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We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging, and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling iid manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient, and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples is considered indicating the wide applicability of our approach. Supplementary materials for this article are available online.

Functional data analysis (FDA) is a fast-growing area of research and development in statistics. While most FDA literature imposes the classical L 2 Hilbert structure on function spaces, there is an emergent need for a different, shape-based approach for analyzing functional data. This paper reviews and develops fundamental geometrical concepts that help connect traditionally diverse fields of shape and functional analyses. It showcases that focusing on shapes is often more appropriate when structural features (number of peaks and valleys and their heights) carry salient information in data. It recaps recent mathematical representations and associated procedures for comparing, summarizing, and testing the shapes of functions. Specifically, it discusses three tasks: shape fitting, shape fPCA, and shape regression models. The latter refers to the models that separate the shapes of functions from their phases and use them individually in regression analysis. The ensuing results provide better interpretations and tend to preserve geometric structures. The paper also discusses an extension where the functions are not real-valued but manifold-valued. The article presents several examples of this shape-centric functional data analysis using simulated and real data.

We express our gratitude to the authors of five comment articles for their valuable contributions, feedback, and recommendations on our discussion document (Wu et al. Test, 2023 ). All the reviewers acknowledged the value of our proposed research direction, which focuses on shape-based functional data analysis. They also provided insightful suggestions to enhance and expand upon these ideas. In this response, we address their comments and provide further insights.

Seasonal influenza viruses are constantly changing and produce a different set of circulating strains each season. Small genetic changes can accumulate over time and result in antigenically different viruses; this may prevent the body’s immune system from recognizing those viruses. Due to rapid mutations, in particular, in the haemagglutinin (HA) gene, seasonal influenza vaccines must be updated frequently. This requires choosing strains to include in the updates to maximize the vaccines’ benefits, according to estimates of which strains will be circulating in upcoming seasons. This is a challenging prediction task. In this paper, we use longitudinally sampled phylogenetic trees based on HA sequences from human influenza viruses, together with counts of epitope site polymorphisms in HA, to predict which influenza virus strains are likely to be successful. We extract small groups of taxa (subtrees) and use a suite of features of these subtrees as key inputs to the machine learning tools. Using a range of training and testing strategies, including training on H3N2 and testing on H1N1, we find that successful prediction of future expansion of small subtrees is possible from these data, with accuracies of 0.71–0.85 and a classifier ‘area under the curve’ 0.75–0.9.

Many summary statistics currently used in population genetics and in phylogenetics depend only on a rather coarse resolution of the underlying tree (the number of extant lineages, for example). Hence, for computational purposes, working directly on these resolutions appears to be much more efficient. However, this approach seems to have been overlooked in the past. In this paper, we describe six different resolutions of the Kingman–Tajima coalescent together with the corresponding Markov chains, which are essential for inference methods. Two of the resolutions are the well-known n -coalescent and the lineage death process due to Kingman. Two other resolutions were mentioned by Kingman and Tajima, but never explicitly formalized. Another two resolutions are novel, and complete the picture of a multi-resolution coalescent. For all of them, we provide the forward and backward transition probabilities, the probability of visiting a given state as well as the probability of a given realization of the full Markov chain. We also provide a description of the state-space that highlights the computational gain obtained by working with lower-resolution objects. Finally, we give several examples of summary statistics that depend on a coarser resolution of Kingman’s coalescent, on which simulations are usually based.

The problem of the definition and estimation of generative models based on deformable templates from raw data is of particular importance for modeling non-aligned data affected by various types of geometric variability. This is especially true in shape modeling in the computer vision community or in probabilistic atlas building in computational anatomy. A first coherent statistical framework modeling geometric variability as hidden variables was described in Allassonnière, Amit and Trouvé [J. R. Stat. Soc. Ser. B Stat. Methodol. 69 (2007) 3-29]. The present paper gives a theoretical proof of convergence of effective stochastic approximation expectation strategies to estimate such models and shows the robustness of this approach against noise through numerical experiments in the context of handwritten digit modeling.

Various approaches to statistical shape analysis exist in current literature. They mainly differ in the representations, metrics, and/or methods for alignment of shapes. One such approach is based on landmarks, that is, mathematically or structurally meaningful points, which ignores the remaining outline information. Elastic shape analysis, a more recent approach, attempts to fix this by using a special functional representation of the parametrically defined outline to perform shape registration, and subsequent statistical analyses. However, the lack of landmark identification can lead to unnatural alignment, particularly in biological and medical applications, where certain features are crucial to shape structure, comparison, and modeling. The main contribution of this work is the definition of a joint landmark-constrained elastic statistical shape analysis framework. We treat landmark points as constraints in the full shape analysis process. Thus, we inherit benefits of both methods: the landmarks help disambiguate shape alignment when the fully automatic elastic shape analysis framework produces unsatisfactory solutions. We provide standard statistical tools on the landmark-constrained shape space including mean and covariance calculation, classification, clustering, and tangent principal component analysis (PCA). We demonstrate the benefits of the proposed framework on complex shapes from the MPEG-7 dataset and two real data examples: mice T2 vertebrae and Hawaiian Drosophila fly wings. Supplementary materials for this article are available online.

Any change in shape of a configuration of landmark points in two or three dimensions includes a uniform component, a component that is a wholly linear (affine) transformation. The formulas for estimating this component have been standardized for two-dimensional data but not for three-dimensional data. We suggest estimating the component by way of the complementarity between the uniform component and the space of partial warps. The component can be estimated by regression in either one space or the other: regression on the partial warps, followed by their removal, or regression on a basis for the uniform component itself. Either of the new methods can be used for both two- and three-dimensional landmark data and thus generalize Bookstein's (1996, pages 153–168 in Advances in morphometrics [L. F. Marcus et al., eds.], Plenum, New York) linearized Procrustes formula for estimating the uniform component in two dimensions.

For two decades, the Colless index has been the most frequently used statistic for assessing the balance of phylogenetic trees. In this article, this statistic is studied under the Yule and uniform model of phylogenetic trees. The main tool of analysis is a coupling argument with another well-known index called the Sackin statistic. Asymptotics for the mean, variance and covariance of these two statistics are obtained, as well as their limiting joint distribution for large phylogenies. Under the Yule model, the limiting distribution arises as a solution of a functional fixed point equation. Under the uniform model, the limiting distribution is the Airy distribution. The cornerstone of this study is the fact that the probabilistic models for phylogenetic trees are strongly related to the random permutation and the Catalan models for binary search trees.

The social aspect is an important but often overlooked part of sustainable development philosophy. In hoping to popularise and show the importance of social sustainable development, this study tries to find a relation between the social environment and urban form. Research in the social capital field provided the methodology to acquire social computational data. The relation between human actions and the environment is noted in many theories, and used in some practices. Human cognition is computationally predictable with natural shape analysis and machine learning methods. In the analysis of shape, a topological skeleton is a proven method to acquire statistical data that correlates with data collected from human experiments. In this study, the analysis of urban form with respect to human cognition was used to acquire computational data for a machine learning model of social capital in counties in the USA Tvarios plėtros teorijoje socialinė aplinka yra pripažinta kaip svarbus veiksnys, tačiau trūksta praktinės metodikos. Ryšio tarp urbanistinės formos ir socialinės aplinkos radimas aktualizuotų ir padėtų populiarinti socialinę tvarią plėtrą. Aplinkos įtaka žmonių tarpusavio elgesiui yra ne kartą aptartas reiškinys, tačiau praktikoje retai taikomas. Ankstesniuose socialinio kapitalo tyrimuose pateikiamos metodologijos ir statistiniai duomenys esamos situacijos analizei atlikti. Kaip žmonės suvokia formas, yra nuspėjama taikant statistinę formos analizę ir dirbtinio intelekto metodologiją – sistemos mokymąsi. Klasifikuojant formas topologinio skeleto metodologija gaunami rezultatai koreliuoja su duomenimis, surinktais per eksperimentą, kuriame žmonės klasifikuoja formas. Taikant žinomas formos analizės metodologijas, atspindinčias suvokimą, buvo surinkti duomenys modeliuoti socialinį kapitalą su sisteminio mokymosi modeliu. Sisteminis mokymasis yra dirbtinio intelekto sritis, kurioje remiantis pateiktais duomenimis automatiškai sukalibruojama kompleksinė matematinė formulė. Modeliuojant socialinį kapitalą su formos skeleto statistiniais duomenimis, geriausi rezultatai pasiekti taikant neuroniniais tinklais pagristą sisteminį mokymąsi.