Wasserstein Regression

Chen, Yaqing; Lin, Zhenhua; Müller, Hans‐Georg

doi:10.1080/01621459.2021.1956937

Cited by 33 publications

(38 citation statements)

References 61 publications

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“…When adopting this metric, the time series is not spherical and needs to be modeled in the Wasserstein manifold, where one can use tangent bundles (Chen et al 2021;Zhang et al 2021) or an intrinsic optimal transport approach (Zhu and Müller 2021). However when dealing with the Wasserstein space for multivariate distributions one faces major hurdles in both theory and computation.…”

Section: Discussionmentioning

confidence: 99%

“…x for the rotation exp(Q) of x ∈ H + , we use projection operators to enforce the constraint; see Chen et al (2021) and Pegoraro and Beraha (2021) for related projections in Wasserstein space.…”

Section: Predictionmentioning

confidence: 99%

“…The distributional time series is then transformed to a functional time series, which have been well investigated (Bosq 2000). Geometric approaches that are based on constructing tangent bundles on the Wasserstein manifold have recently been shown to provide better predictions for autoregressive models (Chen et al 2021;Zhang et al 2021), while an autoregressive model that is intrinsic to the Wasserstein manifold can be based on a recently developed transport algebra (Zhu and Müller 2021). It bears emphasizing that all these methods are limited to the case of distributional time series composed of one-dimensional distributions.…”

Section: Introductionmentioning

confidence: 99%

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Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

Zhu¹,

Müller²

2022

Preprint

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We introduce a new class of autoregressive models for spherical time series, where the dimension of the spheres on which the observations of the time series are situated may be finite-dimensional or infinite-dimensional as in the case of a general Hilbert sphere. Spherical time series arise in various settings. We focus here on distributional and compositional time series. Applying a square root transformation to the densities of the observations of a distributional time series maps the distributional observations to the Hilbert sphere, equipped with the Fisher-Rao metric. Likewise, applying a square root transformation to the components of the observations of a compositional time series maps the compositional observations to a finite-dimensional sphere, equipped with the geodesic metric on spheres. The challenge in modeling such time series lies in the intrinsic non-linearity of spheres and Hilbert spheres, where conventional arithmetic operations such as addition or scalar multiplication are no longer available. To address this difficulty, we consider rotation operators to map observations on the sphere. Specifically, we introduce a class of skew-symmetric operator such that the associated exponential operators are rotation operators that for each given pair of points on the sphere map one of the points to the other one. We exploit the fact that the space of skew-symmetric operators is Hilbertian to develop autoregressive modeling of geometric differences that correspond to rotations of spherical and distributional time series. Differences expressed in terms of rotations can be taken between the Fréchet mean and the observations or between consecutive observations of the time series. We derive theoretical properties of the ensuing autoregressive models and showcase these approaches with several motivating data. These include a time series of yearly observations of bivariate distributions of the minimum/maximum temperatures for a period of 120 days during each summer for the years 1990-2018 at Los Angeles (LAX) and John F. Kennedy (JFK) international airports. A second data application concerns a compositional time series with annual observations of compositions of energy sources for power generation in the U.S..

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Section: Discussionmentioning

confidence: 99%

Section: Predictionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

Zhu¹,

Müller²

2022

Preprint

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“…The ISW 2 can also be useful for machine learning applications where the prediction targets live in a general domain. Given that rigorous Frechét mean-based methodology for such problems has only been proposed recently [9,14,22], development of prediction models for manifold-valued data that are free of restrictive assumptions is an attractive future line of research.…”

Section: Discussionmentioning

confidence: 99%

“…Adopting the transportation theoretic approaches to our problem-particularly the notion of Fréchet mean for averaging the distributions-immediately hits a roadblock beyond the real line case. While Fréchet mean has been at the center of the recent progress in adapting the traditional statistical approaches to the settings of manifolds and graphs, such as PCA [12,7], analysis of variance (ANOVA) [13], and regression [9], it has a number of shortcomings. On general domains, the existence and uniqueness of the Fréchet mean is not guaranteed, and it can be sensitive to small changes in the distributions.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs

Rustamov¹,

Majumdar²

2020

Preprint

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Collections of probability distributions arise in a variety of statistical applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions are represented by histograms over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of histograms over such general domains. To this end, we introduce the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, which allows us to reduce the histogram collection comparison problem to the comparison of means in a high-dimensional Euclidean space. We develop a hypothesis testing procedure based on conducting t-tests on each dimension of this embedding, then combining the resulting p-values using recently proposed p-value combination techniques. Our numerical experiments in a variety of data settings show that the resulting tests are powerful and the p-values are well-calibrated. Example applications to user activity patterns, spatial data, and brain connectomics are provided.

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Differences in functional connectivity distribution after transcranial direct‐current stimulation: A connectivity density point of view

Tang

Zhao

Venkataraman

et al. 2022

Human Brain Mapping

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In this manuscript, we consider the problem of relating functional connectivity measurements viewed as statistical distributions to outcomes. We demonstrate the utility of using the distribution of connectivity on a study of resting‐state functional magnetic resonance imaging association with an intervention. The method uses the estimated density of connectivity between nodes of interest as a functional covariate. Moreover, we demonstrate the utility of the procedure in an instance where connectivity is naturally considered an outcome by reversing the predictor/response relationship using case/control methodology. The method utilizes the density quantile, the density evaluated at empirical quantiles, instead of the empirical density directly. This improved the performance of the method by highlighting tail behavior, though we emphasize that by being flexible and non‐parametric, the technique can detect effects related to the central portion of the density. To demonstrate the method in an application, we consider 47 primary progressive aphasia patients with various levels of language abilities. These patients were randomly assigned to two treatment arms, transcranial direct‐current stimulation and language therapy versus sham (language therapy only), in a clinical trial. We use the method to analyze the effect of direct stimulation on functional connectivity. As such, we estimate the density of correlations among the regions of interest and study the difference in the density post‐intervention between treatment arms. We discover that it is the tail of the density, rather than the mean or lower order moments of the distribution, that demonstrates a significant impact in the classification. The new approach has several benefits. Among them, it drastically reduces the number of multiple comparisons compared with edge‐wise analysis. In addition, it allows for the investigation of the impact of functional connectivity on the outcomes where the connectivity is not geometrically localized.

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Wasserstein Regression

Cited by 33 publications

References 61 publications

Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

Spherical Autoregressive Models, With Application to Distributional and Compositional Time Series

Intrinsic Sliced Wasserstein Distances for Comparing Collections of Probability Distributions on Manifolds and Graphs

Differences in functional connectivity distribution after transcranial direct‐current stimulation: A connectivity density point of view

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