Functional Data Analysis Methods Research Articles

A functional index model for dynamically evaluating China's energy security

With the aim of overcoming the shortcomings of the conventional energy security index (ESI) used in dynamic assessment of energy security, the present study introduces the functional data analysis method into ESI construction and establishes a dynamic energy security index (DESI) model. The DESI is a generalization of the conventional ESI with a dynamic weighting mechanism in a continuous time domain. In the DESI model, every normalized indicator within three dimensions — energy supply, energy consumption and environmental — is smoothed into a continuous curve, and the changes in its importance are reflected objectively by functional information entropy weights with dynamic information updating. Whereas the three sub-index curves with different importance are distinguished via assigning different weights according to the special national circumstances. The methods of generating a curve for single indicator, sub-index curves, and dyna mic weighting curves are discussed in detail, and the robustness of weighting and aggregation choice in DESI is tested using sensitive analysis. Results of applying DESI to evaluate China's energy security show that the overall changes of China's energy security have undergone a W-shaped fluctuation. To ensure consistently high levels of energy security, the government should focus on oil supply and renewable energy technologies.

A comparison of seed germination coefficients using functional regression.

PremiseSeed germination over time is characterized by a sigmoid curve, called a germination curve, in which the percentage (or absolute number) of seeds that have completed germination is plotted against time. A number of individual coefficients have been developed to characterize this germination curve. However, as germination is considered to be a qualitative developmental response of an individual seed that occurs at one time point, but individual seeds within a given treatment respond at different time points, it has proven difficult to develop a single index that satisfactorily incorporates both percentage and rate. The aim of this paper is to develop a new coefficient, the continuous germination index (CGI), which quantifies seed germination as a continuous process, and to compare the CGI with other commonly used indexes.MethodsTo create the new index, the germination curves were smoothed using nondecreasing splines and the CGI was derived as the area under the resulting spline. For the comparison of the CGI with other common indexes, a regression model with functional response was developed.ResultsUsing both an experimentally obtained wild pea (Pisum sativum subsp. elatius) seed data set and a hypothetical data set, we showed that the CGI is able to characterize the germination process better than most other indices. The CGI captures the local behavior of the germination curves particularly well.DiscussionThe CGI can be used advantageously for the characterization of the germination process. Moreover, B‐spline coefficients extracted by its construction can be employed for the further statistical processing of germination curves using functional data analysis methods.

Quantifying uncertainty in brain-predicted age using scalar-on-image quantile regression

Prediction of subject age from brain anatomical MRI has the potential to provide a sensitive summary of brain changes, indicative of different neurodegenerative diseases. However, existing studies typically neglect the uncertainty of these predictions. In this work we take into account this uncertainty by applying methods of functional data analysis. We propose a penalised functional quantile regression model of age on brain structure with cognitively normal (CN) subjects in the Alzheimer’s Disease Neuroimaging Initiative (ADNI), and use it to predict brain age in Mild Cognitive Impairment (MCI) and Alzheimer’s Disease (AD) subjects. Unlike the machine learning approaches available in the literature of brain age prediction, which provide only point predictions, the outcome of our model is a prediction interval for each subject.

Air Quality Analysis of Wuhan from the Perspective of Functional Data

In recent years, with the continuous advancement of Wuhan’s industry, manufacturing and engineering construction, the consumption of energy is also increasing, and the air quality problems caused by it have also received attention from all walks of life. In order to explore the characteristics of air pollution in Wuhan, this paper uses functional data analysis methods to study air quality (AQI) from the perspective of space and time. The results show that 13 prefecture-level cities in Wuhan can be divided into 4 levels according to the quality of air quality. In general, Xinzhou District, Huang pi District, Jiangxia District and Hannan District have the best air quality, while Qingshan District and Dongxihu District have the worst air quality; all have significant seasonal characteristics in the time dimension, that is, winter is a high-frequency band with serious air pollution.

Smoothing and Differentiation of Kinematic Data Using Functional Data Analysis Approach: An Application of Automatic and Subjective Methods

Smoothing is one of the fundamental procedures in functional data analysis (FDA). The smoothing parameter λ influences data smoothness and fitting, which is governed by selecting automatic methods, namely, cross-validation (CV) and generalized cross-validation (GCV) or subjective assessment. However, previous biomechanics research has only applied subjective assessment in choosing optimal λ without using any automatic methods beforehand. None of that research demonstrated how the subjective assessment was made. Thus, the goal of this research was to apply the FDA method to smoothing and differentiating kinematic data, specifically right hip flexion/extension (F/E) angle during the American kettlebell swing (AKS) and determine the optimal λ . CV and GCV were applied prior to the subjective assessment with various values of λ together with cubic and quintic spline (B-spline) bases using the FDA approach. The selection of optimal λ was based on smoothed and well-fitted first and second derivatives. The chosen optimal λ was 1 × 10 − 12 with a quintic spline (B-spline) basis and penalized fourth-order derivative. Quintic spline is a better smoothing and differentiation method compared to cubic spline, as it does not produce zero acceleration at endpoints. CV and GCV did not give optimal λ , forcing subjective assessment to be employed instead.

Functional Soft Sensor Based on Spectra Data for Predicting Multiple Quality Variables

In many complex chemical processes, such as ethanol fermentation process, prediction of the multivariate quality variables based on spectra data presents a great challenge because the dimensions of the spectra far exceed their sampling number. To address this problem, the dimension reduction of the predictors is necessary. It can be conducted either by regressing on components or by smoothing methods with basis functions. Based on the functional data analysis methods, this work introduces a novel wavelet functional partial least squares (WFPLS), which combines both of the foregoing dimension-reduction approaches. The high-dimensional spectra can be well fitted by fewer wavelet basis functions in the proposed method. Using the proposed WFPLS method does not require the measured data to be linear and to be sampled on a regular basis. It will be proved that the proposed WFPLS method can be finally transferred into the traditional PLS method in computation. By comparison with the existing prediction methods, the advantages of the proposed method are well demonstrated via a numerical case and an ethanol fermentation experiment.

Randomized estimation of functional covariance operator via subsampling

Covariance operators are fundamental concepts and modelling tools for many functional data analysis methods, such as functional principal component analysis. However, the empirical (or estimated) covariance operator becomes too costly to compute when the functional dataset gets big. This paper studies a randomized algorithm for covariance operator estimation. The algorithm works by sampling and rescaling observations from the large functional data collection to form a sketch of much smaller size and performs computation on the sketch to obtain the subsampled empirical covariance operator. The proposed algorithm is theoretically justified via nonasymptotic bounds between the subsampled and the full‐sample empirical covariance operator in terms of the Hilbert‐Schmidt norm and the operator norm. It is shown that the optimal sampling probability that minimizes the expected squared Hilbert‐Schmidt norm of the subsampling error is determined by the norm of each function. Simulated and real data examples are used to illustrate the effectiveness of the proposed algorithm.

Analysing Maximum Monthly Temperatures in South Africa for 45 years Using Functional Data Analysis

The paper uses Functional Data Analysis (FDA) to explore space and time variation of monthly maximum temperature data of 16 locations in South Africa for the period 1965 - 2010 at intervals of 5 years. We explore monthly maximum temperature variation by first representing data using the B-spline basis functions. Thereafter registration of the smooth temperature curves was performed. This data was then subjected to analysis using phase-plane plots which revealed the constant shifting of energy over the years analysed. We next applied functional Principal Component Analysis (fPCA) to reduce the dimension of maximum temperature curves by identifying the maximum variation without loss of relevant information, which revealed that the first functional PCA explains mostly summer variation while the second functional PCA explains winter variation. We next explored the functional data using functional clustering using K-means to reveal the spatial location of maximum temperature clusters across the country, which revealed that maximum temperature clusters were not consistent over the 45 years of data analysed, and that the cluster points within a cluster were not necessarily always spatially adjacent. The overall analysis has displayed that maximum temperature clusters have not been static across the country over time. To the best of our knowledge, this the first instance of performing in-depth analysis of maximum temperature data for 16 locations in South Africa using various FDA methods.

Forecasting of electricity price through a functional prediction of sale and purchase curves

AbstractThis work proposes a new approach for the prediction of the electricity price based on forecasting aggregated purchase and sale curves. The basic idea is to model the hourly purchase and the sale curves, to predict them and to find the intersection of the predicted curves in order to obtain the predicted equilibrium market price and volume. Modeling and forecasting of purchase and sale curves is performed by means of functional data analysis methods. More specifically, parametric (FAR) and nonparametric (NPFAR) functional autoregressive models are considered and compared to some benchmarks. An appealing feature of the functional approach is that, unlike other methods, it provides insights into the sale and purchase mechanism connected with the price and demand formation process and can therefore be used for the optimization of bidding strategies. An application to the Italian electricity market (IPEX) is also provided, showing that NPFAR models lead to a statistically significant improvement in the forecasting accuracy.

Profile clustering in clinical trials with longitudinal and functional data methods

ABSTRACTRepeated measurements are widely encountered in medical or pharmaceutical studies, which can be analyzed by both longitudinal data and functional data analysis methods, particularly when the underlying process is continuous and the number of measurement points is not too small. Motivated by real problems of clustering patient profiles in clinical trials, this paper gives an overview of the clustering methods for repeated measurement data and compares three longitudinal data methods and two functional data methods with extensive simulation studies. Methods with appropriate properties are applied to the real data to produce interpretable results.

Functional GARCH models: The quasi-likelihood approach and its applications

The increasing availability of high frequency data has initiated many new research areas in statistics. Functional data analysis (FDA) is one such innovative approach towards modelling time series data. In FDA, densely observed data are transformed into curves and then each (random) curve is considered as one data object. A natural, but still relatively unexplored, context for FDA methods is related to financial data, where high-frequency trading currently takes a significant proportion of trading volumes. Recently, articles on functional versions of the famous ARCH and GARCH models have appeared. Due to their technical complexity, existing estimators of the underlying functional parameters are moment based—an approach which is known to be relatively inefficient in this context. In this paper, we promote an alternative quasi-likelihood approach, for which we derive consistency and asymptotic normality results. We support the relevance of our approach by simulations and illustrate its use by forecasting realised volatility of the S&P100 Index.

Estimation of Rainfall Curve by using Functional Data Analysis and Ordinary Kriging Approach

Rainfall is an interesting phenomenon to investigate since it is directly related to all aspects of life on earth. One of the important studies is to investigate and understand the rainfall patterns that occur throughout the year. To identify the pattern, it requires a rainfall curve to represent daily observation of rainfall received during the year. Functional data analysis methods are capable to convert discrete data intoa function that can represent the rainfall curve and as a result, try to describe the hidden patterns of the rainfall. This study focused on the distribution of daily rainfall amount using functional data analysis. Fourier basis functions are used for periodic rainfall data. Generalized cross-validation showed 123 basis functions were sufficient to describe the pattern of daily rainfall amount. North and west areas of the peninsula show a significant bimodal pattern with the curve decline between two peaks at the mid-year. Meanwhile,the east shows uni-modal patterns that reached a peak in the last three months. Southern areas show more uniform trends throughout the year. Finally, the functional spatial method is introduced to overcome the problem of estimating the rainfall curve in the locations with no data recorded. We use a leave one out cross-validation as a verification method to compare between the real curve and the predicted curve. We used coefficient of basis functions to get the predicted curve. It was foundthatthe methods ofspatial prediction can match up with the existing spatial prediction methods in terms of accuracy,but it is better as the new approach provides a simpler calculation.

Functional Data Analysis in Sport Science: Example of Swimmers’ Progression Curves Clustering

Many data collected in sport science come from time dependent phenomenon. This article focuses on Functional Data Analysis (FDA), which study longitudinal data by modelling them as continuous functions. After a brief review of several FDA methods, some useful practical tools such as Functional Principal Component Analysis (FPCA) or functional clustering algorithms are presented and compared on simulated data. Finally, the problem of the detection of promising young swimmers is addressed through a curve clustering procedure on a real data set of performance progression curves. This study reveals that the fastest improvement of young swimmers generally appears before 16 years old. Moreover, several patterns of improvement are identified and the functional clustering procedure provides a useful detection tool.

FdANOVA: an R software package for analysis of variance for univariate and multivariate functional data

Functional data, i.e., observations represented by curves or functions, frequently arise in various fields. The theory and practice of statistical methods for such data is referred to as functional data analysis (FDA) which is one of major research fields in statistics. The practical use of FDA methods is made possible thanks to availability of specialized and usually free software. In particular, a number of R packages is devoted to these methods. In the paper, we introduce a new R package fdANOVA which provides an access to a broad range of global analysis of variance methods for univariate and multivariate functional data. The implemented testing procedures mainly for homoscedastic case are briefly overviewed and illustrated by examples on a well known functional data set. To reduce the computation time, parallel implementation is developed and its efficiency is empirically evaluated. Since some of the implemented tests have not been compared in terms of size control and power yet, appropriate simulations are also conducted. Their results can help in choosing proper testing procedures in practice.

Nonparametric operator-regularized covariance function estimation for functional data

In functional data analysis (FDA), the covariance function is fundamental not only as a critical quantity for understanding elementary aspects of functional data but also as an indispensable ingredient for many advanced FDA methods. A new class of nonparametric covariance function estimators in terms of various spectral regularizations of an operator associated with a reproducing kernel Hilbert space is developed. Despite their nonparametric nature, the covariance estimators are automatically positive semi-definite, which is an essential property of covariance functions, via a one-step procedure. An unconventional representer theorem is established to provide a finite dimensional representation for this class of covariance estimators based on data, although the solutions are searched over infinite dimensional functional spaces. To further achieve a low-rank representation, another desirable property, e.g., for dimension reduction and easy interpretation, the trace-norm regularization is particularly studied, under which an efficient algorithm is developed based on the accelerated proximal gradient method. The outstanding practical performance of the trace-norm-regularized covariance estimator is demonstrated by a simulation study and the analysis of a traffic dataset. Under both fixed and random designs, an excellent rate of convergence is established for a broad class of operator-regularized covariance function estimators, which generalizes both the trace-norm-regularized covariance estimator and other popular alternatives.

Applying functional data analysis to assess tele-interpersonal psychotherapy's efficacy to reduce depression

ABSTRACTThe use of parametric linear mixed models and generalized linear mixed models to analyze longitudinal data collected during randomized control trials (RCT) is conventional. The application of these methods, however, is restricted due to various assumptions required by these models. When the number of observations per subject is sufficiently large, and individual trajectories are noisy, functional data analysis (FDA) methods serve as an alternative to parametric longitudinal data analysis techniques. However, the use of FDA in RCTs is rare. In this paper, the effectiveness of FDA and linear mixed models (LMMs) was compared by analyzing data from rural persons living with HIV and comorbid depression enrolled in a depression treatment randomized clinical trial. Interactive voice response systems were used for weekly administrations of the 10-item Self-Administered Depression Scale (SADS) over 41 weeks. Functional principal component analysis and functional regression analysis methods detected a statistically significant difference in SADS between telphone-administered interpersonal psychotherapy (tele-IPT) and controls but linear mixed effects model results did not. Additional simulation studies were conducted to compare FDA and LMMs under a different nonlinear trajectory assumption. In this clinical trial with sufficient per subject measured outcomes and individual trajectories that are noisy and nonlinear, we found FDA methods to be a better alternative to LMMs.

A functional data analysis of spatiotemporal trends and variation in fine particulate matter

In this paper we illustrate the application of modern functional data analysis methods to study the spatiotemporal variability of particulate matter components across the United States. The approach models the pollutant annual profiles in a way that describes the dynamic behavior over time and space. This new technique allows us to predict yearly profiles for locations and years at which data are not available and also offers dimension reduction for easier visualization of the data. Additionally it allows us to study changes of pollutant levels annually or for a particular season. We apply our method to daily concentrations of two particular components of PM2.5 measured by two networks of monitoring sites across the United States from 2003 to 2015. Our analysis confirms existing findings and additionally reveals new trends in the change of the pollutants across seasons and years that may not be as easily determined from other common approaches such as Kriging.

Advances in functional magnetic resonance imaging data analysis methods using Empirical Mode Decomposition to investigate temporal changes in early Parkinson's disease

IntroductionPrevious neuroimaging studies of Parkinson's disease (PD) patients have shown changes in whole-brain functional connectivity networks. Whether connectivity changes can be detected in the early stages (first 3 years) of PD by resting-state functional magnetic resonance imaging (fMRI) remains elusive. Research infrastructure including MRI and analytic capabilities is required to investigate this issue. The National Institutes of Health/National Institute of General Medical Sciences Center for Biomedical Research Excellence awards support infrastructure to advance research goals. MethodsStatic and dynamic functional connectivity analyses were conducted on early stage never-medicated PD subjects (N = 18) and matched healthy controls (N = 18) from the Parkinson's Progression Markers Initiative. ResultsAltered static and altered dynamic functional connectivity patterns were found in early PD resting-state fMRI data. Most static networks (with the exception of the default mode network) had a reduction in frequency and energy in specific low-frequency bands. Changes in dynamic networks in PD were associated with a decreased switching rate of brain states. DiscussionThis study demonstrates that in early PD, resting-state fMRI networks show spatial and temporal differences of fMRI signal characteristics. However, the default mode network was not associated with any measurable changes. Furthermore, by incorporating an optimum window size in a dynamic functional connectivity analysis, we found altered whole-brain temporal features in early PD, showing that PD subjects spend significantly more time than healthy controls in a specific brain state. These findings may help in improving diagnosis of early never-medicated PD patients. These key observations emerged in a Center for Biomedical Research Excellence–supported research environment.

Functional data analysis of single-trial auditory evoked potentials recorded in the awake rat

Evoked potentials (EPs) reflect neural processing and are widely used to study sensory perception. However, methods of analyzing EP have been limited mostly to the conventional ensemble averaging of EP response trials to a repeated stimulus, and less so to single-trials analysis. Here we applied a new approach – functional data analysis (FDA) – to study auditory EP in the rat model of tinnitus, in which overdoses of salicylate (SS) are known to alter sound perception characteristically, as the same way as in humans. Single-trial auditory EPs were analyzed, after being collected on a daily basis from an awake rat, which had been surgically implanted with intracranial electrodes over its auditory cortex. Single-trial EP integrals were generated with sound stimuli (tones and clicks) presented systematically over an intensity range. The results were approximated using the cubic spline to give sets of smoothed response-level functions in dependence on the sound intensity. These functional data were analyzed using the methods of FDA. Comparisons between daily intensity series for each sound type were done using cross-distance measures based on the response-level functions in both the original form and the first-derivative form. From the results of FDA, the first-derivative form was found to provide a clearer separation when EP data from control groups were compared to the data from SS groups. This is also true when the daily data were compared within the more variable SS-group itself. In addition, at the high-intensity region where SS-action is presumably strong, we also observed characteristic changes in two statistical parameters, mean and skewness, of the cross-distance representations. Results suggested that FDA is a sensitive approach for EP studies, and it can become a powerful tool for the research in neural science, particularly neuropharmacology.

Search for evergreens in science: A functional data analysis

Evergreens in science are papers that display a continual rise in annual citations without decline, at least within a sufficiently long time period. Aiming to better understand evergreens in particular and patterns of citation trajectory in general, this paper develops a functional data analysis method to cluster citation trajectories of a sample of 1699 research papers published in 1980 in the American Physical Society (APS) journals. We propose a functional Poisson regression model for individual papers' citation trajectories, and fit the model to the observed 30-year citations of individual papers by functional principal component analysis and maximum likelihood estimation. Based on the estimated paper-specific coefficients, we apply the K-means clustering algorithm to cluster papers into different groups, for uncovering general types of citation trajectories. The result demonstrates the existence of an evergreen cluster of papers that do not exhibit any decline in annual citations over 30 years.