skip to main content

Title: Shape-Based Classification of Partially Observed Curves, With Applications to Anthropology
We consider the problem of classifying curves when they are observed only partially on their parameter domains. We propose computational methods for (i) completion of partially observed curves; (ii) assessment of completion variability through a nonparametric multiple imputation procedure; (iii) development of nearest neighbor classifiers compatible with the completion techniques. Our contributions are founded on exploiting the geometric notion of shape of a curve, defined as those aspects of a curve that remain unchanged under translations, rotations and reparameterizations. Explicit incorporation of shape information into the computational methods plays the dual role of limiting the set of all possible completions of a curve to those with similar shape while simultaneously enabling more efficient use of training data in the classifier through shape-informed neighborhoods. Our methods are then used for taxonomic classification of partially observed curves arising from images of fossilized Bovidae teeth, obtained from a novel anthropological application concerning paleoenvironmental reconstruction.  more » « less
Award ID(s):
1812065 2015226 1839252 1740761
Author(s) / Creator(s):
; ; ; ; ;
Date Published:
Journal Name:
Frontiers in Applied Mathematics and Statistics
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. null (Ed.)
    Understanding the properties of dust attenuation curves in galaxies and the physical mechanisms that shape them are among the fundamental questions of extragalactic astrophysics, with great practical significance for deriving the physical properties of galaxies. Attenuation curves result from a combination of dust grain properties, dust content, and the spatial arrangement of dust and different populations of stars. In this review, we assess the state of the field, paying particular attention to extinction curves as the building blocks of attenuation laws. We introduce a quantitative framework to characterize extinction and attenuation curves, present a theoretical foundation for interpreting empirical results, overview an array of observational methods, and review observational results at low and high redshifts. Our main conclusions include the following: ▪  Attenuation curves exhibit a wide range of UV-through-optical slopes, from curves with shallow (Milky Way–like) slopes to those exceeding the slope of the Small Magellanic Cloud extinction curve. ▪  The slopes of the curves correlate strongly with the effective optical opacities, in the sense that galaxies with lower dust column density (lower visual attenuation) tend to have steeper slopes, whereas the galaxies with higher dust column density have shallower (grayer) slopes. ▪  Galaxies exhibit a range of 2175-Å UV bump strengths, including no bump, but, on average, are suppressed compared with the average Milky Way extinction curve. ▪  Theoretical studies indicate that both the correlation between the slope and the dust column as well as variations in bump strength may result from geometric and radiative transfer effects. 
    more » « less
  2. Abstract

    Proliferation of high‐resolution imaging data in recent years has led to substantial improvements in the two popular approaches for analyzing shapes of data objects based on landmarks and/or continuous curves. We provide an expository account of elastic shape analysis of parametric planar curves representing shapes of two‐dimensional (2D) objects by discussing its differences, and its commonalities, to the landmark‐based approach. Particular attention is accorded to the role of reparameterization of a curve, which in addition to rotation, scaling and translation, represents an important shape‐preserving transformation of a curve. The transition to the curve‐based approach moves the mathematical setting of shape analysis from finite‐dimensional non‐Euclidean spaces to infinite‐dimensional ones. We discuss some of the challenges associated with the infinite‐dimensionality of the shape space, and illustrate the use of geometry‐based methods in the computation of intrinsic statistical summaries and in the definition of statistical models on a 2D imaging dataset consisting of mouse vertebrae. We conclude with an overview of the current state‐of‐the‐art in the field.

    This article is categorized under:

    Image and Spatial Data < Data: Types and Structure

    Computational Mathematics < Applications of Computational Statistics

    more » « less
  3. Summary

    In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. ‘damaged’ areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k-levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to ∞ and their connection with Dirichlet process mixtures.

    more » « less
  4. We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology. 
    more » « less
  5. Abstract One of the most common methods for inferring galaxy attenuation curves is via spectral energy distribution (SED) modeling, where the dust attenuation properties are modeled simultaneously with other galaxy physical properties. In this paper, we assess the ability of SED modeling to infer these dust attenuation curves from broadband photometry, and suggest a new flexible model that greatly improves the accuracy of attenuation curve derivations. To do this, we fit mock SEDs generated from the simba cosmological simulation with the prospector SED fitting code. We consider the impact of the commonly assumed uniform screen model and introduce a new nonuniform screen model parameterized by the fraction of unobscured stellar light. This nonuniform screen model allows for a nonzero fraction of stellar light to remain unattenuated, resulting in a more flexible attenuation curve shape by decoupling the shape of the UV attenuation curve from the optical attenuation curve. The ability to constrain the dust attenuation curve is significantly improved with the use of a nonuniform screen model, with the median offset in UV attenuation decreasing from −0.30 dex with a uniform screen model to −0.17 dex with the nonuniform screen model. With this increase in dust attenuation modeling accuracy, we also improve the star formation rates (SFRs) inferred with the nonuniform screen model, decreasing the SFR offset on average by 0.12 dex. We discuss the efficacy of this new model, focusing on caveats with modeling star-dust geometries and the constraining power of available SED observations. 
    more » « less