Amplitude and phase variation of point processes

Panaretos, Victor M.; Zemel, Yoav

doi:10.1214/15-aos1387

Cited by 77 publications

(102 citation statements)

References 45 publications

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“…Another issue that requires further investigation is the fact that often objects are not fully observed but must be estimated from a random sample that is generated by the object, where one assumes that these within-object samples are independent from the functional random objects. This estimation step induces another error that will typically not satisfy assumption (27) but these errors will vanish as the within-object samples become larger and so under reasonable assumptions can be ignored in the asymptotic analysis; this kind of estimation error has been analysed in detail for the case of distributions (Panaretos and Zemel, 2016;Petersen and Müller, 2016), where the distributions are usually not known but must be estimated from samples.…”

Section: Open Problemsmentioning

confidence: 99%

Functional Models for Time-Varying Random Objects

Dubey

Müller

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary Functional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued. In recent years, samples of time‐varying object data such as time‐varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. We propose metric covariance, a novel association measure for paired object data lying in a metric space (Ω,d) that we use to define a metric autocovariance function for a sample of random Ω‐valued curves, where Ω generally will not have a vector space or manifold structure. The proposed metric autocovariance function is non‐negative definite when the squared semimetric d2 is of negative type. Then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for Ω‐valued functional data, including time‐varying probability distributions, covariance matrices and time dynamic networks. Analogues of functional principal components for time‐varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real‐valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space Ω. We establish asymptotic consistency of the sample‐based estimators for the corresponding population targets under mild metric entropy conditions on Ω and continuity of the Ω‐valued random curves. These concepts are illustrated with samples of time‐varying probability distributions for human mortality, time‐varying covariance matrices derived from trading patterns and time‐varying networks that arise from New York taxi trips.

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Section: Open Problemsmentioning

confidence: 99%

Functional Models for Time-Varying Random Objects

Dubey

Müller

2020

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…For a single density process F, the theoretical and practical properties of the Wasserstein mean have been thoroughly investigated (Bolstad et al, 2003;Zhang & Müller, 2011;Panaretos & Zemel, 2016;Bigot et al, 2017), and recently the Wasserstein variance was adopted to quantify variability explained when performing dimension reduction for densities . To quantify the dependence between two random densities, we propose here the extension of these concepts to a Wasserstein covariance measure.…”

Section: Wasserstein Covariance 2·1 the Wasserstein Metric And Geometrymentioning

confidence: 99%

“…While functional principal component analysis using cross-sectional averaging can be directly applied for samples of density functions (Kneip & Utikal, 2001), more recently techniques have been developed that incorporate the geometric constraints inherent to the space of density functions. A popular metric for data where each data atom corresponds to a randomly sampled distribution or density is the Wasserstein metric, both for its theoretical appeal and its convincing empirical performance in various applications (Bolstad et al, 2003;Zhang & Müller, 2011;Bigot et al, 2016Bigot et al, , 2017Panaretos & Zemel, 2016).…”

Section: Introductionmentioning

confidence: 99%

Wasserstein covariance for multiple random densities

Petersen

Müller

2019

Biometrika

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A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fréchet mean is typically used to quantify and visualize the average density from the sample. For one-dimensional densities, the Wasserstein metric is popular due to its theoretical appeal and interpretive value as an optimal transport metric, leading to the Wasserstein-Fréchet mean or barycenter as the mean density. We extend the existing methodology for samples of densities in two key directions. First, motivated by applications in neuroimaging, we consider dependent density data, where a p-vector of univariate random densities is observed for each sampling unit. Second, we introduce a Wasserstein covariance measure and propose intuitively appealing estimators for both fixed and diverging p, where the latter corresponds to continuously-indexed densities. We also give theory demonstrating consistency and asymptotic normality, while accounting for errors introduced in the unavoidable preparatory density estimation step. The utility of the Wasserstein covariance matrix is demonstrated through applications to functional connectivity in the brain using functional magnetic resonance imaging data and to the secular evolution of mortality for various countries.

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“…A future interesting aspect is the exploration of the applicability of the Optimal Transport framework to the registration problem, as suggested in Panaretos and Zemel (2016) in a discrete context, and its links with the diffeomorphic deformation framework. This is of potential interest in the surface registration framework, where we usually lack physical models that can describe the phenomena, and thus a 'least action' approach could well be effective.…”

Section: Conclusion and Prospectivesmentioning

confidence: 99%

Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

Lila

Aston

2019

Journal of the American Statistical Association

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In Functional Data Analysis, data are commonly assumed to be smooth functions on a fixed interval of the real line. In this work, we introduce a comprehensive framework for the analysis of functional data, whose domain is a two-dimensional manifold and the domain itself is subject to variability from sample to sample. We formulate a statistical model for such data, here called Functions on Surfaces, which enables a joint representation of the geometric and functional aspects, and propose an associated estimation framework. We assess the validity of the framework by performing a simulation study and we finally apply it to the analysis of neuroimaging data of cortical thickness, acquired from the brains of different subjects, and thus lying on domains with different geometries.A set of FoSs, such as the ones in Figure 1, can be mathematically formulated as a collection of pairs tpM i , Y i q : i " 1, . . . , nu. The collection tM i : i " 1, . . . , nu is a set of topologically equivalent smooth two-dimensional manifolds, embedded in R 3 , representing the geometry of the data. The functional aspect of the data is represented by the collection tY i : i " 1, . . . , nu, where Y i is an element of the function space L 2 pM i q, i.e. the Hilbert space of square integrable functions on M i with respect to the area measure.Here, we propose a statistical generative model for FoSs, modelled in terms of mathematically more tractable objects. To this end, we define a deformation operator ϕ, such that ϕ v : R 3 Ñ R 3 is parametrized by the elements of a Hilbert space tv : v P Vu. Moreover, we assume ϕ v is an homomorphism of R 3 for all v P V and that ϕ 0 pxq " x for all x P R 3 . For each v P V, ϕ v : R 3 Ñ R 3 represents a deformation of the space R 3 , which means that when ϕ v is applied to a point x P R 3 this is relocated to the location ϕ v pxq P R 3 . In addition, ϕ v being a homomorphism of R 3 implies that, for a fixed v P V, there is a one-to-one correspondence between each element x P R 3 and the relocated element ϕ v pxq P R 3 .Moreover, we introduce M 0 , a smooth two-dimensional manifold topologically equivalent to tM i u, which represents a fixed template geometric object. Given a FoS, the geometric template together with the deformation operator offers an alternative representation of the geometry of the FoS in hand as: ϕ v˝M0 , for a particular choice of v P V.Here, ϕ v˝M0 is the geometric object obtained by deforming M 0 through the map ϕ v , and specifically, by relocating each point x P M 0 to the new location ϕ v pxq, to resemble the target manifold. For this reason, we will informally say that the element v P V encodes the geometry, or the shape, of a FoS, as in fact v defines the deformation ϕ v , which defines the geometry ϕ v˝M0 . The choice of the deformation operator is driven by the particular problem in hand. We first introduce the generative model and subsequently discuss different choices of this operator.

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Amplitude and phase variation of point processes

Cited by 77 publications

References 45 publications

Functional Models for Time-Varying Random Objects

Functional Models for Time-Varying Random Objects

Wasserstein covariance for multiple random densities

Statistical Analysis of Functions on Surfaces, With an Application to Medical Imaging

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