-mean alignment for curve clustering

Clustering Spatially Correlated Functional Data With Multiple Scalar Covariates

We propose a probabilistic model for clustering spatially correlated functional data with multiple scalar covariates. The motivating application is to partition the 29 provinces of the Chinese mainland into a few groups characterized by the epidemic severity of COVID-19, while the spatial dependence and effects of risk factors are considered. It can be regarded as an extension of mixture models, which allows different subsets of covariates to influence the component weights and the component densities by modeling the parameters of the mixture as functions of the covariates. In this way, provinces with similar spatial factors are a priori more likely to be clustered together. Posterior predictive inference in this model formalizes the desired prediction. Further, the identifiability of the proposed model is analyzed, and sufficient conditions to guarantee "generic" identifiability are provided. An L1 -penalized estimator is developed to assist variable selection and robust estimation when the number of explanatory covariates is large. An efficient expectation-minimization algorithm is presented for parameter estimation. Simulation studies and real-data examples are presented to investigate the empirical performance of the proposed method. Finally, it is worth noting that the proposed model has a wide range of practical applications, e.g., health management, environmental science, ecological studies, and so on.

Semi-supervised clustering of quaternion time series: Application to gait analysis in multiple sclerosis using motion sensor data

Recent approaches in gait analysis involve the use of wearable motion sensors to extract spatio-temporal parameters that characterize multiple aspects of an individual's gait. In particular, the medical community could largely benefit from this type of devices as they could provide the clinicians with a valuable tool for assessing gait impairment. Motion sensor data are however complex and there is an urgent unmet need to develop sound statistical methods for analyzing such data and extracting clinically relevant information. In this article, we measure gait by following the hip rotation over time and the resulting statistical unit is a time series of unit quaternions. We explore the possibility to form groups of patients with similar walking impairment by taking into account their walking data and their global decease severity with semi-supervised clustering. We generalize a compromise-based method named hclustcompro to unit quaternion time series by combining it with the proper dissimilarity quaternion dynamic time warping. We apply this method on patients diagnosed with multiple sclerosis to form groups of patients with similar walking deficiencies while accounting for the clinical assessment of their overall disability. We also compare the compromise-based clustering approach with the method mergeTrees that falls into a sub-class of ensemble clustering named collaborative clustering. The results provide a first proof of both the interest of using wearable motion sensors for assessing gait impairment and the use of prior knowledge to guide the clustering process. It also demonstrates that compromise-based clustering is a more appropriate approach in this context.

Boosted-oriented probabilistic smoothing-spline clustering of series

Fuzzy clustering methods allow the objects to belong to several clusters simultaneously, with different degrees of membership. However, a factor that influences the performance of fuzzy algorithms is the value of fuzzifier parameter. In this paper, we propose a fuzzy clustering procedure for data (time) series that does not depend on the definition of a fuzzifier parameter. It comes from two approaches, theoretically motivated for unsupervised and supervised classification cases, respectively. The first is the Probabilistic Distance clustering procedure. The second is the well known Boosting philosophy. Our idea is to adopt a boosting prospective for unsupervised learning problems, in particular we face with non hierarchical clustering problems. The global performance of the proposed method is investigated by various experiments.

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.

Band depth based initialization of K-means for functional data clustering

The k-Means algorithm is one of the most popular choices for clustering data but is well-known to be sensitive to the initialization process. There is a substantial number of methods that aim at finding optimal initial seeds for k-Means, though none of them is universally valid. This paper presents an extension to longitudinal data of one of such methods, the BRIk algorithm, that relies on clustering a set of centroids derived from bootstrap replicates of the data and on the use of the versatile Modified Band Depth. In our approach we improve the BRIk method by adding a step where we fit appropriate B-splines to our observations and a resampling process that allows computational feasibility and handling issues such as noise or missing data. We have derived two techniques for providing suitable initial seeds, each of them stressing respectively the multivariate or the functional nature of the data. Our results with simulated and real data sets indicate that our Functional Data Approach to the BRIK method (FABRIk) and our Functional Data Extension of the BRIK method (FDEBRIk) are more effective than previous proposals at providing seeds to initialize k-Means in terms of clustering recovery.

Associations Between Posterior Communicating Artery Aneurysms and Morphological Characteristics of Surrounding Arteries

Objectives

To explore the associations between posterior communicating artery (PComA) aneurysms and morphological characteristics of arteries upstream of and around the PComA bifurcation site.

Methods

In this study, fifty-seven patients with PComA aneurysms and sixty-two control subjects without aneurysms were enrolled. The centerlines of the internal carotid artery (ICA) and important branches were generated for the measurement and analysis of morphological parameters, such as carotid siphon types, diameters of two fitting circles, and the angle formed by them (D₁, D₂, and ϕ), length (L) and tortuosity (T_L) of ICA segment between an ophthalmic artery and PComA bifurcations, bifurcation angle (θ), tortuosity (T_ICA and T_PComA), and flow direction changes (θ_ICA and θ_PComA) around the PComA bifurcation site.

Results

No significant difference (p > 0.05) was found in the siphon types (p = 0.467) or L (p = 0.114). Significant differences (p < 0.05) were detected in D₁ (p = 0.036), T_L (p < 0.001), D₂ (p = 0.004), ϕ (p = 0.008), θ (p = 0.001), T_ICA (p < 0.001), T_PComA (p = 0.012), θ_ICA (p < 0.001), and θ_PComA (p < 0.001) between the two groups. T_ICA had the largest area under the curve (AUC) (0.843) in the receiver operating characteristic (ROC) analysis in diagnosing the probability of PComA aneurysms presence and was identified as the only potent morphological parameter (OR = 11.909) associated with PComA aneurysms presence.

Conclusions

The high tortuosity of the ICA segment around the PComA bifurcation is associated with PComA aneurysm presence.

Ensemble clustering of longitudinal bivariate HIV biomarker profiles to group patients by patterns of disease progression

This paper describes an ensemble cluster analysis of bivariate profiles of HIV biomarkers, viral load and CD4 cell counts, which jointly measure disease progression. Data are from a prevalent cohort of HIV positive participants in a clinical trial of vitamin supplementation in Botswana. These individuals were HIV positive upon enrollment, but with unknown times of infection. To categorize groups of participants based on their patterns of progression of HIV infection using both biomarkers, we combine univariate shape-based cluster results for multiple biomarkers through the use of ensemble clustering methods. We first describe univariate clustering for each of the individual biomarker profiles, and make use of shape-respecting distances for clustering the longitudinal profile data. In our data, profiles are subject to either missing or irregular measurements as well as unobserved initiation times of the process of interest. Shape-respecting distances that can handle such data issues, preserve time-ordering, and identify similar profile shapes are useful in identifying patterns of disease progression from longitudinal biomarker data. However, their performance with regard to clustering differs by severity of the data issues mentioned above. We provide an empirical investigation of shape-respecting distances (Fréchet and dynamic time warping (DTW)) on benchmark shape data, and use DTW in cluster analysis of biomarker profile observations. These reveal a primary group of ‘typical progressors,’ as well as a smaller group that shows relatively rapid progression. We then refine the analysis using ensemble clustering for both markers to obtain a single classification. The information from joint evaluation of the two biomarkers combined with ensemble clustering reveals subgroups of patients not identifiable through univariate analyses; noteworthy subgroups are those that appear to represent recently and chronically infected subsets.

Breathing patterns recognition: A functional data analysis approach

BACKGROUND AND OBJECTIVE

The ongoing pandemic proved fundamental is to assess a subject's respiratory functionality and breathing pattern measurement during quiet breathing is feasible in almost all patients, even those uncooperative. Breathing pattern consists of tidal volume and respiratory rate in an individual assessed by data tracks of lung or chest wall volume over time. State-of-art analysis of these data requires operator-dependent choices such as individuation of local minima in the track, elimination of anomalous breaths and individuation of breath clusters corresponding to different breathing patterns.

METHODS

A semi-automatic, robust and reproducible procedure was proposed to pre-process and analyse respiratory tracks, based on Functional Data Analysis (FDA) techniques, to identify representative breath curve and the corresponding breathing patterns. This was achieved through three steps: 1) breath separation through precise localization of the minima of the volume trace; 2) functional outlier breaths detection according to time-duration, magnitude and shape; 3) breath clustering to identify different pattern of interest, through K-medoids with Alignment. The method was firstly validated on simulated tracks and then applied to real data in conditions of clinical interest: operational volume change, exercise, mechanical ventilation, paradoxical breathing and age.

RESULTS

The total error in the accuracy of minima detection and in was less than 5%; with the artificial outliers being almost completely removed with an accuracy of 99%. During incremental exercise and independently on the bike resistance level, five clusters were identified (quiet breathing; recovery phase; onset of exercise; maximal and intermediate levels of exercise). During mechanical ventilation, the procedure was able to separate the non-ventilated from the ventilatory-supported breathing and to identify the worsening of paradoxical breathing due to the disease progression and the breathing pattern changes in healthy subjects due to age.

CONCLUSIONS

We proposed a robust validated automatic breathing patterns identification algorithm that extracted representative curves that could be implemented in clinical practice for objective comparison of the breathing patterns within and between subjects. In all case studies the identified patterns proved to be coherent with the clinical conditions and the physiopathology of the subjects, therefore enforcing the potential clinical translational value of the method.

Model-based joint curve registration and classification

In this paper, we consider the problem of classification of misaligned multivariate functional data. We propose to use a model-based approach for the joint registration and classification of such data. The observed functional inputs are modeled as a functional nonlinear mixed effects model containing a nonlinear functional fixed effect constructed upon warping functions to account for curve alignment, and a nonlinear functional random effects component to address the variability among subjects. The warping functions are also modeled to accommodate common effect within groups and the variability between subjects. Then, a functional logistic regression model defined upon the representation of the aligned curves and scalar inputs is used to account for curve classification. EM-based algorithms are developed to perform maximum likelihood inference of the proposed models. The identifiability of the registration model and the asymptotical properties of the proposed method are established. The performance of the proposed procedure is illustrated via simulation studies and an analysis of a hyoid bone movement data application. The statistical developments proposed in this paper were motivated by the hyoid bone movement study, the methodology is designed and presented generality and can be applied to numerous areas of scientific research.

Robust clustering of functional directional data

A robust approach for clustering functional directional data is proposed. The proposal adapts “impartial trimming” techniques to this particular framework. Impartial trimming uses the dataset itself to tell us which appears to be the most outlying curves. A feasible algorithm is proposed for its practical implementation justified by some theoretical properties. A “warping” approach is also introduced which allows including controlled time warping in that robust clustering procedure to detect typical “templates”. The proposed methodology is illustrated in a real data analysis problem where it is applied to cluster aircraft trajectories.

Alzheimer’s Garden: Understanding Social Behaviors of Patients with Dementia to Improve Their Quality of Life

This paper aims at understanding the social behavior of people with dementia through the use of technology, specifically by analyzing localization data of patients of an Alzheimer’s assisted care home in Italy. The analysis will allow to promote social relations by enhancing the facility’s spaces and activities, with the ultimate objective of improving residents’ quality of life. To assess social wellness and evaluate the effectiveness of the village areas and activities, this work introduces measures of sociability for both residents and places. Our data analysis is based on classical statistical methods and innovative machine learning techniques. First, we analyze the correlation between relational indicators and factors such as the outdoor temperature and the patients’ movements inside the facility. Then, we use statistical and accessibility analyses to determine the spaces residents appreciate the most and those in need of enhancements. We observe that patients’ sociability is strongly related to the considered factors. From our analysis, outdoor areas result less frequented and need spatial redesign to promote accessibility and attendance among patients. The data awareness obtained from our analysis will also be of great help to caregivers, doctors, and psychologists to enhance assisted care home social activities, adjust patient-specific treatments, and deepen the comprehension of the disease.

Spatio-Temporal Variation Characteristics of PM2.5 in the Beijing-Tianjin-Hebei Region, China, from 2013 to 2018

Air pollution, including particulate matter (PM_2.5) pollution, is extremely harmful to the environment as well as human health. The Beijing–Tianjin–Hebei (BTH) Region has experienced heavy PM_2.5 pollution within China. In this study, a six-year time series (January 2013–December 2018) of PM_2.5 mass concentration data from 102 air quality monitoring stations were studied to understand the spatio-temporal variation characteristics of the BTH region. The average annual PM_2.5 mass concentration in the BTH region decreased from 98.9 μg/m³ in 2013 to 64.9 μg/m³ in 2017. Therefore, China has achieved its Air Pollution Prevention and Control Plan goal of reducing the concentration of fine particulate matter in the BTH region by 25% by 2017. The PM_2.5 pollution in BTH plain areas showed a more significant change than mountains areas, with the highest PM_2.5 mass concentration in winter and the lowest in summer. The results of spatial autocorrelation and cluster analyses showed that the PM_2.5 mass concentration in the BTH region from 2013–2018 showed a significant spatial agglomeration, and that spatial distribution characteristics were high in the south and low in the north. Changes in PM_2.5 mass concentration in the BTH region were affected by both socio-economic factors and meteorological factors. Our results can provide a point of reference for making PM_2.5 pollution control decisions.

Registration for exponential family functional data

We introduce a novel method for separating amplitude and phase variability in exponential family functional data. Our method alternates between two steps: the first uses generalized functional principal components analysis to calculate template functions, and the second estimates smooth warping functions that map observed curves to templates. Existing approaches to registration have primarily focused on continuous functional observations, and the few approaches for discrete functional data require a pre-smoothing step; these methods are frequently computationally intensive. In contrast, we focus on the likelihood of the observed data and avoid the need for preprocessing, and we implement both steps of our algorithm in a computationally efficient way. Our motivation comes from the Baltimore Longitudinal Study on Aging, in which accelerometer data provides valuable insights into the timing of sedentary behavior. We analyze binary functional data with observations each minute over 24 hours for 592 participants, where values represent activity and inactivity. Diurnal patterns of activity are obscured due to misalignment in the original data but are clear after curves are aligned. Simulations designed to mimic the application indicate that the proposed methods outperform competing approaches in terms of estimation accuracy and computational efficiency. Code for our method and simulations is publicly available.

Functional regression models: Some directions of future research

We congratulate the authors for their excellent work that provides a clear overview of the large and now mature field of regression models for functional data. We here complement their discussion indicating some directions of further research that we deem particularly important.

kmlShape: An Efficient Method to Cluster Longitudinal Data (Time-Series) According to Their Shapes

BACKGROUND

Longitudinal data are data in which each variable is measured repeatedly over time. One possibility for the analysis of such data is to cluster them. The majority of clustering methods group together individual that have close trajectories at given time points. These methods group trajectories that are locally close but not necessarily those that have similar shapes. However, in several circumstances, the progress of a phenomenon may be more important than the moment at which it occurs. One would thus like to achieve a partitioning where each group gathers individuals whose trajectories have similar shapes whatever the time lag between them.

METHOD

In this article, we present a longitudinal data partitioning algorithm based on the shapes of the trajectories rather than on classical distances. Because this algorithm is time consuming, we propose as well two data simplification procedures that make it applicable to high dimensional datasets.

RESULTS

In an application to Alzheimer disease, this algorithm revealed a "rapid decline" patient group that was not found by the classical methods. In another application to the feminine menstrual cycle, the algorithm showed, contrarily to the current literature, that the luteinizing hormone presents two peaks in an important proportion of women (22%).

Clustering multivariate functional data with phase variation

When functional data come as multiple curves per subject, characterizing the source of variations is not a trivial problem. The complexity of the problem goes deeper when there is phase variation in addition to amplitude variation. We consider clustering problem with multivariate functional data that have phase variations among the functional variables. We propose a conditional subject-specific warping framework in order to extract relevant features for clustering. Using multivariate growth curves of various parts of the body as a motivating example, we demonstrate the effectiveness of the proposed approach. The found clusters have individuals who show different relative growth patterns among different parts of the body.

Warped functional analysis of variance

This article presents an Analysis of Variance model for functional data that explicitly incorporates phase variability through a time-warping component, allowing for a unified approach to estimation and inference in presence of amplitude and time variability. The focus is on single-random-factor models but the approach can be easily generalized to more complex ANOVA models. The behavior of the estimators is studied by simulation, and an application to the analysis of growth curves of flour beetles is presented. Although the model assumes a smooth latent process behind the observed trajectories, smootheness of the observed data is not required; the method can be applied to irregular time grids, which are common in longitudinal studies.

Object Oriented Data Analysis: A few methodological challenges

This is a discussion of the paper "Overview of object oriented data analysis" by J. Steve Marron and Andrés M. Alonso.