Functional Principal Component Research Articles

High dimensional variable selection through group Lasso for multiple function‐on‐function linear regression: A case study in PM10 monitoring

SummaryAnalyzing the effect of chemical and local meteorological variables over the behaviour in concentrations in the Abruzzo region (Italy), with the objective of forecasting and controlling air quality, motivates the current work. Given that the available data are curves that represent the day‐to‐day variations, a multiple function‐on‐function linear regression (MFFLR) model is considered. By assuming the Karhunen‐Loève expansion, MFFLR model can be reduced to a classical linear regression model for each principal component of the functional response in terms of all principal components (PCs) of the functional predictors. In this sense, a regularization approach for functional principal component regression based on the merge of functional data analysis with group Lasso is proposed. This novel methodology allows to estimate the model and, simultaneously, select those relevant functional predictors with the functional response, where each functional independent variable is represented by a group of input variables derived by the PCs.

Degradation modeling and remaining useful lifetime prediction based on functional variance process

AbstractDynamic fluctuation is a common phenomenon in degradation processes. Hence, how to model it properly has a great impact on the degradation modeling as well as the remaining useful lifetime prediction. To capture the dynamic features and to avoid the risk of the model mis‐specification, a nonparametric degradation model based on functional variance process is proposed in this article. The model is composed of a unit‐specific mean trend and a degradation fluctuation which follows a stochastic process. The mean trend is estimated by the local smoother method, while the stochastic fluctuation is estimated by the functional principal component analysis method. The asymptotic properties of the estimators are proved. Also, the prediction for the remaining useful lifetime is discussed and the estimator is proved to converge in distribution. Moreover, a Bayesian scheme is developed to forecast the remaining useful lifetime for units with incomplete degradation observations. Simulation results show the superiority of the proposed method by comparing it with some existing methods. Finally, two real data sets are analyzed and used to illustrate the application of the method.

Using functional principal component analysis (FPCA) to quantify sitting patterns derived from wearable sensors

BackgroundSedentary behavior (SB) is a recognized risk factor for many chronic diseases. ActiGraph and activPAL are two commonly used wearable accelerometers in SB research. The former measures body movement and the latter measures body posture. The goal of the current study is to quantify the pattern and variation of movement (by ActiGraph activity counts) during activPAL-identified sitting events, and examine associations between patterns and health-related outcomes, such as systolic and diastolic blood pressure (SBP and DBP).MethodsThe current study included 314 overweight postmenopausal women, who were instructed to wear an activPAL (at thigh) and ActiGraph (at waist) simultaneously for 24 hours a day for a week under free-living conditions. ActiGraph and activPAL data were processed to obtain minute-level time-series outputs. Multilevel functional principal component analysis (MFPCA) was applied to minute-level ActiGraph activity counts within activPAL-identified sitting bouts to investigate variation in movement while sitting across subjects and days. The multilevel approach accounted for the nesting of days within subjects.ResultsAt least 90% of the overall variation of activity counts was explained by two subject-level principal components (PC) and six day-level PCs, hence dramatically reducing the dimensions from the original minute-level scale. The first subject-level PC captured patterns of fluctuation in movement during sitting, whereas the second subject-level PC delineated variation in movement during different lengths of sitting bouts: shorter (< 30 minutes), medium (30 -39 minutes) or longer (> 39 minute). The first subject-level PC scores showed positive association with DBP (standardized β^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hat{\\beta }$$\\end{document}: 2.041, standard error: 0.607, adjusted p = 0.007), which implied that lower activity counts (during sitting) were associated with higher DBP.ConclusionIn this work we implemented MFPCA to identify variation in movement patterns during sitting bouts, and showed that these patterns were associated with cardiovascular health. Unlike existing methods, MFPCA does not require pre-specified cut-points to define activity intensity, and thus offers a novel powerful statistical tool to elucidate variation in SB patterns and health.Trial registrationClinicalTrials.gov NCT03473145; Registered 22 March 2018; https://clinicaltrials.gov/ct2/show/NCT03473145; International Registered Report Identifier (IRRID): DERR1-10.2196/28684

Minimax estimation of functional principal components from noisy discretized functional data

AbstractFunctional Principal Component Analysis is a reference method for dimension reduction of curve data. Its theoretical properties are now well understood in the simplified case where the sample curves are fully observed without noise. However, functional data are noisy and necessarily observed on a finite discretization grid. Common practice consists in smoothing the data and then to compute the functional estimates, but the impact of this denoising step on the procedure's statistical performance are rarely considered. Here we prove new convergence rates for functional principal component estimators. We introduce a double asymptotic framework: one corresponding to the sampling size and a second to the size of the grid. We prove that estimates based on projection onto histograms show optimal rates in a minimax sense. Theoretical results are illustrated on simulated data and the method is applied to the visualization of genomic data.

Different PCA approaches for vector functional time series with applications to resistive switching processes

This paper is motivated by modeling the cycle-to-cycle variability associated with the resistive switching operation behind memristors. Although the data generated by this stochastic process are by nature current–voltage curves associated with the creation (set process) and destruction (reset process) of a conductive filament, the statistical analysis is usually based on analyzing only the scalar time series related to the reset and set voltages/currents in consecutive cycles. As the data are by nature curves, functional principal component analysis is a suitable candidate to explain the main modes of variability associated with these processes. Taking into account this data-driven motivation, in this paper we propose two new forecasting approaches based on studying the sequential cross-dependence between and within a multivariate functional time series in terms of vector autoregressive modeling of the most explicative functional principal component scores. The main difference between the two methods lies in whether a univariate or multivariate PCA is performed so that we have a different set of principal component scores for each functional time series or the same one for all of them. Finally, the sample performance of the proposed methodologies is illustrated by an application on a bivariate functional time series of reset/set curves.

Longitudinal study of paralytic shellfish toxins along Canada's coast

Paralytic shellfish toxins (PST) in shellfish products have led to severe risks to human health. To monitor the risk, the Canadian Shellfish Sanitation Program has been collecting longitudinal PST measurements in blue mussel (Mytilus edulis) and soft-shell clam (Mya arenaria) samples in six coastal provinces of Canada. The spatial distributions of major temporal variation patterns were studied via Functional Principal Component Analysis. Seasonal increases in PST contamination were found to vary the most in terms of magnitude along the coastlines, which provides support for location-specific management of the time-sensitive PST contamination. In British Columbia, the first functional principal component (FPC1) indicated the variance among the magnitudes, while FPC2 indicated the seasonality of the PST levels. The temporal variations tended to be positively correlated with the abundance of dianoflagellates Alexandrium spp., and negatively with precipitation and inorganic nutrients. These findings indicate the underlying mechanism of PST variation in various geographical settings. In New Brunswick, Prince Edward, and Nova Scotia, the top FPCs indicated that the PST contamination differed mostly in the seasonal increase of the PST level during summer.

Language-Specific Prosody in Statements of Palenquero/Spanish Bilinguals

This study explores the extent to which Palenquero/Spanish bilinguals, a population that is said to have a residual high tone of African origin, keep their two languages temporally and intonationally distinct across statements. While creole languages that emerged from the contact of African and European languages, such as Palenquero, may develop hybrid prosodic systems with tones from substrate languages, and stress from the majority language, language-specific prosody might be expected to converge or simplify over the course of time. As prosodic convergence seems to be inescapable under Palenquero’s circumstances, which factors could support language-specific prosody in this population, if there are any? Two-hundred and thirty-four five-syllable statements were elicited through a discourse completion task, with the participation of ten Palenquero/Spanish bilinguals, in two unilingual sessions. Both phrase-final lengthening and F0 contours were assessed using linear mixed-effects models testing their association with final stress, language, and generation. F0 contours were dimensionally reduced using functional principal component analysis. Despite the strong similarities between the two languages, results indicate that both groups keep their two languages intonationally distinct using plateau-shaped contours in Palenquero initial rises followed by steeper declinations in Spanish. However, elderly bilinguals implement penultimate lengthening language-specifically, being more pronounced in Palenquero. Adults, in contrast, do not show this distinction. In addition to this, elderly speakers show hyperarticulation in Spanish intonation, increasing the difference between their languages. This leads us to believe that adults exhibit a more simplified prosodic system between their languages, relative to elderly bilinguals. In spite of such differences, both generations seem to have the same underlying process (perhaps a substrate effect) driving plateau-shaped intonation in Palenquero, which enhances language differentiation.

Functional sufficient dimension reduction through information maximization with application to classification

Considering the case where the response variable is a categorical variable and the predictor is a random function, two novel functional sufficient dimensional reduction (FSDR) methods are proposed based on mutual information and square loss mutual information. Compared to the classical FSDR methods, such as functional sliced inverse regression and functional sliced average variance estimation, the proposed methods are appealing because they are capable of estimating multiple effective dimension reduction directions in the case of a relatively small number of categories, especially for the binary response. Moreover, the proposed methods do not require the restrictive linear conditional mean assumption and the constant covariance assumption. They avoid the inverse problem of the covariance operator which is often encountered in the functional sufficient dimension reduction. The functional principal component analysis with truncation be used as a regularization mechanism. Under some mild conditions, the statistical consistency of the proposed methods is established. Simulation studies and real data analyzes are used to evaluate the finite sample properties of our methods.

Impaired Neuromotor Control During Gait in Concussed Adolescents-A Frequency Analysis.

Disruptions in gait function are common after concussion in adolescents; however, the neuromotor control deficits driving these gait disruptions are not well known. Fifteen concussed (age mean [SD]): 17.4 [0.6], 13 females, days since injury: 26.3 [9.9]) and 17 uninjured (age: 18.0 [0.7], 10 females) adolescents completed 3 trials each of single-task gait and dual-task gait (DT). During DT, participants simultaneously walked while completing a serial subtraction task. Gait metrics and variability in instantaneous mean frequency in lower extremity muscles were captured by inertial sensors and surface electromyography, respectively. A 2-way analysis of covariance was used to compare gait metrics across groups and conditions. Functional principal components analysis was used to identify regions of variability in instantaneous mean frequency curves. Functional principal component scores were compared across groups using a Welch statistic. Both groups displayed worse performance on gait metrics during DT condition compared to single-task, with no differences between groups (P < .001). Concussed adolescents displayed significantly greater instantaneous mean frequency, indicated by functional principal component 1, in the tibialis anterior, biceps femoris, and semitendinosus (P < .05) during single-task and DT compared with uninjured adolescents. Our observations suggest that concussed adolescents display inefficient motor unit recruitment lasting longer than 2weeks following injury, regardless of the addition of a secondary task.

Quantifying Apathy in Late-Life Depression: Unraveling Neurobehavioral Links through Daily Activity Patterns and Brain Connectivity Analysis

Better understanding apathy in late-life depression (LLD) would help predicting poor prognosis of the disease such as dementia. Actimetry provides an objective and ecological measure of apathy from patients' daily motor activity. We aimed to determine if patterns of motor activity were associated with apathy and brain connectivity in networks underlying goal-directed behaviors. Resting-state functional MRI and diffusion MRI were collected from 38 non-demented LLD subjects. Apathy was evaluated using the diagnostic criteria for apathy, the apathy evaluation scale (AES) and the apathy motivation index (AMI). Functional principal components (fPC) of motor activity were derived from actimetry recordings of 72 hours. Associations between fPC and apathy were estimated by linear regression. Subnetworks whose connectivity was significantly associated with fPC were identified via the threshold-free network-based statistics. The relationship between apathy and microstructure metrics was estimated along fibers by diffusion tensor imaging and a multicompartment model called neurite orientation dispersion and density imaging via tractometry. We found two fPC associated with apathy: mean diurnal activity, negatively associated with AES, and an early chronotype, negatively associated with AMI. Mean diurnal activity was associated with increased connectivity in the default-mode, the cingulo-opercular and the frontoparietal networks, while chronotype was associated with a more heterogenous connectivity pattern in the same networks. We did not find significant associations between microstructural metrics and fPCs. Our findings suggest that mean diurnal activity and chronotype could provide indirect ambulatory measures of apathy in LLD, associated with modified functional connectivity of brain networks underlying goal-directed behaviors.

Temporal and spatial variation of domoic acid along Canada's coast

Shellfish poisonings have posed severe risks to human health globally. The Canadian Shellfish Sanitation Program was established in 1948 to monitor the toxin levels at shellfish harvesting sites along the coast of six provinces in Canada. Domoic acid has been a causal toxin for amnesic shellfish poisoning, and a macro-scale analysis of the temporal and spatial variation of domoic acid along Canada's coast were conducted in this study. We aggregated the toxin levels by week in blue mussel (Mytilus edulis) and soft-shell clam (Mya arenaria) samples, respectively, over a one-year scale. The subsequent application of Functional Principal Component Analysis unveiled that magnitudes of seasonal variation and peaked DA levels around early summer, spring, or mid-fall formed the largest variation in the toxin levels in blue mussels along coastlines of British Columbia and Prince Edward Island and in soft-shell calms along those of New Brunswick and Nova Scotia. In Quebec, the DA levels were low and varied mostly in terms of the overall magnitude across spring to fall. Downstream correlation analyses in British Columbia further discovered that, at most sites, the strongest correlations were negative between precipitation as well as inorganic nutrients (including nitrate, nitrite, phosphate, and silicate) on one side and DA a few weeks afterward on the other. These findings indicated associations between amnesic shellfish poisoning and environmental stresses.

A sustainable model for the simultaneous determination of tadalafil and dapoxetine hydrochloride in their binary mixture: Overcoming the challenges of environmental, operational, and instrumental variability for spectral data

This study employed functional principal component analysis (FPCA) and covariate design of experiments (DoE) to mitigate the susceptibility of chemometric methods to unexpected variations stemming from operational and instrumental factors. A comparative analysis with partial least squares (PLS) revealed that our proposed approach effectively reduced variability across different analysts, days, and instruments. Specifically, FPCA was utilized to compress available spectral wavelength information, while covariate DoE aided in selecting an optimal training set within the experimental space. Subsequently, PLS was applied for the simultaneous determination of tadalafil (TD) and dapoxetine hydrochloride (DP) in their binary mixture. Validation of the proposed method through accuracy profiles demonstrated its reliability, paving the way for its application in pharmaceutical analysis.

Evaluation and Selection of Multi-Spectral Indices to Classify Vegetation Using Multivariate Functional Principal Component Analysis

The identification, classification and mapping of different plant communities and habitats is of fundamental importance for defining biodiversity monitoring and conservation strategies. Today, the availability of high temporal, spatial and spectral data from remote sensing platforms provides dense time series over different spectral bands. In the case of supervised mapping, time series based on classical vegetation indices (e.g., NDVI, GNDVI, …) are usually input characteristics, but the selection of the best index or set of indices (which guarantees the best performance) is still based on human experience and is also influenced by the study area. In this work, several different time series, based on Sentinel-2 images, were created exploring new combinations of bands that extend the classic basic formulas as the normalized difference index. Multivariate Functional Principal Component Analysis (MFPCA) was used to contemporarily decompose the multiple time series. The principal multivariate seasonal spectral variations identified (MFPCA scores) were classified by using a Random Forest (RF) model. The MFPCA and RF classifications were nested into a forward selection strategy to identify the proper and minimum set of indices’ (dense) time series that produced the most accurate supervised classification of plant communities and habitat. The results we obtained can be summarized as follows: (i) the selection of the best set of time series is specific to the study area and the habitats involved; (ii) well-known and widely used indices such as the NDVI are not selected as the indices with the best performance; instead, time series based on original indices (in terms of formula or combination of bands) or underused indices (such as those derivable with the visible bands) are selected; (iii) MFPCA efficiently reduces the dimensionality of the data (multiple dense time series) providing ecologically interpretable results representing an important tool for habitat modelling outperforming conventional approaches that consider only discrete time series.

Multivariate Functional Clustering with Variable Selection and Application to Sensor Data from Engineering Systems

Multisensor data that track system operating behaviors are widely available nowadays from various engineering systems. Measurements from each sensor over time form a curve and can be viewed as functional data. Clustering of these multivariate functional curves is important for studying the operating patterns of systems. One complication in such applications is the possible presence of sensors whose data do not contain relevant information. Hence, it is desirable for the clustering method to equip with an automatic sensor selection procedure. Motivated by a real engineering application, we propose a functional data clustering method that simultaneously removes noninformative sensors and groups functional curves into clusters using informative sensors. Functional principal component analysis is used to transform multivariate functional data into a coefficient matrix for data reduction. We then model the transformed data by a Gaussian mixture distribution to perform model-based clustering with variable selection. Three types of penalties, the individual, variable, and group penalties, are considered to achieve automatic variable selection. Extensive simulations are conducted to assess the clustering and variable selection performance of the proposed methods. The application of the proposed methods to an engineering system with multiple sensors shows the promise of the methods and reveals interesting patterns in the sensor data. History: Kwok Tsui served as the senior editor for this article. Funding: The research by J. Min and Y. Hong was partially supported by the National Science Foundation [Grant CMMI-1904165] to Virginia Tech. The work by Y. Hong was partially supported by the Virginia Tech College of Science Research Equipment Fund. Data Ethics &amp; Reproducibility Note: The original dataset is proprietary and cannot be shared. The full code to replicate the results in this paper, based on summary statistics of the original data, is available at https://github.com/jiem3/MultiFuncClustering . The code applied to a simplified version is available at https://codeocean.com/capsule/4041000/tree/v1 , which covers the data analysis and part of the simulation scenarios with a single dataset under each scenario using a fixed set of hyperparameters, for reducing computation time, and at https://doi.org/10.1287/ijds.2022.0034 .

Joint analysis of multivariate longitudinal, imaging, and time-to-event data

Abstract Alzheimer’s (AD) is a progressive neurodegenerative disease frequently associated with memory deficits and cognitive decline. Despite its irreversible once onset, some discoveries revealed the existence of a certain percentage of people who are non-susceptible to AD. This study proposes a joint analysis of multivariate longitudinal data, survival data with a non-susceptible fraction, and ultrahigh-dimensional imaging data. The proposed model comprises three major components. The first component is a mixture proportional hazards cure model with images to examine the potential predictors of the non-susceptible probability and hazards of interest. The second component is a dynamic factor analysis model with images to characterize group-specific latent factors through multiple observed variables. The last component is a semiparametric trajectory model to reveal the change patterns of the dynamic latent factors in the ‘non-susceptible’ and ‘susceptible’ groups. A two-stage approach is developed for statistical inference. The first stage manages the imaging data through high-dimensional functional principal component analysis. The second stage develops a Bayesian approach coupled with penalized splines, data augmentation, and Markov chain Monte Carlo techniques to perform estimation. The application to the Alzheimer’s Disease Neuroimaging Initiative dataset sheds new insight into the pathology of AD.

Predicting health outcomes with intensive longitudinal data collected by mobile health devices: a functional principal component regression approach

BackgroundIntensive longitudinal data (ILD) collected in near real time by mobile health devices provide a new opportunity for monitoring chronic diseases, early disease risk prediction, and disease prevention in health research. Functional data analysis, specifically functional principal component analysis, has great potential to abstract trends in ILD but has not been used extensively in mobile health research.ObjectiveTo introduce functional principal component analysis (fPCA) and demonstrate its potential applicability in estimating trends in ILD collected by mobile heath devices, assessing longitudinal association between ILD and health outcomes, and predicting health outcomes.MethodsfPCA and scalar-to-function regression models were reviewed. A case study was used to illustrate the process of abstracting trends in intensively self-measured blood glucose using functional principal component analysis and then predicting future HbA1c values in patients with type 2 diabetes using a scalar-to-function regression model.ResultsBased on the scalar-to-function regression model results, there was a slightly increasing trend between daily blood glucose measures and HbA1c. 61% of variation in HbA1c could be predicted by the three preceding months’ blood glucose values measured before breakfast (P < 0.0001, \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{adjusted}^{2}=0.61$$\\end{document}).ConclusionsFunctional data analysis, specifically fPCA, offers a unique tool to capture patterns in ILD collected by mobile health devices. It is particularly useful in assessing longitudinal dynamic association between repeated measures and outcomes, and can be easily integrated in prediction models to improve prediction precision.

Functional principal component analysis for incomplete space–time data

AbstractEnvironmental signals, acquired, e.g., by remote sensing, often present large gaps of missing observations in space and time. In this work, we present an innovative approach to identify the main variability patterns, in space–time data, when data may be affected by complex missing data structures. We formalize the problem in the framework of functional data analysis, proposing an innovative method of functional principal component analysis (fPCA) for incomplete space–time data. The functional nature of the proposed method permits to borrow information from measurements observed at nearby spatio-temporal locations. The resulting functional principal components are smooth fields over the considered spatio-temporal domain, and can lead to interesting insights in the spatio-temporal dynamic of the phenomenon under study. Moreover, they can be used to provide a reconstruction of the missing entries, also under severe missing data patterns. The proposed model combines a weighted rank-one approximation of the data matrix with a roughness penalty. We show that the estimation problem can be solved using a majorize–minimization approach, and provide a numerically efficient algorithm for its solution. Thanks to a discretization based on finite elements in space and B-splines in time, the proposed method can handle multidimensional spatial domains with complex shapes, such as water bodies with complicated shorelines, or curved spatial regions with complex orography. As shown by simulation studies, the proposed space–time fPCA is superior to alternative techniques for Principal Component Analysis with missing data. We further highlight the potentiality of the proposed method for environmental problems, by applying space–time fPCA to the study of the lake water surface temperature (LWST) of Lake Victoria, in Central Africa, starting from satellite measurements with large gaps. LWST is considered one of the fundamental indicators of how climate change is affecting the environment, and is recognized as an essential climate variable.

Functional support vector machine.

Linear and generalized linear scalar-on-function modeling have been commonly used to understand the relationship between a scalar response variable (e.g. continuous, binary outcomes) and functional predictors. Such techniques are sensitive to model misspecification when the relationship between the response variable and the functional predictors is complex. On the other hand, support vector machines (SVMs) are among the most robust prediction models but do not take account of the high correlations between repeated measurements and cannot be used for irregular data. In this work, we propose a novel method to integrate functional principal component analysis with SVM techniques for classification and regression to account for the continuous nature of functional data and the nonlinear relationship between the scalar response variable and the functional predictors. We demonstrate the performance of our method through extensive simulation experiments and two real data applications: the classification of alcoholics using electroencephalography signals and the prediction of glucobrassicin concentration using near-infrared reflectance spectroscopy. Our methods especially have more advantages when the measurement errors in functional predictors are relatively large.

Demonstrating the relevance of spatial-functional statistical analysis in marine ecological studies: The case of environmental variations in micronektonic layers

In this study, we conducted an analysis of a multifrequency acoustics dataset acquired from scientific echosounders in the West African water. Our objective was to explore the spatial arrangement of marine organism aggregations. We investigated various attributes of these intricate biological entities, such as thickness, relative density, and depth, in relation to their surroundings. These environmental conditions were represented at a fine scale using a towed multiparameter system. This study is closely intertwined with two key domains: Fisheries acoustics techniques and functional data analysis. Fisheries acoustics techniques facilitate the collection of high-resolution spatial and temporal data concerning marine organisms at various depths and spatial scales, all without causing any disturbance. On the other hand, spatial-functional data analysis is a statistical approach for examining data characterised by functional attributes distributed across a spatial domain. This analysis encompasses dimension reduction techniques, as well as supervised and unsupervised methods, which take into consideration spatial dependencies within extensive datasets.We began by applying multivariate statistical techniques and subsequently employed Functional Data Analysis (FDA). In the modeling section, we introduced the spatial dimension with the spatial coordinates as covariates in the General Additive Model (GAM) and Functional Generalized Spectral Additive Model (FGSAM) models, aiming to underscore its relevance in those contexts. In an exploratory phase, Multivariate Functional Principal Component Analysis provided detailed insights into the variations of parameters at different depths, a capability not offered by traditional Principal Component Analysis. When it came to regression tasks, we explored the interactions between descriptors of Sound Scattering Layers and key environmental variables, both with and without considering spatial dimensions. Our findings revealed significant distinctions between northern and southern Sound Scattering Layers, as well as between coastal and high-sea regions. The use of the spatial locations enhanced the performance of GAM and FGSAM, particularly in the case of salinity, reflecting the influence of water mixing and seawater temperature. The multifaceted effects of environmental variations on Sound Scattering Layers underscore the importance of spatial-functional statistical analysis in ecological studies involving complex, spatially functional objects. Beyond the scope of this specific case study, the application of functional data analysis shows promise for a wide array of ecological studies dealing with extensive spatial datasets.

Integrating Functional Principal Component Analysis with Data-Rich Experimentation for Enhanced Drug Substance Development

Integrating Functional Principal Component Analysis with Data-Rich Experimentation for Enhanced Drug Substance Development