Functional data analysis on nonlinear manifolds has drawn recent interest. We propose an intrinsic principal component analysis for smooth Riemannian manifold-valued functional data and study its asymptotic properties. The proposed Riemannian functional principal component analysis (RFPCA) is carried out by first mapping the manifold-valued data through Riemannian logarithm maps to tangent spaces around the time-varying Fr echet mean function, and then performing a classical multivariate functional principal component analysis on the linear tangent spaces. Representations of the Riemannian manifold-valued functions and the eigenfunctions on the original manifold are then obtained by mapping back with exponential maps. The tangent-space approximation through functional principal component analysis is shown to be well-behaved in terms of controlling the residual variation if the Riemannian manifold has nonnegative curvature. We derive uniform convergence rates for the model components, including the mean function, covariance function, eigenfunctions, and functional principal component scores and apply the proposed methodology to obtain a novel framework for the analysis of longitudinal compositional data. This is achieved by mapping longitudinal compositional data to trajectories on the sphere, and we illustrate this approach with longitudinal fruit fly behavior patterns. The proposed Riemannian functional principal component analysis is shown to be superior in terms of trajectory recovery in comparison to an unrestricted functional principal component analysis in applications that include flight trajectories, as well as in simulations. RFPCA is also found to produce principal component scores that are better predictors for classification compared to traditional functional functional principal component scores.

Bayes classifiers for functional data pose a challenge. This is because probability density functions do not exist for functional data. As a consequence, the classical Bayes classifier using density quotients needs to be modified. We propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified. The density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one-dimensional density estimates. This is an extension to functional data of some of the very earliest nonparametric Bayes classifiers that were based on simple density ratios in the one-dimensional case. By means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect Bayes classifiers for functional data can be avoided. We demonstrate that in the case of Gaussian functional data, the proposed functional Bayes classifier reduces to a functional version of the classical quadratic discriminant. A study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero, a phenomenon that has been referred to as "perfect classification". The proposed classifiers also perform favorably in finite sample applications, as we demonstrate in comparisons with other functional classifiers in simulations and various data applications, including wine spectral data, functional magnetic resonance imaging (fMRI) data for attention deficit hyperactivity disorder (ADHD) patients, and yeast gene expression data.

From birth to 5 years of age, brain structure matures and evolves alongside emerging cognitive and behavioral abilities. In relating concurrent cognitive functioning and measures of brain structure, a major challenge that has impeded prior investigation of their time-dynamic relationships is the sparse and irregular nature of most longitudinal neuroimaging data. We demonstrate how this problem can be addressed by applying functional concurrent regression models (FCRMs) to longitudinal cognitive and neuroimaging data. The application of FCRM in neuroimaging is illustrated with longitudinal neuroimaging and cognitive data acquired from a large cohort (n = 210) of healthy children, 2-48 months of age. Quantifying white matter myelination by using myelin water fraction (MWF) as imaging metric derived from MRI scans, application of this methodology reveals an early period (200-500 days) during which whole brain and regional white matter structure, as quantified by MWF, is positively associated with cognitive ability, while we found no such association for whole brain white matter volume. Adjusting for baseline covariates including socioeconomic status as measured by maternal education (SES-ME), infant feeding practice, gender, and birth weight further reveals an increasing association between SES-ME and cognitive development with child age. These results shed new light on the emerging patterns of brain and cognitive development, indicating that FCRM provides a useful tool for investigating these evolving relationships.

Long terminal repeat-retrotransposons (LTR-RTs) are a major component of all flowering plant genomes. To analyze the time dynamics of LTR-RTs, we modeled the insertion rates of the 35 most abundant LTR-RT families in the genome of Aegilops tauschii, one of the progenitors of wheat. Our model of insertion rate (birth) takes into account random variation in LTR divergence and the deletion rate (death) of LTR-RTs. Modeling the death rate is crucial because ignoring it would underestimate insertion rates in the distant past. We rejected the hypothesis of constancy of insertion rates for all 35 families and showed by simulations that our hypothesis test controlled the false-positive rate. LTR-RT insertions peaked from 0.064 to 2.39 MYA across the 35 families. Among other effects, the average age of elements within a family was negatively associated with recombination rate along a chromosome, with proximity to the closest gene, and weakly associated with the proximity to its 5' end. Elements within a family that were near genes colinear with genes in the genome of tetraploid emmer wheat tended to be younger than those near noncolinear genes. We discuss these associations in the context of genome evolution and stability of genome sizes in the tribe Triticeae. We demonstrate the general utility of our models by analyzing the two most abundant LTR-RT families in Arabidopsis lyrata, and show that these families differed in their insertion dynamics. Our estimation methods are available in the R package TE on CRAN.

The maturation of the myelinated white matter throughout childhood is a critical developmental process that underlies emerging connectivity and brain function. In response to genetic influences and neuronal activities, myelination helps establish the mature neural networks that support cognitive and behavioral skills. The emergence and refinement of brain networks, traditionally investigated using functional imaging data, can also be interrogated using longitudinal structural imaging data. However, few studies of structural network development throughout infancy and early childhood have been presented, likely owing to the sparse and irregular nature of most longitudinal neuroimaging data, which complicates dynamic analysis. Here, we overcome this limitation and investigate through concurrent correlation the co-development of white matter myelination and volume, and structural network development of white matter myelination between brain regions as a function of age, using statistically well-supported methods. We show that the concurrent correlation of white matter myelination and volume is overall positive and reaches a peak at 580 days. Brain regions are found to differ in overall magnitudes and patterns of time-varying association throughout early childhood. We introduce time-dynamic developmental networks based on temporal similarity of association patterns in the levels of myelination across brain regions. These networks reflect groups of brain regions that share similar patterns of evolving intra-regional connectivity, as evidenced by levels of myelination, are biologically interpretable and provide novel visualizations of brain development. Comparing the constructed networks between different maternal education groups, we found that children with higher and lower maternal education differ significantly in the overall magnitude of the time-dynamic correlations.

Numerous quantitative trait loci (QTL) have been mapped in tetraploid and hexaploid wheat and wheat relatives, mostly with simple sequence repeat (SSR) or single nucleotide polymorphism (SNP) markers. To conduct meta-analysis of QTL requires projecting them onto a common genomic framework, either a consensus genetic map or genomic sequence. The latter strategy is pursued here. Of 774 QTL mapped in wheat and wheat relatives found in the literature, 585 (75.6%) were successfully projected onto the Aegilops tauschii pseudomolecules. QTL mapped with SNP markers were more successfully projected (92.2%) than those mapped with SSR markers (66.2%). The QTL were not distributed homogeneously along chromosome arms. Their frequencies increased in the proximal-to-distal direction but declined in the most distal regions and were weakly correlated with recombination rates along the chromosome arms. Databases for projected SSR markers and QTL were constructed and incorporated into the Ae. tauschii JBrowse. To facilitate meta-QTL analysis, eight clusters of QTL were used to estimate standard deviations ([Formula: see text]) of independently mapped QTL projected onto the Ae. tauschii genome sequence. The standard deviations [Formula: see text] were modeled as an exponential decay function of recombination rates along the Ae. tauschii chromosomes. We implemented four hypothesis tests for determining the membership of query QTL. The hypothesis tests and estimation procedure for [Formula: see text] were implemented in a web portal for meta-analysis of projected QTL. Twenty-one QTL for Fusarium head blight resistance mapped on wheat chromosomes 3A, 3B, and 3D were analyzed to illustrate the use of the portal for meta-QTL analyses.

Aegilops tauschii is the diploid progenitor of the D genome of hexaploid wheat (Triticum aestivum, genomes AABBDD) and an important genetic resource for wheat. The large size and highly repetitive nature of the Ae. tauschii genome has until now precluded the development of a reference-quality genome sequence. Here we use an array of advanced technologies, including ordered-clone genome sequencing, whole-genome shotgun sequencing, and BioNano optical genome mapping, to generate a reference-quality genome sequence for Ae. tauschii ssp. strangulata accession AL8/78, which is closely related to the wheat D genome. We show that compared to other sequenced plant genomes, including a much larger conifer genome, the Ae. tauschii genome contains unprecedented amounts of very similar repeated sequences. Our genome comparisons reveal that the Ae. tauschii genome has a greater number of dispersed duplicated genes than other sequenced genomes and its chromosomes have been structurally evolving an order of magnitude faster than those of other grass genomes. The decay of colinearity with other grass genomes correlates with recombination rates along chromosomes. We propose that the vast amounts of very similar repeated sequences cause frequent errors in recombination and lead to gene duplications and structural chromosome changes that drive fast genome evolution.