which allows to forecast the entire function without the need of post-processing procedures to grant positiveness and unit integral. We exploit a suitable isometry that allows to transfer the analysis in a subset of the space of square integrable functions, where we build on nonparametric functional data analysis techniques to perform the analysis. The framework does not assume the densities to belong to any parametric family and it can be successfully applied also to general multivariate probability density functions with bounded or unbounded support. Finally, a noteworthy field of application pertains the study of time varying networks represented through vine copula models. We apply the proposed methodology for estimating and forecasting the time varying dependence structure between the S&P500 and NASDAQ indices.

Application and interpretation of functional data analysis techniques to differential scanning calorimetry data from lupus patients.

BackgroundDSC is used to determine thermally-induced conformational changes of biomolecules within a blood plasma sample. Recent research has indicated that DSC curves (or thermograms) may have different characteristics based on disease status and, thus, may be useful as a monitoring and diagnostic tool for some diseases. Since thermograms are curves measured over a range of temperature values, they are considered functional data. In this paper we apply functional data analysis techniques to analyze differential scanning calorimetry (DSC) data from individuals from the Lupus Family Registry and Repository (LFRR). The aim was to assess the effect of lupus disease status as well as additional covariates on the thermogram profiles, and use FD analysis methods to create models for classifying lupus vs. control patients on the basis of the thermogram curves.MethodsThermograms were collected for 300 lupus patients and 300 controls without lupus who were matched with diseased individuals based on sex, race, and age. First, functional regression with a functional response (DSC) and categorical predictor (disease status) was used to determine how thermogram curve structure varied according to disease status and other covariates including sex, race, and year of birth. Next, functional logistic regression with disease status as the response and functional principal component analysis (FPCA) scores as the predictors was used to model the effect of thermogram structure on disease status prediction. The prediction accuracy for patients with Osteoarthritis and Rheumatoid Arthritis but without Lupus was also calculated to determine the ability of the classifier to differentiate between Lupus and other diseases. Data were divided 1000 times into separate 2/3 training and 1/3 test data for evaluation of predictions. Finally, derivatives of thermogram curves were included in the models to determine whether they aided in prediction of disease status.ResultsFunctional regression with thermogram as a functional response and disease status as predictor showed a clear separation in thermogram curve structure between cases and controls. The logistic regression model with FPCA scores as the predictors gave the most accurate results with a mean 79.22% correct classification rate with a mean sensitivity = 79.70%, and specificity = 81.48%. The model correctly classified OA and RA patients without Lupus as controls at a rate of 75.92% on average with a mean sensitivity = 79.70% and specificity = 77.6%. Regression models including FPCA scores for derivative curves did not perform as well, nor did regression models including covariates.ConclusionChanges in thermograms observed in the disease state likely reflect covalent modifications of plasma proteins or changes in large protein-protein interacting networks resulting in the stabilization of plasma proteins towards thermal denaturation. By relating functional principal components from thermograms to disease status, our Functional Principal Component Analysis model provides results that are more easily interpretable compared to prior studies. Further, the model could also potentially be coupled with other biomarkers to improve diagnostic classification for lupus.

Bi-level heterogeneity modeling of functional performance degradation for the aging population

ABSTRACTAs the majority of service recipients of the U.S. long-term care system, the elderly with disabilities experience heterogeneous functional limitation and performance degradation. Successful modeling of heterogeneous degradation becomes of great importance for long-term care decision making. Existing heterogeneity modeling approaches are mainly restricted to single-level modeling of functional performance degradation for older adults. This article proposes a novel bi-level heterogeneity modeling and quantification framework to systematically investigate heterogeneous performance degradation at both sub-population and individual levels. Specifically, a Bayesian non-parametric modeling and inference framework is first proposed to characterize sub-population-level degradation heterogeneity. It relaxes the conventional assumption of pre-specifying a fixed number of sub-populations and allows joint model estimation and selection. Functional data analysis techniques are further considered to quantify individual heterogeneity nested within sub-populations. Based on the extracted temporal variation signatures of degradation profiles, post-analysis is performed to identify relevant risk/protective factors in explaining individual heterogeneity. To illustrate the proposed work and demonstrate its validity, the heterogeneous cognitive degradation quantification of a sample of community-dwelling older adults from the Health and Retirement Study is provided.