Generative models for functional data using phase and amplitude separation

Time-warping analysis for biological signals: methodology and application

Any set of biological signals has variability, both in the temporal and spatial domains. To extract characteristic features of the ensemble, these spatiotemporal profiles are typically summarized by their mean and variance, often requiring prior padding or resampling of the data to equalize signal length. Such compression can conceal essential information in the signal. This work presents the method of time-warping, reformulated as elastic functional data analysis (EFDA), in an accessible way. This powerful approach rescales the temporal evolution of signals, aligns them accurately, decouples their spatial and temporal variability, and faithfully extracts their characteristics. This technique was compared to conventional methods of normalizing or padding data followed by averaging, using synthetized signals with controlled variability and real human data from a complex manipulation task. Comparative analysis demonstrates that EFDA successfully reveals otherwise concealed features and teases apart temporal and spatial variability. Critical advances to the more common method of dynamic time-warping (DTW) are discussed. Application of EFDA and potential new insights are illustrated in the context of human motor neuroscience. Annotated code to facilitate the use of this technique is provided.

Topology-preserving contourwise shape fusion

The preservation of morphological features, such as protrusions and concavities, and of the topology of input shapes is important when establishing reference data for benchmarking segmentation algorithms or when constructing a mean or median shape. We present a contourwise topology-preserving fusion method, called shape-aware topology-preserving means (SATM), for merging complex simply connected shapes. The method is based on key point matching and piecewise contour averaging. Unlike existing pixelwise and contourwise fusion methods, SATM preserves topology and does not smooth morphological features. We also present a detailed comparison of SATM with state-of-the-art fusion techniques for the purpose of benchmarking and median shape construction. Our experiments show that SATM outperforms these techniques in terms of shape-related measures that reflect shape complexity, manifesting itself as a reliable method for both establishing a consensus of segmentation annotations and for computing mean shapes.

Motif clustering and digital biomarker extraction for free-living physical activity analysis

BACKGROUND

Analyzing free-living physical activity (PA) data presents challenges due to variability in daily routines and the lack of activity labels. Traditional approaches often rely on summary statistics, which may not capture the nuances of individual activity patterns. To address these limitations and advance our understanding of the relationship between PA patterns and health outcomes, we propose a novel motif clustering algorithm that identifies and characterizes specific PA patterns.

METHODS

This paper proposes an elastic distance-based motif clustering algorithm for identifying specific PA patterns (motifs) in free-living PA data. The algorithm segments long-term PA curves into short-term segments and utilizes elastic shape analysis to measure the similarity between activity segments. This enables the discovery of recurring motifs through pattern clustering. Then, functional principal component analysis (FPCA) is then used to extract digital biomarkers from each motif. These digital biomarkers can subsequently be used to explore the relationship between PA and health outcomes of interest.

RESULTS

We demonstrate the efficacy of our method through three real-world applications. Results show that digital biomarkers derived from these motifs effectively capture the association between PA patterns and disease outcomes, improving the accuracy of patient classification.

CONCLUSIONS

This study introduced a novel approach to analyzing free-living PA data by identifying and characterizing specific activity patterns (motifs). The derived digital biomarkers provide a more nuanced understanding of PA and its impact on health, with potential applications in personalized health assessment and disease detection, offering a promising future for healthcare.

Timing based clustering of childhood BMI trajectories reveals differential maturational patterns; Study in the Northern Finland Birth Cohorts 1966 and 1986

BACKGROUND/OBJECTIVES

Children's biological age does not always correspond to their chronological age. In the case of BMI trajectories, this can appear as phase variation, which can be seen as shift, stretch, or shrinking between trajectories. With maturation thought of as a process moving towards the final state - adult BMI, we assessed whether children can be divided into latent groups reflecting similar maturational age of BMI. The groups were characterised by early factors and time-related features of the trajectories.

SUBJECTS/METHODS

We used data from two general population birth cohort studies, Northern Finland Birth Cohorts 1966 and 1986 (NFBC1966 and NFBC1986). Height (n = 6329) and weight (n = 6568) measurements were interpolated in 34 shared time points using B-splines, and BMI values were calculated between 3 months to 16 years. Pairwise phase distances of 2999 females and 3163 males were used as a similarity measure in k-medoids clustering.

RESULTS

We identified three clusters of trajectories in females and males (Type 1: females, n = 1566, males, n = 1669; Type 2: females, n = 1028, males, n = 973; Type 3: females, n = 405, males, n = 521). Similar distinct timing patterns were identified in males and females. The clusters did not differ by sex, or early growth determinants studied.

CONCLUSIONS

Trajectory cluster Type 1 reflected to the shape of what is typically illustrated as the childhood BMI trajectory in literature. However, the other two have not been identified previously. Type 2 pattern was more common in the NFBC1966 suggesting a generational shift in BMI maturational patterns.

Topo-Geometric Analysis of Variability in Point Clouds Using Persistence Landscapes

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects known as persistence diagrams. To better aid statistical analysis, a functional representation of the diagrams, known as persistence landscapes, enable use of functional data analysis and machine learning tools. Topological and geometric variabilities inherent in point clouds are confounded in both persistence diagrams and landscapes, and it is important to distinguish topological signal from noise to draw reliable conclusions on the structure of the point clouds when using persistence homology. We develop a framework for decomposing variability in persistence diagrams into topological signal and topological noise through alignment of persistence landscapes using an elastic Riemannian metric. Aligned landscapes (amplitude) isolate the topological signal. Reparameterizations used for landscape alignment (phase) are linked to a resolution parameter used to generate persistence diagrams, and capture topological noise in the form of geometric, global scaling and sampling variabilities. We illustrate the importance of decoupling topological signal and topological noise in persistence diagrams (landscapes) using several simulated examples. We also demonstrate that our approach provides novel insights in two real data studies.

Reliability testing of an IMU-based 2-segment foot model for clinical gait analysis

BACKGROUND

The one of the most commonly used reference system for clinical gait analysis is an optical motion capture system (OMC) using a multi-segment foot model. A time- and cost-efficient alternative could be an inertial measurement unit (IMU)-based systems. However, these are limited to a single segment approach for the foot and ankle. Therefore, the current setup was modified to be based on a 2-segment foot model, allowing for a separate analysis of the hind- and midfoot. The study aimed to evaluate the reliability (inter-rater, intra-rater, and test-retest reliability) of an IMU-based 2-segment foot model.

MATERIAL AND METHODS

Twelve healthy subjects were recruited to test the inter-rater, intra-rater, and test-retest reliability of the new IMU based 2-segment foot model. Gait analysis was performed on a treadmill at a constant speed of 4 km/h. Kinematic data of the tibia/hindfoot, tibia/forefoot and hindfoot/forefoot over 100 % gait cycle were analyzed. The reliability was tested by using statistical parametric mapping (SPM) and the intraclass correlation coefficient (ICC).

RESULTS

The SPM showed no significant difference for inter-, intra-rater, and test-retest reliability, but for a small segment of tibia/forefoot dorsiflexion test-retest reliability (2.1° difference). The single standard deviation measurement error for the sagittal and transverse plane was <5° and worse for the frontal plane.

CONCLUSION

The new 2-segment foot model revealed a high inter-rater, intra-rater, and test-retest reliability. It is suitable for use in adult clinical practice. Still, comparative data to the OMC system using a multi-segment foot model are missing.

Gait phase normalization resolves the problem of different phases being compared in gait cycle normalization

For time-continuous analysis of gait, the problem of variations in cycle durations is resolved by normalizing to the gait cycle, but results depend on the definition of the cycle start. Gait cycle normalization ignores variations in gait phase durations, which results in averaging and comparing data across different phases. We propose gait phase normalization as part of a comprehensive method for independently analyzing magnitude and timing differences. First, gait phases are identified and differences in absolute and/or relative timing of phase durations or any point of interest between conditions or groups are analyzed using standard statistics. Next, time-continuous gait data is normalized to gait phases, and statistical parametric mapping (SPM) is used to assess magnitude differences in gait data. This approach is demonstrated on data recorded from ten young healthy adults walking on a treadmill at five different speeds. Sagittal knee angle was normalized to gait cycle or gait phase using five different gait cycle start events. Walking at different speeds resulted in significant changes in gait phase durations, highlighting a problem ignored by gait cycle normalization. SPM results for knee angle normalized to gait cycle varied from normalization to gait phases. Gait phase normalized SPM results were robust to the definition of the cycle start, in contrast to gait cycle normalized data. The approach of analyzing phase durations and normalizing data to gait phases overcomes previous limitations and enables a comprehensive analysis of magnitude and timing differences in time-continuous gait data and could be readily adapted to other tasks.

Improved data quality and statistical power of trial-level event-related potentials with Bayesian random-shift Gaussian processes

Studies of cognitive processes via electroencephalogram (EEG) recordings often analyze group-level event-related potentials (ERPs) averaged over multiple subjects and trials. This averaging procedure can obscure scientifically relevant variability across subjects and trials, but has been necessary due to the difficulties posed by inference of trial-level ERPs. We introduce the Bayesian Random Phase-Amplitude Gaussian Process (RPAGP) model, for inference of trial-level amplitude, latency, and ERP waveforms. We apply RPAGP to data from a study of ERP responses to emotionally arousing images. The model estimates of trial-specific signals are shown to greatly improve statistical power in detecting significant differences in experimental conditions compared to existing methods. Our results suggest that replacing the observed data with the de-noised RPAGP predictions can potentially improve the sensitivity and accuracy of many of the existing ERP analysis pipelines.

Time-Warping Analysis of the T-Wave Peak-to-End Interval to Quantify Ventricular Repolarization Dispersion During Ischemia

Variations in the dispersion of ventricular repolarization can be quantified by T-wave time-warping based index, dw. However, the early phase of the T-wave can be affected by ST-segment changes during ischemia. We hypothesized that restricting dw to the T-wave peak-to-end ( Tpe) would circumvent this limitation while still quantifying variations in repolarization dispersion. A total of 101 ECG recordings from patients undergoing coronary occlusion, together with their control recordings, were analyzed. A series of dw values was calculated by quantifying the Tpe morphological variations between the T-waves at different occlusion stages and a baseline T-wave. We introduced a normalized version of dw, Rd, reflecting variations of dw during occlusion relative to control recordings ( Rd = 1 corresponds to the same level of variation). The dw series followed a gradually increasing trend with occlusion time, reaching median [range] Rd values of 9.44 [1.01, 80.74] at the occlusion end. Rd at occlusion end was significantly higher than threshold values of 1, 2, 5, and 10 in 94.1%, 85.11%, 64.4% and 48.5% of patients, respectively. The spatial lead-wise analysis of dw showed distinct distributions depending on the occluded artery, suggesting a relation with the ischemia location. The relative variation R with ischemia of index dw (9.4) is greater than that of the T-wave amplitude (7.7), Tpe interval (2.7) and T-wave width (3.0). In conclusion, dw tracks ischemic-induced variations in repolarization dispersion in a more robust manner than classical indexes, avoiding the impact of ST segment elevation/depression or early T-wave distortions, thus warranting further clinical studies.

Associations of interleukin-6 with functional trajectories in older adults with cancer: Findings from the Health, Aging, and Body Composition Study

BACKGROUND

Elevated markers of inflammation, such as interleukin-6 (IL-6), are associated with aging, cancer, and functional decline. We assessed the association of pre-diagnosis IL-6 levels with post-diagnosis functional trajectories among older adults with cancer. Black and White participants experience different social structures, therefore we sought to understand whether these associations differ between Black and White participants.

METHODS

We conducted secondary analysis of the Health Aging, Body, and Composition (ABC) prospective longitudinal cohort study. Participants were recruited from 4/1997 to 6/1998. We included 179 participants with a new cancer diagnosis and IL-6 level measured within 2 years before diagnosis. Primary endpoint was functional measures (self-reported ability to walk 1/4, 20-meter gait speed). Nonparametric longitudinal models were used to cluster the trajectories; multinomial and logistic regressions to model associations.

FINDINGS

Mean age was 74 (SD 2.9); 36 % identified as Black. For self-reported functional status, we identified 3 clusters: high stable, decline, low stable. For gait speed, we identified 2 clusters: resilient, decline. The relationship between cluster trajectory and IL-6 was different between Black and White participants (p for interaction<0.05). For gait speed, among White participants, a greater log IL-6 level was associated with greater odds of being in the decline vs. resilient cluster [Adjusted Odds Ratio (AOR): 4.31, 95 % CI: 1.43, 17.46]. Among Black participants, a greater log IL-6 levels were associated with lower odds of being in the decline vs. resilient cluster (AOR: 0.49, 95 % CI: 0.10, 2.08). Directionality was similar for self-reported ability to walk ¼ mile (high stable vs. low stable). Among White participants, a higher log IL-6 level was associated numerically with greater odds of being in the low stable vs. high stable cluster (AOR: 1.99, 95 % CI: 0.82, 4.85). Among Black participants, a higher log IL-6 level was associated numerically with lower odds of being in the low stable cluster vs. high stable cluster (AOR: 0.78, 95 % CI: 0.30, 2.00).

INTERPRETATION

The association between IL-6 levels and functional trajectories of older adults differed by race. Future analyses exploring stressors faces by other minoritized racial backgrounds are needed to determine the association between IL-6 and functional trajectories.

PANEL

RESEARCH IN CONTEXT: Evidence before this study: Previous research has shown that aging is the greatest risk factor for cancer and older adults with cancer experience a higher burden of comorbidities, increasing their risk of functional decline. Race has also been shown to be associated with increased risk for functional decline. Black individuals are exposed to more chronic negative social determinants, compared to White individuals. Previous work has shown that chronic exposure to negative social determinants leads to elevated levels of inflammatory markers, such as IL-6, but studies investigating the relationship between inflammatory markers and functional decline are limited. Added value of this study: Authors of this study sought to understand the association between pre-diagnosis IL-6 levels and functional trajectories post-diagnosis in older adults with cancer, and whether these associations differed between Black and White participants with cancer. Authors decided to utilize the data from the Health, Aging and Body Composition (Health ABC) Study. The Health ACB study was a prospective longitudinal cohort study that has a high representation of Black older adults and collected inflammatory cytokines and physical function data over time. Implications of all available evidence: This work adds to the literature by providing an opportunity to study the difference in the relationships between IL-6 levels and functional trajectories between older Black and White participants with cancer. Identifying factors associated with functional decline and its trajectories may inform treatment decision making and guide development of supportive care interventions to prevent functional decline. Additionally, given the disparities in clinical outcomes for Black individuals, a better understanding of the difference in functional decline based on race will allow more equitable care to be distributed.

Shape Analysis of Functional Data With Elastic Partial Matching

Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. In practice, functional data often comes with unmatched boundaries. It happens, for example, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering of functions with phase variability and uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods.

Bayesian Framework for Simultaneous Registration and Estimation of Noisy, Sparse and Fragmented Functional Data

In many applications, smooth processes generate data that is recorded under a variety of observational regimes, including dense sampling and sparse or fragmented observations that are often contaminated with error. The statistical goal of registering and estimating the individual underlying functions from discrete observations has thus far been mainly approached sequentially without formal uncertainty propagation, or in an application-specific manner by pooling information across subjects. We propose a unified Bayesian framework for simultaneous registration and estimation, which is flexible enough to accommodate inference on individual functions under general observational regimes. Our ability to do this relies on the specification of strongly informative prior models over the amplitude component of function variability using two strategies: a data-driven approach that defines an empirical basis for the amplitude subspace based on training data, and a shape-restricted approach when the relative location and number of extrema is well-understood. The proposed methods build on the elastic functional data analysis framework to separately model amplitude and phase variability inherent in functional data. We emphasize the importance of uncertainty quantification and visualization of these two components as they provide complementary information about the estimated functions. We validate the proposed framework using multiple simulation studies and real applications.

Describing Droplet Motion on Surface-Textured Ratchet Tracks with an Inverted Double Pendulum Model

We describe the motion of a droplet on a textured ratchet track using a nonlinear resonator model. A textured ratchet track is composed of a semicircular pillar array that induces a net surface tension local gradient on a droplet placed on it. When a vertical vibration is applied, hysteresis is overcome, and the droplet moves toward the local lower energy barrier; however, due to the repetitive structure of texture, it keeps moving until the end of the track. The droplet motion depends on the amplitude and frequency of the vertical oscillation, and this dependence is nonlinear. Therefore, finding a fully analytic solution to represent this motion is not trivial. Consequently, the droplet motion remains poorly understood. In this study, we elaborate on the utility of a double pendulum as a basis for modeling the droplet motion on surfaces inducing asymmetric force. Similar to the droplet motion, resonators, such as a double pendulum, are simple, yet nonlinear systems. Moreover, an inverted double pendulum motion has key characteristics such as the two-phase motion and the double peak motion, which are also observed in the droplet motion. We use various data-processing methods to highlight the similarity between these two systems both qualitatively and quantitatively. After establishing this comparison, we propose a model that utilizes an inverted double pendulum mounted on a moving cart to successfully simulate the motion of a droplet on a ratchet track. This methodology will lead to the development of an accurate droplet-motion modeling approach, and we believe that it will be useful to understand droplet dynamics more deeply.

Monitoring blood potassium concentration in hemodialysis patients by quantifying T-wave morphology dynamics

We investigated the ability of time-warping-based ECG-derived markers of T-wave morphology changes in time ([Formula: see text]) and amplitude ([Formula: see text]), as well as their non-linear components ([Formula: see text] and [Formula: see text]), and the heart rate corrected counterpart ([Formula: see text]), to monitor potassium concentration ([Formula: see text]) changes ([Formula: see text]) in end-stage renal disease (ESRD) patients undergoing hemodialysis (HD). We compared the performance of the proposed time-warping markers, together with other previously proposed [Formula: see text] markers, such as T-wave width ([Formula: see text]) and T-wave slope-to-amplitude ratio ([Formula: see text]), when computed from standard ECG leads as well as from principal component analysis (PCA)-based leads. 48-hour ECG recordings and a set of hourly-collected blood samples from 29 ESRD-HD patients were acquired. Values of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] were calculated by comparing the morphology of the mean warped T-waves (MWTWs) derived at each hour along the HD with that from a reference MWTW, measured at the end of the HD. From the same MWTWs [Formula: see text] and [Formula: see text] were also extracted. Similarly, [Formula: see text] was calculated as the difference between the [Formula: see text] values at each hour and the [Formula: see text] reference level at the end of the HD session. We found that [Formula: see text] and [Formula: see text] showed higher correlation coefficients with [Formula: see text] than [Formula: see text]-Spearman's ([Formula: see text]) and Pearson's (r)-and [Formula: see text]-Spearman's ([Formula: see text])-in both SL and PCA approaches being the intra-patient median [Formula: see text] and [Formula: see text] in SL and [Formula: see text] and [Formula: see text] in PCA respectively. Our findings would point at [Formula: see text] and [Formula: see text] as the most suitable surrogate of [Formula: see text], suggesting that they could be potentially useful for non-invasive monitoring of ESRD-HD patients in hospital, as well as in ambulatory settings. Therefore, the tracking of T-wave morphology variations by means of time-warping analysis could improve continuous and remote [Formula: see text] monitoring of ESRD-HD patients and flagging risk of [Formula: see text]-related cardiovascular events.

Robust Functional Manifold Clustering

In machine learning, it is common to interpret each data sample as a multivariate vector disregarding the correlations among covariates. However, the data may actually be functional, i.e., each data point is a function of some variable, such as time, and the function is discretely sampled. The naive treatment of functional data as traditional multivariate data can lead to poor performance due to the correlations. In this article, we focus on subspace clustering for functional data or curves and propose a new method robust to shift and rotation. The idea is to define a function or curve and all its versions generated by shift and rotation as an equivalent class and then to find the subspace structure among all equivalent classes as the surrogate for all curves. Experimental evaluation on synthetic and real data reveals that this method massively outperforms prior clustering methods in both speed and accuracy when clustering functional data.

Modeling and analyzing neural signals with phase variability using Fisher-Rao registration

BACKGROUND

The dynamic time warping (DTW) has recently been introduced to analyze neural signals such as EEG and fMRI where phase variability plays an important role in the data.

NEW METHOD

In this study, we propose to adopt a more powerful method, referred to as the Fisher-Rao Registration (FRR), to study the phase variability.

COMPARISON WITH EXISTING METHODS

We systematically compare FRR with DTW in three aspects: (1) basic framework, (2) mathematical properties, and (3) computational efficiency.

RESULTS

We show that FRR has superior performance in all these aspects and the advantages are well illustrated with simulation examples.

CONCLUSIONS

We then apply the FRR method to two real experimental recordings - one fMRI and one EEG data set. It is found the FRR method properly removes the phase variability in each set. Finally, we use the FRR framework to examine brain networks in these two data sets and the result demonstrates the effectiveness of the new method.

A geometric approach for computing tolerance bounds for elastic functional data

We develop a method for constructing tolerance bounds for functional data with random warping variability. In particular, we define a generative, probabilistic model for the amplitude and phase components of such observations, which parsimoniously characterizes variability in the baseline data. Based on the proposed model, we define two different types of tolerance bounds that are able to measure both types of variability, and as a result, identify when the data has gone beyond the bounds of amplitude and/or phase. The first functional tolerance bounds are computed via a bootstrap procedure on the geometric space of amplitude and phase functions. The second functional tolerance bounds utilize functional Principal Component Analysis to construct a tolerance factor. This work is motivated by two main applications: process control and disease monitoring. The problem of statistical analysis and modeling of functional data in process control is important in determining when a production has moved beyond a baseline. Similarly, in biomedical applications, doctors use long, approximately periodic signals (such as the electrocardiogram) to diagnose and monitor diseases. In this context, it is desirable to identify abnormalities in these signals. We additionally consider a simulated example to assess our approach and compare it to two existing methods.

Reproducibility of Brain Responses: High for Speech Perception, Low for Reading Difficulties

Neuroscience findings have recently received critique on the lack of replications. To examine the reproducibility of brain indices of speech sound discrimination and their role in dyslexia, a specific reading difficulty, brain event-related potentials using EEG were measured using the same cross-linguistic passive oddball paradigm in about 200 dyslexics and 200 typically reading 8-12-year-old children from four countries with different native languages. Brain responses indexing speech and non-speech sound discrimination were extremely reproducible, supporting the validity and reliability of cognitive neuroscience methods. Significant differences between typical and dyslexic readers were found when examined separately in different country and language samples. However, reading group differences occurred at different time windows and for different stimulus types between the four countries. This finding draws attention to the limited generalizability of atypical brain response findings in children with dyslexia across language environments and raises questions about a common neurobiological factor for dyslexia. Our results thus show the robustness of neuroscience methods in general while highlighting the need for multi-sample studies in the brain research of language disorders.

NITPicker: selecting time points for follow-up experiments

Background

The design of an experiment influences both what a researcher can measure, as well as how much confidence can be placed in the results. As such, it is vitally important that experimental design decisions do not systematically bias research outcomes. At the same time, making optimal design decisions can produce results leading to statistically stronger conclusions. Deciding where and when to sample are among the most critical aspects of many experimental designs; for example, we might have to choose the time points at which to measure some quantity in a time series experiment. Choosing times which are too far apart could result in missing short bursts of activity. On the other hand, there may be time points which provide very little information regarding the overall behaviour of the quantity in question.

Results

In this study, we develop a tool called NITPicker (Next Iteration Time-point Picker) for selecting optimal time points (or spatial points along a single axis), that eliminates some of the biases caused by human decision-making, while maximising information about the shape of the underlying curves. NITPicker uses ideas from the field of functional data analysis. NITPicker is available on the Comprehensive R Archive Network (CRAN) and code for drawing figures is available on Github (https://github.com/ezer/NITPicker).

Conclusions

NITPicker performs well on diverse real-world datasets that would be relevant for varied biological applications, including designing follow-up experiments for longitudinal gene expression data, weather pattern changes over time, and growth curves.

Electronic supplementary material

The online version of this article (10.1186/s12859-019-2717-5) contains supplementary material, which is available to authorized users.

Functional Data Analyses of Gait Data Measured Using In-Shoe Sensors

In studies of gait, continuous measurement of force exerted by the ground on a body, or ground reaction force (GRF), provides valuable insights into biomechanics, locomotion, and the possible presence of pathology. However, gold-standard measurement of GRF requires a costly in-lab observation obtained with sophisticated equipment and computer systems. Recently, in-shoe sensors have been pursued as a relatively inexpensive alternative to in-lab measurement. In this study, we explore the properties of continuous in-shoe sensor recordings using a functional data analysis approach. Our case study is based on measurements of three healthy subjects, with more than 300 stances (defined as the period between the foot striking and lifting from the ground) per subject. The sensor data show both phase and amplitude variabilities; we separate these sources via curve registration. We examine the correlation of phase shifts across sensors within a stance to evaluate the pattern of phase variability shared across sensors. Using the registered curves, we explore possible associations between in-shoe sensor recordings and GRF measurements to evaluate the in-shoe sensor recordings as a possible surrogate for in-lab GRF measurements.

Dissimilarity for functional data clustering based on smoothing parameter commutation

Many studies measure the same type of information longitudinally on the same subject at multiple time points, and clustering of such functional data has many important applications. We propose a novel and easy method to implement dissimilarity measure for functional data clustering based on smoothing splines and smoothing parameter commutation. This method handles data observed at regular or irregular time points in the same way. We measure the dissimilarity between subjects based on varying curve estimates with pairwise commutation of smoothing parameters. The intuition is that smoothing parameters of smoothing splines reflect the inverse of the signal-to-noise ratios and that when applying an identical smoothing parameter the smoothed curves for two similar subjects are expected to be close. Our method takes into account the estimation uncertainty using smoothing parameter commutation and is not strongly affected by outliers. It can also be used for outlier detection. The effectiveness of our proposal is shown by simulations comparing it to other dissimilarity measures and by a real application to methadone dosage maintenance levels.

Separating Timing, Movement Conditions and Individual Differences in the Analysis of Human Movement

A central task in the analysis of human movement behavior is to determine systematic patterns and differences across experimental conditions, participants and repetitions. This is possible because human movement is highly regular, being constrained by invariance principles. Movement timing and movement path, in particular, are linked through scaling laws. Separating variations of movement timing from the spatial variations of movements is a well-known challenge that is addressed in current approaches only through forms of preprocessing that bias analysis. Here we propose a novel nonlinear mixed-effects model for analyzing temporally continuous signals that contain systematic effects in both timing and path. Identifiability issues of path relative to timing are overcome by using maximum likelihood estimation in which the most likely separation of space and time is chosen given the variation found in data. The model is applied to analyze experimental data of human arm movements in which participants move a hand-held object to a target location while avoiding an obstacle. The model is used to classify movement data according to participant. Comparison to alternative approaches establishes nonlinear mixed-effects models as viable alternatives to conventional analysis frameworks. The model is then combined with a novel factor-analysis model that estimates the low-dimensional subspace within which movements vary when the task demands vary. Our framework enables us to visualize different dimensions of movement variation and to test hypotheses about the effect of obstacle placement and height on the movement path. We demonstrate that the approach can be used to uncover new properties of human movement.

When you move a cup to a new location on a table, the movement of lifting, transporting, and setting down the cup appears to be completely automatic. Although the hand could take continuously many different paths and move on any temporal trajectory, real movements are highly regular and reproducible. From repetition to repetition movements vary, and the pattern of variance reflects movement conditions and movement timing. If another person performs the same task, the movement will be similar. When we look more closely, however, there are systematic individual differences. Some people will overcompensate when avoiding an obstacle and some people will systematically move slower than others. When we want to understand human movement, all these aspects are important. We want to know which parts of a movement are common across people and we want to quantify the different types of variability. Thus, the models we use to analyze movement data should contain all the mentioned effects. In this work, we developed a framework for statistical analysis of movement data that respects these structures of movements. We showed how this framework modeled the individual characteristics of participants better than other state-of-the-art modeling approaches. We combined the timing-and-path-separating model with a novel factor analysis model for analyzing the effect of obstacles on spatial movement paths. This combination allowed for an unprecedented ability to quantify and display different sources of variation in the data. We analyzed data from a designed experiment of arm movements under various obstacle avoidance conditions. Using the proposed statistical models, we documented three findings: a linearly amplified deviation in mean path related to increase in obstacle height; a consistent asymmetric pattern of variation along the movement path related to obstacle placement; and the existence of obstacle-distance invariant focal points where mean trajectories intersect in the frontal and vertical planes.

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.

Unifying Amplitude and Phase Analysis: A Compositional Data Approach to Functional Multivariate Mixed-Effects Modeling of Mandarin Chinese

Mandarin Chinese is characterized by being a tonal language; the pitch (or F0) of its utterances carries considerable linguistic information. However, speech samples from different individuals are subject to changes in amplitude and phase, which must be accounted for in any analysis that attempts to provide a linguistically meaningful description of the language. A joint model for amplitude, phase, and duration is presented, which combines elements from functional data analysis, compositional data analysis, and linear mixed effects models. By decomposing functions via a functional principal component analysis, and connecting registration functions to compositional data analysis, a joint multivariate mixed effect model can be formulated, which gives insights into the relationship between the different modes of variation as well as their dependence on linguistic and nonlinguistic covariates. The model is applied to the COSPRO-1 dataset, a comprehensive database of spoken Taiwanese Mandarin, containing approximately 50,000 phonetically diverse sample F0 contours (syllables), and reveals that phonetic information is jointly carried by both amplitude and phase variation. Supplementary materials for this article are available online.

Pairwise alignment of chromatograms using an extended Fisher-Rao metric

A conceptually new approach for aligning chromatograms is introduced and applied to examples of metabolite identification in human blood plasma by liquid chromatography-mass spectrometry (LC-MS). A square-root representation of the chromatogram's derivative coupled with an extended Fisher-Rao metric enables the computation of relative differences between chromatograms. Minimization of these differences using a common dynamic programming algorithm brings the chromatograms into alignment. Application to a complex sample, National Institute of Standards and Technology (NIST) Standard Reference Material 1950, Metabolites in Human Plasma, analyzed by two different LC-MS methods having significantly different ranges of elution time is described.