Methods For Functional Data Research Articles

Variograms for kriging and clustering of spatial functional data with phase variation

Spatial, amplitude and phase variations in spatial functional data are confounded. Conclusions from the popular functional trace-variogram, which quantifies spatial variation, can be misleading when analyzing misaligned functional data with phase variation. To remedy this, we describe a framework that extends amplitude-phase separation methods in functional data to the spatial setting, with a view towards performing clustering and spatial prediction. We propose a decomposition of the trace-variogram into amplitude and phase components, and quantify how spatial correlations between functional observations manifest in their respective amplitude and phase. This enables us to generate separate amplitude and phase clustering methods for spatial functional data, and develop a novel spatial functional interpolant at unobserved locations based on combining separate amplitude and phase predictions. Through simulations and real data analyses, we demonstrate advantages of our approach when compared to standard ones that ignore phase variation, through more accurate predictions and more interpretable clustering results.

Angle-specific torque profiles of concentric and eccentric thigh muscle strength 20 years after anterior cruciate ligament injury

ABSTRACT Thigh muscle weakness prevails following anterior cruciate ligament (ACL) injury, as usually evaluated by peak concentric quadriceps strength. Assessment throughout the range of motion (ROM), and for antagonists may provide more comprehensive information. We evaluated angle-specific torque profiles and ratios of isokinetic thigh muscle strength in 70 individuals 23 ± 2 years post-ACL injury (44males, 46.9 ± 5.4 years); 33 treated with ACL-reconstruction (ACL-R), and 37 treated only with physiotherapy (ACL-PT), and 33 controls. Quadriceps and hamstrings torques for concentric/eccentric contractions (90°/s) and ratios between hamstrings/quadriceps strength (HQ) were compared between and within groups using inferential functional data methods. The injured ACL-R leg had lower concentric and eccentric quadriceps strength compared to non-injured leg throughout the ROM, and lower concentric (interval 70–79°) and eccentric (64–67°) quadriceps strength compared to controls. The injured ACL-PT leg showed lower eccentric quadriceps strength (53–77°) than non-injured leg and lower concentric (41–79°) and eccentric (52–81°) quadriceps and eccentric hamstrings (30–77°) strength than controls. There were no group differences for HQ-ratios. The injured ACL-R leg had higher HQ-ratio (34–37°) than non-injured leg. Angle-specific torque profiles revealed strength deficits, masked if using only peak values, and seem valuable for ACL-injury rehabilitation.

Functional data analysis for longitudinal data with informative observation times.

In functional data analysis for longitudinal data, the observation process is typically assumed to be noninformative, which is often violated in real applications. Thus, methods that fail to account for the dependence between observation times and longitudinal outcomes may result in biased estimation. For longitudinal data with informative observation times, we find that under a general class of shared random effect models, a commonly used functional data method may lead to inconsistent model estimation while another functional data method results in consistent and even rate-optimal estimation. Indeed, we show that the mean function can be estimated appropriately via penalized splines and that the covariance function can be estimated appropriately via penalized tensor-product splines, both with specific choices of parameters. For the proposed method, theoretical results are provided, and simulation studies and a real data analysis are conducted to demonstrate itsperformance.

A time-varying distance based interval-valued functional principal component analysis method – A case study of consumer price index

Functional principal component analysis (FPCA) is an extension of conventional principal component analysis (PCA) that allows the processing of functional data. Besides the reduction in dimensionality that is inherent to PCA, FPCA relies on fewer assumptions and offers a greater ability to visualize the functional data. Thus, FPCA can be used in, for example, social, economic, and medical research. However, the existing FPCA methods are sensitive to outliers, and underperform when extracting features from interval-valued functional data. At the same time, the existing PCA methods for interval-valued functional data suffer from inconsistency in the interpretation of the principal components, and substantial information loss. Therefore, this paper proposes an interval-valued functional principal component analysis (IFPCA) method based on the time-varying distance function. The time-varying distance function containing information on the midpoint and radius is constructed to mitigate information loss. The novel IFPCA method is also able to solve the problem of the inconsistent interpretation of the principal components. The effectiveness of the method is verified by considering the case of the consumer price index.

Semi-supervised Clustering Method for Non-negative Functional Data

Semi-supervised Clustering Method for Non-negative Functional Data

A Wilcoxon-Mann-Whitney spatial scan statistic for functional data

A nonparametric scan method for functional data indexed in space is introduced. The associated scan statistic is derived from the Wilcoxon-Mann-Whitney test statistic defined for infinite dimensional data. It is completely nonparametric as it does not assume any distribution concerning the functional marks. Whatever the clustering scenario, this scan test seems to be efficient to detect and locate the cluster. This method is applied to a data set for extracting features in Spanish province population growth. A significant spatial cluster of low demographic evolution rates is found, exhibiting a specific phenomenon in the North-West of Spain.

FUNCTIONAL PREDICTION REGRESSION VIA BOOSTING UNTUK PEMODELAN CURAH HUJAN DI PROVINSI JAWA TIMUR

Natural phenomena that can cause natural changes on earth caused by increasing greenhouse gases and decreasing landthat absorbs carbon dioxide are called climate change. The elements that cause changes include rainfall and temperature.The constantly rising temperature of the earth results in changing rainfall patterns and can have various effects on theenvironment. Therefore, research on rainfall modeling with annual average temperature and rainfall data from theprovince of East Java from 2006 to 2017 which was taken from the official website of the Central Statistics Agency ofEast Java Province makes sense. This data is a data multivariate time series that is approached with Functional DataAnalysis and modeled using Functional Prediction Regression. Functional Prediction Regression is a form of modelingwith functional data that can test the overall model for high-dimensional data and one of the improved methods ofregression methods for functional data. One way to model Functional Prediction Regression is through boosting. Thisresearch conducted rainfall modeling with Functional Prediction Regression through boosting from East Java Provinceand obtained modeling results using an additional predictor model with 5-fold bootstrap and adjustment of the splineregression (knots) used, namely 16 indicated by the iteration value-boosting bootstrap where the model used to have alinear functional effect of 𝑚𝑠𝑡𝑜𝑝 for the mu parameter and 100 for the sigma parameter.

Detecting and classifying outliers in big functional data

We propose two new outlier detection methods, for identifying and classifying different types of outliers in (big) functional data sets. The proposed methods are based on an existing method called Massive Unsupervised Outlier Detection (MUOD). MUOD detects and classifies outliers by computing for each curve, three indices, all based on the concept of linear regression and correlation, which measure outlyingness in terms of shape, magnitude and amplitude, relative to the other curves in the data. 'Semifast-MUOD', the first method, uses a sample of the observations in computing the indices, while 'Fast-MUOD', the second method, uses the point-wise or $L_1$ median in computing the indices. The classical boxplot is used to separate the indices of the outliers from those of the typical observations. Performance evaluation of the proposed methods using simulated data show significant improvements compared to MUOD, both in outlier detection and computational time. We show that Fast-MUOD is especially well suited to handling big and dense functional datasets with very small computational time compared to other methods. Further comparisons with some recent outlier detection methods for functional data also show superior or comparable outlier detection accuracy of the proposed methods. We apply the proposed methods on weather, population growth, and video data.

Ordinal classification of 3D brain structures by functional data analysis

We introduce several ordinal classification methods for functional data, specifically multiargument and multivariate functional data. Their performance is analyzed in four real data sets that belong to a neuroeducational problem and a neuropathological problem.

Canonical Correlation Analysis for Market and Environmental Factors of Vegetable using Functional Data Methods

최근 5년 동안 5개의 광역시도의 대표 과채류 도매 시장에서 일 단위로 갱신된 자료들을 함수형 자료로 간주하여 분석하고자 한다. 주된 관심은 최근 5년간 주키니 호박의 가격 함수, 거래량 함수, 환경 인자와 관련한 함수 간의 연관성을 발견하고자 한다. 하지만 다루고자 하는 대상은 스칼라값을 가지는 변수들이 아니기 때문에 피어슨 상관계수와 같은 널리 알려진 방법을 사용하는 것은 적당하지 않다. 이러한 이유로 FDA(functional data analysis)에서 사용되는 FCCA(functional canonical correlation analysis) 방법을 사용하여 분석하고자 한다. 구체적으로 주키니 호박의 가격, 반입량, 기온, 습도, 일사량을 365일 동안 관측되어지는 함수로 생각하고 개별 단위에 대하여 평활화를 진행할 것이다. 그리고 나서 이들 함수들에 대한 기저 함수를 FPCA(functional principal components analysis) 방법으로 유도하고 유도된 기저 함수로 FCCA를 사용하고자 한다. 또한 FD(functional data)에 대한 모형 중 하나인 fully functional linear model을 사용하여 1st functional canonical value가 가장 높았던 쌍인 가격과 일사량을 반응변수와 설명변수로 두고 실제로 효과가 있는지 검정을 통해 확인하였다.

A Bayesian multiple imputation approach to bivariate functional data with missing components.

Existing missing data methods for functional data mainly focus on reconstructing missing measurements along a single function-a univariate functional data setting. Motivated by a renal study, we focus on a bivariate functional data setting, where each sampling unit is a collection of two distinct component functions, one of which may be missing. Specifically, we propose a Bayesian multiple imputation approach based on a bivariate functional latent factor model that exploits the joint changing patterns of the component functions to allow accurate and stable imputation of one component given the other. We further extend the framework to address multilevel bivariate functional data with missing components by modeling and exploiting inter-component and intra-subject correlations. We develop a Gibbs sampling algorithm that simultaneously generates multiple imputations of missing component functions and posterior samples of model parameters. For multilevel bivariate functional data, a partially collapsed Gibbs sampler is implemented to improve computational efficiency. Our simulation study demonstrates that our methods outperform other competing methods for imputing missing components of bivariate functional data under various designs and missingness rates. The motivating renal study aims to investigate the distribution and pharmacokinetic properties of baseline and post-furosemide renogram curves that provide further insights into the underlying mechanism of renal obstruction, with post-furosemide renogram curves missing for some subjects. We apply the proposed methods to impute missing post-furosemide renogram curves and obtain more refined insights.

Analysis Choices Impact Movement Evaluation: A Multi-Aspect Inferential Method Applied to Kinematic Curves of Vertical Hops in Knee-Injured and Asymptomatic Persons.

Three-dimensional human motion analysis provides in-depth understanding in order to optimize sports performance or rehabilitation following disease or injury. Recent developments of statistical methods for functional data allow for novel ways to analyze often complex biomechanical data. Even so, for such methods as well as for traditional well-established statistical methods, the interpretations of the results may be influenced by analysis choices made prior to the analysis. We evaluated the consequences of three such choices when comparing one-leg vertical hop (OLVH) performance in individuals who had ruptured their anterior cruciate ligament (ACL), to that of asymptomatic controls, and also athletes. Kinematic data were analyzed using a statistical approach for functional data, targeting entire curve data. This was done not only for one joint at a time but also for multiple lower limb joints and movement planes simultaneously using a multi-aspect methodology, testing for group differences while also accounting for covariates. We present the results of when an individual representative curve out of three available was either: (1) a mean curve (Mean), (2) a curve from the highest hop (Max), or (3) a curve describing the variability (Var), as a representation of performance stability. We also evaluated choice of sample leg comparison; e.g., ACL-injured leg compared to either the dominant or non-dominant leg of asymptomatic groups. Finally, we explored potential outcome effects of different combinations of included joints. There were slightly more pronounced group differences when using Mean compared to Max, while the specifics of the observed differences depended on the outcome variable. For Var there were less significant group differences. Generally, there were more disparities throughout the hop movement when comparing the injured leg to the dominant leg of controls, resulting in e.g., group differences for trunk and ankle kinematics, for both Mean and Max. When the injured leg was instead compared to the non-dominant leg of controls, there were trunk, hip and knee joint differences. For a more stringent comparison, we suggest considering to compare the injured leg to the non-dominant leg. Finally, the multiple-joint analyses were coherent with the single-joint analyses. The direct effects of analysis choices can be explored interactively by the reader in the Supplementary Material. To summarize, the choices definitively have an impact on the interpretation of a hop test results commonly used in rehabilitation following knee injuries. We therefore strongly recommend well-documented methodological analysis choices with regards to comparisons and representative values of the measures of interests.

Functional data clustering using principal curve methods

In this paper we propose a novel clustering method for functional data based on the principal curve clustering approach. By this method functional data are approximated using functional principal component analysis (FPCA) and the principal curve clustering is then performed on the principal scores. The proposed method makes use of the nonparametric principal curves to summarize the features of the principal scores extracted from the original functional data, and a probabilistic model combined with Bayesian Information Criterion is employed to automatically and simultaneously find the appropriate number of features, the optimal degree of smoothing and the corresponding cluster members. The simulation studies show that the proposed method outperforms the existing functional clustering approaches considered. The capability of this method is also demonstrated by the applications in the human mortality and fertility data.

Nonparametric clustering for longitudinal functional data with the application to H-NMR spectra of kidney transplant patients. Longitudinal functional data clustering.

Longitudinal functional data are increasingly common in the health domain. The motivated dataset for this paper comprises H-NMR spectra of kidney transplant patients [8]. Our aim is to cluster patients into different clinical outcome subgoups to reveal the success of the transplantation. The NMR spectra of each patient at each time point are functional data and the data are longitudinally collected at up to nine different time points. Existing methods are available for functional data collected at one time point, but not for longitudinal functional data collected at a grid of time points subject to missingness. We therefore first apply a method to extract the same number of functional feactures for each subject. Next we propose a novel nonparametric clustering method for mulitivariate functional data. We applied our proposed clustering method to the kidney transplant dataset both to a subset of the raw data with only two time points and the extacted functional features. It appeared that the proposed method achieves better clustering performance on the extracted functional features than on the subset of raw data. A data simulation study was performed to further evaluate the method. The design mimiced the kidney transplant dataset but with a larger sample size. Scenarios which have different levels of noise were considered. The simulation study showed the accuarcy of our proposed method.

Functional Kernel Density Estimation: Point and Fourier Approaches to Time Series Anomaly Detection.

We present an unsupervised method to detect anomalous time series among a collection of time series. To do so, we extend traditional Kernel Density Estimation for estimating probability distributions in Euclidean space to Hilbert spaces. The estimated probability densities we derive can be obtained formally through treating each series as a point in a Hilbert space, placing a kernel at those points, and summing the kernels (a “point approach”), or through using Kernel Density Estimation to approximate the distributions of Fourier mode coefficients to infer a probability density (a “Fourier approach”). We refer to these approaches as Functional Kernel Density Estimation for Anomaly Detection as they both yield functionals that can score a time series for how anomalous it is. Both methods naturally handle missing data and apply to a variety of settings, performing well when compared with an outlyingness score derived from a boxplot method for functional data, with a Principal Component Analysis approach for functional data, and with the Functional Isolation Forest method. We illustrate the use of the proposed methods with aviation safety report data from the International Air Transport Association (IATA).

A study of longitudinal mobile health data through fuzzy clustering methods for functional data: The case of allergic rhinoconjunctivitis in childhood.

The use of mobile communication devices in health care is spreading worldwide. A huge amount of health data collected by these devices (mobile health data) is nowadays available. Mobile health data may allow for real-time monitoring of patients and delivering ad-hoc treatment recommendations. This paper aims at showing how this may be done by exploiting the potentialities of fuzzy clustering techniques. In fact, such techniques can be fruitfully applied to mobile health data in order to identify clusters of patients for diagnostic classification and cluster-specific therapies. However, since mobile health data are full of noise, fuzzy clustering methods cannot be directly applied to mobile health data. Such data must be denoised prior to analyzing them. When longitudinal mobile health data are available, functional data analysis represents a powerful tool for filtering out the noise in the data. Fuzzy clustering methods for functional data can then be used to determine groups of patients. In this work we develop a fuzzy clustering method, based on the concept of medoid, for functional data and we apply it to longitudinal mHealth data on daily symptoms and consumptions of anti-symptomatic drugs collected by two sets of patients in Berlin (Germany) and Ascoli Piceno (Italy) suffering from allergic rhinoconjunctivitis. The studies showed that clusters of patients with similar changes in symptoms were identified opening the possibility of precision medicine.

Cross‐component registration for multivariate functional data, with application to growth curves

AbstractMultivariate functional data are becoming ubiquitous with advances in modern technology and are substantially more complex than univariate functional data. We propose and study a novel model for multivariate functional data where the component processes are subject to mutual time warping. That is, the component processes exhibit a similar shape but are subject to systematic phase variation across their time domains. To address this previously unconsidered mode of warping, we propose new registration methodology that is based on a shift‐warping model. Our method differs from all existing registration methods for functional data in a fundamental way. Namely, instead of focusing on the traditional approach to warping, where one aims to recover individual‐specific registration, we focus on shift registration across the components of a multivariate functional data vector on a population‐wide level. Our proposed estimates for these shifts are identifiable, enjoy parametric rates of convergence, and often have intuitive physical interpretations, all in contrast to traditional curve‐specific registration approaches. We demonstrate the implementation and interpretation of the proposed method by applying our methodology to the Zürich Longitudinal Growth data and study its finite sample properties in simulations.

A validation framework for neuroimaging software: The case of populationreceptive fields.

Neuroimaging software methods are complex, making it a near certainty that some implementations will contain errors. Modern computational techniques (i.e., public code and data repositories, continuous integration, containerization) enable the reproducibility of the analyses and reduce coding errors, but they do not guarantee the scientific validity of the results. It is difficult, nay impossible, for researchers to check the accuracy of software by reading the source code; ground truth test datasets are needed. Computational reproducibility means providing software so that for the same input anyone obtains the same result, right or wrong. Computational validity means obtaining the right result for the ground-truth test data. We describe a framework for validating and sharing software implementations, and we illustrate its usage with an example application: population receptive field (pRF) methods for functional MRI data. The framework is composed of three main components implemented with containerization methods to guarantee computational reproducibility. In our example pRF application, those components are: (1) synthesis of fMRI time series from ground-truth pRF parameters, (2) implementation of four public pRF analysis tools and standardization of inputs and outputs, and (3) report creation to compare the results with the ground truth parameters. The framework was useful in identifying realistic conditions that lead to imperfect parameter recovery in all four pRF implementations, that would remain undetected using classic validation methods. We provide means to mitigate these problems in future experiments. A computational validation framework supports scientific rigor and creativity, as opposed to the oft-repeated suggestion that investigators rely upon a few agreed upon packages. We hope that the framework will be helpful to validate other critical neuroimaging algorithms, as having a validation framework helps (1) developers to build new software, (2) research scientists to verify the software's accuracy, and (3) reviewers to evaluate the methods used in publications and grants.

Application of Functional Data Analysis to Identify Patterns of Malaria Incidence, to Guide Targeted Control Strategies.

We introduce an approach based on functional data analysis to identify patterns of malaria incidence to guide effective targeting of malaria control in a seasonal transmission area. Using functional data method, a smooth function (functional data or curve) was fitted from the time series of observed malaria incidence for each of 575 villages in west-central Senegal from 2008 to 2012. These 575 smooth functions were classified using hierarchical clustering (Ward’s method), and several different dissimilarity measures. Validity indices were used to determine the number of distinct temporal patterns of malaria incidence. Epidemiological indicators characterizing the resulting malaria incidence patterns were determined from the velocity and acceleration of their incidences over time. We identified three distinct patterns of malaria incidence: high-, intermediate-, and low-incidence patterns in respectively 2% (12/575), 17% (97/575), and 81% (466/575) of villages. Epidemiological indicators characterizing the fluctuations in malaria incidence showed that seasonal outbreaks started later, and ended earlier, in the low-incidence pattern. Functional data analysis can be used to identify patterns of malaria incidence, by considering their temporal dynamics. Epidemiological indicators derived from their velocities and accelerations, may guide to target control measures according to patterns.

Pseudo-quantile functional data clustering

This paper studies the problem of functional data clustering. Functional data have their own characteristics and contain rich information that cannot be obtained when regarding the data as multivariate data. Functional data are inherently infinite-dimensional, so classical clustering techniques for finite-dimensional data may not be suitable for functional data. There are several clustering methods for functional data based on probabilistic models or basis expansion approaches. However, most of these depend on the symmetric structure of the model or the mean response; hence, these cannot reflect characteristics of the distribution of data beyond the mean, such as behavior at the extremes. In this paper, we propose a new approach for functional data clustering based on the concept of an asymmetric norm. For this purpose, we consider pseudo-quantiles, such as M-quantiles and expectiles, and their corresponding curves that can provide rich distributional information about hidden structures in the data at various levels. Moreover, as a theoretical justification for the proposed method, a strong consistency property is investigated. Results from numerical examples, including real data analysis, demonstrate the promising empirical properties of the proposed approach.