Functional Data Analysis Methods Research Articles

Statistics in Phonetics

Phonetics is the scientific field concerned with the study of how speech is produced, heard, and perceived. It abounds with data, such as acoustic speech recordings, neuroimaging data, or articulatory data. In this article, we provide an introduction to different areas of phonetics (acoustic phonetics, sociophonetics, speech perception, articulatory phonetics, speech inversion, sound change, and speech technology), an overview of the statistical methods for analyzing their data, and an introduction to the signal processing methods commonly applied to speech recordings. A major transition in the statistical modeling of phonetic data has been the shift from fixed effects to random effects regression models, the modeling of curve data (for instance, via generalized additive mixed models or functional data analysis methods), and the use of Bayesian methods. This shift has been driven in part by the increased focus on large speech corpora in phonetics, which has arisen from machine learning methods such as forced alignment. We conclude by identifying opportunities for future research.

Roughness regularization for functional data analysis with free knots spline estimation

In the era of big data, an ever-growing volume of information is recorded, either continuously over time or sporadically, at distinct time intervals. Functional Data Analysis (FDA) stands at the cutting edge of this data revolution, offering a powerful framework for handling and extracting meaningful insights from such complex datasets. The currently proposed FDA methods can often encounter challenges, especially when dealing with curves of varying shapes. This can largely be attributed to the method’s strong dependence on data approximation as a key aspect of the analysis process. In this work, we propose a free knots spline estimation method for functional data with two penalty terms and demonstrate its performance by comparing the results of several clustering methods on simulated and real data.

A Study of Pupil Response to Light as a Digital Biomarker of Recent Cannabis Use

Introduction: Given the traffic safety and occupational injury prevention implications associated with cannabis impairment, there is a need for objective and validated measures of recent cannabis use. Pupillary light response may offer an approach for detection. Method: Eighty-four participants (mean age: 32, 42% female) with daily, occasional, and no-use cannabis use histories participated in pupillary light response tests before and after smoking cannabis ad libitum or relaxing for 15 min (no use). The impact of recent cannabis consumption on trajectories of the pupillary light response was modeled using functional data analysis tools. Logistic regression models for detecting recent cannabis use were compared, and average pupil trajectories across cannabis use groups and times since light test administration were estimated. Results: Models revealed small, significant differences in pupil response to light after cannabis use comparing the occasional use group to the no-use control group, and similar statistically significant differences in pupil response patterns comparing the daily use group to the no-use comparison group. Trajectories of pupillary light response estimated using functional data analysis found that acute cannabis smoking was associated with less initial and sustained pupil constriction compared to no cannabis smoking. Conclusion: These analyses show the promise of pairing pupillary light response and functional data analysis methods to assess recent cannabis use.

Predicting Driver's mental workload using physiological signals: A functional data analysis approach

This study investigates the impact of advanced driver-assistance systems on drivers' mental workload. Using a combination of physiological signals including ECG, EMG, EDA, EEG (af4 and fc6 channels from the theta band), and eye diameter data, this study aims to predict and categorize drivers’ mental workload into low, adequate, and high levels. Data were collected from five different driving situations with varying cognitive demands. A functional linear regression model was employed for prediction, and the accuracy rate was calculated. Among the 31 tested combinations of physiological variables, 9 combinations achieved the highest accuracy result of 90%. These results highlight the potential benefits of utilizing raw physiological signal data and employing functional data analysis methods to understand and assess driver mental workload. The findings of this study have implications for the design and improvement of driver-assistance systems to optimize safety and performance.

Functional Data Clustering Via Functional Mahalanobis Distance

As an exploratory data analysis method, functional data clustering aims to identify the underlying features of the observed data. In this context, this paper proposes a functional data clustering method based on functional Mahalanobis distance. As a distance-based non-parametric clustering model, the proposed method can effectively avoid the disadvantages of generative models and has excellent properties of decoupling and dimension standardization. Compared with other functional data clustering models, this method has lower computational complexity. In addition, the proposed method can be applied to any distance-based multivariate clustering method, thus generalizing it to the case of functional data. In practical data analysis, this paper compares the performance of this method with some other functional clustering methods, using k-means clustering as an example, and finds that it has better performance in terms of purity, adjusted Rand index, and computational speed. Finally, the idea of using Mahalanobis distance for functional data distance measurement can also be extended to construct kernel functions for measuring similarity between functional data samples, thus developing non-linear functional data analysis methods based on reproducing kernel theory.

Asymmetric loading strategies during squats following anterior cruciate ligament reconstruction: A longitudinal investigation with curve analyses throughout and after rehabilitation.

Investigations of kinetic asymmetries during bilateral squats following anterior cruciate ligament reconstruction (ACLR) are limited to mainly cross-sectional studies and discrete value data extracted at specific knee angles. We assessed loading asymmetries during squats longitudinally throughout rehabilitation using curve analysis and compared patient-reported outcome measures (PROMs) between those with and without asymmetry. Bodyweight squats were performed by 24 individuals (13 females) post-ACLR on three occasions: (1) Early 2.9 (1.1) months; (2) Mid 8.8 (3.1) months; (3) at Return to Sport (RTS) 13.1 (3.6) months; and 29 asymptomatic controls (22 females) once. Time-normalized between-leg asymmetry curves of sagittal plane hip, knee, and ankle moments and vertical ground reaction forces were compared using functional data analysis methods. Individual asymmetrical loading for ACLR was classified when exceeding the 95% confidence interval of controls during ≥50% of the squat. At Early, ACLR had greater asymmetry than controls for knee (15%-100% eccentric phase; 0%-100% concentric) and ankle flexion moments (56%-65% concentric). At Mid, ACLR had greater asymmetry for knee (41%-72% eccentric) and ankle flexion moments (56%-69% concentric). No between-group differences were found at RTS. From Early to RTS, ACLR reduced asymmetry for hip (21%-46% eccentric), knee (27%-58% concentric), and ankle flexion moments (21%-57% eccentric). At Early, 11/24 underloaded their ACLR knee and 1 overloaded compared with controls. At RTS, 4 underloaded and 6 overloaded. No differences in PROMs were found based on loading asymmetry. Beyond the early phase of rehabilitation from ACLR, individual-level analyses are required to reveal differing loading strategies during bilateral squats.

Cannulation Skill Assessment Using Functional Data Analysis.

A clinician's operative skill-the ability to safely and effectively perform a procedure-directly impacts patient outcomes and well-being. Therefore, it is necessary to accurately assess skill progression during medical training as well as develop methods to most efficiently train healthcare professionals. In this study, we explore whether time-series needle angle data recorded during cannulation on a simulator can be analyzed using functional data analysis methods to (1) identify skilled versus unskilled performance and (2) relate angle profiles to degree of success of the procedure. Our methods successfully differentiated between types of needle angle profiles. In addition, the identified profile types were associated with degrees of skilled and unskilled behavior of subjects. Furthermore, the types of variability in the dataset were analyzed, providing particular insight into the overall range of needle angles used as well as the rate of change of angle as cannulation progressed in time. Finally, cannulation angle profiles also demonstrated an observable correlation with degree of cannulation success, a metric that is closely related to clinical outcome. In summary, the methods presented here enable rich assessment of clinical skill since the functional (i.e., dynamic) nature of the data is duly considered.

Compositional functional regression andisotemporal substitution analysis: Methods and application in time-use epidemiology.

The distribution of time that people spend in physical activity of various intensities has important health implications. Physical activity (commonly categorised by the intensity into light, moderate and vigorous physical activity), sedentary behaviour and sleep, should not be analysed separately, because they are parts of a time-use composition with a natural constraint of h/day. To find out how are relative reallocations of time between physical activity of various intensities associated with health, herewith we describe compositional scalar-on-function regression and a newly developed compositional functional isotemporal substitution analysis. Physical activity intensity data can be considered as probability density functions, which better reflects the continuous character of their measurement using accelerometers. These probability density functions are characterised by specific properties, such as scale invariance and relative scale, and they are geometrically represented using Bayes spaces with the Hilbert space structure. This makes possible to process them using standard methods of functional data analysis in the space, via centred logratio (clr) transformation. The scalar-on-function regression with clr transformation of the explanatory probability density functions and compositional functional isotemporal substitution analysis were applied to a dataset from a cross-sectional study on adiposity conducted among school-aged children in the Czech Republic. Theoretical reallocations of time to physical activity of higher intensities were found to be associated with larger and more progressive expected decreases in adiposity. We obtained a detailed insight into the dose-response relationship between physical activity intensity and adiposity, which was enabled by using the compositional functional approach.

A data-driven soft-sensing approach using probabilistic latent variable model with functional data framework

Functional principal component analysis (FPCA) and functional partial least squares (FPLS) are two mainstream functional data analysis (FDA) methods, which have been commonly used to extract deep information hidden in the original data space. However, the process data always contain random noise, which affects the performance of FDA models. To overcome this issue, two functional probabilistic latent variable models (FPLVMs), including functional probabilistic principal component analysis (FPPCA) and functional probabilistic partial least squares (FPPLS) are proposed in this work. First, the process data are converted into functional data using the FDA. Subsequently, a log-likelihood function considering the noise factor and functional latent variables is designed. Finally, the regression model parameters are estimated using an expectation–maximisation algorithm. In contrast to FPPCA, FPPLS decomposes the process data and the key variable with constrained latent variables, which is similar to the partial least squares (PLS) and the principal component analysis (PCA). Moreover, the degeneration mechanism from FPLVMs into probabilistic latent variable models and latent variable models is discussed. An adaptive strategy with functional covariance is used to satisfy the online predictive capabilities of the model. Finally, the proposed approach is validated using a numerical case, the Tennessee Eastman process and an industrial o-xylene distillation column for evaluation.

THE APPLIED OF FUNCTIONAL DATA ANALYSIS TO COMPARISON 100 TIMES SIMULATED MONTHLY RAINFALL USING SOME TWO PARAMETERS DISTRIBUTION

Three probability models of mothly rainfall such as Gamma, Weibull and Log Normal distribution are evaluated in terms of their ability to reproduce the mean statistics derived from 100 times simulation of monthly rainfall in the Pekanbaru City, Indonesia. One of the important studies is to investigate and understand the simulate mean monthly rainfall patterns that occur throughout the year. To identify the pattern, it requires a rainfall curve to represent monthly observation of rainfall received during the year. Functional data analysis (FDA) methods are capable to convert discrete data into a function that can represent the rainfall curve and as a result, try to describe the hidden patterns of the rainfall. This study is focused on investigating 100 curve average monthly rainfall simulatated by three different quantile functions using the FDA. The mean and standard deviation of FDA for average monthly precipitation are obtained. Through these two statistics a the confidence interval curves of the mean curve are presented represent 95% pointwise confidence intervals. In this study, most of the monthly average rainfall from the actual data were around the FDA mean and the monthly average rainfall was within the FDA confidence interval. In this study, 100 times monthly rainfall simulations using the quantile function of the gamma and log normal distributions found that the mean FDA can capture most of the mean monthly rainfall from historical data, and within the FDA interval. The contradictory results shown by the monthly rainfall simulations using the Weibull distribution, most of the monthly average rainfall historical data cannot be captured by the FDA mean and are outside the FDA confidence interval. Based on the Mean Absolute Error (MAE) value of the average monthly rainfall of historical data and the monthly average of the FDA, it can be concluded that the gamma distribution can produce simulated rain better than the log normal distribution.

Has China's carbon market stress released? Measurement and comparison of national and pilot carbon markets' stress.

This paper constructs a novel stress measurement system of carbon market from the perspective of trading, emission reduction, and external shocks and simulates the stress indices of national and pilot carbon markets of China with the methods of functional data analysis and criteria importance through intercriteria correlation. It concludes that the overall carbon market stress is in the shape of "W" and still at a high level, with frequent fluctuations and an upward trend. In addition, the stress of Hubei, Beijing, and Shanghai carbon market fluctuates and rises, while the stress of Guangdong carbon market decreases. Moreover, carbon market stress mainly comes from trading and emission reduction. Furthermore, stress fluctuation of Guangdong and Beijing carbon market is more prone to "big waves," indicating that the two markets are sensitive to big events. Finally, the pilot carbon markets are divided into stress-driven and stress-release market and the type of which keeps change in different period.

Analysis of air pollution in Fenwei Plain in China based on functional spatial autoregressive combined model.

The Fenwei Plain is listed as one of the most serious air pollution regions in China, along with Beijing-Tianjin-Hebei and Yangtze River Delta regions. This paper proposed a functional data analysis method to study the environmental pollution problem in the Fenwei Plain of China. Functional spatial autoregressive combined (FSAC) model with spatial autocorrelation of both the response variable and error term is developed. The model takes the SO2 concentration of Fenwei Plain as the dependent variable and the dew point temperature as the independent variable and realizes the maximum likelihood estimation using functional principal component analysis to obtain the asymptotic properties of parameter estimation and the confidence interval of the slope function. According to the findings of the empirical analysis of the Fenwei Plain, the SO2 concentration has significant seasonal characteristics and has decreased year over year for three years in a row. Winter is the season with the highest concentration on the Fenwei Plain, followed by spring and autumn, while summer is the season with the lowest concentration. Winter also has a high spatial autocorrelation. The FSAC model is more effective at fitting the concentration and dew point temperature of the Fenwei Plain in China because its mean square error (MSE) is significantly lower than that of the other models. As a result, this paper can more thoroughly study the pollution problem on the Fenwei Plain and offer guidance for prevention and control.

Exploring rainfall variabilities using statistical functional data analysis

Functional data analysis (FDA) has been widely applied in various scientific fields, including climatological, hydrological, environmental, and biomedical. The flexibility of the FDA in incorporating temporal elements into the statistical analysis makes the method highly demanded compared to the conventional statistical approach. This study introduces FDA methods to investigate the variations and patterns of rainfall throughout Peninsular Malaysia, which includes 16 rain gauge stations in Peninsular Malaysia from 1999 to 2019. A descriptive statistic of the functional data depicted the mean and variation of the rainfall curve over time, while the functional principal component analysis measured the temporal variability of the rainfall curve. According to the findings, the first and second principal components accounted for 87.4% of all variations. The first principal component was highly characterised by the stations over the eastern region during the northeast monsoon since the highest variability was observed from November to January. On the other hand, the stations impacted by the inter-monsoon season were best described by the second principal component. Based on the factor scores derived from the functional principal component, those rain gauge stations with comparable features were then clustered. Overall, the results showed that the rainfall pattern is strongly influenced by their geographical and topographical features and the seasonal monsoon effect.

Functional Data Analysis on Global COVID-19 Data

No one can deny that COVID-19 has spread around the world and is still emerging as a new variation in some areas. Since the time of its breakout, three coronavirus waves have emerged. Daily cases of COVID-19 from 79 countries around the world were selected for the study. The main objective of this research was to model and analyze the behavior of the disease's first wave in the world and on the Asian continent using functional data analysis methods. Functional models were fitted to the data using B-spline basis functions at different orders, and the best-fit curves were further analyzed with respect to their functional behavior and the rate of change. These curves were visualized during the preliminary analysis and later clustered within each continent using functional cluster analysis. The results indicated that all continents, apart from Asia, had two clusters based on their functional behavior, whereas the world data had three clusters. All the continents except the Asian continent had different functional forms and numbers of peaks, but they all had the same number of clusters. The world's 18 countries were divided into two categories, with the remaining 61 countries clustered into a single group. The identified cluster indices were further modeled using multinomial logistic regression models with six popular health index variables. In the world, people over the age of 70 made a significant contribution to selecting cluster 2 over cluster 1 and cluster 3 over cluster 2. On the Asian continent, the female smoker variable preferred cluster 2 over cluster 1, while cluster 3 over cluster 1 could be determined by the median age variable. The findings of the overall study would be helpful for the researchers to understand the spread of the disease and the impact of the health indices on its functional behavior.

CellDrift: inferring perturbation responses in temporally sampled single-cell data.

Cells and tissues respond to perturbations in multiple ways that can be sensitively reflected in the alterations of gene expression. Current approaches to finding and quantifying the effects of perturbations on cell-level responses over time disregard the temporal consistency of identifiable gene programs. To leverage the occurrence of these patterns for perturbation analyses, we developed CellDrift (https://github.com/KANG-BIOINFO/CellDrift), a generalized linear model-based functional data analysis method that is capable of identifying covarying temporal patterns of various cell types in response to perturbations. As compared to several other approaches, CellDrift demonstrated superior performance in the identification of temporally varied perturbation patterns and the ability to impute missing time points. We applied CellDrift to multiple longitudinal datasets, including COVID-19 disease progression and gastrointestinal tract development, and demonstrated its ability to identify specific gene programs associated with sequential biological processes, trajectories and outcomes.

Functional data analysis of the relationship between electricity consumption and climate change drivers

Climate change has become increasingly important in recent years. It is the outcome of the burning of fossil fuels that increased the concentration of atmospheric carbon dioxide (CO), over the last century. Mitigating the impacts of climate change requires a better understanding and assessment of the countries' economic decisions on the amount of CO emissions. This paper assesses the variability between the different countries in the trends of CO emissions and electricity consumption from 1975 to 2014, while identifying clusters of countries of similar trends over time. The novel methodology applied in this paper enables us to assess long-debated issues in climate literature. The temporal dynamic effects of electricity consumption and economic growth on CO emissions across countries are studied using functional data analysis (FDA) methods. The latter have proven to be useful tools for visualising similarities and differences in the non-linear trends of CO emissions without forcing linear trends and stationary relationships which can be unrealistic and misleading. The results indicate the possibility of identifying changes in the trends of CO emissions and electricity consumption for a wide range of heterogeneous countries over the study period. The findings also reveal that economic growth puts a strain on the environment, where many high-income countries are still away from attaining economic-energy sustainability.

Dynamic assessment of China’s re-electrification level considering carbon peaking and carbon neutrality goals: A study based on functional data analysis

Electrification is currently the most mature technological path to carbon neutrality. However, the traditional measurement methods of the electrification levels cannot meet the development requirements of China’s electrification under the dual-carbon strategy. Thus, this study incorporated carbon constraints into the re-electrification-level evaluation system and evaluated the re-electrification level of 30 provinces in China from 2007 to 2019 using the entropy weight method. Then, the functional data analysis method was employed to further explore the dynamic variation rule of the regional re-electrification level. The evaluation results show that the top three provinces in the mean value of the re-electrification index are Guangdong (0.628), Jiangsu (0.617), and Zhejiang (0.573). The clustering results show that there are great spatial differences in the development of electrification in China. According to the fitted function curve, China’s electrification development can be roughly divided into the following four stages: uptrend, brief decline, sharp uptrend, and slight and steady decline. At present, China’s electrification is entering a new stage. To implement re-electrification further, the government and the power grid enterprises need to find the priority and breakthroughs under carbon emissions and environmental constraints. Our research results provide reference for the improvement of regional re-electrification levels in China.

The dynamics of ideology drift among U.S. Supreme Court justices: A functional data analysis.

We study the U.S. Supreme Court dynamics by analyzing the temporal evolution of the underlying policy positions of the Supreme Court Justices as reflected by their actual voting data, using functional data analysis methods. The proposed fully flexible nonparametric method makes it possible to dissect the time-dynamics of policy positions at the level of individual Justices, as well as providing a comprehensive view of the ideology evolution over the history of Supreme Court since its establishment. In addition to quantifying individual Justice’s policy positions, we uncover average changes over time and also the major patterns of change over time. Additionally, our approach allows for representing highly complex dynamic trajectories by a few principal components which complements other models of analyzing and predicting court behavior.

Ensemble calibration model of near-infrared spectroscopy based on functional data analysis

As a nondestructive detection technology, near-infrared spectroscopy has been widely applied in various fields. With the wide application of near-infrared spectroscopy, the research on data processing has attracted more attention. Different from the existing discrete data model and based on the functional data analysis method, an ensemble calibration model FDA-EM-PLS (functional data analysis-ensemble learning-partial least squares) of near-infrared spectroscopy is proposed in this paper. Firstly, the near-infrared spectroscopy of each sample is divided into several intervals, and the functional data analysis is carried out on each interval. Then, the samples are clustered according to the generated functions, which can not only reduce the influence of noise, but also provide a theoretical basis for selecting variables. Further, Monte Carlo sampling is used to generate training subsets from clustering samples for ensemble learning, which not only solves the problem of small samples, but also improves the robustness of the model. The relevant experimental results show that the absolute relative error of FDA-EM-PLS for the corn and soil data are both less than 10%.

Functional data analysis: An application to COVID‐19 data in the United States in 2020

BackgroundIn this paper, we conduct an analysis of the COVID‐19 data in the United States in 2020 via functional data analysis methods. Through this research, we investigate the effectiveness of the practice of public health measures, and assess the correlation between infections and deaths caused by the COVID‐19. Additionally, we look into the relationship between COVID‐19 spread and geographical locations, and propose a forecasting method to predict the total number of confirmed cases nationwide.MethodsThe functional data analysis methods include functional principal analysis methods, functional canonical correlation analysis methods, an expectation‐maximization (EM) based clustering algorithm and a functional time series model used for forecasting.ResultsIt is evident that the practice of public health measures helps to reduce the growth rate of the epidemic outbreak over the nation. We have observed a high canonical correlation between confirmed and death cases. States that are geographically close to the hot spots are likely to be clustered together, and population density appears to be a critical factor affecting the cluster structure. The proposed functional time series model gives more reliable and accurate predictions of the total number of confirmed cases than standard time series methods.ConclusionsThe results obtained by applying the functional data analysis methods provide new insights into the COVID‐19 data in the United States. With our results and recommendations, the health professionals can make better decisions to reduce the spread of the epidemic, and mitigate its negative effects to the national public health.