Functional Principal Component Analysis for Extrapolating Multistream Longitudinal Data

Functional principal component analysis (FPCA) is a commonly used nonparametric degradation modeling technique. A basic requirement of FPCA is that different units must share the same scale in X-axis. A FPCA-based degradation modeling method has been specially designed for truncated degradation signals whose amplitudes are truncated at the failure threshold. This method formulates the degradation time as a function of state level by axis rotation. Since the degradation signals of different units are truncated at the same failure threshold level, the axis rotation strategy ensures the same scale in X-axis. However, in cases of noisy signals, one state level may correspond to multiple time points, which means that the time function depending on state level does not exist. In addition, this method assumes that the noise term as well as the functional principal component (FPC) scores follow the Gaussian distribution, which restricts the flexibility in real practice. Aiming at the above two issues, this paper develops a new degradation modeling method based on the FPCA. The proposed method formulates the degradation model on the definition of first pass time (FPT), which ensures the existence of the time function depending on state level. The noise term and the FPC scores are not restricted to the Gaussian distribution. A particle filtering based updating algorithm is developed to deal with the non-Gaussian state estimation issue of the FPC scores according to the online condition monitoring signals. The effectiveness of the proposed method is verified using an experiment of milling cutter wear.

PurposeThe purpose of this paper is to analyze and forecast the Chinese term structure of interest rates using functional principal component analysis (FPCA).Design/methodology/approachThe authors propose an FPCA-K model using FPCA. The forecasting of the yield curve is based on modeling functional principal component (FPC) scores as standard scalar time series models. The authors evaluate the out-of-sample forecast performance using the root mean square and mean absolute errors.FindingsMonthly yield data from January 2002 to December 2016 are used in this paper. The authors find that in the full sample, the first two FPCs account for 98.68 percent of the total variation in the yield curve. The authors then construct an FPCA-K model using the leading principal components. The authors find that the FPCA-K model compares favorably with the functional signal plus noise model, the dynamic Nelson-Siegel models and the random walk model in the out-of-sample forecasting.Practical implicationsThe authors propose a functional approach to analyzing and forecasting the yield curve, which effectively utilizes the smoothness assumption and conveniently addresses the missing-data issue.Originality/valueTo the best knowledge, the authors are the first to use FPCA in the modeling and forecasting of yield curves.

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.

Background: A shift towards the dynamic measurement of physiologic resilience and improved technology incorporated into experimental paradigms in aging research is producing high-resolution data. Identifying the most appropriate analysis method for this type of data is a challenge. In this work, the functional principal component analysis (fPCA) was employed to demonstrate a data-driven approach to the analysis of high-resolution data in aging research.Methods: Cerebral oxygenation during standing was measured in a large cohort [The Irish Longitudinal Study on Aging (TILDA)]. FPCA was performed on tissue saturation index (TSI) data. A regression analysis was then conducted with the functional principal component (fPC) scores as the explanatory variables and transition time as the response.Results: The mean ± SD age of the analysis sample was 64 ± 8 years. Females made up 54% of the sample and overall, 43% had tertiary education. The first PC explained 96% of the variance in cerebral oxygenation upon standing and was related to a baseline shift. Subsequent components described the recovery to before-stand levels (fPC2), drop magnitude and initial recovery (fPC3 and fPC4) as well as a temporal shift in the location of the minimum TSI value (fPC5). Transition time was associated with components describing the magnitude and timing of the nadir.Conclusions: Application of fPCA showed utility in reducing a large amount of data to a small number of parameters which summarize the inter-participant variation in TSI upon standing. A demonstration of principal component regression was provided to allow for continued use and development of data-driven approaches to high-resolution data analysis in aging research.

SummaryThe transitional phase of menopause induces significant hormonal fluctuations, exerting a profound influence on the long-term well-being of women. In an extensive longitudinal investigation of women’s health during mid-life and beyond, known as the Study of Women’s Health Across the Nation (SWAN), hormonal biomarkers are repeatedly assessed, following an asynchronous schedule compared to other error-prone covariates, such as physical and cardiovascular measurements. We conduct a subgroup analysis of the SWAN data employing a semiparametric mixture regression model, which allows us to explore how the relationship between hormonal responses and other time-varying or time-invariant covariates varies across subgroups. To address the challenges posed by asynchronous scheduling and measurement errors, we model the time-varying covariate trajectories as functional data with reduced-rank Karhunen–Loéve expansions, where splines are employed to capture the mean and eigenfunctions. Treating the latent subgroup membership and the functional principal component (FPC) scores as missing data, we propose an Expectation-Maximization algorithm to effectively fit the joint model, combining the mixture regression for the hormonal response and the FPC model for the asynchronous, time-varying covariates. In addition, we explore data-driven methods to determine the optimal number of subgroups within the population. Through our comprehensive analysis of the SWAN data, we unveil a crucial subgroup structure within the aging female population, shedding light on important distinctions and patterns among women undergoing menopause.

We propose a nonparametric method to explicitly model and represent the derivatives of smooth underlying trajectories for longitudinal data. This representation is based on a direct Karhunen--Lo\`eve expansion of the unobserved derivatives and leads to the notion of derivative principal component analysis, which complements functional principal component analysis, one of the most popular tools of functional data analysis. The proposed derivative principal component scores can be obtained for irregularly spaced and sparsely observed longitudinal data, as typically encountered in biomedical studies, as well as for functional data which are densely measured. Novel consistency results and asymptotic convergence rates for the proposed estimates of the derivative principal component scores and other components of the model are derived under a unified scheme for sparse or dense observations and mild conditions. We compare the proposed representations for derivatives with alternative approaches in simulation settings and also in a wallaby growth curve application. It emerges that representations using the proposed derivative principal component analysis recover the underlying derivatives more accurately compared to principal component analysis-based approaches especially in settings where the functional data are represented with only a very small number of components or are densely sampled. In a second wheat spectra classification example, derivative principal component scores were found to be more predictive for the protein content of wheat than the conventional functional principal component scores.

We propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data, where both the predictor and response are functions of a covariate such as time. Predictor and response processes have smooth random trajectories, and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects. In longitudinal studies, the number of repeated measurements per subject is often small and may be modeled as a discrete random number and, accordingly, only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit. We propose a functional regression approach for this situation, using functional principal component analysis, where we estimate the functional principal component scores through conditional expectations. This allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory. The resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories. Asymptotic properties for a sample of n subjects are investigated under mild conditions, as n→∞, and we obtain consistent estimation for the regression function. Besides convergence results for the components of functional linear regression, such as the regression parameter function, we construct asymptotic pointwise confidence bands for the predicted trajectories. A functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced, extending the standard R2 to the functional case. The proposed methods are illustrated with a simulation study, longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index.

We present the application of a nonparametric method to performing functional principal component analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatterplot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covariance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal component scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at minimizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing of healthy adults with a small tracer dose of 14C-folic acid. A time transformation was incorporated to handle design irregularity concerning the time points on which the measurements were taken. The proposed methodology, incorporating shrinkage and data-adaptive features, is seen to be well suited for describing population kinetics of 14C-folate-specific activity and random effects, and can also be applied to other functional data analysis problems.

BackgroundFetal movement counting has long been suggested as a screening tool to identify impaired placental function. However, quantitative limits for decreased fetal movement perform poorly for screening purposes, indicating the need for methodological refinement. We aimed to identify the main individual temporal patterns in fetal movement counting charts, and explore their associations with pregnancy characteristics.MethodsIn a population-based prospective cohort in Norway, 2009–2011, women with singleton pregnancies counted fetal movements daily from pregnancy week 24 until delivery using a modified "count-to-ten” procedure. To account for intra-woman correlation of observations, we used functional data analysis and corresponding functional principal component analysis to identify the main individual temporal patterns in fetal movement count data. The temporal patterns are described by continuous functional principal component (FPC) curves, with an individual score on each FPC for each woman. These scores were later used as outcome variables in multivariable linear regression analyses, with pregnancy characteristics as explanatory variables.ResultsFetal movement charts from 1086 pregnancies were included. Three FPC curves explained almost 99% of the variation in the temporal data, with the first FPC, representing the individual overall counting time, accounting for 91% alone. There were several statistically significant associations between the FPCs and various pregnancy characteristics. However, the effects were small and of limited clinical value.ConclusionsThis statistical approach for analyzing fetal movement counting data successfully captured clinically meaningful individual temporal patterns and how these patterns vary between women. Maternal body mass index, gestational age and placental site explained little of the variation in the temporal fetal movement counting patterns. Thus, a perceived decrease in fetal movement should not be attributed to a woman’s basic pregnancy characteristics, but assessed as a potential marker of risk.

Primary open angle glaucoma (POAG) is a chronic, progressive, irreversible, and potentially blinding optic neuropathy. The risk of blindness due to progressive visual field (VF) loss varies substantially from patient to patient. Early identification of those patients destined to rapid progressive visual loss is crucial to prevent further damage. In this article, a latent class growth model (LCGM) was developed to predict the binary outcome of VF progression using longitudinal mean deviation (MD) and pattern standard deviation (PSD). Specifically, the trajectories of MD and PSD were summarized by a functional principal component (FPC) analysis, and the estimated FPC scores were used to identify subgroups (latent classes) of individuals with distinct patterns of MD and PSD trajectories. Probability of VF progression for an individual was then estimated as weighted average across latent classes, weighted by posterior probability of class membership given baseline covariates and longitudinal MD/PSD series. The model was applied to the participants with newly diagnosed POAG from the Ocular Hypertension Treatment Study (OHTS), and the OHTS data was best fit by a model with 4 latent classes. Using the resultant optimal LCGM, the OHTS participants with and without VF progression could be accurately differentiated by incorporating longitudinal MD and PSD.

Background:Knee osteoarthritis (OA) is a leading cause of pain and disability globally. Pain from knee OA leads to functional limitation. Physical activity (PA), e.g., taking more steps/day, reduces the risk...

Chapter 32 - StatStream: Statistical Monitoring of Thousands of Data Streams in Real Time

A new methodology based on functional principal component analysis to study postural stability post-stroke

We propose an international yield curve predictive model, where common factors are identified using the common functional principal component (CFPC) method that enables a comparison of the variation patterns across different economies with heterogeneous covariances. The dynamics of the international yield curves are further forecasted based on the data-driven common factors in an autoregression framework. For the 1-day ahead out-of-sample forecasts of the US, Sterling, Euro and Japanese yield curve from 07 April 2014 to 06 April 2015, the CFPC factor model is compared with an alternative factor model based on the functional principal component analysis.

In this paper, we introduce a computational framework for recovering a high-resolution approximation of an unknown function from its low-resolution indirect measurements as well as high-resolution training observations by merging the frameworks of generalized sampling and functional principal component analysis. In particular, we increase the signal resolution via a data-driven approach, which models the function of interest as a realization of a random field and leverages a training set of observations generated via the same underlying random process. We study the performance of the resulting estimation procedure and show that high-resolution recovery is indeed possible provided appropriate low rank and angle conditions hold and provided the training set is sufficiently large relative to the desired resolution. Moreover, we show that the size of the training set can be reduced by leveraging sparse representations of the functional principal components. Furthermore, the effectiveness of the proposed reconstruction procedure is illustrated by various numerical examples.