Latent deformation models for multivariate functional data and time-warping separability

Multivariate functional data present theoretical and practical complications that are not found in univariate functional data. One of these is a situation where the component functions of multivariate functional data are positive and are subject to mutual time warping. That is, the component processes exhibit a common shape but are subject to systematic phase variation across their domains in addition to subject-specific time warping, where each subject has its own internal clock. This motivates a novel model for multivariate functional data that connect such mutual time warping to a latent-deformation-based framework by exploiting a novel time-warping separability assumption. This separability assumption allows for meaningful interpretation and dimension reduction. The resulting latent deformation model is shown to be well suited to represent commonly encountered functional vector data. The proposed approach combines a random amplitude factor for each component with population-based registration across the components of a multivariate functional data vector and includes a latent population function, which corresponds to a common underlying trajectory. We propose estimators for all components of the model, enabling implementation of the proposed data-based representation for multivariate functional data and downstream analyses such as Fréchet regression. Rates of convergence are established when curves are fully observed or observed with measurement error. The usefulness of the model, interpretations, and practical aspects are illustrated in simulations and with application to multivariate human growth curves and multivariate environmental pollution data.

Arc-Length Re-Parametrization and Signal Registration to Determine a Characteristic Average and Statistical Response Corridors of Biomechanical Data

A characteristic average and biofidelity response corridors are commonly used to represent the average behaviour and variability of biomechanical signal data for analysis and comparison to surrogates such as anthropometric test devices and computational models. However, existing methods for computing the characteristic average and corresponding response corridors of experimental data are often customized to specific types or shapes of signal and therefore limited in general applicability. In addition, simple methods such as point-wise averaging can distort or misrepresent important features if signals are not well aligned and highly correlated. In this study, an improved method of computing the characteristic average and response corridors of a set of experimental signals is presented based on arc-length re-parameterization and signal registration. The proposed arc-length corridor method was applied to three literature datasets demonstrating a range of characteristics common to biomechanical data, such as monotonic increasing force-displacement responses with variability, oscillatory acceleration-time signals, and hysteretic load-unload data. The proposed method addresses two challenges in assessing experimental data: arc-length re-parameterization enables the assessment of complex-shaped signals, including hysteretic load-unload data, while signal registration aligned signal features such as peaks and valleys to prevent distortion when determining the characteristic average response. The arc-length corridor method was shown to compute the characteristic average and response corridors for a wide range of biomechanical data, while providing a consistent statistical framework to characterize variability in the data. The arc-length corridor method is provided to the community in the freely available and open-source software package, ARCGen.

Nonparametric Bayesian Regression and Classification on Manifolds, With Applications to 3D Cochlear Shapes

Advanced shape analysis studies such as regression and classification need to be performed on curved manifolds, where often, there is a lack of standard statistical formulations. To overcome these limitations, we introduce a novel machine-learning method on the shape space of curves that avoids direct inference on infinite-dimensional spaces and instead performs Bayesian inference with spherical Gaussian processes decomposition. As an application, we study the shape of the cochlear spiral-shaped cavity within the petrous part of the temporal bone. This problem is particularly challenging due to the relationship between shape and gender, especially in children. Experimental results for both synthetic and real data show improved performance compared to state-of-the-art 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.

Automated extraction of auditory brainstem response latencies and amplitudes by means of non-linear curve registration

BACKGROUND AND OBJECTIVES

Animal results have suggested that auditory brainstem responses (ABRs) to transient sounds presented at supra-threshold levels may be useful for measuring hearing damage that is hidden to current audiometric tests. Evaluating such ABRs requires extracting the latencies and amplitudes of relevant deflections, or "waves". Currently, this is mostly done by human observers manually picking the waves' peaks and troughs in each individual response - a process that is both time-consuming and requiring of expert experience. Here, we propose a highly automated procedure for extracting individual ABR wave latencies and amplitudes based on the well-established methodology of non-linear curve registration.

METHODS

First, the to-be-analysed individual ABRs are temporally aligned - either with one another or, if available, with a pre-existing template - by locally compressing or stretching their time axes with smooth and invertible time warping functions. Then, the individual latencies and amplitudes of relevant ABR waves are obtained by picking the latencies of the waves' peaks and troughs on the common (aligned) time axis and combining these with the individual aligned responses and inverse time warping functions.

RESULTS

Using an example ABR data set with a wide range of response latencies and signal-to-noise ratios (SNRs), we test different choices for fitting the time warping functions. We cross-validate the warping results using independent response replicates and compare automatically and manually extracted latencies and amplitudes for ABR waves I and V. Using a Bayesian approach, we show that, for the best registration condition, automatic and manual data were statistically similar.

CONCLUSIONS

Non-linear curve registration can be used to temporally align individual ABRs and extract their wave latencies and amplitudes in a way that closely matches results from manual picking.

Analogies can speed up the motor learning process

Analogies have been shown to improve motor learning in various tasks and settings. In this study we tested whether applying analogies can shorten the motor learning process and induce insight and skill improvement in tasks that usually demand many hours of practice. Kinematic measures were used to quantify participant’s skill and learning dynamics. For this purpose, we used a drawing task, in which subjects drew lines to connect dots, and a mirror game, in which subjects tracked a moving stimulus. After establishing a baseline, subjects were given an analogy, explicit instructions or no further instruction. We compared their improvement in skill (quantified by coarticulation or smoothness), accuracy and movement duration. Subjects in the analogy and explicit groups improved their coarticulation in the target task, while significant differences were found in the mirror game only at a slow movement frequency between analogy and controls.

We conclude that a verbal analogy can be a useful tool for rapidly changing motor kinematics and movement strategy in some circumstances, although in the tasks selected it did not produce better performance in most measurements than explicit guidance. Furthermore, we observed that different movement facets may improve independently from others, and may be selectively affected by verbal instructions. These results suggest an important role for the type of instruction in motor learning.

Distribution on Warp Maps for Alignment of Open and Closed Curves

Alignment of curve data is an integral part of their statistical analysis, and can be achieved using modelor optimization-based approaches. The parameter space is usually the set of monotone, continuous warp maps of a domain. Infinite-dimensional nature of the parameter space encourages sampling based approaches, which require a distribution on the set of warp maps. Moreover, the distribution should also enable sampling in the presence of important landmark information on the curves which constrain the warp maps. For alignment of closed and open curves in ℝ d , d = 1, 2, 3, possibly with landmark information, we provide a constructive, point-process based definition of a distribution on the set of warp maps of [0, 1] and the unit circle S , that is, (1) simple to sample from, and (2) possesses the desiderata for decomposition of the alignment problem with landmark constraints into multiple unconstrained ones. For warp maps on [0, 1], the distribution is related to the Dirichlet process. We demonstrate its utility by using it as a prior distribution on warp maps in a Bayesian model for alignment of two univariate curves, and as a proposal distribution in a stochastic algorithm that optimizes a suitable alignment functional for higher-dimensional curves. Several examples from simulated and real datasets are provided.

Observation of an expert model induces a skilled movement coordination pattern in a single session of intermittent practice

We tested how observation of a skilled pattern of planar movements can assist in the learning of a new motor skill, which otherwise requires rigorous long-term practice to achieve fast and smooth performance. Sixty participants performed a sequence of planar hand movements on pre-test, acquisition, post-test and 24 h post-training blocks, under 1 of 4 conditions: an observation group (OG), a slowed observation group (SOG), a random motion control group (RMCG) and a double physical training control group (DPTCG). The OG and SOG observed an expert model's right hand performing the study task intermittently throughout acquisition, RMCG observed random dots movement instead of a model. Participants in the DPTCG received extra physical practice trials instead of the visually observed trials. Kinematic analysis revealed that only in conditions with observation of an expert model there was an instant robust improvement in motor planning of the task. This step-wise improvement was not only persistent in post-training retests but was also apparently implicit and subject to further incremental improvements in movement strategy over the period of 24 hours. The rapid change in motor strategy was accompanied by a transient within-session increase in spatial error for the observation groups, but this went away by 24 h post-training. We suggest that observation of hand movements of an expert model coaligned with self-produced movements during training can significantly condense the time-course of ecologically relevant drawing/writing skill mastery.

Measuring Brain Connectivity via Shape Analysis of fMRI Time Courses and Spectra

We present a shape matching approach for functional magnetic resonance imaging (fMRI) time course and spectral alignment. We use ideas from differential geometry and functional data analysis to define a functional representation for fMRI signals. The space of fMRI functions is then equipped with a reparameterization invariant Riemannian metric that enables elastic alignment of both amplitude and phase of the fMRI time courses as well as their power spectral densities. Experimental results show significant increases in pairwise node to node correlations and coherences following alignment. We apply this method for finding group differences in connectivity between patients with major depression and healthy controls.

A time warping approach to multiple sequence alignment

We propose an approach for multiple sequence alignment (MSA) derived from the dynamic time warping viewpoint and recent techniques of curve synchronization developed in the context of functional data analysis. Starting from pairwise alignments of all the sequences (viewed as paths in a certain space), we construct a median path that represents the MSA we are looking for. We establish a proof of concept that our method could be an interesting ingredient to include into refined MSA techniques. We present a simple synthetic experiment as well as the study of a benchmark dataset, together with comparisons with 2 widely used MSA softwares.

Synchrony in Joint Action Is Directed by Each Participant's Motor Control System

In this work, we ask how the probability of achieving synchrony in joint action is affected by the choice of motion parameters of each individual. We use the mirror game paradigm to study how changes in leader's motion parameters, specifically frequency and peak velocity, affect the probability of entering the state of co-confidence (CC) motion: a dyadic state of synchronized, smooth and co-predictive motions. In order to systematically study this question, we used a one-person version of the mirror game, where the participant mirrored piece-wise rhythmic movements produced by a computer on a graphics tablet. We systematically varied the frequency and peak velocity of the movements to determine how these parameters affect the likelihood of synchronized joint action. To assess synchrony in the mirror game we used the previously developed marker of co-confident (CC) motions: smooth, jitter-less and synchronized motions indicative of co-predicative control. We found that when mirroring movements with low frequencies (i.e., long duration movements), the participants never showed CC, and as the frequency of the stimuli increased, the probability of observing CC also increased. This finding is discussed in the framework of motor control studies showing an upper limit on the duration of smooth motion. We confirmed the relationship between motion parameters and the probability to perform CC with three sets of data of open-ended two-player mirror games. These findings demonstrate that when performing movements together, there are optimal movement frequencies to use in order to maximize the possibility of entering a state of synchronized joint action. It also shows that the ability to perform synchronized joint action is constrained by the properties of our motor control systems.

Growth curve registration for evaluating salinity tolerance in barley

BACKGROUND

Smarthouses capable of non-destructive, high-throughput plant phenotyping collect large amounts of data that can be used to understand plant growth and productivity in extreme environments. The challenge is to apply the statistical tool that best analyzes the data to study plant traits, such as salinity tolerance, or plant-growth-related traits.

RESULTS

We derive family-wise salinity sensitivity (FSS) growth curves and use registration techniques to summarize growth patterns of HEB-25 barley families and the commercial variety, Navigator. We account for the spatial variation in smarthouse microclimates and in temporal variation across phenotyping runs using a functional ANOVA model to derive corrected FSS curves. From FSS, we derive corrected values for family-wise salinity tolerance, which are strongly negatively correlated with Na but not significantly with K, indicating that Na content is an important factor affecting salinity tolerance in these families, at least for plants of this age and grown in these conditions.

CONCLUSIONS

Our family-wise methodology is suitable for analyzing the growth curves of a large number of plants from multiple families. The corrected curves accurately account for the spatial and temporal variations among plants that are inherent to high-throughput experiments.

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.

Automatic cycle averaging for denoising approximately periodic spatiotemporal signals

Optical mapping has become a common tool for cardiovascular research, providing high resolution spatiotemporal data of cardiac action potential propagation. However, noise in cardiac optical mapping (COM) data hampers quantitative measurements and analyses. Spatial and temporal filters have been used to increase the signal-to-noise ratio (SNR) of COM data. Although these help reduce noise and increase SNR, they also lower the spatial and temporal resolution of the data, which is not desirable. This paper utilizes the approximate periodicity of COM data to perform denoising by averaging cycles. We consider the entire approximately periodic spatiotemporal (APST) COM data as a concatenation of random samples generated from a deterministic single cycle spatiotemporal signal. The image difference signal (IDS) was defined and calculated to provide "global" periodicity information. Parameters for cycle segmentation, scaling and alignment were estimated based on the IDS. Finally, these parameters were used to segment, align and average cycles from each individual signal, which produces a clean and denoised single cycle spatiotemporal signal. The novel, fully automated pipeline was validated both qualitatively and quantitatively on zebrafish heart optical mapping data.

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.

Attention to eyes is present but in decline in 2-6-month-old infants later diagnosed with autism

Deficits in eye contact have been a hallmark of autism since the condition's initial description. They are cited widely as a diagnostic feature and figure prominently in clinical instruments; however, the early onset of these deficits has not been known. Here we show in a prospective longitudinal study that infants later diagnosed with autism spectrum disorders (ASDs) exhibit mean decline in eye fixation from 2 to 6 months of age, a pattern not observed in infants who do not develop ASD. These observations mark the earliest known indicators of social disability in infancy, but also falsify a prior hypothesis: in the first months of life, this basic mechanism of social adaptive action--eye looking--is not immediately diminished in infants later diagnosed with ASD; instead, eye looking appears to begin at normative levels prior to decline. The timing of decline highlights a narrow developmental window and reveals the early derailment of processes that would otherwise have a key role in canalizing typical social development. Finally, the observation of this decline in eye fixation--rather than outright absence--offers a promising opportunity for early intervention that could build on the apparent preservation of mechanisms subserving reflexive initial orientation towards the eyes.